Post on 06-Sep-2018
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12440
Nanofluids Slip Mechanisms on Hydromagnetic Flow of Nanofluids over a
Nonlinearly Stretching Sheet under Nonlinear Thermal Radiation
1S.P.Anjali Devi and 2Mekala Selvaraj
1Former Professor & Head, Department of Applied Mathematics, Bharathiar University, Coimbatore-46 Tamilnadu, India. 2Research Scholar,Department of Mathematics, Bharathiar University, Coimbatore-641046 Tamilnadu, India.
Orcid Id:0000-0001-7485-5114 Abstract
In this paper, Heat transfer characteristics of two
dimensional, steady hydromagnetic boundary layer flow of
water based nanofluids containing metallic nanoparticles
such as copper (Cu) and Silver (Ag) over a nonlinearly
stretching surface taking into account the effects of
nonlinear thermal radiation and viscous dissipation has been
investigated numerically. The model used for the nanofluids
incorporates the effects of Brownian motion and
thermophoresis. The governing nonlinear partial differential
equations were transformed into nonlinear ordinary
differential equations using similarity transformations and
then are solved numerically subject to the transformed
boundary conditions by most efficient Nachtsheim- Swigert
shooting iteration scheme for satisfaction of asymptotic
boundary conditions along with fourth order Runge-Kutta
Integration method. Numerical computations are carried out
for distributions of velocity, temperature and nanoparticles
volume fraction by means of graphs for different values of
physical parameters such as magnetic interaction parameter,
nonlinear stretching parameter, Eckert number, temperature
ratio parameter, radiation parameter, Prandtl number,
Brownian motion parameter, thermophoresis parameter and
Lewis number. The numerical results of the problem are
validated by comparing with previously published results in
the literature. Numerical values of skin friction coefficient
and Nusselt number at the wall are also obtained and given
in tabular form. Sherwood number is vanished due to new
mass flux condition.
Key words: Nanofluid, Stretching Sheet, MHD, Radiation.
Nomenclature
c stretching coefficient
B0 magnetic induction
nanoparticle volume fraction
∞ ambient nanoparticle volume fraction
DB Brownian diffusion coefficient
DT thermophoretic diffusion coefficient
f dimensionless stream function
Ec Eckert number
k* Rosseland mean absorption coefficient
Le Lewis number
M2 magnetic field parameter
n nonlinear stretching parameter
Nb Brownian motion parameter
Nt thermophoresis parameter
Nux local Nusselt number
Pr Prandtl number
qr radiative heat flux
Rexlocal Reynolds number
T temperature of the nanofluid within the boundary layer
Twtemperature at the surface of the sheet
T∞temperature of the ambient nanofluid
u velocity along the surface of the sheet
v velocity normal to the surface of the sheet
(x, y) Cartesian coordinates
Greek symbols
nfthermal diffusivity of the
nanofluid
ρnf density of the nanofluid
(cp)nf heat capacity of the nanofluid
μnf viscosity of the nanofluid
υnf kinematic viscosity of the nanofluid
ψ stream function
η similarity variable
θ dimensionless temperature
θw surface wall temperature
φ dimensionless rescaled nanoparticle volume fraction
κnf thermal conductivity of the
nanofluid
τ nanoparticle heat capacity ratio
σ magnetic permeability
σ* Stefan-Boltzmann constant Subscripts
w surface conditions
∞ conditions far away from the surface
Superscripts
differentiation with respect to η
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12441
INTRODUCTION
Ultrahigh performance cooling is one of the most vital needs
of many industrial technologies. Nanofluids which exhibit
ultra high performance cooling are engineered by
suspending nanoparticles with average size below 100nm in
traditional heat transfer fluids such as water, oil and
ethylene glycol. Nanofluid is the term coined by Choi
(1995) [5] to describe the new class of nanotechnology
based heat transfer fluids that exhibit thermal properties
superior to those of their host fluids or conventional particle
fluid suspensions. A comprehensive study on the nanofluids
characteristics is documented by Das et al.(2007)[8]. Kaufui
V. Wong and Omar De Leon (2010)[11] presented the wide
range of applications of nanofluids in current and future
such as nuclear reactors, transportation, electronics cooling,
biomedicine and food. Ahmad et al. (2011)[2] presented a
numerical study of the Blasious and Sakiadis flows in
nanofluids under isothermal condition. Their results
revealed that solid volume fraction affects the fluid flow and
heat transfer characteristics of nanofluids. An analytical
derivation of effective thermal conductivity of nanofluids
which incorporates the contribution of interfacial layer as
well as the Brownian motion was solved by Ritu Pasrija and
Sunita Srivastava (2013)[22]. Sandeep Pal et al.(2014)[24]
has presented a review on enhanced thermal conductivity of
colloidal suspension of nanosized particles (nanofluids).The
recent literature of nanofluids was reviewed by Mohameed
Saad Kamel et al.(2016)[13].
