Research Article An Exact Analysis of Heat and Mass ...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 434571 9 pageshttpdxdoiorg1011552013434571

Research ArticleAn Exact Analysis of Heat and Mass Transfer Past a VerticalPlate with Newtonian Heating

Abid Hussanan Ilyas Khan and Sharidan Shafie

Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia (UTM) 81310 Skudai Johor Malaysia

Correspondence should be addressed to Sharidan Shafie ridafieyahoocom

Received 3 February 2013 Revised 30 April 2013 Accepted 15 May 2013

Academic Editor Mehmet Sezer

Copyright copy 2013 Abid Hussanan et alThis 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

An exact analysis of heat and mass transfer past an oscillating vertical plate with Newtonian heating is presented Equationsare modelled and solved for velocity temperature and concentration using Laplace transforms The obtained solutions satisfygoverning equations and conditions Expressions of skin friction Nusselt number and Sherwood number are obtained andpresented in tabular forms The results show that increasing the Newtonian heating parameter leads to increase velocity andtemperature distributions whereas skin friction decreases and rate of heat transfer increases

1 Introduction

Generally the problems of free convection flows are usuallymodeled under the assumptions of constant surface tem-perature ramped wall temperature or constant surface heatflux [1ndash7] However in many practical situations where theheat transfer from the surface is taken to be proportionalto the local surface temperature the above assumptionsfail to work Such types of flows are termed as conjugateconvective flows and the proportionally condition of theheat transfer to the local surface temperature is termed asNewtonian heating This work was pioneered by Merkin[8] for the free convection boundary layer flow over avertical flat plate immersed in a viscous fluid Howeverdue to numerous practical applications in many importantengineering devices several other researchers are gettinginterested to consider the Newtonian heating condition intheir problems Few of these applications are found in heatexchanger heat management in electrical appliances (suchas computer power supplies or substation transformer) andengine cooling (such as thin fins in car radiator) Moreoverthe flow over an oscillating plate with Newtonian heating alsooccurs in the conjugate heat transfer around fins where theconduction within the fin and the convection in the fluidsurrounding it must be simultaneously analyzed in order toobtain the vital design information and in also convection

flows set up when the bounding surface absorb heat by solarradiation The literature survey shows that much attentionto the problems of free convection flow with Newtonianheating is given by numerical solvers as we can see [9ndash16] andthe references therein However the exact solutions of theseproblems are very few [17ndash21] Exact solutions on the otherhand can provide an important check for numerical methodsthat are used to study such flows in more complex domains

Furthermore the free convection flows together with heatand mass transfer are of great importance in geophysicsaeronautics and engineering In several process such asdrying evaporation of water at body surface energy transferin a wet cooling tower and flow in a desert cooler heatand mass transfer occurs simultaneously In view of suchapplications several authors investigated free convectionflows with simultaneous heat andmass transfer phenomenon[22ndash25]

To the best of authorsrsquo knowledge so far no studyhas been reported in the literature which investigates theunsteady free convection flow of an incompressible viscousfluid past an oscillating vertical plate with Newtonian heatingand constant mass diffusion The present study is an attemptin this direction to fill this space In this study the equationsof the problem are first formulated and transformed into theirdimensionless forms where the Laplace transform method isapplied to find the exact solutions for velocity temperature

2 Journal of Applied Mathematics

Momentum boundary layer

Thermal boundary layer

Concentration boundary layer

0

T998400 = Tinfin C998400 = Cinfin

g

y998400

x998400u998400=U0

cos(120596998400 t998400 )120597T998400

120597y998400=minushsT

998400 C998400=C120596

Figure 1 Physical model and coordinate system

and concentration Moreover expressions for skin frictionNusselt number and Sherwood number are obtained andpresented in tabular forms Finally the obtained results areplotted graphically and discussed for the pertinent flowparameters

2 Mathematical Formulation

Consider the unsteady free convection flow of a viscousincompressible fluid past a vertical plate The 1199091015840-axis is takenalong the vertical plate and the 1199101015840-axis is taken normal tothe plate Initially for time 1199051015840 le 0 both the plate and fluidare at stationary condition with the constant temperature 119879

infin

and concentration 119862infin At time 1199051015840 = 0

+ the plate started anoscillatory motion in its plane with the velocity 119880

0cos (12059610158401199051015840)

against the gravitational field where1198800is the amplitude of the

plate oscillations At the same time the heat transfer from theplate to the fluid is directly proportional to the local surfacetemperature 1198791015840 and the concentration level near the plate israised from 119862

infinto 119862119908 As the plate is considered infinite in

the 1199091015840-axis therefore all physical variables are independent of1199091015840 and are functions of 1199101015840 and 1199051015840 onlyThe physical model and

coordinate system are presented in Figure 1Under the above assumptions the governing equations of

the free convective flow with Boussinesqrsquos approximation areas follows

1205971199061015840

1205971199051015840= ]

12059721199061015840

12059711991010158402+ 119892120573 (119879

1015840minus 119879infin) + 119892120573

lowast(1198621015840minus 119862infin) (1)

