Research Article Homotopy Simulation of Nonlinear Unsteady...
16
Research Article Homotopy Simulation of Nonlinear Unsteady Rotating Nanofluid Flow from a Spinning Body O. Anwar Bég, 1 F. Mabood, 2 and M. Nazrul Islam 3 1 Gort Engovation (Engineering Sciences Research), 11 Rooley Croſt, Bradford BD6 1FA, UK 2 Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa 25120, Pakistan 3 Aerospace, Department of Engineering and Mathematics, Sheaf Building, Sheffield Hallam University, Sheffield S1 1WB, UK Correspondence should be addressed to O. Anwar B´ eg; [email protected] Received 10 May 2015; Revised 16 August 2015; Accepted 16 August 2015 Academic Editor: Jos` e A. Tenereiro Machado Copyright © 2015 O. Anwar B´ eg et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e development of new applications of nanofluids in chemical engineering and other technologies has stimulated significant interest in computational simulations. Motivated by coating applications of nanomaterials, we investigate the transient nanofluid flow from a time-dependent spinning sphere using laminar boundary layer theory. e free stream velocity varies continuously with time. e unsteady conservations equations are normalized with appropriate similarity transformations and rendered into a ninth-order system of nonlinear coupled, multidegree ordinary differential equations. e transformed nonlinear boundary value problem is solved using the homotopy analysis method (HAM), a semicomputational procedure achieving fast convergence. Computations are verified with an Adomian decomposition method (ADM). e influence of acceleration parameter, rotational body force parameter, Brownian motion number, thermophoresis number, Lewis number, and Prandtl number on surface shear stress, heat, and mass (nanoparticle volume fraction) transfer rates is evaluated. e influence on boundary layer behavior is also investigated. HAM demonstrates excellent stability and leads to highly accurate solutions. 1. Introduction Nanofluids continue to stimulate significant interest in mod- ern engineering and medical sciences [1]. ese fluids offer significant thermal enhancement characteristics. Nanofluids contain suspended metallic nanoparticles, which increase the thermal conductivity of the base fluid by a substantial amount. Fabrication techniques for nanofluids are also being continuously refined [2]. e heat transfer coefficient of nanofluids increases with volume concentration and they offer other advantages. Nanofluids have been deployed in a tremendous spectrum of applications including sterilization of medical suspensions [3], nanomaterial processing [4, 5], automotive coolants [6], microbial fuel cell technology [7], polymer coating [8], intelligent building design [9], microfluid delivery devices [10], and aerospace tribology [11]. Although initially nanofluid dynamics research con- centrated on quantification of fundamental thermophysical properties of nanofluids (including thermal conductivity, density, viscosity, and heat transfer coefficient), over the past few years the focus has become increasingly orien- tated towards modelling and simulation. In this regard building on mathematical models of nanofluid transport, an extensive range of numerical approaches have been implemented to simulate increasingly more practical prob- lems of nanofluids. e inherent nonlinearity of nanofluid flows which involve momentum, heat, and mass transfer necessitates very robust computational algorithms for their analysis. e approaches range from intensive molecular dynamics methods [12, 13] which can model interfacial tension between nanoparticles, control volume methods [14], genetic algorithms [15], control volume finite elements [16], Nakamura finite difference codes [17], homotopy analysis techniques [18], ANSYS finite element commercial code [19], differential transform methods [20], MAPLE integration quadrature routines [21], Crank-Nicolson finite difference Hindawi Publishing Corporation International Journal of Engineering Mathematics Volume 2015, Article ID 272079, 15 pages http://dx.doi.org/10.1155/2015/272079
Transcript of Research Article Homotopy Simulation of Nonlinear Unsteady...
Research ArticleHomotopy Simulation of Nonlinear Unsteady RotatingNanofluid Flow from a Spinning Body
O Anwar Beacuteg1 F Mabood2 and M Nazrul Islam3
1Gort Engovation (Engineering Sciences Research) 11 Rooley Croft Bradford BD6 1FA UK2Department of Mathematics University of Peshawar Khyber Pakhtunkhwa 25120 Pakistan3Aerospace Department of Engineering and Mathematics Sheaf Building Sheffield Hallam University Sheffield S1 1WB UK
Correspondence should be addressed to O Anwar Beg gortoabgmailcom
Received 10 May 2015 Revised 16 August 2015 Accepted 16 August 2015
Academic Editor Jose A Tenereiro Machado
Copyright copy 2015 O Anwar Beg et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The development of new applications of nanofluids in chemical engineering and other technologies has stimulated significantinterest in computational simulations Motivated by coating applications of nanomaterials we investigate the transient nanofluidflow from a time-dependent spinning sphere using laminar boundary layer theory The free stream velocity varies continuouslywith time The unsteady conservations equations are normalized with appropriate similarity transformations and rendered intoa ninth-order system of nonlinear coupled multidegree ordinary differential equations The transformed nonlinear boundaryvalue problem is solved using the homotopy analysis method (HAM) a semicomputational procedure achieving fast convergenceComputations are verified with an Adomian decomposition method (ADM) The influence of acceleration parameter rotationalbody force parameter Brownian motion number thermophoresis number Lewis number and Prandtl number on surface shearstress heat and mass (nanoparticle volume fraction) transfer rates is evaluated The influence on boundary layer behavior is alsoinvestigated HAM demonstrates excellent stability and leads to highly accurate solutions
1 Introduction
Nanofluids continue to stimulate significant interest in mod-ern engineering and medical sciences [1] These fluids offersignificant thermal enhancement characteristics Nanofluidscontain suspended metallic nanoparticles which increasethe thermal conductivity of the base fluid by a substantialamount Fabrication techniques for nanofluids are also beingcontinuously refined [2] The heat transfer coefficient ofnanofluids increases with volume concentration and theyoffer other advantages Nanofluids have been deployed in atremendous spectrum of applications including sterilizationof medical suspensions [3] nanomaterial processing [45] automotive coolants [6] microbial fuel cell technology[7] polymer coating [8] intelligent building design [9]microfluid delivery devices [10] and aerospace tribology[11] Although initially nanofluid dynamics research con-centrated on quantification of fundamental thermophysical
properties of nanofluids (including thermal conductivitydensity viscosity and heat transfer coefficient) over thepast few years the focus has become increasingly orien-tated towards modelling and simulation In this regardbuilding on mathematical models of nanofluid transportan extensive range of numerical approaches have beenimplemented to simulate increasingly more practical prob-lems of nanofluids The inherent nonlinearity of nanofluidflows which involve momentum heat and mass transfernecessitates very robust computational algorithms for theiranalysis The approaches range from intensive moleculardynamics methods [12 13] which can model interfacialtension between nanoparticles control volumemethods [14]genetic algorithms [15] control volume finite elements [16]Nakamura finite difference codes [17] homotopy analysistechniques [18] ANSYS finite element commercial code[19] differential transformmethods [20]MAPLE integrationquadrature routines [21] Crank-Nicolson finite difference
Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2015 Article ID 272079 15 pageshttpdxdoiorg1011552015272079
2 International Journal of Engineering Mathematics
schemes [22] Keller box implicit methods [23] Mathematicaintegration subroutines [24] Blottner difference methods[25] and dual reciprocity boundary element methods [26]These studies have generally examined Brownianmotion andthermophoresis effects for various nanoparticle suspensionsand considered two-dimensional boundary layer channeland cavity flows They have ignored Coriolis body forceeffects which arise in rotational fluid mechanics Externalflows from spinning bodies are significant in many branchesof chemical and industrial engineering including electrolysistreatments [27] and polymer deposition on components [28]In such systems the rotation strongly influences boundarylayer growth and structure on the body periphery This inturn controls heat and mass transfer rates whether flowsare laminar transitional or indeed fully turbulent [29] Thebodies may be conical spherical elliptical disk-shaped andindeed concentric and eccentric systemsManymathematicaland numerical studies of such flows have been presentedFaltas and Saad [30] used a collocation method to analyzesteady axisymmetric flow between two spinning eccentricspheres with a linear slip of Basset-type boundary conditionat both surfaces Andersson and Rousselet [31] studied partialmomentum and thermal slip from a rotating disk witha Runge-Kutta method Niazmand and Renksizbulut [32]employed a finite volume code to investigate unsteady heattransfer and thermal patterns around a rotating sphere (as amodel of a particle) with surface blowing delineating threedistinct wake regimes namely steady and axisymmetricalsteady but nonsymmetrical and unsteady with vortex shed-ding They found that although rotation strongly induceslocal modifications in flow patterns the surface-averagedheat transfer rates were not altered markedly even at largerotational speeds Roy and Anilkumar [33] used the Kellerbox method to simulate transient free and forced convectionboundary layer flow from a rotating cone for the case whenthe free stream angular velocity and the angular velocityof the cone vary arbitrarily with the time Subhashini etal [34] used the Bellman-Kalaba quasilinearization methodto address uniform slot injectionsuction and nonuniformtotal enthalpy wall effects on steady nonsimilar laminarcompressible boundary layer flowover a rotating sphereTheyshowed that greater rotation and total enthalpy at the wallencourages earlier flow separation whereas cooling delaysthis and furthermore that greater Mach number displacesthe point of separation upstream as a result of the adversepressure gradient
The focus of the present work is to analyse rotatingnanofluid boundary layer flows at the stagnation point on theexternal surface of a spinning sphere Computational rotatingnanofluid dynamics has recently attracted some interestsince the deployment of nanofluids in revolving chemicalengineering devices offers significant improvements overexisting designs Rana et al [35] used a variational finiteelement algorithm to study unsteady magnetonanofluidtransport from a rotating stretching continuous sheet Theyshowed that greater rotational parameter reduces primary
y
Nanofluid
Stagnation point
Isothermal sphere
z
x
Ω
r(x)
Uprop
Figure 1 Physical model
and secondary velocities temperature and nanoparticleconcentration They further showed that reduced Nusseltnumber (wall temperature gradient) was suppressed withboth Brownian motion and thermophoresis effects whereasreduced Sherwood number (wall mass transfer gradient) wasenhanced Further studies of swirling nanofluid dynamicshave been reported by Nadeem and Saleem [36] for a verticalcone and Malvandi [37] for stagnation point nanofluidflow from a spinning sphere In the present study we useboth a homotopy analysis method (HAM) and an Ado-mian decomposition method (ADM) to simulate stagnationpoint nanofluid flow from a rotating sphere The influenceof acceleration parameter rotational body force parameterLewis number Brownianmotion number and thermophore-sis parameter on velocity temperature and nanoparticledistributions is examined This work is relevant to coatingapplications in the polymeric industry
2 Mathematical Model
The physical regime under investigation is illustrated inFigure 1 in (119909 119910 119911) coordinate system Transient lami-nar boundary layer flow of an incompressible Newtoniannanofluid is studied in the vicinity of the stagnation pointregion of an isothermal rotating sphere of radius 119903 rotat-ing with angular velocity Ω Soret and Dufour effects areneglected Following Anilkumar and Roy [38] the freestream and angular velocities depend on time in the formof 119880119890(119909 119905) = 119860119909119905 and Ω(119905) = 119861119905 where 119860 and 119861
are arbitrary constants both greater than zero The flowfield is assumed to be axisymmetric and the fluid possessesconstant thermophysical properties with the exception ofthose caused by density changeswhich generate the buoyancyforce under the Boussinesq approximationThe Buonjiornionanofluid model is adopted prioritizing Brownian motionand thermophoresis effects [39] In light of these approx-imations the time-dependent conservation equations formass momentum energy and species (nanoparticle volume
International Journal of Engineering Mathematics 3
fraction) may be presented as follows as documented byMalvandi [37]
120597 (119903119906)
120597119909+
120597 (119903V)120597119910
= 0 (1)
120597119906
120597119905+ 119906
120597119906
120597119909+ V
120597119906
120597119910minus (
1199082
119903)
119889119903
119889119909
=120597119880119890
120597119905+ 119906119890
120597119880119890
120597119909+ ](
1205972119906
1205971199102)
(2)
120597119908
120597119905+ 119906
120597119908
120597119909+ V
120597119908
120597119910+ (
119906119908
119903)
119889119903
119889119909= ](
1205972119908
1205971199102) (3)
120597119879
120597119905+ 119906
120597119879
120597119909+ V
120597119879
120597119910
= 120572(1205972119879
1205971199102) + 120591(119863
119861
120597119862
120597119910
120597119879
120597119910+
119863119879
119879infin
120597119879
120597119910
120597119879
120597119910)
(4)
120597119862
120597119905+ 119906
120597119862
120597119909+ V
120597119862
120597119910= 119863119861(1205972119862
1205971199102) +
119863119879
119879infin
1205972119879
1205971199102 (5)
The appropriate initial and boundary conditions take theform
119906 (0 119909 119910) = 119906119894(119909 119910)
V (0 119909 119910) = V119894(119909 119910)
119908 (0 119909 119910) = 119908119894(119909 119910)
119879 (0 119909 119910) = 119879119894(119909 119910)
119862 (0 119909 119910) = 119862119894(119909 119910)
119906 (119905 119909 0) = 0
V (119905 119909 0) = 0
119908 (0 119909 119910) = Ω (119905 119903) 119903
119879 (119905 119909 0) = 119879119908
119908 (119909 119905infin) = 0
119906 (119905 119909 119908) = 119880119890(119905 119909) =
119860119909
119905
119879 (119905 119909infin) = 119879infin
= const