Steady boundary layer flow of incompressible fluids over a
stretching sheet has considerable bearing on various
technological processes. The flow over a stretching plate
was first considered by Crane (1970)[7] who found a closed
form analytic solution of the self-similar equation for steady
boundary layer flow of a Newtonian fluid. MHD was
initially known in the field of astrophysics and geophysics
and later becomes very important in engineering and
industrial processes. Pavlov (1974)[16] gave an exact
similarity solution of the MHD boundary layer equations for
the steady two-dimensional flow of an electrically
conducting fluid due to the stretching of a plane elastic
surface in the presence of a uniform transverse magnetic
field. Anjali Devi and Thiyagarajan (2006)[9] solved the
problem of steady nonlinear MHD flow of an
incompressible, viscous and electrically conducting fluid
with heat transfer over a surface of variable temperature
stretching with a power law velocity in the presence of
variable transverse magnetic field.
The role of thermal radiation is of major importance in some
industrial applications such as glass production, furnace
design, nuclear power plants space technology such as in
comical flight aerodynamics rocket, propulsion systems,
plasma physics and space craft reentry aerodynamics which
operates high temperatures. The effect of thermal radiation
on the boundary layer flow has been investigated by Rafael
Cortell (2008)[6].
Viscous dissipation plays an important role in changing the
temperature distribution which affects the heat transfer rates
considerably. The thermal radiation and viscous dissipation
effects on the laminar boundary layer about a flat plate in a
uniform stream of fluid (Blasius flow), and about a moving
plate in a quiescent ambient fluid (Sakiadis flow) both under
convective boundary condition is presented by
Olanrewaju.P.O et al. (2011)[14].Similarity solutions to
boundary layer flow and heat transfer of nanofluid over
nonlinearly stretching sheet with viscous dissipation effects
was studied by Hamad and M.Ferdows (2012)[10]. The
effect of variable viscosity on the flow and heat transfer of a
viscous Ag- water and Cu-water nanofluids was investigated
by Vajravelu (2012)[28].Convective-radiation effects on
stagnation point flow of nanofluids over a
stretching/shrinking surface with viscous dissipation was
studied by Pal et al.(2014)[15]. The radiating and
electrically conducting fluid over a porous stretching surface
with the effect of viscous dissipation was researched by
Sreenivasalu et al. (2016)[26].
Buongiorno (2006)[4] proposed a mathematical nanofluid
model by taking into account the Brownian motion and
thermophoresis effects on flow and heat transfer fields.In his
work he has considered seven slip mechanisms those affect
nanofluid flow such as inertia, Brownian diffusion,
thermophoresis, diffusiophoresis, Magnus effect, fluid
drainage and gravity. He indicated that of those seven only
Brownian diffusion and thermophoresis are important slip
mechanisms in nanofluids. Reza Azizian et al. (2012)[21]
has investigated the effect of nanoconvection caused by
Brownian motion on the enhancement of thermal
conductivity in nanofluids. The non-linear stretching of a
flat surface in a nanofluid with Brownian motion and
thermophoresis effects was investigated by Rana and
Bhargava (2012)[18].
The temperature-dependent thermo-physical properties on
the boundary layer flow and heat transfer of a nanofluid past
a moving semi-infinite horizontal flat plate in a uniform free
stream with the effects of Brownian motion, thermophoresis
and viscous dissipation due to frictional heating are
analyzed by Vajravelu and Prasad (2012)[29].The effects
of thermal radiation and viscous dissipation on
magnetohydrodynamic (MHD) stagnation point flow and
heat transfer of nanofluids towards a stretching sheet are
investigated by Yohannes Yirga and Bandari Shankar
(2013)[30]. The problem of laminar fluid flow which results
from a permeable stretching of a flat surface in a nanofluid
with the effects of heat radiation, magnetic field, velocity
slip, brownian motion and thermophoresis parameters and
convective boundary conditions have been examined by
Reddy (2014)[20].Sumalatha et al.(2016)[27] published the
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12442
mixed convection flow of nanofluids past a nonlinear
stretching sheet in the existence of nanofluids important slip
mechanisms with MHD, variable surface temperature and
volume fraction.
Motivated by the above discussed investigations and
applications, in this present work mainly concentrate on the
effects of nonlinear thermal radiation, viscous dissipation
and variable magnetic field on heat transfer flow of
nanofluids (Cu Water nanofluid and Ag Water nanofluid)
over a nonlinearly stretching sheet with variable surface
temperature. And also the model includes the effects of
Brownian motion and Thermophoresis effects.