120588119888119901

1205971198791015840

1205971199051015840= 119896

12059721198791015840

12059711991010158402minus

120597119902119903

1205971199101015840 (2)

1205971198621015840

1205971199051015840= 119863

12059721198621015840

12059711991010158402 (3)

The initial and boundary conditions are1199051015840le 0 119906

1015840= 0 119879

1015840= 119879infin 1198621015840= 119862infin

forall1199101015840ge 0

1199051015840gt 0 119906

1015840= 1198800cos (12059610158401199051015840)

1205971198791015840

1205971199101015840= minusℎ1199041198791015840 1198621015840= 119862119908

at 1199101015840 = 0

1199061015840997888rarr 0 119879

1015840997888rarr 119879infin 1198621015840997888rarr 119862

infinas 1199101015840 997888rarr infin

(4)

The radiation heat flux under Rosseland approximation[26] is expressed by

119902119903= minus

4120590

3119896lowast

12059711987910158404

1205971199101015840 (5)

It should be noted that by using the Rosseland approx-imation we limit our analysis to optically thick fluids It isassumed that the temperature difference within the flow aresufficiently small and then (5) can be linearized by expanding11987910158404 into the Taylor series about 119879

infin which after neglecting

higher order terms takes the form

11987910158404cong 41198793

infin1198791015840minus 31198794

infin (6)

In view of (5) and (6) (2) reduces to

120588119888119901

1205971198791015840

1205971199051015840= 119896(1 +

161205901198793

infin

3119896119896lowast)1198791015840 (7)

To reduce the above equations into their nondimensionalforms we introduce the following nondimensional quanti-ties

119910 =

11991010158401198800

] 119905 =

11990510158401198802

0

] 119906 =

1199061015840

1198800

120579 =

1198791015840minus 119879infin

119879infin

119862 =

1198621015840minus 119862infin

119862119908minus 119862infin

120596 =

1205961015840]

1198802

0

(8)

Substituting (8) into (1) (7) and (3) we obtain thefollowing nondimensional Partial Differential Equations

120597119906

120597119905

=

1205972119906

1205971199102+ Gr 120579 + Gm119862

Pr120597120579120597119905

= (1 + 119877)

1205972120579

1205971199102

Sc120597119862120597119905

=

1205972119862

1205971199102

(9)

where

Gr =]119892120573119879infin

1198803

0

Gm =

]119892120573lowast (119862119908minus 119862infin)

1198803

0

Pr =120583119888119901

119896

119877 =

161205901198793

infin

3119896119896lowast Sc = ]

119863

(10)

are the Grashof number modified Grashof number Prandtlnumber radiation parameter and Schmidt number respec-tively The corresponding initial and boundary conditions innondimensional form are

119905 le 0 119906 = 0 120579 = 0 119862 = 0 forall119910 ge 0 (11)

119905 gt 0 119906 = cos (120596119905) 120597120579120597119910

= minus120574 (1 + 120579) 119862 = 1 at 119910 = 0

(12)

119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin (13)Here 120574 = ℎ

119904]1198800is theNewtonian heating parameterWe

note that (12) gives 120579 = 0when 120574 = 0 which physically meansthat no heating from the plate exists [12 16]

Journal of Applied Mathematics 3

3 Method of Solution

The Laplace transform method solves differential equationsand corresponding initial and boundary value problemsTheLaplace transform has the advantage that it solves problemsdirectly initial value problems without determining first ageneral solution and nonhomogeneous differential equationswithout solving first the corresponding homogeneous equa-tions In order to obtain the exact solution of the presentproblem wewill use the Laplace transform technique Apply-ing the Laplace transformwith respect to time 119905 to the systemof (9) we get

119902119906 (119910 119902) minus 119906 (119910 0) =

1198892119906 (119910 119902)

1198891199102

+ Gr 120579 (119910 119902) + Gm119862 (119910 119902)

Pr [119902120579 (119910 119902) minus 120579 (119910 0)] = (1 + 119877)1198892120579 (119910 119902)

1198891199102

Sc [119902119862 (119910 119902) minus 119862 (119910 0)] =1198892119862 (119910 119902)

1198891199102

(14)

Here 119906(119910 119902) = int

infin

0119890minus119902119905119906(119910 119905)119889119905 120579(119910 119902) = int

infin

0119890minus119902119905120579(119910 119905)119889119905

and 119862(119910 119902)=intinfin0119890minus119902119905119862(119910 119905)119889119905 denote the Laplace transforms

of 119906(119910 119905) 120579(119910 119905) and 119862(119910 119905) respectivelyUsing the initial condition (11) we get

1198892119906 (119910 119902)

1198891199102

minus 119902119906 (119910 119902) + Gr 120579 (119910 119902) + Gm119862 (119910 119902) = 0

1198892120579 (119910 119902)

1198891199102

minus 119902 (

Pr1 + 119877

) 120579 (119910 119902) = 0

1198892119862 (119910 119902)

1198891199102

minus 119902 Sc119862 (119910 119902) = 0

(15)

The corresponding transformed boundary conditions are

119906 (119910 119902) =

119902

1199022+ 1205962

119889120579 (119910 119902)