119862 (119905 119909infin) = 119862infin
= const
(6)
Here 119906 V and 119908 denote velocity components along the 119909119910 and 119911 coordinates where these coordinates are orientatedrespectively from the forward stagnation point along the sur-face (see Figure 1) normal to the surface and in the rotatingdirections respectively 119903(119909) is the radial distance from asurface element to the axis of symmetry 119905 is time 119894 is initialcondition 120572 is the thermal diffusivity of the nanofluid ] iskinematic viscosity of nanofluid 119896 is the thermal conductivityof nanofluid 119863
119861is the Brownian diffusion coefficient (a
measure of the species diffusivity of nanoparticles) 119863119879is
the thermophoretic diffusion coefficient 120591 = (120588119888119901)119901(120588119888119901)119891
defines the ratio of effective heat capacity of the nanoparticles(eg titanium oxide) to the heat capacity of the fluid 119888
119901
is isothermal specific heat capacity and 119879 is the nanofluidtemperature119879
infinis the free stream temperature (at the edge of
the boundary layer) and 119862infin
is the free stream nanoparticleconcentration It is assumed that the base fluid and thenanoparticles are in thermal equilibrium and no slip occursbetween them We note that in (4) which is a statement ofFickrsquos law of mass (species) diffusion for nanoparticles thefirst term on the left hand side is the transient concentrationgradient and the second and third terms are the convectivemass transfer terms The first term on the right hand sidedenotes the species diffusion and the last term is the relativecontribution of thermophoresis to Brownian motion Theseeffects have also been considered in detail by Dib et al[40] and Gupta and Saha Ray [41] The boundary valueproblem defined in terms of primitive variables may besolved using numerical methods However to provide a moreamenable solution and one which enables the use of scalingparameters it is advantageous to introduce a set of similaritytransformation variables We define the following
120578 = (V119905)12 119910
120595 = 119860119909(V119905)12
119865 (120578)
119908 = (119861119909
119897)119866 (120578)
119903 sim 119909
119889119903
119889119909= 1
120582 = (Ω119905
119886)
119867 (120578) =119879 minus 119879infin
119879119882
minus 119879infin
119873 (120578) =119862 minus 119862
infin
119862119908minus 119862infin
Pr =]120572
Le =]119863119861
Nb =(120588119888119901)119901
119863119861(119862119882
minus 119862infin)
(120588119888119901)119891
]
Nt =(120588119888119901)119901
119863119879(119879119882
minus 119879infin)
(120588119888119901)119891
]119879infin
(7)
Here 119860 is the acceleration (unsteadiness) parameter 120578 isthe transformed normal coordinate (perpendicular to spheresurface) 120582 is the normalized rotation parameter 119886 is sphere
4 International Journal of Engineering Mathematics
radius 120595 is dimensional stream function 119865 is dimensionlessstream function 119866 is the dimensionless secondary velocityfunction 119867 is the dimensionless temperature function 119873 isdimensionless nanoparticle concentration function (volumefraction) Pr is Prandtl number Le is Lewis number Nb isBrownian motion parameter and Nt is the thermophoresisparameter Introducing these transformations into (1)ndash(5)the partial differential boundary layer equations contractto the following nonlinear coupled system of self-similarordinary differential equations namely
1198893119865
1198891205783+
1198892119865
1198891205782(119860119865 +
120578
2) +
119889119865
119889120578(1 minus 119860
119889119865
119889120578)
+ 119860 (1205821198662
+ 1) minus 1 = 0
(8)
1198892119866
1198891205782+
119889119866
119889120578(119860119865 +
120578
2) + 119866(1 minus 2119860
119889119865
119889120578) = 0 (9)
1
Pr1198892119867
1198891205782+ Nb119889119867
119889120578
119889119873
119889120578+Nt(119889119867
119889120578)
2
+ 119860119865119889119867
119889120578
+120578
2
119889119867
119889120578= 0
(10)
1198892119873
1198891205782+ Le119889119873
119889120578(119860119865 +
120578
2) +
NtNb
1198892119867
1198891205782= 0 (11)
The corresponding transformed boundary conditions arespecified as follows
At 120578 = 0
119865 = 0
119889119865
119889120578= 0
119866 = 1
119867 = 1
119873 = 1
(12a)
As 120578 997888rarr infin
119889119865
119889120578= 1
119866 = 0
119867 = 0
119873 = 0
(12b)
In engineering simulations of nanofluid flows not onlythe velocity temperature and nanoparticle volume fractiondistributions but also primary (119909-) skin friction 119862
119891119909and
secondary (119911-) skin friction coefficients 119862119891119911 and local Nusselt
number function are important We define these as
119862119891119909
=2120583 (120597119906120597119910)
119910=0
120588119880119890
2= Re119909
12
119860minus12
1198892119865 (0)
1198891205782
119862119891119911
=2120583 (120597119908120597119910)
119910=0
120588119880119890
2= minusRe
119909
12
12058212
119860minus12
119889119866 (0)
119889120578
119873119906 =minus119909 (120597119879120597119910)
119910=0
(119879119882
minus 119879infin)
= minusRe119909
12
119860minus12
119889119867 (0)
119889120578
(13)
Here Re119909
= 119880119890119909] = 1198601199092]119905 is the local Reynolds number
The set of ordinary differential equations (8)ndash(11) are highlynonlinear and purely analytical solutions are difficult if notintractable An efficient homotopy analysismethod (HAM) istherefore adopted Solutions are validated with the Adomiandecompositionmethod (ADM) Although full solutions weregiven based on a shooting algorithm in Malvandi [37]unfortunately the exact data needed for a comparison is notavailable in that work Therefore another objective of thepresent study is to provide a dual approach for validatedsolutions both with HAM and ADM techniques in order todocument correct solutions for other researchers to utilizeThis therefore allows future researchers who may wish toextend the model to for example magnetohydrodynamicsproper benchmark data with which to validate their ownnumerical methods
3 Homotopy Analysis Method (HAM)Simulation
HAM has emerged as a significant alternative to conven-tional numerical methods for nonlinear systems of partialor ordinary differential equations Liao [42] employed thebasic ideas of homotopy in topology to propose an alterna-tive and general analytical-numerical method for nonlinearproblems namely the Homotopy Analysis Method (HAM)The validity of HAM is independent of whether or notthere exist small parameters in the considered equation(s)Therefore HAM can overcome the foregoing restrictions ofperturbation methods In recent years HAM has been suc-cessfully employed to solvemany types of nonlinear problemsin engineering sciences including thin membrane reaction-diffusion phenomena [43] and porousmedia convection [44]We provided details of the application of HAM to the systemof (8)ndash(11) in the next section
31 Homotopy Analysis Method (HAM) We write the initialguesses and linear operators as
1198650(120578) = 120578 minus 1 + 119890
minus120578
1198660(120578) = 119890
minus120578
1198670(120578) = 119890
minus120578
1198730(120578) = 119890
minus120578
(14a)
International Journal of Engineering Mathematics 5
119871119865=
1198893119865
1198891205783minus
119889119865
119889120578
119871119866
=1198892119866
1198891205782minus 119866
119871119867
=1198892119867
1198891205782minus 119867
119871119873
=1198892119873
1198891205782minus 119873
(14b)
with the following properties
119871119865(1198621+ 1198622119890120578
+ 1198623119890minus120578
) = 0
119871119866(1198624119890120578
+ 1198625119890minus120578
) = 0
119871119866(1198626119890120578
+ 1198627119890minus120578
) = 0
119871119873
(1198628119890120578
+ 1198629119890minus120578
) = 0
(15)
where 119862119894(119894 = 1ndash9) are arbitrary constants Let 119902 isin [0 1]
represent an embedding parameter and ℏ119865 ℏ119866 ℏ119867 ℏ119873denote
the nonzero auxiliary linear operators and construct thefollowing zeroth-order deformation equations
(1 minus 119902) 119871119891[119865 (120578 119902) minus 119865
0(120578)] = 119902ℏ
119865119873lowast
119865[119865 (120578 119902)] (16)
(1 minus 119902) 119871119866[119866 (120578 119902) minus 119866
0(120578)]
= 119902ℏ119866119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
(17)
(1 minus 119902) 119871119867[ (120578 119902) minus 119867
0(120578)]
= 119902ℏ119867119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
(18)
(1 minus 119902) 119871119873
[ (120578 119902) minus 1198730(120578)]
= 119902ℏ119873119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
(19)
subject to the boundary conditions
119865 (0 119902) = 0
1198651015840
(0 119902) = 0
1198651015840
(infin 119902) = 1
119866 (0 119902) = 1
119866 (infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(20)
where the nonlinear operators are defined as
119873lowast
119865[119865 (120578 119902) 119866 (120578 119902)]
=1205973
119865 (120578 119902)
1205971205783+
1205972
119865 (120578 119902)
1205971205782(119860119865 (120578 119902) +
120578
2)
+120597119865 (120578 119902)
120597120578(1 minus 119860
120597119865 (120578 119902)
120597120578)
+ 119860 (120582119866 (120578 119902)2
+ 1) minus 1
119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
=1205972119866 (120578 119902)
1205971205782+
119889119866 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+ 119866 (120578 119902) (1 minus 2119860119889119865 (120578 119902)
119889120578)
119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
=1
Pr1205972 (120578 119902)
1205971205782+Nb
119889 (120578 119902)
119889120578
119889 (120578 119902)
119889120578
+Nt(119889 (120578 119902)
119889120578)
2
+ 119860119865 (120578 119902)119889 (120578 119902)
119889120578
+120578
2
119889 (120578 119902)
119889120578
119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
=1205972 (120578 119902)
1205971205782+ Le
119889 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+NtNb
1198892 (120578 119902)
1198891205782
(21)
Setting 119902 = 0 and 119902 = 1 we obtain from (16)ndash(18)
119865 (120578 0) = 1198650(120578)
119865 (120578 1) = 119865 (120578)
119866 (120578 0) = 1198660(120578)
119866 (120578 1) = 119866 (120578)
(120578 0) = 1198670(120578)
(120578 1) = 119867 (120578)
(120578 0) = 1198730(120578)
(120578 1) = 119873 (120578)
(22)
6 International Journal of Engineering Mathematics
We further define
119865119898(120578) =
1
119898
120597119898119865 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119866119898(120578) =
1
119898
120597119898119866 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119867119898(120578) =
1
119898
120597119898119867(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119873119898(120578) =
1
119898
120597119898119873(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
(23)
and expanding119865(119902 120578)119866(119902 120578) (119902 120578) and (119902 120578) bymeansof Taylorrsquos theorem with respect to 119902 we obtain
119865 (119902 120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578) 119902119898
119866 (119902 120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578) 119902119898
(119902 120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578) 119902119898
(119902 120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578) 119902119898
(24)
The auxiliary parameters are properly chosen so that series(24) converge at 119902 = 1 and thus
119865 (120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578)
119866 (120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578)
119867 (120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578)
119873 (120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578)
(25)
The resulting problems at the119898th order deformation are
119871119865[119865119898(120578) minus 119883
119898119865119898minus1
(120578)] = ℏ119865119877119865
119898(120578)
119871119866[119866119898(120578) minus 119883
119898119866119898minus1
(120578)] = ℏ119866119877119866
119898(120578)
119871119867[119867119898(120578) minus 119883
119898119867119898minus1
(120578)] = ℏ119867119877119867
119898(120578)
119871119873
[119873119898(120578) minus 119883
119898119873119898minus1
(120578)] = ℏ119873119877119873
119898(120578)
(26)
subject to boundary conditions
119865119898(0) = 0
1198651015840
119898(0) = 0
1198651015840
119898(infin) = 0
119866119898(0) = 0
119866119898(infin) = 0
119867119898(0) = 0
119867119898(infin) = 0
119873119898(0) = 0
119873119898(infin) = 0
(27)
119877119865
119898(120578) = 119865
101584010158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
11986511989611986510158401015840
119898minus1minus119896+
120578
211986510158401015840
119898minus1
minus 119860
119898minus1
sum119896=0
1198651015840
1198961198651015840
119898minus1minus119896+ (119860 minus 1) (1 minus 120594
119898minus1)
+ 119860120582
119898minus1
sum119896=0
119866119896119866119898minus1minus119896
+ 1198651015840
119898minus1
(28)
119877119866
119898(120578) = 119866
10158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
1198651198961198661015840
119898minus1minus119896+
120578
21198661015840
119898minus1
minus 2119860
119898minus1
sum119896=0
1198651015840
119896119866119898minus1minus119896
+ 119866119898minus1
(29)
119877119867
119898(120578) =
1
Pr11986710158401015840
119898minus1(120578) +Nb
119898minus1
sum119896=0
1198671015840
1198961198731015840
119898minus1minus119896
+ Nt119898minus1
sum119896=0
1198671015840
1198961198671015840
119898minus1minus119896+ 119860
119898minus1
sum119896=0
1198651198961198671015840
119898minus1minus119896
+120578
21198671015840
119898minus1
(30)
119877119873
119898(120578) = 119873
10158401015840
119898minus1(120578) + 119860Le
119898minus1
sum119896=0
1198651198961198731015840
119898minus1minus119896+ Le
120578
21198731015840
119898minus1
+NtNb
11986710158401015840
119898minus1
(31)
119883119898
=
0 119898 le 1
1 119898 gt 1(32)
The general solution of (26) is
119865119898(120578) = 119865
lowast
119898(120578) + 119862
1+ 1198622exp (120578) + 119862
3exp (minus120578)
119866119898(120578) = 119866
lowast
119898(120578) + 119862
4exp (120578) + 119862
5exp (minus120578)
International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Engineering Mathematics