MATHEMATICAL FORMULATION
Consider two-dimensional, hydromagnetic flow over a
nonlinearly stretching sheet with convective heat transfer in
water based nanofluids containing copper (Cu) and Silver
(Ag) nanoparticles and the Cartesian coordinates such as x -
axis runs along the direction of the continuous stretching
surface and the y - axis is measured normal to the surface of
the sheet. It is also considered that the sheet is stretching
with velocity Uw = cxn, where c > 0.Let us assume, the base
fluid (water) and the nanoparticles are in equilibrium and the
nanofluids is viscous and incompressible.(See Fig. i).
Figure i: Physical model of the problem
Taking into account the effects of Brownian motion and
thermophoresis and based on model developed by
Buongiorno [4]. The basic steady boundary-layer equations
in the presence of variable magnetic field, nonlinear thermal
radiation and viscous dissipation are given by
0
yv
xu
(1)
22
2
( )nf
nf
B xu u uu v ux y y
(2)
2 22
2nfnf
T rp B nfsp
D qT T T T T uc u v k c D yx y y y T y yy
(3)
2 2
2 2
TB
D Tu v Dx y y T y
(4)
The boundary conditions are given by
u = uw(x) = cxn, v =0,T = Tw (x) = T∞ + bxm ,
0TB
D TDy T y
at y = 0
u=0,T→T∞,, as y→∞ (5)
In the above boundary conditions, assume m = 2n is a
surface temperature parameter and the nanoparticle mass
flux due to the Brownian motion and thermophoresis effects
tends to zero at the boundary(y=0)[A.V.Kuznetsov and
D.A.Nield [12]].
where the symbols are as defined in the nomenclature.
The variable magnetic field B(x) = B0 x (n-1)/2 (Afzal 1993)[1]
is applied in the transverse direction. The magnetic
Reynolds number is assumed to be small so that the induced
magnetic field is negligible in comparison with the applied
magnetic field. Since the induced magnetic field is neglected
and B0 is independent of time, 0curl E . Also,
0Ediv
in the absence of surface charge density. Hence
0E
.
The Rosseland approximation [Rosseland (1936)[23],Raptis
(1998)[19], Sparrow and Cess(1978)[25], Brewster
(1992)[3]] is used to describe the radiative heat flux which
is negligible in x direction in comparison to that in y
direction. Full radiation term has been taken into account.
Employing the Rosseland diffusion approximation, the
radiative heat flux is modeled as
yTT
kq *
*
r3
3
16σ
(6)
Hence
2* 22 3
* 2
163
3r
T Tq T Ty k y y
(7)
where σ* is the Stefan Boltzmann constant, k* is the
Rosseland mean absorption coefficient.
The nonlinear governing equations (1) to (4) with the
boundary conditions (5) are solved by employing the
similarity transformations which are given below.
1
22
1
nfc
x fn
, 1
21
2
n
f
c ny x
,
w
T TT T
,
(8)
Where is the similarity space variable and f is the
dimensionless stream function.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12443
Using the Stream function
uy
and vx
The velocity components are expressed as follows
nxu c f ,
1
2
1
2
-1-
1
nc n f x nv f fn
(9)
Using the similarity transformations (9), equation of
continuity (1) is automatically satisfied and the equations
(7),(8) and (9), the nonlinear partial differential equation (2),
(3) and (4) with boundary conditions (5) are reduced to the
following nonlinear ordinary differential equations
2 2
1 1
20
1
nf b c ff f M fn
(10)
3
2
1
2
2 2
4 4 21 1 1 1 1 13
4 Pr.Pr.
1
f fw
R Rnf nf
f
nf
k kw wN k N k
k n Ecc f f f Nb Ntk n b
(11)
Pr 0NtLe fNb
(12)
Here, b1, c1 and c2 are constants whose values are given in
Appendix.
The appropriate boundary conditions are
0 0f , 0 1f
, 0 1
0 0 0Nb Nt at =0,
0,f 0 0 as (13)
The nondimensional parameters appeared in Equations (10)
to (12) are defined as follows
22 02
1 fcBM
n
is the
magnetic interaction parameter, 3
*
4 *
fR
k kN
T
is the
radiation parameter Pr
pf f
f
c
k
is the Prandtl
number ,
TTw
w is the Temperature ratio parameter,
2
w
p wf
uEc
c T T
is the Eckert number,
( )
( )
p s B
p f f
c DNb
c
is the Brownian motion parameter,
( )
( )
p s T w
p f f
c D T TNt
c T
is the thermophoresis
parameter and f
B
LeD
is the Lewis number.
Skin-friction coefficient
The skin friction coefficient (rate of shear stress) is defined
as
2
wf
f w
CU
, where
0
w nfy
uy
(14)
Substituting equations (8) and (9) into equation (14),
1/2
2.5
1Re = 0
1x
n fC f
Nusselt number
The Nusselt number (rate of heat transfer) is defined as
w
xf w
q xNu
k T T
, where surface heat flux is
3
0
16
3w nf
y
Tq k Tk y
(15)
Using equations (8) and (9), equation (15) can be written as
341 1 (0)
Re 3
nf fxw
f nfx R
k kNun
k k N
Here,
12
Ren
xf
c x
Due to the effects of Brownian motion and
thermophoresis at the boundary, the Sherwood number
vanishes because which characteristics the mass flux is zero
at y=0.
Numerical Solutions
In this work, steady, two dimensional, hydromagnetic
boundary layer flow of nonlinearly stretching surface over
two types of nanofluids namely Cu – Water nanofluid and
Ag – Water nanofluid in the presence of viscous dissipation
and nonlinear thermal radiation and also the effects of
Brownian motion and thermophoresis has been investigated.
The governing nonlinear partial differential equations are
converted to nonlinear ordinary differential equations by
similarity transformations incorporating the necessary
similarity variables. The resulting nonlinear ordinary
differential equations (10) to (12) along with the relevant
boundary conditions (13) constitute a nonlinear boundary
value problem which is difficult to solve analytically.
Hence, these equations are solved using the most efficient
shooting method such as the Nachtsheim-Swigert shooting
iteration scheme for satisfaction of the asymptotic boundary
conditions along with the Fourth-order Runge Kutta
integration method. The difficulty lies in guessing the values
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12444
for ''(0),f (0) and (0) properly to get the
convergence and solution. The level of accuracy for
convergence is chosen as 10-5.
RESULTS AND DISCUSSIONS
The numerical and graphical results for two types of water
based nanofluids such as Cu-water nanofluid and Ag-water
nanofluid are presented. The value of the Prandtl number for
the base fluid (water) is kept to be the constant Pr = 6.2.
In order to verify the accuracy of the present method, we
have compared our results with those of Cortell [17] and
Hamad et al.[10] for the Skin friction coefficient –f (0) and
nondimensional rate of heat transfer -0 in the absence of
nanoparticles ( = 0), Magnetic interaction parameter and
viscous dissipation parameter and without thermal radiation
parameter ( NR ) , Brownian motion and thermophoresis
which is shown in Table 2 and Table 3. It is clearly note that
our results are good agreement with that of Cortell and
Hamad et al.
Table 2: Comparison of results for −f(0) when = 0 and
M2 = 0.0
n Cortell Hamad et al. Present work
0.0
0.2
0.5
1.0
3.0
10.0
20.0
0.6276
0.7668
0.8895
1.0000
1.1486
1.2349
1.2574
0.6369
0.7659
0.8897
1.0043
1.1481
1.2342
1.2574
0.6276
0.7668
0.8895
1.0000
1.1486
1.2348
1.2574
Table 3: Comparison of results for−θ(0) when = 0,
Pr = 5.0,Ec = 0.0 and NR
n Cortell Hamad et al. Present work
0.75
1.5
7.0
10.0
3.1250
3.5677
4.1854
4.2560
3.1246
3.5672
4.1848
4.2560
3.1251
3.5679
4.1854
4.2558
Fig.1 to Fig.13 demonstrate the influence of
Magnetic interaction parameter, nonlinear stretching
parameter, viscous dissipation parameter, surface
temperature parameter, radiation parameter, Lewis number,
Brownian motion and thermophoresis parameter
respectively on velocity distribution, temperature
distribution and nanoparticle volume fraction of two types
of nanofluids such as copper water nanofluid and silver
water nanofluid.