schemes [22] Keller box implicit methods [23] Mathematicaintegration subroutines [24] Blottner difference methods[25] and dual reciprocity boundary element methods [26]These studies have generally examined Brownianmotion andthermophoresis effects for various nanoparticle suspensionsand considered two-dimensional boundary layer channeland cavity flows They have ignored Coriolis body forceeffects which arise in rotational fluid mechanics Externalflows from spinning bodies are significant in many branchesof chemical and industrial engineering including electrolysistreatments [27] and polymer deposition on components [28]In such systems the rotation strongly influences boundarylayer growth and structure on the body periphery This inturn controls heat and mass transfer rates whether flowsare laminar transitional or indeed fully turbulent [29] Thebodies may be conical spherical elliptical disk-shaped andindeed concentric and eccentric systemsManymathematicaland numerical studies of such flows have been presentedFaltas and Saad [30] used a collocation method to analyzesteady axisymmetric flow between two spinning eccentricspheres with a linear slip of Basset-type boundary conditionat both surfaces Andersson and Rousselet [31] studied partialmomentum and thermal slip from a rotating disk witha Runge-Kutta method Niazmand and Renksizbulut [32]employed a finite volume code to investigate unsteady heattransfer and thermal patterns around a rotating sphere (as amodel of a particle) with surface blowing delineating threedistinct wake regimes namely steady and axisymmetricalsteady but nonsymmetrical and unsteady with vortex shed-ding They found that although rotation strongly induceslocal modifications in flow patterns the surface-averagedheat transfer rates were not altered markedly even at largerotational speeds Roy and Anilkumar [33] used the Kellerbox method to simulate transient free and forced convectionboundary layer flow from a rotating cone for the case whenthe free stream angular velocity and the angular velocityof the cone vary arbitrarily with the time Subhashini etal [34] used the Bellman-Kalaba quasilinearization methodto address uniform slot injectionsuction and nonuniformtotal enthalpy wall effects on steady nonsimilar laminarcompressible boundary layer flowover a rotating sphereTheyshowed that greater rotation and total enthalpy at the wallencourages earlier flow separation whereas cooling delaysthis and furthermore that greater Mach number displacesthe point of separation upstream as a result of the adversepressure gradient
The focus of the present work is to analyse rotatingnanofluid boundary layer flows at the stagnation point on theexternal surface of a spinning sphere Computational rotatingnanofluid dynamics has recently attracted some interestsince the deployment of nanofluids in revolving chemicalengineering devices offers significant improvements overexisting designs Rana et al [35] used a variational finiteelement algorithm to study unsteady magnetonanofluidtransport from a rotating stretching continuous sheet Theyshowed that greater rotational parameter reduces primary
y
Nanofluid
Stagnation point
Isothermal sphere
z
x
Ω
r(x)
Uprop
Figure 1 Physical model
and secondary velocities temperature and nanoparticleconcentration They further showed that reduced Nusseltnumber (wall temperature gradient) was suppressed withboth Brownian motion and thermophoresis effects whereasreduced Sherwood number (wall mass transfer gradient) wasenhanced Further studies of swirling nanofluid dynamicshave been reported by Nadeem and Saleem [36] for a verticalcone and Malvandi [37] for stagnation point nanofluidflow from a spinning sphere In the present study we useboth a homotopy analysis method (HAM) and an Ado-mian decomposition method (ADM) to simulate stagnationpoint nanofluid flow from a rotating sphere The influenceof acceleration parameter rotational body force parameterLewis number Brownianmotion number and thermophore-sis parameter on velocity temperature and nanoparticledistributions is examined This work is relevant to coatingapplications in the polymeric industry
2 Mathematical Model
The physical regime under investigation is illustrated inFigure 1 in (119909 119910 119911) coordinate system Transient lami-nar boundary layer flow of an incompressible Newtoniannanofluid is studied in the vicinity of the stagnation pointregion of an isothermal rotating sphere of radius 119903 rotat-ing with angular velocity Ω Soret and Dufour effects areneglected Following Anilkumar and Roy [38] the freestream and angular velocities depend on time in the formof 119880119890(119909 119905) = 119860119909119905 and Ω(119905) = 119861119905 where 119860 and 119861
are arbitrary constants both greater than zero The flowfield is assumed to be axisymmetric and the fluid possessesconstant thermophysical properties with the exception ofthose caused by density changeswhich generate the buoyancyforce under the Boussinesq approximationThe Buonjiornionanofluid model is adopted prioritizing Brownian motionand thermophoresis effects [39] In light of these approx-imations the time-dependent conservation equations formass momentum energy and species (nanoparticle volume
International Journal of Engineering Mathematics 3
fraction) may be presented as follows as documented byMalvandi [37]
120597 (119903119906)
120597119909+
120597 (119903V)120597119910
= 0 (1)
120597119906
120597119905+ 119906
120597119906
120597119909+ V
120597119906
120597119910minus (
1199082
119903)
119889119903
119889119909
=120597119880119890
120597119905+ 119906119890
120597119880119890
120597119909+ ](
1205972119906
1205971199102)
(2)
120597119908
120597119905+ 119906
120597119908
120597119909+ V
120597119908
120597119910+ (
119906119908
119903)
119889119903
119889119909= ](
1205972119908
1205971199102) (3)
120597119879
120597119905+ 119906
120597119879
120597119909+ V
120597119879
120597119910
= 120572(1205972119879
1205971199102) + 120591(119863
119861
120597119862
120597119910
120597119879
120597119910+
119863119879
119879infin
120597119879
120597119910
120597119879
120597119910)
(4)
120597119862
120597119905+ 119906
120597119862
120597119909+ V
120597119862
120597119910= 119863119861(1205972119862
1205971199102) +
119863119879
119879infin
1205972119879
1205971199102 (5)
The appropriate initial and boundary conditions take theform
119906 (0 119909 119910) = 119906119894(119909 119910)
V (0 119909 119910) = V119894(119909 119910)
119908 (0 119909 119910) = 119908119894(119909 119910)
119879 (0 119909 119910) = 119879119894(119909 119910)
119862 (0 119909 119910) = 119862119894(119909 119910)
119906 (119905 119909 0) = 0
V (119905 119909 0) = 0
119908 (0 119909 119910) = Ω (119905 119903) 119903
119879 (119905 119909 0) = 119879119908
119908 (119909 119905infin) = 0
119906 (119905 119909 119908) = 119880119890(119905 119909) =
119860119909
119905
119879 (119905 119909infin) = 119879infin
= const
119862 (119905 119909infin) = 119862infin
= const
(6)
Here 119906 V and 119908 denote velocity components along the 119909119910 and 119911 coordinates where these coordinates are orientatedrespectively from the forward stagnation point along the sur-face (see Figure 1) normal to the surface and in the rotatingdirections respectively 119903(119909) is the radial distance from asurface element to the axis of symmetry 119905 is time 119894 is initialcondition 120572 is the thermal diffusivity of the nanofluid ] iskinematic viscosity of nanofluid 119896 is the thermal conductivityof nanofluid 119863
119861is the Brownian diffusion coefficient (a
measure of the species diffusivity of nanoparticles) 119863119879is
the thermophoretic diffusion coefficient 120591 = (120588119888119901)119901(120588119888119901)119891
defines the ratio of effective heat capacity of the nanoparticles(eg titanium oxide) to the heat capacity of the fluid 119888
119901
is isothermal specific heat capacity and 119879 is the nanofluidtemperature119879
infinis the free stream temperature (at the edge of
the boundary layer) and 119862infin
is the free stream nanoparticleconcentration It is assumed that the base fluid and thenanoparticles are in thermal equilibrium and no slip occursbetween them We note that in (4) which is a statement ofFickrsquos law of mass (species) diffusion for nanoparticles thefirst term on the left hand side is the transient concentrationgradient and the second and third terms are the convectivemass transfer terms The first term on the right hand sidedenotes the species diffusion and the last term is the relativecontribution of thermophoresis to Brownian motion Theseeffects have also been considered in detail by Dib et al[40] and Gupta and Saha Ray [41] The boundary valueproblem defined in terms of primitive variables may besolved using numerical methods However to provide a moreamenable solution and one which enables the use of scalingparameters it is advantageous to introduce a set of similaritytransformation variables We define the following
120578 = (V119905)12 119910
120595 = 119860119909(V119905)12
119865 (120578)
119908 = (119861119909
119897)119866 (120578)
119903 sim 119909
119889119903
119889119909= 1
120582 = (Ω119905
119886)
119867 (120578) =119879 minus 119879infin
119879119882
minus 119879infin
119873 (120578) =119862 minus 119862
infin
119862119908minus 119862infin
Pr =]120572
Le =]119863119861
Nb =(120588119888119901)119901
119863119861(119862119882
minus 119862infin)
(120588119888119901)119891
]
Nt =(120588119888119901)119901
119863119879(119879119882
minus 119879infin)
(120588119888119901)119891
]119879infin
(7)
Here 119860 is the acceleration (unsteadiness) parameter 120578 isthe transformed normal coordinate (perpendicular to spheresurface) 120582 is the normalized rotation parameter 119886 is sphere
4 International Journal of Engineering Mathematics
radius 120595 is dimensional stream function 119865 is dimensionlessstream function 119866 is the dimensionless secondary velocityfunction 119867 is the dimensionless temperature function 119873 isdimensionless nanoparticle concentration function (volumefraction) Pr is Prandtl number Le is Lewis number Nb isBrownian motion parameter and Nt is the thermophoresisparameter Introducing these transformations into (1)ndash(5)the partial differential boundary layer equations contractto the following nonlinear coupled system of self-similarordinary differential equations namely
1198893119865
1198891205783+
1198892119865
1198891205782(119860119865 +
120578
2) +
119889119865
119889120578(1 minus 119860
119889119865
119889120578)
+ 119860 (1205821198662
+ 1) minus 1 = 0
(8)
1198892119866
1198891205782+
119889119866
119889120578(119860119865 +
120578
2) + 119866(1 minus 2119860
119889119865
119889120578) = 0 (9)
1
Pr1198892119867
1198891205782+ Nb119889119867
119889120578
119889119873
119889120578+Nt(119889119867
119889120578)
2
+ 119860119865119889119867
119889120578
+120578
2
119889119867
119889120578= 0
(10)
1198892119873
1198891205782+ Le119889119873
119889120578(119860119865 +
120578
2) +
NtNb
1198892119867
1198891205782= 0 (11)
The corresponding transformed boundary conditions arespecified as follows
At 120578 = 0
119865 = 0
119889119865
119889120578= 0
119866 = 1
119867 = 1
119873 = 1
(12a)
As 120578 997888rarr infin
119889119865
119889120578= 1
119866 = 0
119867 = 0
119873 = 0
(12b)
In engineering simulations of nanofluid flows not onlythe velocity temperature and nanoparticle volume fractiondistributions but also primary (119909-) skin friction 119862
119891119909and
secondary (119911-) skin friction coefficients 119862119891119911 and local Nusselt
number function are important We define these as
119862119891119909
=2120583 (120597119906120597119910)
119910=0
120588119880119890
2= Re119909
12
119860minus12
1198892119865 (0)
1198891205782
119862119891119911
=2120583 (120597119908120597119910)
119910=0
120588119880119890
2= minusRe
119909
12
12058212
119860minus12
119889119866 (0)
119889120578
119873119906 =minus119909 (120597119879120597119910)
119910=0
(119879119882
minus 119879infin)
= minusRe119909
12
119860minus12
119889119867 (0)
119889120578
(13)
Here Re119909
= 119880119890119909] = 1198601199092]119905 is the local Reynolds number
The set of ordinary differential equations (8)ndash(11) are highlynonlinear and purely analytical solutions are difficult if notintractable An efficient homotopy analysismethod (HAM) istherefore adopted Solutions are validated with the Adomiandecompositionmethod (ADM) Although full solutions weregiven based on a shooting algorithm in Malvandi [37]unfortunately the exact data needed for a comparison is notavailable in that work Therefore another objective of thepresent study is to provide a dual approach for validatedsolutions both with HAM and ADM techniques in order todocument correct solutions for other researchers to utilizeThis therefore allows future researchers who may wish toextend the model to for example magnetohydrodynamicsproper benchmark data with which to validate their ownnumerical methods
3 Homotopy Analysis Method (HAM)Simulation
HAM has emerged as a significant alternative to conven-tional numerical methods for nonlinear systems of partialor ordinary differential equations Liao [42] employed thebasic ideas of homotopy in topology to propose an alterna-tive and general analytical-numerical method for nonlinearproblems namely the Homotopy Analysis Method (HAM)The validity of HAM is independent of whether or notthere exist small parameters in the considered equation(s)Therefore HAM can overcome the foregoing restrictions ofperturbation methods In recent years HAM has been suc-cessfully employed to solvemany types of nonlinear problemsin engineering sciences including thin membrane reaction-diffusion phenomena [43] and porousmedia convection [44]We provided details of the application of HAM to the systemof (8)ndash(11) in the next section
31 Homotopy Analysis Method (HAM) We write the initialguesses and linear operators as
1198650(120578) = 120578 minus 1 + 119890
minus120578
1198660(120578) = 119890
minus120578
1198670(120578) = 119890
minus120578
1198730(120578) = 119890
minus120578
(14a)
International Journal of Engineering Mathematics 5
119871119865=
1198893119865
1198891205783minus
119889119865
119889120578
119871119866
=1198892119866
1198891205782minus 119866
119871119867
=1198892119867
1198891205782minus 119867
119871119873
=1198892119873
1198891205782minus 119873
(14b)
with the following properties
119871119865(1198621+ 1198622119890120578
+ 1198623119890minus120578
) = 0
119871119866(1198624119890120578
+ 1198625119890minus120578
) = 0
119871119866(1198626119890120578
+ 1198627119890minus120578
) = 0
119871119873
(1198628119890120578
+ 1198629119890minus120578
) = 0
(15)
where 119862119894(119894 = 1ndash9) are arbitrary constants Let 119902 isin [0 1]
represent an embedding parameter and ℏ119865 ℏ119866 ℏ119867 ℏ119873denote
the nonzero auxiliary linear operators and construct thefollowing zeroth-order deformation equations
(1 minus 119902) 119871119891[119865 (120578 119902) minus 119865
0(120578)] = 119902ℏ
119865119873lowast
119865[119865 (120578 119902)] (16)
(1 minus 119902) 119871119866[119866 (120578 119902) minus 119866
0(120578)]
= 119902ℏ119866119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
(17)
(1 minus 119902) 119871119867[ (120578 119902) minus 119867
0(120578)]
= 119902ℏ119867119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
(18)
(1 minus 119902) 119871119873
[ (120578 119902) minus 1198730(120578)]
= 119902ℏ119873119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
(19)
subject to the boundary conditions
119865 (0 119902) = 0
1198651015840
(0 119902) = 0
1198651015840
(infin 119902) = 1
119866 (0 119902) = 1
119866 (infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(20)
where the nonlinear operators are defined as
119873lowast
119865[119865 (120578 119902) 119866 (120578 119902)]
=1205973