Figure 1: Velocity profiles for various values of M2
Figure 2: Effect of M2 on Temperature profiles
Figure 3: Nanoparticle volume faction for various values of
M2
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
f ' ()
M2 = 0.5, 1.0, 1.5, 2.0
Cu - Water
Ec = 1.0
Pr = 6.2
NR = 1.0
Le = 0.6
Nt = 0.5
Nb = 0.5
n = 10.0
Ag - Water
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
M2 = 0.5, 1.0, 1.5, 2.0
Cu - Water
Ec = 1.0
Pr = 6.2
NR = 1.0
Le = 0.6
Nt = 0.5
Nb = 0.5
n = 10.0
Ag - Water
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
M2 = 0.5, 1.0, 1.5, 2.0
Ec = 1.0
Pr = 6.2
NR = 1.0
Le = 0.6
Nt = 0.5
Nb = 0.5
n = 10.0
Cu - Water
Ag - Water
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12445
Figure 4: Velocity and volume fraction profiles for n
for Cu – water nanofluid
Fig.1 shows the plot of dimensionless velocity for different
values of magnetic interaction parameter. It is noted that as
magnetic interaction parameter increases, f decreases,
elucidating the fact that the effect of magnetic field is to
decelerate the velocity. This result qualitatively agrees with
the expectation since the Lorentz force which opposes the
flow field increases as M2 increases and leads to enhanced
deceleration of the flow. Further the effect of magnetic field
is to reduce the boundary layer thickness.
Fig.2 represents the graph of dimensionless temperature for
different values of magnetic interaction parameter. Increase
in M2 which enhances the dimensionless temperature
distribution. The influence of magnetic interaction
parameter on the dimensionless volume fraction is plotted in
Fig.3.The figure reveals that the volume fraction of the
nanofluids boosts for increasing values of M2.
Fig.4 and fig.5 respectively is a graphical representation of
dimensionless velocity, volume fraction for Cu water
nanofluid and temperature of both nanofluids for various
values of nonlinear stretching parameter. It is noted that as
the nonlinear stretching parameter increases, f(),and
diminishes. Consequently the effect of nonlinear
stretching parameter over momentum boundary layer
thickness becomes significantly less, for cu - water
nanofluid.
Figure 5:Dimensionless Temperature profiles for n
In Fig.6, the effect of Eckert number on temperature
distribution is displayed. It implied that the Eckert number
enhances temperature and contributes to the thickening of
thermal boundary layer thickness.
Figure 6:Dimensionless temperature distribution
at different values of Ec
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
n = 1.0, 2.0, 3.0, 10.0
M2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
Nb = 0.5
Nt = 0.5
Le = 0.6
w = 0.80
f ' ()
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
n = 1.0, 2.0, 3.0, 4.0
Ag - Water
M2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
Nt = 0.5
Nb = 0.5
Le = 0.6
Cu - Water
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Ec = 0.7, 0.8, 0.9, 1.0
Ag - Water
Cu - Water
M2 = 1.0
Pr = 6.2
NR = 1.0
Nt = 0.5
Nb = 0.5
Le = 0.6
n = 10.0
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12446
Figure 7:Dimensionless Temperature Profiles for w
Figure 8: Radiation parameter effect on
Dimensionless Temperature profiles
Fig.7 depicts the effect of changing temperature ratio
parameter on temperature distribution. The thermal
boundary layer thickness increases with increasing surface
temperature. This can be explained by the statement the
effect of temperature ratio parameter is to increase the rate
of energy transport to the nanofluid and accordingly
increase the temperature. An increase in the radiation
parameter causes a decrease in the temperature and the
thermal boundary layer thickness as displayed in Fig.8.The
values of radiation parameter will cause no change in the
velocity profiles of the nanofluids because the transformed
momentum equation (10) is uncoupled from the energy
equation (12).
Fig.9 shows the effect of Lewis number on the volume
fraction profiles. It illustrates that the volume fraction
decreases as the Lewis number increases. This is because as
the values of Lewis number gets larger the molecular
diffusivity gets smaller thereby causes a decrease in the
volume fraction field.
Figure 9: Dimensionless nanoparticle volume fraction for
Lewis number
Figure 10: Temperature profiles for various values
of Brownian motion parameter
Figure 11: Effect of Brownian motion parameter
on volume fraction distribution
The effect of Brownian motion parameter on temperature
and volume fraction is shown in Fig.10 and Fig.11. The
temperature in the boundary layer has the less result due to
the influence of Brownian motion parameter whereas the
volume fraction decreases with the increasing values of
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
w = 0.8, 0.85, 0.9, 1.0
Cu - Water
Ag - Water
M2 = 1.0
Pr = 6.2
Ec = 1.0
NR = 1.0
Nt = 0.5
Nb = 0.5
Le = 0.6
n = 10.0
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
NR = 0.5, 1.0, 1.5, 2.0
Cu - Water
Ag - Water
M2 = 1.0
Pr = 6.2
Ec = 1.0
Nt = 0.5
Nb = 0.5
Le = 0.6
n = 10.0
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Le = 0.3, 0.6, 0.9, 1.0
Cu - Water
Ag - Water
M2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
Nt = 0.5
Nb = 0.5
= 0.1
n = 10.0
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cu - Water
Nb = 0.2, 0.4, 0.5, 1.0
M2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
n = 10.0
Nt = 0.5
Le = 0.6
Ag - Water
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Nb = 0.2, 0.4, 0.5, 1.0
Cu - Water
M2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
Nt = 0.5
Le = 0.6
= 0.1
n = 10.0
Ag - Water
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
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12447
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
M2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
n = 10.0
Nb = 0.5
Le = 0.6
Cu - Water
Nt = 0.2, 0.4, 0.5, 0.7
Ag - Water
Brownian motion parameter. Brownian motion serves to just
warm the boundary layer.
Figure 12: Temperature profiles for different values
of Thermophoresis parameter
Thermophoresis parameter plays an important key role in
temperature distribution and nanoparticle volume fraction
which is demonstrated through Fig.12 & Fig.13
respectively. It is noticed that the dimensionless temperature
as well as the dimensionless volume fraction increases by
the increase of the values of the thermophoresis parameter.
Increase in Nt causes the increment in the thermophoresis
force which tends to move nanoparticles from hot to cold
areas and consequently it enhances the magnitude for
temperature and nanoparticle volume fraction profiles.
Figure 13: Effect of Thermophoresis parameter on
volume fraction distribution
The numerical results of the skin friction co
efficient and nondimensional rate of heat transfer are
presented in table 4 and table 5 for both cu - water nanofluid
and silver water nanofluid. In Table 4, skin friction
coefficient increases due to the influence of magnetic
interaction parameter and nonlinear stretching parameter in
magnitude. Table 5 illustrates the effect of all the physical
parameters on nondimensional rate of heat transfer. For
increasing values of nonlinear stretching parameter and
surface temperature ratio parameter, the nondimensional rate
of heat transfer enhances meanwhile the physical parameters
such as magnetic interaction parameter, radiation parameter,
thermophoresis parameter, Brownian motion parameter and
Eckert number diminishes the nondimensional rate of heat
transfer.
Table 4: Skin friction coefficient for different values of M2
and n
0 1 2 3 4 5 6 7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Nt = 0.2, 0.4, 0.5, 0.7
Cu - WaterM
2 = 1.0
Pr = 6.2
NR = 1.0
Ec = 1.0
n = 10.0
Nb = 0.5
Le = 0.6
Ag - Water
n
M2
Cu - water Ag - Water
2.5
10
1
n f
2.5
10
1
n f
0.1
10.0
0.0
0.5
1.0
1.5
-6.26091
-6.81681
-7.32783
-7.80406
-6.52906
-7.06412
-7.55865
-8.02142
1.0
2.0
3.0
10.0
1.0 -2.69755
-3.52561
-4.19368
-7.32783
-2.77233
-3.62968
-4.32062
-7.55865
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12448
Table 5: Nondimensional Heat transfer rate for different values of
M2, n, NR, Nb , Nt, Ec and w when = 0.1, Le = 0.6 and Pr = 6.2
M2
n
NR
Nb
Nt
Ec
w
Cu – Water Ag – Water
341 1 (0)
3
nf fw
f nfR
k kn
k k N
341 1 (0)
3
nf fw
f nfR
k kn
k k N
0.0
0.5
1.0
1.5
10.0 1.0 0.5 0.5 1.0 1.1 11.11955
8.83978
7.48352
6.19943
11.22666
9.74846
8.36668
7.06079
1.0 1.0
2.0
3.0
10.0
1.0 0.5 0.5 1.0 1.1 1.51643
2.77657
3.68201
7.48352
1.82964
3.19403
4.18242
8.36668
1.0 10.0 0.5
1.0
1.5
2.0
0.5 0.5 1.0 1.1 10.02533
7.48352
6.39860
5.79354
11.07347
8.36668
7.21558
6.57398
1.0 10.0 1.0 0.2
0.4
0.5
0.7
0.5 1.0 1.1 7.48446
7.48368
7.48352
7.48342
8.36734
8.36690
8.36668
8.36654
1.0 10.0 1.0 0.5 0.2
0.4
0.5
0.7
1.0 1.1 7.56653
7.51081
7.48352
7.42892
8.44682
8.39184
8.36668
8.35694
1.0 10.0 1.0 0.5 0.5 0.7
0.8
0.9
1.0
1.1 10.55091
9.59959
8.54185
7.48352
11.28963
10.38284
9.37520
8.36668
1.0 10.0 1.0 0.5 0.5 1.0 0.8
0.85
0.9
1.1
5.76352
6.01590
6.28266
7.48352
6.54953
6.81637
7.09819
8.36668
CONCLUSION
A role of Brownian motion and thermophoresis effects on
hydromagnetic flow of nanofluids past a nonlinearly
stretching sheet under consideration of viscous dissipation
and nonlinear thermal radiation have been investigated in
this work for two types of nanofluid Cu water nanofluid and
silver water nanofluid. Using similarity transformations the
governing equations of the problem are transformed into
nonlinear ordinary differential equations and solved
numerically by using most efficient Nachtsheim- Swigert
shooting iteration scheme for satisfaction of asymptotic
boundary conditions along with fourth order Runge-Kutta
Integration method (FORTRAN package). Numerical
solutions of the problem are obtained for various physical
parameters.