119865 (120578 119902)
1205971205783+
1205972
119865 (120578 119902)
1205971205782(119860119865 (120578 119902) +
120578
2)
+120597119865 (120578 119902)
120597120578(1 minus 119860
120597119865 (120578 119902)
120597120578)
+ 119860 (120582119866 (120578 119902)2
+ 1) minus 1
119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
=1205972119866 (120578 119902)
1205971205782+
119889119866 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+ 119866 (120578 119902) (1 minus 2119860119889119865 (120578 119902)
119889120578)
119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
=1
Pr1205972 (120578 119902)
1205971205782+Nb
119889 (120578 119902)
119889120578
119889 (120578 119902)
119889120578
+Nt(119889 (120578 119902)
119889120578)
2
+ 119860119865 (120578 119902)119889 (120578 119902)
119889120578
+120578
2
119889 (120578 119902)
119889120578
119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
=1205972 (120578 119902)
1205971205782+ Le
119889 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+NtNb
1198892 (120578 119902)
1198891205782
(21)
Setting 119902 = 0 and 119902 = 1 we obtain from (16)ndash(18)
119865 (120578 0) = 1198650(120578)
119865 (120578 1) = 119865 (120578)
119866 (120578 0) = 1198660(120578)
119866 (120578 1) = 119866 (120578)
(120578 0) = 1198670(120578)
(120578 1) = 119867 (120578)
(120578 0) = 1198730(120578)
(120578 1) = 119873 (120578)
(22)
6 International Journal of Engineering Mathematics
We further define
119865119898(120578) =
1
119898
120597119898119865 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119866119898(120578) =
1
119898
120597119898119866 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119867119898(120578) =
1
119898
120597119898119867(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119873119898(120578) =
1
119898
120597119898119873(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
(23)
and expanding119865(119902 120578)119866(119902 120578) (119902 120578) and (119902 120578) bymeansof Taylorrsquos theorem with respect to 119902 we obtain
119865 (119902 120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578) 119902119898
119866 (119902 120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578) 119902119898
(119902 120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578) 119902119898
(119902 120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578) 119902119898
(24)
The auxiliary parameters are properly chosen so that series(24) converge at 119902 = 1 and thus
119865 (120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578)
119866 (120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578)
119867 (120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578)
119873 (120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578)
(25)
The resulting problems at the119898th order deformation are
119871119865[119865119898(120578) minus 119883
119898119865119898minus1
(120578)] = ℏ119865119877119865
119898(120578)
119871119866[119866119898(120578) minus 119883
119898119866119898minus1
(120578)] = ℏ119866119877119866
119898(120578)
119871119867[119867119898(120578) minus 119883
119898119867119898minus1
(120578)] = ℏ119867119877119867
119898(120578)
119871119873
[119873119898(120578) minus 119883
119898119873119898minus1
(120578)] = ℏ119873119877119873
119898(120578)
(26)
subject to boundary conditions
119865119898(0) = 0
1198651015840
119898(0) = 0
1198651015840
119898(infin) = 0
119866119898(0) = 0
119866119898(infin) = 0
119867119898(0) = 0
119867119898(infin) = 0
119873119898(0) = 0
119873119898(infin) = 0
(27)
119877119865
119898(120578) = 119865
101584010158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
11986511989611986510158401015840
119898minus1minus119896+
120578
211986510158401015840
119898minus1
minus 119860
119898minus1
sum119896=0
1198651015840
1198961198651015840
119898minus1minus119896+ (119860 minus 1) (1 minus 120594
119898minus1)
+ 119860120582
119898minus1
sum119896=0
119866119896119866119898minus1minus119896
+ 1198651015840
119898minus1
(28)
119877119866
119898(120578) = 119866
10158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
1198651198961198661015840
119898minus1minus119896+
120578
21198661015840
119898minus1
minus 2119860
119898minus1
sum119896=0
1198651015840
119896119866119898minus1minus119896
+ 119866119898minus1
(29)
119877119867
119898(120578) =
1
Pr11986710158401015840
119898minus1(120578) +Nb
119898minus1
sum119896=0
1198671015840
1198961198731015840
119898minus1minus119896
+ Nt119898minus1
sum119896=0
1198671015840
1198961198671015840
119898minus1minus119896+ 119860
119898minus1
sum119896=0
1198651198961198671015840
119898minus1minus119896
+120578
21198671015840
119898minus1
(30)
119877119873
119898(120578) = 119873
10158401015840
119898minus1(120578) + 119860Le
119898minus1
sum119896=0
1198651198961198731015840
119898minus1minus119896+ Le
120578
21198731015840
119898minus1
+NtNb
11986710158401015840
119898minus1
(31)
119883119898
=
0 119898 le 1
1 119898 gt 1(32)
The general solution of (26) is
119865119898(120578) = 119865
lowast
119898(120578) + 119862
1+ 1198622exp (120578) + 119862
3exp (minus120578)
119866119898(120578) = 119866
lowast
119898(120578) + 119862
4exp (120578) + 119862
5exp (minus120578)
International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 3
fraction) may be presented as follows as documented byMalvandi [37]
120597 (119903119906)
120597119909+
120597 (119903V)120597119910
= 0 (1)
120597119906
120597119905+ 119906
120597119906
120597119909+ V
120597119906
120597119910minus (
1199082
119903)
119889119903
119889119909
=120597119880119890
120597119905+ 119906119890
120597119880119890
120597119909+ ](
1205972119906
1205971199102)
(2)
120597119908
120597119905+ 119906
120597119908
120597119909+ V
120597119908
120597119910+ (
119906119908
119903)
119889119903
119889119909= ](
1205972119908
1205971199102) (3)
120597119879
120597119905+ 119906
120597119879
120597119909+ V
120597119879
120597119910
= 120572(1205972119879
1205971199102) + 120591(119863
119861
120597119862
120597119910
120597119879
120597119910+
119863119879
119879infin
120597119879
120597119910
120597119879
120597119910)
(4)
120597119862
120597119905+ 119906
120597119862
120597119909+ V
120597119862
120597119910= 119863119861(1205972119862
1205971199102) +
119863119879
119879infin
1205972119879
1205971199102 (5)
The appropriate initial and boundary conditions take theform
119906 (0 119909 119910) = 119906119894(119909 119910)
V (0 119909 119910) = V119894(119909 119910)
119908 (0 119909 119910) = 119908119894(119909 119910)
119879 (0 119909 119910) = 119879119894(119909 119910)
119862 (0 119909 119910) = 119862119894(119909 119910)
119906 (119905 119909 0) = 0
V (119905 119909 0) = 0
119908 (0 119909 119910) = Ω (119905 119903) 119903
119879 (119905 119909 0) = 119879119908
119908 (119909 119905infin) = 0
119906 (119905 119909 119908) = 119880119890(119905 119909) =
119860119909
119905
119879 (119905 119909infin) = 119879infin
= const
119862 (119905 119909infin) = 119862infin
= const
(6)
Here 119906 V and 119908 denote velocity components along the 119909119910 and 119911 coordinates where these coordinates are orientatedrespectively from the forward stagnation point along the sur-face (see Figure 1) normal to the surface and in the rotatingdirections respectively 119903(119909) is the radial distance from asurface element to the axis of symmetry 119905 is time 119894 is initialcondition 120572 is the thermal diffusivity of the nanofluid ] iskinematic viscosity of nanofluid 119896 is the thermal conductivityof nanofluid 119863
119861is the Brownian diffusion coefficient (a
measure of the species diffusivity of nanoparticles) 119863119879is
the thermophoretic diffusion coefficient 120591 = (120588119888119901)119901(120588119888119901)119891
defines the ratio of effective heat capacity of the nanoparticles(eg titanium oxide) to the heat capacity of the fluid 119888
119901
is isothermal specific heat capacity and 119879 is the nanofluidtemperature119879
infinis the free stream temperature (at the edge of
the boundary layer) and 119862infin
is the free stream nanoparticleconcentration It is assumed that the base fluid and thenanoparticles are in thermal equilibrium and no slip occursbetween them We note that in (4) which is a statement ofFickrsquos law of mass (species) diffusion for nanoparticles thefirst term on the left hand side is the transient concentrationgradient and the second and third terms are the convectivemass transfer terms The first term on the right hand sidedenotes the species diffusion and the last term is the relativecontribution of thermophoresis to Brownian motion Theseeffects have also been considered in detail by Dib et al[40] and Gupta and Saha Ray [41] The boundary valueproblem defined in terms of primitive variables may besolved using numerical methods However to provide a moreamenable solution and one which enables the use of scalingparameters it is advantageous to introduce a set of similaritytransformation variables We define the following
120578 = (V119905)12 119910
120595 = 119860119909(V119905)12
119865 (120578)
119908 = (119861119909
119897)119866 (120578)
119903 sim 119909
119889119903
119889119909= 1
120582 = (Ω119905
119886)
119867 (120578) =119879 minus 119879infin
119879119882
minus 119879infin
119873 (120578) =119862 minus 119862
infin
119862119908minus 119862infin
Pr =]120572
Le =]119863119861
Nb =(120588119888119901)119901
119863119861(119862119882
minus 119862infin)
(120588119888119901)119891
]
Nt =(120588119888119901)119901
119863119879(119879119882
minus 119879infin)
(120588119888119901)119891
]119879infin
(7)
Here 119860 is the acceleration (unsteadiness) parameter 120578 isthe transformed normal coordinate (perpendicular to spheresurface) 120582 is the normalized rotation parameter 119886 is sphere
4 International Journal of Engineering Mathematics
radius 120595 is dimensional stream function 119865 is dimensionlessstream function 119866 is the dimensionless secondary velocityfunction 119867 is the dimensionless temperature function 119873 isdimensionless nanoparticle concentration function (volumefraction) Pr is Prandtl number Le is Lewis number Nb isBrownian motion parameter and Nt is the thermophoresisparameter Introducing these transformations into (1)ndash(5)the partial differential boundary layer equations contractto the following nonlinear coupled system of self-similarordinary differential equations namely
1198893119865
1198891205783+
1198892119865
1198891205782(119860119865 +
120578
2) +
119889119865
119889120578(1 minus 119860
119889119865
119889120578)
+ 119860 (1205821198662
+ 1) minus 1 = 0
(8)
1198892119866
1198891205782+
119889119866
119889120578(119860119865 +
120578
2) + 119866(1 minus 2119860
119889119865
119889120578) = 0 (9)
1
Pr1198892119867
1198891205782+ Nb119889119867
119889120578
119889119873
119889120578+Nt(119889119867
119889120578)
2
+ 119860119865119889119867
119889120578
+120578
2
119889119867
119889120578= 0
(10)
1198892119873
1198891205782+ Le119889119873
119889120578(119860119865 +
120578
2) +
NtNb
1198892119867
1198891205782= 0 (11)
The corresponding transformed boundary conditions arespecified as follows
At 120578 = 0
119865 = 0
119889119865
119889120578= 0
119866 = 1
119867 = 1
119873 = 1
(12a)
As 120578 997888rarr infin
119889119865
119889120578= 1
119866 = 0
119867 = 0
119873 = 0
(12b)
In engineering simulations of nanofluid flows not onlythe velocity temperature and nanoparticle volume fractiondistributions but also primary (119909-) skin friction 119862
119891119909and
secondary (119911-) skin friction coefficients 119862119891119911 and local Nusselt
number function are important We define these as
119862119891119909
=2120583 (120597119906120597119910)
119910=0
120588119880119890
2= Re119909
12
119860minus12
1198892119865 (0)
1198891205782
119862119891119911
=2120583 (120597119908120597119910)
119910=0
120588119880119890
2= minusRe
119909
12
12058212
119860minus12
119889119866 (0)
119889120578
119873119906 =minus119909 (120597119879120597119910)
119910=0
(119879119882
minus 119879infin)
= minusRe119909
12
119860minus12
119889119867 (0)
119889120578
(13)
Here Re119909
= 119880119890119909] = 1198601199092]119905 is the local Reynolds number
The set of ordinary differential equations (8)ndash(11) are highlynonlinear and purely analytical solutions are difficult if notintractable An efficient homotopy analysismethod (HAM) istherefore adopted Solutions are validated with the Adomiandecompositionmethod (ADM) Although full solutions weregiven based on a shooting algorithm in Malvandi [37]unfortunately the exact data needed for a comparison is notavailable in that work Therefore another objective of thepresent study is to provide a dual approach for validatedsolutions both with HAM and ADM techniques in order todocument correct solutions for other researchers to utilizeThis therefore allows future researchers who may wish toextend the model to for example magnetohydrodynamicsproper benchmark data with which to validate their ownnumerical methods
3 Homotopy Analysis Method (HAM)Simulation
HAM has emerged as a significant alternative to conven-tional numerical methods for nonlinear systems of partialor ordinary differential equations Liao [42] employed thebasic ideas of homotopy in topology to propose an alterna-tive and general analytical-numerical method for nonlinearproblems namely the Homotopy Analysis Method (HAM)The validity of HAM is independent of whether or notthere exist small parameters in the considered equation(s)Therefore HAM can overcome the foregoing restrictions ofperturbation methods In recent years HAM has been suc-cessfully employed to solvemany types of nonlinear problemsin engineering sciences including thin membrane reaction-diffusion phenomena [43] and porousmedia convection [44]We provided details of the application of HAM to the systemof (8)ndash(11) in the next section
31 Homotopy Analysis Method (HAM) We write the initialguesses and linear operators as
1198650(120578) = 120578 minus 1 + 119890
minus120578
1198660(120578) = 119890
minus120578
1198670(120578) = 119890
minus120578
1198730(120578) = 119890
minus120578
(14a)
International Journal of Engineering Mathematics 5
119871119865=
1198893119865
1198891205783minus
119889119865
119889120578
119871119866
=1198892119866
1198891205782minus 119866
119871119867
=1198892119867
1198891205782minus 119867
119871119873
=1198892119873
1198891205782minus 119873
(14b)
with the following properties
119871119865(1198621+ 1198622119890120578
+ 1198623119890minus120578
) = 0
119871119866(1198624119890120578
+ 1198625119890minus120578
) = 0
119871119866(1198626119890120578
+ 1198627119890minus120578
) = 0
119871119873
(1198628119890120578
+ 1198629119890minus120578
) = 0
(15)
where 119862119894(119894 = 1ndash9) are arbitrary constants Let 119902 isin [0 1]
represent an embedding parameter and ℏ119865 ℏ119866 ℏ119867 ℏ119873denote
the nonzero auxiliary linear operators and construct thefollowing zeroth-order deformation equations
(1 minus 119902) 119871119891[119865 (120578 119902) minus 119865
0(120578)] = 119902ℏ