From the obtained numerical results and discussion
presented in the previous section, the following conclusions
are drawn
An increase in magnetic interaction parameter and
nonlinear stretching parameter decreases the nanofluid
velocity but opposite trend is occurred in skin friction
coefficient.
A rise in the magnetic interaction parameter,
thermophoresis parameter, temperature ratio parameter
and viscous dissipation parameter raises the temperature
distribution. In the mean while nonlinear stretching
parameter and radiation parameter decreases the
temperature distribution. Also the temperature has very
less effect due to Brownian motion parameter.
Nanoparticle volume fraction decelerates with an
increase in the values nonlinear stretching parameter,
Brownian motion parameter and Lewis number.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12449
Nanoparticle volume fraction accelerates for the
increasing values of thermophoresis parameter and
magnetic interaction parameter.
Nondimensional heat transfer rate enhances by means
of rise in the values of nonlinear stretching parameter
and surface temperature parameter but the
nondimensional rate of heat transfer decelerates with an
increasing value of magnetic interaction parameter and
radiation parameter, thermophoresis parameter,
Brownian motion parameter and Eckert number.
Sherwood number vanishes for nanofluids two phase
model with new type of boundary condition.
Finally, the numerical values of nondimensional rate of
heat transfer of Ag - water nanofluid is higher than the
Cu - water nanofluid.
Appendix
The expressions for the physical quantities,,nf nf nfk ,
nf,and p nf
c are given through the following lines
[Ahmad et al. (2011)],
1nf f s ,
2.5
1
fnf
,
2 2
2
s f f snf f
s f f s
k k k kk k
k k k k
,
nfnf
nf
,
1p nf p pf sc c c
The constants values are as follows,
1
2.51
11 ,s
f
bc
,
2
1p
p
s
f
cc
c
Table 1:Thermo-physical properties of fluid and
nanoparticles at 25C
Physical properties Water fluid Cu Ag
CP 4179 385 235
997.1 8933 10500
K 0.613 400 429
REFERENCES
[1] N.Afzal,“Heat transfer from a stretching surface”, Int. J. Heat Mass Transfer, vol.36,pp.1128-1131, 1993.
[2] S.Ahmad,A.M.Rohni, I.Pop,“Blasious and Sakiadis
problems in nanofluids”,Acta Mechanica,
vol.218,pp.195-204, 2011.
[3] M.Q.Brewster,“Thermal Radiative Transfer and
Properties”John Wiley and sons Inc, 1992.
[4] J.Buongiorna, “Convective transport in nanofluids”, Jl. of heat transfer,vol. 128(3), pp.240-250,2006.
[5] S.Choi,“Enhancing thermal conductivity of fluids with
nanoparticles. I sidiner DA, Wang HP (eds)
Developments and applications of non-Newtonian
flows”,ASMEFED, 231/MD, pp.99-105,1995.
[6] R.Cortell,“Effects of viscous dissipation on and
radiation on the thermal boundary layer over a
nonlinearly stretching sheet”, Physics Letters A,
vol.372(5), pp.631-636, 2008.
[7] L.J.Crane,”Flow past a stretching plate”Zeitschrift für Angewandte Mathematik und Physik,
vol.21(4),pp.645–647, 1970.
[8] S.K.Das,S.Choi,W.Yu&T.Pradet,“Nanofluids: science
and Technology,” Wiley, New Jersey, 2007.
[9] S.P.A.Devi, M.Thiyagarajan, “Steady nonlinear
hydromagnetic flow and heat transfer over a stretching
surface of variable temperature”,Heat Mass Transfer,vol.42,pp.671–677,2006.
[10] M.A.A.Hamad,M.Ferdows, “Similarity solutions to
viscous flow and heat transfer of nanofluid over
nonlinearly stretching sheet”, Appl. Math. Mech. Engl. Ed., vol.33(7), pp. 923–930, 2012.
[11] V.Kaufui, Wong and Omar De Leon, “Applications of
nanofluids:Current and Future”, Advances in mechanical engineering,pp.1-11,2010.