119865119873lowast
119865[119865 (120578 119902)] (16)
(1 minus 119902) 119871119866[119866 (120578 119902) minus 119866
0(120578)]
= 119902ℏ119866119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
(17)
(1 minus 119902) 119871119867[ (120578 119902) minus 119867
0(120578)]
= 119902ℏ119867119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
(18)
(1 minus 119902) 119871119873
[ (120578 119902) minus 1198730(120578)]
= 119902ℏ119873119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
(19)
subject to the boundary conditions
119865 (0 119902) = 0
1198651015840
(0 119902) = 0
1198651015840
(infin 119902) = 1
119866 (0 119902) = 1
119866 (infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(20)
where the nonlinear operators are defined as
119873lowast
119865[119865 (120578 119902) 119866 (120578 119902)]
=1205973
119865 (120578 119902)
1205971205783+
1205972
119865 (120578 119902)
1205971205782(119860119865 (120578 119902) +
120578
2)
+120597119865 (120578 119902)
120597120578(1 minus 119860
120597119865 (120578 119902)
120597120578)
+ 119860 (120582119866 (120578 119902)2
+ 1) minus 1
119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
=1205972119866 (120578 119902)
1205971205782+
119889119866 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+ 119866 (120578 119902) (1 minus 2119860119889119865 (120578 119902)
119889120578)
119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
=1
Pr1205972 (120578 119902)
1205971205782+Nb
119889 (120578 119902)
119889120578
119889 (120578 119902)
119889120578
+Nt(119889 (120578 119902)
119889120578)
2
+ 119860119865 (120578 119902)119889 (120578 119902)
119889120578
+120578
2
119889 (120578 119902)
119889120578
119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
=1205972 (120578 119902)
1205971205782+ Le
119889 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+NtNb
1198892 (120578 119902)
1198891205782
(21)
Setting 119902 = 0 and 119902 = 1 we obtain from (16)ndash(18)
119865 (120578 0) = 1198650(120578)
119865 (120578 1) = 119865 (120578)
119866 (120578 0) = 1198660(120578)
119866 (120578 1) = 119866 (120578)
(120578 0) = 1198670(120578)
(120578 1) = 119867 (120578)
(120578 0) = 1198730(120578)
(120578 1) = 119873 (120578)
(22)
6 International Journal of Engineering Mathematics
We further define
119865119898(120578) =
1
119898
120597119898119865 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119866119898(120578) =
1
119898
120597119898119866 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119867119898(120578) =
1
119898
120597119898119867(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119873119898(120578) =
1
119898
120597119898119873(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
(23)
and expanding119865(119902 120578)119866(119902 120578) (119902 120578) and (119902 120578) bymeansof Taylorrsquos theorem with respect to 119902 we obtain
119865 (119902 120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578) 119902119898
119866 (119902 120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578) 119902119898
(119902 120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578) 119902119898
(119902 120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578) 119902119898
(24)
The auxiliary parameters are properly chosen so that series(24) converge at 119902 = 1 and thus
119865 (120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578)
119866 (120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578)
119867 (120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578)
119873 (120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578)
(25)
The resulting problems at the119898th order deformation are
119871119865[119865119898(120578) minus 119883
119898119865119898minus1
(120578)] = ℏ119865119877119865
119898(120578)
119871119866[119866119898(120578) minus 119883
119898119866119898minus1
(120578)] = ℏ119866119877119866
119898(120578)
119871119867[119867119898(120578) minus 119883
119898119867119898minus1
(120578)] = ℏ119867119877119867
119898(120578)
119871119873
[119873119898(120578) minus 119883
119898119873119898minus1
(120578)] = ℏ119873119877119873
119898(120578)
(26)
subject to boundary conditions
119865119898(0) = 0
1198651015840
119898(0) = 0
1198651015840
119898(infin) = 0
119866119898(0) = 0
119866119898(infin) = 0
119867119898(0) = 0
119867119898(infin) = 0
119873119898(0) = 0
119873119898(infin) = 0
(27)
119877119865
119898(120578) = 119865
101584010158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
11986511989611986510158401015840
119898minus1minus119896+
120578
211986510158401015840
119898minus1
minus 119860
119898minus1
sum119896=0
1198651015840
1198961198651015840
119898minus1minus119896+ (119860 minus 1) (1 minus 120594
119898minus1)
+ 119860120582
119898minus1
sum119896=0
119866119896119866119898minus1minus119896
+ 1198651015840
119898minus1
(28)
119877119866
119898(120578) = 119866
10158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
1198651198961198661015840
119898minus1minus119896+
120578
21198661015840
119898minus1
minus 2119860
119898minus1
sum119896=0
1198651015840
119896119866119898minus1minus119896
+ 119866119898minus1
(29)
119877119867
119898(120578) =
1
Pr11986710158401015840
119898minus1(120578) +Nb
119898minus1
sum119896=0
1198671015840
1198961198731015840
119898minus1minus119896
+ Nt119898minus1
sum119896=0
1198671015840
1198961198671015840
119898minus1minus119896+ 119860
119898minus1
sum119896=0
1198651198961198671015840
119898minus1minus119896
+120578
21198671015840
119898minus1
(30)
119877119873
119898(120578) = 119873
10158401015840
119898minus1(120578) + 119860Le
119898minus1
sum119896=0
1198651198961198731015840
119898minus1minus119896+ Le
120578
21198731015840
119898minus1
+NtNb
11986710158401015840
119898minus1
(31)
119883119898
=
0 119898 le 1
1 119898 gt 1(32)
The general solution of (26) is
119865119898(120578) = 119865
lowast
119898(120578) + 119862
1+ 1198622exp (120578) + 119862
3exp (minus120578)
119866119898(120578) = 119866
lowast
119898(120578) + 119862
4exp (120578) + 119862
5exp (minus120578)
International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Engineering Mathematics
radius 120595 is dimensional stream function 119865 is dimensionlessstream function 119866 is the dimensionless secondary velocityfunction 119867 is the dimensionless temperature function 119873 isdimensionless nanoparticle concentration function (volumefraction) Pr is Prandtl number Le is Lewis number Nb isBrownian motion parameter and Nt is the thermophoresisparameter Introducing these transformations into (1)ndash(5)the partial differential boundary layer equations contractto the following nonlinear coupled system of self-similarordinary differential equations namely
1198893119865
1198891205783+
1198892119865
1198891205782(119860119865 +
120578
2) +
119889119865
119889120578(1 minus 119860
119889119865
119889120578)
+ 119860 (1205821198662
+ 1) minus 1 = 0
(8)
1198892119866
1198891205782+
119889119866
119889120578(119860119865 +
120578
2) + 119866(1 minus 2119860
119889119865
119889120578) = 0 (9)
1
Pr1198892119867
1198891205782+ Nb119889119867
119889120578
119889119873
119889120578+Nt(119889119867
119889120578)
2
+ 119860119865119889119867
119889120578
+120578
2
119889119867
119889120578= 0
(10)
1198892119873
1198891205782+ Le119889119873
119889120578(119860119865 +
120578
2) +
NtNb
1198892119867
1198891205782= 0 (11)
The corresponding transformed boundary conditions arespecified as follows
At 120578 = 0
119865 = 0
119889119865
119889120578= 0
119866 = 1
119867 = 1
119873 = 1
(12a)
As 120578 997888rarr infin
119889119865
119889120578= 1
119866 = 0
119867 = 0
119873 = 0
(12b)
In engineering simulations of nanofluid flows not onlythe velocity temperature and nanoparticle volume fractiondistributions but also primary (119909-) skin friction 119862
119891119909and
secondary (119911-) skin friction coefficients 119862119891119911 and local Nusselt
number function are important We define these as
119862119891119909
=2120583 (120597119906120597119910)
119910=0
120588119880119890
2= Re119909
12
119860minus12
1198892119865 (0)
1198891205782
119862119891119911
=2120583 (120597119908120597119910)
119910=0
120588119880119890
2= minusRe
119909
12
12058212
119860minus12
119889119866 (0)
119889120578
119873119906 =minus119909 (120597119879120597119910)
119910=0
(119879119882
minus 119879infin)
= minusRe119909
12
119860minus12
119889119867 (0)
119889120578
(13)
Here Re119909
= 119880119890119909] = 1198601199092]119905 is the local Reynolds number
The set of ordinary differential equations (8)ndash(11) are highlynonlinear and purely analytical solutions are difficult if notintractable An efficient homotopy analysismethod (HAM) istherefore adopted Solutions are validated with the Adomiandecompositionmethod (ADM) Although full solutions weregiven based on a shooting algorithm in Malvandi [37]unfortunately the exact data needed for a comparison is notavailable in that work Therefore another objective of thepresent study is to provide a dual approach for validatedsolutions both with HAM and ADM techniques in order todocument correct solutions for other researchers to utilizeThis therefore allows future researchers who may wish toextend the model to for example magnetohydrodynamicsproper benchmark data with which to validate their ownnumerical methods
3 Homotopy Analysis Method (HAM)Simulation
HAM has emerged as a significant alternative to conven-tional numerical methods for nonlinear systems of partialor ordinary differential equations Liao [42] employed thebasic ideas of homotopy in topology to propose an alterna-tive and general analytical-numerical method for nonlinearproblems namely the Homotopy Analysis Method (HAM)The validity of HAM is independent of whether or notthere exist small parameters in the considered equation(s)Therefore HAM can overcome the foregoing restrictions ofperturbation methods In recent years HAM has been suc-cessfully employed to solvemany types of nonlinear problemsin engineering sciences including thin membrane reaction-diffusion phenomena [43] and porousmedia convection [44]We provided details of the application of HAM to the systemof (8)ndash(11) in the next section
31 Homotopy Analysis Method (HAM) We write the initialguesses and linear operators as
1198650(120578) = 120578 minus 1 + 119890
minus120578
1198660(120578) = 119890
minus120578
1198670(120578) = 119890
minus120578
1198730(120578) = 119890
minus120578
(14a)
International Journal of Engineering Mathematics 5
119871119865=
1198893119865
1198891205783minus
119889119865
119889120578
119871119866
=1198892119866
1198891205782minus 119866
119871119867
=1198892119867
1198891205782minus 119867
119871119873
=1198892119873
1198891205782minus 119873
(14b)
with the following properties
119871119865(1198621+ 1198622119890120578
+ 1198623119890minus120578
) = 0
119871119866(1198624119890120578
+ 1198625119890minus120578
) = 0
119871119866(1198626119890120578
+ 1198627119890minus120578
) = 0
119871119873
(1198628119890120578
+ 1198629119890minus120578
) = 0
(15)
where 119862119894(119894 = 1ndash9) are arbitrary constants Let 119902 isin [0 1]
represent an embedding parameter and ℏ119865 ℏ119866 ℏ119867 ℏ119873denote
the nonzero auxiliary linear operators and construct thefollowing zeroth-order deformation equations
(1 minus 119902) 119871119891[119865 (120578 119902) minus 119865
0(120578)] = 119902ℏ
119865119873lowast
119865[119865 (120578 119902)] (16)
(1 minus 119902) 119871119866[119866 (120578 119902) minus 119866
0(120578)]
= 119902ℏ119866119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
(17)
(1 minus 119902) 119871119867[ (120578 119902) minus 119867
0(120578)]
= 119902ℏ119867119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
(18)
(1 minus 119902) 119871119873
[ (120578 119902) minus 1198730(120578)]
= 119902ℏ119873119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
(19)
subject to the boundary conditions
119865 (0 119902) = 0
1198651015840
(0 119902) = 0
1198651015840
(infin 119902) = 1
119866 (0 119902) = 1
119866 (infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(20)
where the nonlinear operators are defined as
119873lowast
119865[119865 (120578 119902) 119866 (120578 119902)]
=1205973
119865 (120578 119902)
1205971205783+
1205972
119865 (120578 119902)
1205971205782(119860119865 (120578 119902) +
120578
2)
+120597119865 (120578 119902)
120597120578(1 minus 119860
120597119865 (120578 119902)
120597120578)
+ 119860 (120582119866 (120578 119902)2
+ 1) minus 1
119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
=1205972119866 (120578 119902)
1205971205782+
119889119866 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+ 119866 (120578 119902) (1 minus 2119860119889119865 (120578 119902)
119889120578)
119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
=1
Pr1205972 (120578 119902)
1205971205782+Nb
119889 (120578 119902)
119889120578
119889 (120578 119902)
119889120578
+Nt(119889 (120578 119902)
119889120578)
2
+ 119860119865 (120578 119902)119889 (120578 119902)
119889120578
+120578
2
119889 (120578 119902)
119889120578
119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
=1205972 (120578 119902)
1205971205782+ Le
119889 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+NtNb
1198892 (120578 119902)
1198891205782
(21)
Setting 119902 = 0 and 119902 = 1 we obtain from (16)ndash(18)
119865 (120578 0) = 1198650(120578)
119865 (120578 1) = 119865 (120578)
119866 (120578 0) = 1198660(120578)
119866 (120578 1) = 119866 (120578)
(120578 0) = 1198670(120578)
(120578 1) = 119867 (120578)
(120578 0) = 1198730(120578)
(120578 1) = 119873 (120578)
(22)
6 International Journal of Engineering Mathematics
We further define
119865119898(120578) =
1
119898
120597119898119865 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119866119898(120578) =
1
119898
120597119898119866 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119867119898(120578) =
1
119898
120597119898119867(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119873119898(120578) =
1
119898
120597119898119873(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
(23)
and expanding119865(119902 120578)119866(119902 120578) (119902 120578) and (119902 120578) bymeansof Taylorrsquos theorem with respect to 119902 we obtain
119865 (119902 120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578) 119902119898
119866 (119902 120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578) 119902119898
(119902 120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578) 119902119898
(119902 120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578) 119902119898
(24)
The auxiliary parameters are properly chosen so that series(24) converge at 119902 = 1 and thus
119865 (120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578)
119866 (120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578)
119867 (120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578)
119873 (120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578)
(25)
The resulting problems at the119898th order deformation are
119871119865[119865119898(120578) minus 119883
119898119865119898minus1
(120578)] = ℏ119865119877119865
119898(120578)
119871119866[119866119898(120578) minus 119883
119898119866119898minus1
(120578)] = ℏ119866119877119866
119898(120578)
119871119867[119867119898(120578) minus 119883
119898119867119898minus1
(120578)] = ℏ119867119877119867
119898(120578)
119871119873
[119873119898(120578) minus 119883
119898119873119898minus1
(120578)] = ℏ119873119877119873
119898(120578)
(26)
subject to boundary conditions
119865119898(0) = 0
1198651015840
119898(0) = 0
1198651015840
119898(infin) = 0
119866119898(0) = 0
119866119898(infin) = 0
119867119898(0) = 0
119867119898(infin) = 0
119873119898(0) = 0
119873119898(infin) = 0
(27)
119877119865
119898(120578) = 119865
101584010158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
11986511989611986510158401015840
119898minus1minus119896+
120578
211986510158401015840
119898minus1
minus 119860
119898minus1
sum119896=0
1198651015840
1198961198651015840
119898minus1minus119896+ (119860 minus 1) (1 minus 120594
119898minus1)
+ 119860120582
119898minus1
sum119896=0
119866119896119866119898minus1minus119896
+ 1198651015840
119898minus1
(28)
119877119866
119898(120578) = 119866
10158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
1198651198961198661015840
119898minus1minus119896+
120578
21198661015840
119898minus1
minus 2119860
119898minus1
sum119896=0
1198651015840
119896119866119898minus1minus119896
+ 119866119898minus1
(29)
119877119867
119898(120578) =
1
Pr11986710158401015840
119898minus1(120578) +Nb
119898minus1
sum119896=0
1198671015840
1198961198731015840
119898minus1minus119896
+ Nt119898minus1
sum119896=0
1198671015840
1198961198671015840
119898minus1minus119896+ 119860
119898minus1
sum119896=0
1198651198961198671015840
119898minus1minus119896
+120578
21198671015840
119898minus1
(30)
119877119873
119898(120578) = 119873
10158401015840
119898minus1(120578) + 119860Le
119898minus1
sum119896=0
1198651198961198731015840
119898minus1minus119896+ Le
120578
21198731015840
119898minus1
+NtNb
11986710158401015840
119898minus1
(31)
119883119898
=
0 119898 le 1
1 119898 gt 1(32)
The general solution of (26) is
119865119898(120578) = 119865
lowast
119898(120578) + 119862
1+ 1198622exp (120578) + 119862
3exp (minus120578)
119866119898(120578) = 119866
lowast
119898(120578) + 119862
4exp (120578) + 119862
5exp (minus120578)
International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 5
119871119865=
1198893119865
1198891205783minus
119889119865
119889120578
119871119866
=1198892119866
1198891205782minus 119866
119871119867
=1198892119867
1198891205782minus 119867
119871119873
=1198892119873
1198891205782minus 119873
(14b)
with the following properties
119871119865(1198621+ 1198622119890120578
+ 1198623119890minus120578
) = 0
119871119866(1198624119890120578
+ 1198625119890minus120578
) = 0
119871119866(1198626119890120578
+ 1198627119890minus120578
) = 0
119871119873
(1198628119890120578
+ 1198629119890minus120578
) = 0
(15)
where 119862119894(119894 = 1ndash9) are arbitrary constants Let 119902 isin [0 1]
represent an embedding parameter and ℏ119865 ℏ119866 ℏ119867 ℏ119873denote
the nonzero auxiliary linear operators and construct thefollowing zeroth-order deformation equations
(1 minus 119902) 119871119891[119865 (120578 119902) minus 119865
0(120578)] = 119902ℏ
119865119873lowast
119865[119865 (120578 119902)] (16)
(1 minus 119902) 119871119866[119866 (120578 119902) minus 119866
0(120578)]
= 119902ℏ119866119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
(17)
(1 minus 119902) 119871119867[ (120578 119902) minus 119867
0(120578)]
= 119902ℏ119867119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
(18)
(1 minus 119902) 119871119873
[ (120578 119902) minus 1198730(120578)]
= 119902ℏ119873119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
(19)
subject to the boundary conditions
119865 (0 119902) = 0
1198651015840
(0 119902) = 0
1198651015840
(infin 119902) = 1
119866 (0 119902) = 1
119866 (infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(0 119902) = 1
(infin 119902) = 0
(20)
where the nonlinear operators are defined as
119873lowast
119865[119865 (120578 119902) 119866 (120578 119902)]
=1205973
119865 (120578 119902)
1205971205783+
1205972
119865 (120578 119902)
1205971205782(119860119865 (120578 119902) +
120578
2)
+120597119865 (120578 119902)
120597120578(1 minus 119860
120597119865 (120578 119902)
120597120578)
+ 119860 (120582119866 (120578 119902)2
+ 1) minus 1
119873lowast
119866[119866 (120578 119902) 119865 (120578 119902)]
=1205972119866 (120578 119902)
1205971205782+
119889119866 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+ 119866 (120578 119902) (1 minus 2119860119889119865 (120578 119902)
119889120578)
119873lowast
119867[ (120578 119902) 119865 (120578 119902)]
=1
Pr1205972 (120578 119902)
1205971205782+Nb
119889 (120578 119902)
119889120578
119889 (120578 119902)
119889120578
+Nt(119889 (120578 119902)
119889120578)
2
+ 119860119865 (120578 119902)119889 (120578 119902)
119889120578
+120578
2
119889 (120578 119902)
119889120578
119873lowast
119873[ (120578 119902) 119865 (120578 119902)]
=1205972 (120578 119902)
1205971205782+ Le
119889 (120578 119902)
119889120578(119860119865 (120578 119902) +
120578
2)
+NtNb
1198892 (120578 119902)
1198891205782
(21)
Setting 119902 = 0 and 119902 = 1 we obtain from (16)ndash(18)
119865 (120578 0) = 1198650(120578)
119865 (120578 1) = 119865 (120578)
119866 (120578 0) = 1198660(120578)
119866 (120578 1) = 119866 (120578)
(120578 0) = 1198670(120578)
(120578 1) = 119867 (120578)
(120578 0) = 1198730(120578)
(120578 1) = 119873 (120578)
(22)
6 International Journal of Engineering Mathematics
We further define
119865119898(120578) =
1
119898
120597119898119865 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119866119898(120578) =
1
119898
120597119898119866 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119867119898(120578) =
1
119898
120597119898119867(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119873119898(120578) =
1
119898
120597119898119873(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
(23)
and expanding119865(119902 120578)119866(119902 120578) (119902 120578) and (119902 120578) bymeansof Taylorrsquos theorem with respect to 119902 we obtain
119865 (119902 120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578) 119902119898
119866 (119902 120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578) 119902119898
(119902 120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578) 119902119898
(119902 120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578) 119902119898
(24)
The auxiliary parameters are properly chosen so that series(24) converge at 119902 = 1 and thus
119865 (120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578)
119866 (120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578)
119867 (120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578)
119873 (120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578)
(25)
The resulting problems at the119898th order deformation are
119871119865[119865119898(120578) minus 119883
119898119865119898minus1
(120578)] = ℏ119865119877119865
119898(120578)
119871119866[119866119898(120578) minus 119883
119898119866119898minus1
(120578)] = ℏ119866119877119866
119898(120578)
119871119867[119867119898(120578) minus 119883
119898119867119898minus1
(120578)] = ℏ119867119877119867
119898(120578)
119871119873
[119873119898(120578) minus 119883
119898119873119898minus1
(120578)] = ℏ119873119877119873
119898(120578)
(26)
subject to boundary conditions
119865119898(0) = 0
1198651015840
119898(0) = 0
1198651015840
119898(infin) = 0
119866119898(0) = 0
119866119898(infin) = 0
119867119898(0) = 0
119867119898(infin) = 0
119873119898(0) = 0
119873119898(infin) = 0
(27)
119877119865
119898(120578) = 119865
101584010158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
11986511989611986510158401015840
119898minus1minus119896+
120578
211986510158401015840
119898minus1
minus 119860
119898minus1
sum119896=0
1198651015840
1198961198651015840
119898minus1minus119896+ (119860 minus 1) (1 minus 120594
119898minus1)
+ 119860120582
119898minus1
sum119896=0
119866119896119866119898minus1minus119896
+ 1198651015840
119898minus1
(28)
119877119866
119898(120578) = 119866
10158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
1198651198961198661015840
119898minus1minus119896+
120578
21198661015840
119898minus1
minus 2119860
119898minus1
sum119896=0
1198651015840
119896119866119898minus1minus119896
+ 119866119898minus1
(29)
119877119867
119898(120578) =
1
Pr11986710158401015840
119898minus1(120578) +Nb
119898minus1
sum119896=0
1198671015840
1198961198731015840
119898minus1minus119896
+ Nt119898minus1
sum119896=0
1198671015840
1198961198671015840
119898minus1minus119896+ 119860
119898minus1
sum119896=0
1198651198961198671015840
119898minus1minus119896
+120578
21198671015840
119898minus1
(30)
119877119873
119898(120578) = 119873
10158401015840
119898minus1(120578) + 119860Le
119898minus1
sum119896=0
1198651198961198731015840
119898minus1minus119896+ Le
120578
21198731015840
119898minus1
+NtNb
11986710158401015840
119898minus1
(31)
119883119898
=
0 119898 le 1
1 119898 gt 1(32)
The general solution of (26) is
119865119898(120578) = 119865
lowast
119898(120578) + 119862
1+ 1198622exp (120578) + 119862
3exp (minus120578)
119866119898(120578) = 119866
lowast
119898(120578) + 119862
4exp (120578) + 119862
5exp (minus120578)
International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Engineering Mathematics
We further define
119865119898(120578) =
1
119898
120597119898119865 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119866119898(120578) =
1
119898
120597119898119866 (120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119867119898(120578) =
1
119898
120597119898119867(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
119873119898(120578) =
1
119898
120597119898119873(120578 119902)
120597119902119898
100381610038161003816100381610038161003816100381610038161003816119902=0
(23)
and expanding119865(119902 120578)119866(119902 120578) (119902 120578) and (119902 120578) bymeansof Taylorrsquos theorem with respect to 119902 we obtain
119865 (119902 120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578) 119902119898
119866 (119902 120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578) 119902119898
(119902 120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578) 119902119898
(119902 120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578) 119902119898
(24)
The auxiliary parameters are properly chosen so that series(24) converge at 119902 = 1 and thus
119865 (120578) = 1198650(120578) +
+infin
sum119898=1
119865119898(120578)
119866 (120578) = 1198660(120578) +
+infin
sum119898=1
119866119898(120578)
119867 (120578) = 1198670(120578) +
+infin
sum119898=1
119867119898(120578)
119873 (120578) = 1198730(120578) +
+infin
sum119898=1
119873119898(120578)
(25)
The resulting problems at the119898th order deformation are
119871119865[119865119898(120578) minus 119883
119898119865119898minus1
(120578)] = ℏ119865119877119865
119898(120578)
119871119866[119866119898(120578) minus 119883
119898119866119898minus1
(120578)] = ℏ119866119877119866
119898(120578)
119871119867[119867119898(120578) minus 119883
119898119867119898minus1
(120578)] = ℏ119867119877119867
119898(120578)
119871119873
[119873119898(120578) minus 119883
119898119873119898minus1
(120578)] = ℏ119873119877119873
119898(120578)
(26)
subject to boundary conditions
119865119898(0) = 0
1198651015840
119898(0) = 0
1198651015840
119898(infin) = 0
119866119898(0) = 0
119866119898(infin) = 0
119867119898(0) = 0
119867119898(infin) = 0
119873119898(0) = 0
119873119898(infin) = 0
(27)
119877119865
119898(120578) = 119865
101584010158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
11986511989611986510158401015840
119898minus1minus119896+
120578
211986510158401015840
119898minus1
minus 119860
119898minus1
sum119896=0
1198651015840
1198961198651015840
119898minus1minus119896+ (119860 minus 1) (1 minus 120594
119898minus1)
+ 119860120582
119898minus1
sum119896=0
119866119896119866119898minus1minus119896
+ 1198651015840
119898minus1
(28)
119877119866
119898(120578) = 119866
10158401015840
119898minus1(120578) + 119860
119898minus1
sum119896=0
1198651198961198661015840
119898minus1minus119896+
120578
21198661015840
119898minus1
minus 2119860
119898minus1
sum119896=0
1198651015840
119896119866119898minus1minus119896
+ 119866119898minus1
(29)
119877119867
119898(120578) =
1
Pr11986710158401015840
119898minus1(120578) +Nb
119898minus1
sum119896=0
1198671015840
1198961198731015840
119898minus1minus119896
+ Nt119898minus1
sum119896=0
1198671015840
1198961198671015840
119898minus1minus119896+ 119860
119898minus1
sum119896=0
1198651198961198671015840
119898minus1minus119896
+120578
21198671015840
119898minus1
(30)
119877119873
119898(120578) = 119873
10158401015840
119898minus1(120578) + 119860Le
119898minus1
sum119896=0
1198651198961198731015840
119898minus1minus119896+ Le
120578
21198731015840
119898minus1
+NtNb
11986710158401015840
119898minus1
(31)
119883119898
=
0 119898 le 1
1 119898 gt 1(32)
The general solution of (26) is
119865119898(120578) = 119865
lowast
119898(120578) + 119862
1+ 1198622exp (120578) + 119862
3exp (minus120578)
119866119898(120578) = 119866
lowast
119898(120578) + 119862
4exp (120578) + 119862
5exp (minus120578)
International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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International Journal of Engineering Mathematics 7
minus15 minus10 minus05minus20 00ℏ
minus15
minus10
minus05
00
Figure 2 ℏ-curves for 11986510158401015840 (solid line) 1198661015840 (dash dot line) 1198671015840 (dashline)1198731015840 (dash dot dot line)
119867119898(120578) = 119867
lowast
119898(120578) + 119862
6exp (120578) + 119862
7exp (minus120578)
119873119898(120578) = 119873
lowast
119898(120578) + 119862
8exp (120578) + 119862
9exp (minus120578)
(33)
where 119865lowast119898(120578) 119866lowast
119898(120578) 119867lowast
119898(120578) and 119873lowast
119898(120578) are the particular
solutions and the constants are to be determined by boundaryconditions (27)
32 Convergence of the HAM Solution Equations (25) givesan analytical solution of the problem in series form Theconvergence of the series solution given by HAM dependsstrongly upon auxiliary parameters ℏ
119865 ℏ119866 ℏ119867 and ℏ
119873These
parameters provide a convenientmechanism for adjusting andcontrolling the convergence region and convergence rate ofthe series solution Therefore in order to select appropriatevalues for these auxiliary parameters the so-called ℏ
119865 ℏ119866 ℏ119867
and ℏ119873
curves are displayed at 20th-order approximationsas shown in Figure 2 This achieves excellent accuracy and istherefore adopted in allHAMnumerical computationsHAMis executed in a symbolic code to investigate the influence ofthe following five control parameters for the present nonlinearboundary value problem namely Le (Lewis number) 119860