[12] A.V.Kuznetsov,D.A.Nield, “Natural convective
boundary layer flow of a nanofluid past a vertical
plate: A revised model,”Int. J. of thermal sciences,
vol.77, 126-129,2014.
[13] Mohammed Saad Kamel, Raheem Abed
Syeal,Abdulameer Amdulhussein, “Heat transfer
enhancement using nanofluid: A review of the recent
literature,”American Jour. of Nano Research and Applications, vol.24(1), pp.1-5,2016.
[14] P.O.Olanrewaju,J.A.Gbadeyan,O.O.Agboola, and
S.O.Abah, “Radiation and viscous dissipation effects
for the Blasius and Sakiadis flows with a convective
surface boundary condition”, International Journal of Advances in Science and Technology,vol.2(4),2011.
[15] D.Pal, G.Mandal, and K.Vajravelu, “Convective-
Radiation Effects on Stagnation Point Flow of
Nanofluids Over a Stretching/Shrinking Surface with
Viscous Dissipation”,Journal of Mechanics,
vol.8,pp.1-9,2014.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 22 (2017) pp. 12440-12450
© Research India Publications. http://www.ripublication.com
12450
[16] K.B.Pavlov,“Magnetohydrodynamic flow of an
incompressible viscous fluid caused by the
deformation of a plane surface”. Magnytnaya Gidrodinamika, vol.4, pp.146–147, 1974.
[17] Rafael Cortell, “Viscous flow and heat transfer over a
nonlinearly stretching sheet”, Applied Mathematics and Computation,vol.184,pp.864–873,2007.
[18] P.Rana,R.Bhargava, “Flow and heat transfer of a
nanofluid over a nonlinearly stretching sheet: A
numerical study”,Commun Nonlinear Sci Numer Simulat,vol.17,212–pp.226,2012.
[19] A.Raptis.“Flow of a micripolarfluid past a
continuously moving plate by the presence of
Radiation”, Int. Jour. of Heat & Mass
transfer,vol.41(18),pp.2865-2866,1998.
[20] M.G.Reddy,“Influence of Magnetohydrodynamic and
Thermal Radiation Boundary Layer Flow of a
Nanofluid Past a Stretching Sheet”,J. Sci. Res.Vol.6(2), pp.257-272,2014.
[21] Reza Azizian, Elham Doroodchi, and Behdad
Moghtaderi,“Effect of nanoconvection caused by
brownian motion on the enhancement of thermal
conductivity in nanofluids”, Ind. Eng. Chem. Res.,vol.51, pp.1782–1789,2012.
[22] Ritu Pasrija and Sunita Srivastava, “On the Effective
Thermal Conductivity of metallic and oxide
Nanofluids”,Int. Jour. of NanoScience and Nanotechnology,vol.4,pp.131-143,2013.
[23] S.Rosseland,“Theoretical Astrophysics”, Clarendon Press, Oxford, 1936.
[24] Sandeep Pal, Tikamchand Soni Akriti Agrawala and
Deepak Sharma, “Review on Enhanced Thermal
Conductivity of Colloidal Suspension of Nanosized
Particles (Nanofluids)”,Int. Jour of Advanced Mechanical Engineering,vol.4,pp.199-214,2014.
[25] E,M,Sparrow, R.D.Cess,“Radiation heat transfer
hemisphere”, Washington(Chaps. 7 & 10), 1978.
[26] P.Sreenivasulu,T.Poornima, Bhaskar
N.Reddy.,“Thermal radian effects on MHD boundary
layer slip flow past a permeable exponential stretching
sheet in the presence of Joule heating and viscous
dissipation”, JAFM, vol.9(1), pp.267-278,2016.
[27] Sumalatha, Chenna Shanker, Bandari,“MHD Mixed
Convection Flow of a Nanofluid Over a Nonlinear
Stretching Sheet with Variable Wall Temperature and
Volume fraction”,Journal of
Nanofluids,vol.5(5), pp.707-712,2016.
[28] K.Vajravelu, “The effect of variable viscosity on the
flow and heat transfer of a viscous Ag- water and Cu-
water nanofluids”, Journal of Hydrodynamics,vol.25,
pp.1-9,2012.
[29] K.Vajravelu and K.V.Prasad,“Heat transfer
phenomena in a moving nanofluid over a horizontal
surface”, Journal of Mechanics, vol.28, pp.579-
588,2012.
[30] Yohannes Yirga and Bandari Shankar, “Effects of
thermal radiation and viscous dissipation on
magnetohydrodynamic stagnation point flow and heat
transfer of nanofluid towards a stretching
sheet,”Journal of Nanofluids, vol.2, pp.283–291, 2013.