(acceleration (unsteadiness) parameter) Nb (Brownianmotionparameter) Nt (thermophoresis parameter) and 120582 (rotationparameter) Prandtl number is assigned unity value Theeffects of the other parameters on primary velocity (1198651015840(120578)ie 119889119865119889120578) secondary velocity (119866) temperature (119867) andnanoparticle concentration (119873) functions versus transversecoordinate (120578) are depicted in Figures 3ndash15
Further computations for primary skin friction sec-ondary skin friction wall heat transfer andwallmass transferrate are presented in Tables 1ndash3 where a comparison is alsogiven with the ADM algorithm [45] discussed in the nextsection
135
710
Le
N(120578)
0
02
04
06
08
1
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = A = 1
Figure 3 Effects of Lewis number (Le) on119873(120578)
0512
35
A
0
02
04
06
08
1
F998400 (120578)
1 2 3 4 50120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 4 Effects of unsteadiness that is acceleration parameter (119860)on 1198651015840(120578)
4 Validation with Adomian DecompositionMethod (ADM)
Validation of the HAM computations is achieved with ADMa seminumerical technique which employs Adomian poly-nomials to achieve very accurate solutions which may beevaluated using symbolic packages such as Mathematica
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Engineering Mathematics
0512
35
A
G(120578)
1 2 3 40120578
0
02
04
06
08
1Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 5 Effects of unsteadiness that is acceleration parameter (119860)on 119866(120578)
0512
35
A
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 6 Effects of unsteadiness that is acceleration parameter (119860)on 119867(120578)
Introduced by American mathematician George Adomian[46] it has been embraced extensively in computationalengineering sciences over the past two decades Interestingstudies using ADM include enzyme kinetics in biologicalengineering [47] heat transfer [48] structural dampingsystems [49] non-Newtonian foam drainage problems [50]
N(120578)
0
02
04
06
08
1
1 2 3 40120578
0512
35
A
Nb = Nt = 02 120582 = Pr = 1 and Le = 5
Figure 7 Effects of unsteadiness that is acceleration parameter (119860)on 119873(120578)
012
45
0
02
04
06
08
1
120582
F998400 (120578)
1 2 3 40120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 8 Effects of rotation parameter (120582) on 1198651015840(120578)
and most recently magnetic biotribology [51] and nanofluidsqueezing flows [40 41 52] ADM [46] deploys an infiniteseries solution for the unknown functions that is 119865 119866 119867and 119873 and utilizes recursive relations The present ordinarydifferential nonlinear boundary value problem (BVP) is
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 9
175
176
177
178
179 1
81
81
012
45
120582
1 2 3 40120578
0
02
04
06
08
1
G(120578)
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 9 Effects of rotation parameter (120582) on 119866(120578)
0 1 2 3 4
199 2
201
H(120578)
0
02
04
06
08
1
012
45
120582
120578
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 10 Effects of rotation parameter (120582) on 119867(120578)
rewritten using the standard operator following Beg et al[51]
119871119906 + 119877119906 + 119873119906 = 119892 (34)
where 119906 is the unknown function 119871 is the highest-orderderivative (assumed to be easily invertible) 119877 is a lineardifferential operator of order less than 119871 119873 designates thenonlinear terms and 119892 is the source term Applying the
N(120578)
1 2 3 40120578
0
02
04
06
08
1
012
45
120582
178 179
Nb = Nt = 02 A = Pr = 1 and Le = 5
Figure 11 Effects of rotation parameter (120582) on 119873(120578)
0 1 2 3 4
010305
Nb
H(120578)
120578
0
02
04
06
08
1
Nt = 02 A = Pr = 120582 = 1 and Le = 5
199
6
199
8 2
200
2
200
4
Figure 12 Effects of Brownian motion parameter (Nb) on 119867(120578)
inverse operator 119871minus1 to both sides of (34) and using the givenconditions we obtain
119906 = V minus 119871minus1
(119877119906) minus 119871minus1
(119873) (35)
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Engineering MathematicsN(120578)
0
02
04
06
08
1
1 2 3 40120578
010305
Nb
Nt = 02 A = Pr = 120582 = 1 and Le = 5
Figure 13 Effects of Brownian motion parameter (Nb) on 119873(120578)
Nt
199
81
999 2
200
12
002
010305
H(120578)
0
02
04
06
08
1
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 14 Effects of thermophoresis parameter (Nt) on 119867(120578)
where V represents the terms arising from integrating thesource term 119892 and from the auxiliary conditions ADMdefines solution 119906 by the series
119906 =
infin
sum119899=0
119906119899 (36)
N(120578)
0
02
04
06
08
1
Nt010305
1 2 3 40120578
Nb = 02 A = Pr = 120582 = 1 and Le = 5
Figure 15 Effects of thermophoresis parameter (Nt) on 119873(120578)
The solution for the nonlinear terms is
119873 =
infin
sum119899=0
119860119899 (37)
Here 119860119899are the Adomian polynomials which are evaluated
via the following relation [51]
119860119899=
1
119899
119889119899
119889120582119899[119873
infin
sum119894=0
120582119894
119906119894]
120582=0
(38)
If the nonlinear term is expressed as a nonlinear function119891(119906) the Adomian polynomials are arranged into the form
119860 = 119891 (1199060)
1198601= 1199061119891(1)
(1199060)
1198602= 1199062119891(1)
(1199060) +
1
21199062
1119891(2)
(1199060)
1198603= 1199063119891(1)
(1199060) + 11990611199062119891(2)
(1199060) +
1
31199063
1119891(3)
(1199060)
(39)
Components 1199060 1199061 1199062 are then determined recursively by
using the relation
1199060= V
119906119896+1
= minus119871minus1
119877119906119896minus 119871minus1
119860119896 119896 ge 0
(40)
where 1199060is referred to as the zeroth component An 119899-
components truncated series solution is finally obtained as
119878119899=
infin
sum119899=0
119906119894 (41)
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 11
Table 1 HAM and ADM solutions compared for surface functions with Nt = 01 Nb = 01 Le = 2 Pr = 1 and 120582 = 5 for various values of119860
119860
1198892
119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
ADM
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
minus119889119867 (0)
119889120578(HAM)
minus119889119867 (0)
119889120578(ADM)
minus119889119873 (0)
119889120578(HAM)
minus119889119873 (0)
119889120578(ADM)
1 247946 247941 085624 085619 075182 075180 089386 0893842 354449 354446 140650 140647 094597 094595 112428 1124303 435908 435905 180054 180051 110755 110752 131608 131603
Table 2 HAM and ADM solutions compared with Nt = Nb = Le =
0 Pr = 07 and 120582 = 1 for various values of 119860
119860
1198892119865 (0)
1198891205782
(HAM)
1198892119865 (0)
1198891205782
(ADM)
minus119889119866 (0)
119889120578(HAM)
minus119889119866 (0)
119889120578(ADM)
05 073558 073553 019557 0195521 128493 128490 067240 0672372 198443 198441 119932 119935
Table 3 HAM solutions for various values of the thermophysicalparameters
119860 120582 Pr Nb Nt Le 11986510158401015840
(0) minus1198661015840
(0) minus1198671015840
(0) minus1198731015840
(0)
1 1 1 01 01 1 128493 067240 071326 0492872 198443 119932 089135 0630863 249736 156508 104043 0743901 2 161277 072937 072816 050037
3 191755 077737 074107 0506855 247946 085623 076299 0517801 2 128493 067240 091031 035517
5 128493 067240 119335 01388210 128493 067240 137458 0002911 02 128493 067240 067740 065038
03 128493 067240 064282 07024705 128493 067240 057742 07433901 02 128493 067240 068814 024902
03 128493 067240 066395 00433005 128493 067240 061822 -02638601 2 128493 067240 070306 083627
5 128493 067240 069208 1431767 128493 067240 068886 17062510 128493 067240 068591 204036
Decomposition series (41) converges exceptionally fast inparticular on high memory dual processor machines [53]The rapid convergence means that relatively few terms arerequired to obtain an approximate analytical solution Thisis a considerable advantage of the ADM approach comparedwith other semianalytical methods such as perturbationexpansionsThe present HAM accuracy is compared with theADM solutions in Tables 1-2 Excellent agreement guaranteesconfidence in the HAM computations Further HAM com-putations are given in Table 3 for variation of all the controlparameters
5 Discussion and Interpretation of Results
Figure 3 illustrates the influence of the Lewis number (Le)on the nanoparticle concentration distribution Althoughthe effects of this parameter were investigated on velocityfunctions and temperature no tangible modifications wereobserved and therefore are not discussed further Lewisnumber quantifies the ratio the thermal diffusivity to themass diffusivity An increase of Lewis number correspondsto a lower species diffusivity of the nanoparticles (119863
119861) for
a prescribed thermal diffusivity (120572) For this reason a risein Le induces a significant reduction in the dimensionlessnanoparticle volume fraction For Le gt 1 thermal diffusivityexceeds the species diffusivity and vice versa for Le lt 1For Le = 1 both thermal and species diffusivity will be thesame and thermal and nanoparticle concentration boundarylayer thicknesses will be equal Concentration boundary layerthickness for the nanoparticle species is significantly reducedwith greater Lewis number With greater Lewis number thedecay in concentration profiles also progressively evolvesfrom a linear descent (from the maximum at the sphere sur-face to zero in the free stream) to a more monotonic profileIn all cases asymptotically smooth profiles are achieved withHAM testifying to the prescription of a suitably large valuefor infinity that is 5
Figures 4ndash7 depict the influence of the unsteadinessparameter (119860) on flow characteristics Primary velocity1198651015840(120578)(Figure 4) is observed to be strongly accelerated with greater119860 values The rotation of the sphere draws momentum fromthe 119910-direction and redistributes this in the 119909-directionSecondary velocity (Figure 5) is therefore strongly decreasedwith greater unsteadiness parameter The primary and sec-ondary profiles are also very different Primary velocitygrows with greater distance from the sphere surface attainingmaxima in the free stream Secondary velocity119866(120578) howeverdecays from amaximumat the sphere surface (wall) to vanishin the free stream The rotation of the sphere acts like a fandrawing momentum from one direction and channeling itinto another Effectively as the 119909-direction flow is acceleratedthe momentum boundary layer thickness is decreasedThesecomputations concur well with the trends of Malvandi [37]although an erroneous interpretation is given in that paperIn Figures 6 and 7 both temperature and nanoparticle con-centration are found to be strongly depressed with increasingacceleration parameter Although 119860 arises multiple times inthe primary and secondary momenta equations (8) and (9)it features only in a single term in each of the energy andconcentration equations (10) and (11) specifically 119860119865119867
in
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 International Journal of Engineering Mathematics
(10) and Le119873119860119865 in (11) These terms couple the thermaland species diffusion to the primary momentum field onlyThe key influence from increasing unsteadiness is thereforean acceleration in primary flow which will counteract bothheat and nanoparticle diffusion Effectively as the primarymomentum boundary layer is thickened the decrease intemperatureswill cool the boundary layer and reduce thermalboundary layer thickness Species (nanoparticle) boundarylayer thickness will also be reduced Both temperatureand concentration distributions exhibit a consistent descentfrom the wall (sphere surface) to the free stream Howeverthe decay in temperatures is more gradual compared withconcentrations which plummet more sharply Generally theunsteadiness is found to induce a nontrivial influence on allflow characteristics
In Figures 8ndash11 the effects of spin (rotation) parameter120582(= Ω119905119886) on velocity functions temperature and nanopar-ticle concentration are depicted This parameter embodiesthe influence of the secondary velocity field on the primaryvelocity field that is via the swirl effect It is directlyproportional to the rotational velocity of the sphere and arisesin the coupling term 120582119866
2 in (8) For the case of a stationarysphere Ω rarr 0 and 120582 rarr 0 and for this scenario theprimary flow (Figure 8) is weakest and the secondary flow(Figure 9) is strongest As 120582 is increased the rotation becomesmore intense and this boosts primary momentum leading toescalation in 119865
values The converse response is computedfor secondary flow which is suppressed with greater 120582 valuesThe reduction in secondary flow however is weaker thanthe growth in the primary flow Temperatures (Figure 10)are found to be weakly reduced with greater rotation effectimplying a slight thinning in thermal boundary layers Simi-larly nanoparticle concentration (Figure 11) is alsomarginallydecreased with increasing 120582 values Heat and mass transferare therefore weakly resisted with greater rotation They aremaximized for the stationary sphere case Better control ofthermal and species diffusion is achieved with rotation ofthe sphere This may be beneficial therefore in spin coatingoperations employing nanomaterials
Figures 12 and 13 illustrate the response of temperature(119867) and species concentration (119873) to a change in Brownianmotion parameter (Nb) Temperature is slightly increased asNb is increased The reverse trend is noticed in the case ofconcentration Physically smaller nanoparticles yield higherNb values which assist in thermal diffusion in the boundarylayer via increased thermal conduction On the contrarylarger nanoparticles show lower Nb values and this depressesthermal conduction Higher Nb values will conversely stiflethe diffusion of nanoparticles away from the surface into thefluid regime which will manifest in a decrease in nanoparticleconcentration values in the boundary layer The distributionof nanoparticles in the boundary layer regime can thereforebe regulated via the Brownianmotionmechanism (higherNbvalues) and cooling of the regime can also be achieved vialarger Nb values Heat transfer from the fluid to the spheresurface (wall) is promoted with higher Nb values Thickerthermal boundary layers are produced with higher Nb valueswhereas larger concentration boundary layer thickness is
associated with lower Nb values The influence of Brownianmotion on the velocity fields was found to be inconsequentialand these plots are therefore excluded here
Finally Figures 14 and 15 illustrate the effects of ther-mophoresis parameter (Nt) on temperature and nanoparticleconcentration distributions Increasing thermophoresis effect(greater Nt values) slightly elevates nanofluid temperatures(Figure 14) Higher Nt values also increase nanoparticleconcentrations since lesser particle deposition will occurat the wall and greater migration of nanoparticles fromthe wall to the fluid regime will result Thermal boundarylayer thickness is slightly increased with thermophoresiswhereas concentration boundary layer thickness is moresignificantly enhanced It is further noted that the strongestinfluence of thermophoresis on nanoparticle distribution isat intermediate distances from the sphere transverse to thesphere surface
Table 3 documents the influence of many parameterson the skin friction components heat and mass transferrates With greater rotation effects (120582) primary skin friction(11986510158401015840(0)) is strongly elevated whereas secondary skin friction(minus1198661015840(0)) is weakly elevated There is also a weak increasein the surface heat (minus1198671015840(0)) and mass transfer (minus1198731015840(0))rates With an increase in Prandtl number (Pr) skin fric-tion components are unaffected whereas heat transfer rateis strongly increased and mass transfer rate (nanoparticlediffusion rate at the sphere surface) is decreased Cooling istherefore achieved successfully in the rotating boundary layerregime with larger Prandtl number Pr (decreasing nanofluidthermal conductivity) since more heat is conducted awayfrom the fluid to the sphereThis is one of themain attractionsof nanofluids Greater thermophoresis (Nt) boosts the heattransfer rate whereas it decreases the mass transfer rate Itexerts no tangible influence on the skin friction magnitudesGreater Brownian motion effect (Nb) decreases wall heattransfer rate but elevates the mass transfer rate Increasingunsteadiness parameter (119860) enhances both primary andsecondary friction and furthermore increases both heat andmass transfer rates Greater Lewis number (Le) results in areduction in the surface heat transfer rate and increase in thesurface mass transfer rate but does not alter the primary orsecondary skin friction components
6 Conclusions
Computational algorithms have been developed to study thetransient nanofluid flow in the stagnation region from aspinning spherical body The Buongiorno model has beenemployed to simulate nanoparticle Brownian motion andthermophoresis effects for the case of dilute nanofluidsThe nonlinear boundary value problem has been solvedwith HAM ADM has also been used to verify the HAMsolutions The computations have shown that with greaterrotation effect the primary flow is enhanced whereas thesecondary flow is weakened With increasing unsteadinessboth primary and secondary velocity fields are aided as arethewall heat andmass transfer rates An increase in nanoscaleparameters (Brownian motion and thermophoresis) is found
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 13
to mainly influence the temperature and nanoparticle distri-butions although a slight alteration is computed in surfaceskin friction components Thermophoresis tends to enhancethe wall heat transfer rate and reduces the mass transferrate Brownian motion exerts the opposite influence to ther-mophoresisThe current study is relevant to nanotechnologi-cal coating applications in the polymer industry In this studywe have employed a Newtonian nanofluid model Futureinvestigations will use non-Newtonian nanofluidmodels (egmicropolar theory) [8] and will be communicated immi-nently Furthermore the current study it is envisaged hasdemonstrated the advantage of HAM in being able to achievevery high order approximations in symbolic packages It isa computer-extended series expansion method a modernanalogy to Van Dykes asymptotic expansionperturbationseries method of the 1970s (which was used in inviscid andviscous supersonic flows) The popularity of this methodamong Eastern researchers is immense However very fewBritish researchers have explored this technique AlthoughHAM is algebraically laborious it is nevertheless an elegantapproach and avoids the traditional pitfall of other numericalschemes namely the time-consuming nature of discretiza-tion processes We hope that the present paper will furtherpopularize the scheme with British researchers who may nothave encountered it thus far
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors are grateful to the reviewer for hisher construc-tive comments which have served to improve the presentpaper
References
[1] R Taylor S Coulombe T Otanicar et al ldquoSmall particles bigimpacts a review of the diverse applications of nanofluidsrdquoJournal of Applied Physics vol 113 Article ID 011301 2013
[2] H Chang C S Jwo C H Lo et al ldquoProcess developmentand photocatalytic property of nanofluid prepared by combinedASNSSrdquoMaterials Science andTechnology vol 21 no 6 pp 671ndash677 2005
[3] L Zhang Y Li X Liu et al ldquoThe properties of ZnO nanofluidsand the role of H
2O2in the disinfection activity against
Escherichia colirdquo Water Research vol 47 no 12 pp 4013ndash40212013
[4] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 10 pages 2013
[5] L Zheng C Zhang X Zhang and J Zhang ldquoFlow andradiation heat transfer of a nanofluid over a stretching sheetwith velocity slip and temperature jump in porous mediumrdquoJournal of the Franklin Institute vol 350 no 5 pp 990ndash10072013
[6] K S Suganthi V Leela Vinodhan and K S Rajan ldquoHeattransfer performance and transport properties of ZnO-ethyleneglycol and ZnO-ethylene glycol-water nanofluid coolantsrdquoApplied Energy vol 135 pp 548ndash559 2014
[7] K Zaimi A Ishak and I Pop ldquoStagnation-point flow towarda stretchingshrinking sheet in a nanofluid containing bothnanoparticles and gyrotactic microorganismsrdquo Journal of HeatTransfer vol 136 no 4 Article ID 041705 2014
[8] H Y Lee H K Park Y M Lee K Kim and S B Park ldquoApractical procedure for producing silver nanocoated fabric andits antibacterial evaluation for biomedical applicationsrdquo Chem-ical Communications vol 2007 no 28 pp 2959ndash2961 2007
[9] D P Kulkarni D K Das and R S Vajjha ldquoApplication ofnanofluids in heating buildings and reducing pollutionrdquoAppliedEnergy vol 86 no 12 pp 2566ndash2573 2009
[10] S Sarkar and S Ganguly ldquoFully developed thermal transportin combined pressure and electroosmotically driven flow ofnanofluid in a microchannel under the effect of a magneticfieldrdquoMicrofluidics and Nanofluidics vol 18 no 4 pp 623ndash6362015
[11] A H Battez R Gonzalez J L Viesca et al ldquoCuO ZrO2and
ZnO nanoparticles as antiwear additive in oil lubricantsrdquoWearvol 265 no 3-4 pp 422ndash428 2008
[12] K-L Liu K Kondiparty A D Nikolov and D WasanldquoDynamic spreading of nanofluids on solids part II modelingrdquoLangmuir vol 28 no 47 pp 16274ndash16284 2012
[13] J Eapen J Li and S Yip ldquoProbing transport mechanisms innanofluids by molecular dynamics simulationsrdquo Tech RepMITCenter for Nanofluids Technology Department of NuclearScience and Engineering Massachusetts Institute of Technol-ogy Cambridge Mass USA 2007
[14] K Das ldquoSlip flow and convective heat transfer of nanofluidsover a permeable stretching surfacerdquo Computers amp Fluids vol64 pp 34ndash42 2012
[15] H Karimi F Yousefi and M R Rahimi ldquoCorrelation ofviscosity in nanofluids using genetic algorithm-neural network(GA-NN)rdquoHeat andMass Transfer vol 47 no 11 pp 1417ndash14252011
[16] M Sheikholeslami R Ellahi M Hassan and S Soleimani ldquoAstudy of natural convection heat transfer in a nanofluid filledenclosure with elliptic inner cylinderrdquo International Journal ofNumerical Methods for Heat amp Fluid Flow vol 24 no 8 pp1906ndash1927 2014
[17] O A Beg V R Prasad and B Vasu ldquoNumerical study of mixedbioconvection in porous media saturated with nanofluid con-taining oxytactic microorganismsrdquo Journal of Mechanics inMedicine and Biology vol 13 no 4 Article ID 1350067 2013
[18] W A Khan M J Uddin and A I M Ismail ldquoFree convectionof non-Newtonian nanofluids in porous media with gyrotacticmicroorganismsrdquo Transport in Porous Media vol 97 no 2 pp241ndash252 2013
[19] B Vasu and R S R Gorla ldquoTwo-phase laminar mixed convec-tion Al
2O3water nanofluid in elliptic ductrdquo in Nanoscale Flow
Advances Modeling and Applications S M Musa Ed chapter4 pp 101ndash120 2015
[20] MM Rashidi O A BegM Asadi andM T Rastegari ldquoDTM-Pade modeling of natural convective boundary layer flow ofa nanofluid past a vertical surfacerdquo International Journal ofThermal and Environmental Engineering vol 4 no 1 pp 13ndash242011
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 International Journal of Engineering Mathematics
[21] M J Uddin W A Khan and N S Amin ldquoG-Jitter mixed con-vective slip flow of nanofluid past a permeable stretching sheetembedded in a Darcian porous media with variable viscosityrdquoPLoS ONE vol 9 no 6 Article ID e99384 2014
[22] O Anwar Beg R S R Gorla V R Prasad B Vasu andD RanaldquoComputational study of mixed thermal convection nanofluidflow in a porousmediumrdquo inProceedings of the 12thUKNationalHeat Transfer Conference University of Leeds School of ProcessEngineering (Energy Institute) Leeds UK August-September2011
[23] A Raees H Xu Q Sun and I Pop ldquoMixed convection ingravity-driven nano-liquid film containing both nanoparticlesand gyrotactic microorganismsrdquo Applied Mathematics andMechanics vol 36 no 2 pp 163ndash178 2015
[24] S Shaw P Sibanda A Sutradhar and P V S N MurthyldquoMagnetohydrodynamics and soret effects on bioconvectionin a porous medium saturated with a nanofluid containinggyrotactic microorganismsrdquo Journal of Heat Transfer vol 136no 5 Article ID 052601 2014
[25] A Noghrehabadi R Pourrajab and M Ghalambaz ldquoEffect ofpartial slip boundary condition on the flow and heat transferof nanofluids past stretching sheet prescribed constant walltemperaturerdquo International Journal of Thermal Sciences vol 54pp 253ndash261 2012
[26] S Gumgum and M Tezer-Sezgin ldquoDRBEM solution of mixedconvection flow of nanofluids in enclosures with moving wallsrdquoJournal of Computational and AppliedMathematics vol 259 pp730ndash740 2014
[27] C Y Cheng and D-T Chin ldquoMass transfer in ac electrolysisextension of a film model to turbulent flow on a rotatinghemisphererdquo Chemical Engineering Communications vol 36no 1ndash6 pp 17ndash26 1985
[28] N S Berman and M A Pasch ldquoLaser doppler velocity mea-surements for dilute polymer solutions in the laminar boundarylayer of a rotating diskrdquo Journal of Rheology vol 30 no 3 pp441ndash458 1986
[29] M G Morsy F M Wassef V H Morcos and H A MEl Biblawy ldquoOverall heat transfer coefficient for a multi-tuberotating condenserrdquo Chemical Engineering Communicationsvol 57 no 1ndash6 pp 41ndash49 2007
[30] M S Faltas and E I Saad ldquoStokes flow between eccentricrotating spheres with slip regimerdquo Zeitschrift fur angewandteMathematik und Physik vol 63 no 5 pp 905ndash919 2012
[31] H I Andersson and M Rousselet ldquoSlip flow over a lubricatedrotating diskrdquo International Journal of Heat and Fluid Flow vol27 no 2 pp 329ndash335 2006
[32] H Niazmand and M Renksizbulut ldquoTransient three-dimen-sional heat transfer from rotating spheres with surface blowingrdquoChemical Engineering Science vol 58 no 15 pp 3535ndash35542003
[33] S Roy and D Anilkumar ldquoUnsteady mixed convection from arotating cone in a rotating fluid due to the combined effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 47 no 8-9 pp 1673ndash1684 2004
[34] S V Subhashini H S Takhar andGNath ldquoNon-uniformmasstransfer or wall enthalpy into a compressible flow over a rotatingsphererdquo Heat and Mass Transfer vol 43 no 11 pp 1133ndash11412007
[35] P Rana R Bhargava and O A Beg ldquoFinite element simulationof unsteady magneto-hydrodynamic transport phenomena on
a stretching sheet in a rotating nanofluidrdquo Proceedings of theInstitution of Mechanical Engineers Part N Journal of Nanoengi-neering and Nanosystems vol 227 no 2 pp 77ndash99 2013
[36] S Nadeem and S Saleem ldquoAn optimized study of mixed con-vection flow of a rotating Jeffrey nanofluid on a rotating verticalconerdquo Journal of Computational and Theoretical Nanosciencevol 12 pp 1ndash8 2015
[37] A Malvandi ldquoThe unsteady flow of a nanofluid in the stagna-tion point region of a time-dependent rotating sphererdquoThermalScience 2013
[38] D Anilkumar and S Roy ldquoSelf-similar solution of the unsteadymixed convection flow in the stagnation point region of arotating sphererdquo Heat and Mass Transfer vol 40 no 6-7 pp487ndash493 2004
[39] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[40] A Dib A Haiahem and B Bou-said ldquoApproximate analyticalsolution of squeezing unsteady nanofluid flowrdquo Powder Technol-ogy vol 269 pp 193ndash199 2015
[41] A K Gupta and S Saha Ray ldquoNumerical treatment forinvestigation of squeezing unsteady nanofluid flowbetween twoparallel platesrdquo Powder Technology vol 279 pp 282ndash289 2015
[42] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman HallCRC Press Boca Raton FlaUSA 2003
[43] V Ananthaswamy A Eswari and L Rajendran ldquoNonminuslinearreactionminusdiffusion process in a thin membrane and homotopyanalysis methodrdquo International Journal of Automation andControl Engineering vol 2 pp 10ndash17 2013
[44] F Mabood and W A Khan ldquoHomotopy analysis method forboundary layer flow and heat transfer over a permeable flat platein a Darcian porous medium with radiation effectsrdquo Journal ofthe Taiwan Institute of Chemical Engineers vol 45 no 4 pp1217ndash1224 2014
[45] M A Abdou ldquoNew analytic solution of von Karman swirlingviscous flowrdquoActa ApplicandaeMathematicae vol 111 no 1 pp7ndash13 2010
[46] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer Academic Boston Mass USA 1994
[47] J R Sonnad and C T Goudar ldquoSolution of the Haldane equa-tion for substrate inhibition enzyme kinetics using the decom-position methodrdquo Mathematical and Computer Modellingvol 40 no 5-6 pp 573ndash582 2004
[48] P Vadasz and S Olek ldquoConvergence and accuracy of Adomianrsquosdecomposition method for the solution of Lorenz equationsrdquoInternational Journal of Heat and Mass Transfer vol 43 no 10pp 1715ndash1734 2000
[49] S S Ray B P Poddar and R K Bera ldquoAnalytical solution of adynamic system containing fractional derivative of order one-half by adomian decomposition methodrdquo Journal of AppliedMechanics TransactionsASME vol 72 no 2 pp 290ndash295 2005
[50] A M Siddiqui A Hameed T Haroon and AWalait ldquoAnalyticsolution for the drainage of Sisko fluid film down a vertical beltrdquoApplications amp Applied Mathematics vol 8 pp 465ndash470 2013
[51] O A Beg D Tripathi T Sochi and P K Gupta ldquoAdomiandecomposition method (ADM) simulation of magneto-bio-tribological squeeze film with magnetic induction effectsrdquoJournal of Mechanics in Medicine and Biology 2015
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 15
[52] M Sheikholeslami D D Ganji and H R Ashorynejad ldquoInves-tigation of squeezing unsteady nanofluid flow using ADMrdquoPowder Technology vol 239 pp 259ndash265 2013
[53] O A Beg ldquoADSIMNANmdasha program for Adomian simulationof nanofluid problemsrdquo Tech Rep NANO-5613 GORT Brad-ford UK 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![A homotopy perturbation analysis of nonlinear free ...°±_.… · namical, and vibrational behavior of beams. Pirbodaghi et al. [4] used the homotopy analysis method (HAM) to investigate](https://static.fdocuments.net/doc/165x107/5fe94cbfaf11be272232348c/a-homotopy-perturbation-analysis-of-nonlinear-free-namical-and-vibrational.jpg)