119889119910

= minus120574 [

1

119902

+ 120579 (119910 119902)]

119862 (119910 119902) =

1

119902

at 119910 = 0

119906 (119910 119902) 997888rarr 0 120579 (119910 119902) 997888rarr0 119862 (119910 119902) 997888rarr 0

as 119910 997888rarrinfin

(16)

The solutions of (15) subject to the boundary conditions(16) are

119906 (119910 119902) =

1

2 (119902 + 119894120596)

119890minus119910radic119902

+

1

2 (119902 minus 119894120596)

119890minus119910radic119902

+

119886119888

1199022(radic119902 minus 119888)

119890minus119910radic119902

minus

119886119888

1199022(radic119902 minus 119888)

119890minus119910radic119902Preff

+

119887

1199022119890minus119910radic119902

minus

119887

1199022119890minus119910radic119902Sc

120579 (119910 119902) =

119888

119902 (radic119902 minus 119888)


119862 (119910 119902) =

1

119902

119890minus119910radic119902Sc

(17)

where 119886 = Gr(Preff minus 1) 119887 = Gm(Sc minus 1) 119888 = 120574radicPreffand Preff = Pr(1+119877) is the effective Prandtl number definedby Magyari and Pantokratoras [27] By taking the inverseLaplace transform of (17) and use formulae from Appendixwe obtain

120579 (119910 119905) = 1198654(119910radicPreff 119905 119888) (18)

119862 (119910 119905) = 1198651(119910radicSc 119905) (19)

119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

minus

119886

1198882[1198654(119910radicPreff 119905 119888) minus 1198654 (119910 119905 119888)]

+

119886

119888

[1198652(119910radicPreff 119905) minus 1198652 (119910 119905)]

+ 119886 [1198653(119910radicPreff 119905) minus 1198653 (119910 119905)]

minus 119887 [1198653(119910radicSc 119905) minus 119865

3(119910 119905)]

(20)

Note that the above solution is valid only for Preff = 1 andSc = 1 Moreover the other solutions are

Case 1 When Sc = 1 and Preff = 1

119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

minus

119886


+

119886

119888


+ 119886 [1198653(119910radicPreff 119905) minus 1198653 (119910 119905)] +

119910Gm2

1198652(119910 119905)

(21)



119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]


3(119910 119905)] +

119910Gr2120574

[1198654(119910 119905 120574)]

(22)


119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

+

119910Gr2120574

[1198654(119910 119905 120574)] minus

119910Gr2

times [1198652(119910 119905)] +

119910Gm2

[1198652(119910 119905)]

(23)

Here

1198651(V 119905) = erf 119888 ( V

2radic119905

)

1198652(V 119905) = 2radic

119905

120587

119890minusV24119905

minus V erf 119888 ( V

2radic119905

)

1198653(V 119905) = (

V2

2

+ 119905) erf 119888 ( V

2radic119905

)

minus Vradic119905

120587

119890minusV24119905

1198654(V 119905 120572) = 119890(120572

2119905minus120572V) erf 119888 ( V

2radic119905

minus 120572radic119905)

minus erf 119888 ( V

2radic119905

)

1198655(V 119905 120572) =

1

2

119890120572119905[119890minusVradic120572 erf 119888 ( V

2radic119905

minus radic120572119905)

+119890Vradic120572 erf 119888 ( V

2radic119905

+ radic120572119905)]

(24)

where erf 119888 (sdot) is the complementary error function V and120572 are dummy variables and 119865

1 1198652 1198653 1198654 and 119865

5are dummy

functionsThe dimensionless expression for skin friction evaluated

from (20) is given by

120591 =

1205911015840

1205881198802

0

= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=

1

2

[119890minus119894120596119905

radicminus119894120596 erf(radicminus119894120596119905)

+119890119894120596119905radic119894120596 erf(radic119894120596119905)]

+

119886

119888

[1 minus 1198901198882119905(1 + erf(119888radic119905))]

times (radicPreff minus 1) +1

radic120587119905

+ 2radicPreff119905120587

minus 2119886radic119905

120587


120587

(radicSc minus 1)

(25)

0 1 2 3 40

05

1

15

2

25

3

Velo

city

y

t = 01 02 03 04

Figure 2 Velocity profiles for different values of 119905 when 119877 = 2Pr =071Gr = 5Gm = 1 Sc = 078 120574 = 1 and 120596 = 1205876

0 05 1 15 2 25 30

02

04

06

08

1

Velo

city

y

R = 0 1 2 3

Figure 3 Velocity profiles for different values of 119877 when 119905 =

02Pr = 071Gr = 2Gm = 3 Sc = 094 120574 = 1 and 120596 = 1205874

The dimensionless expression of Nusselt number is givenby

Nu = minus

]

1198800(1198791015840minus 119879infin)

1205971198791015840

1205971199101015840

1003816100381610038161003816100381610038161003816100381610038161199101015840=0

=

1

120579 (0 119905)

+ 1

= 119888radicPreff(1 +1

1198901198882119905[1 + erf (119888radic119905)] minus 1

)

(26)

The dimensionless expression of Sherwood number isgiven by

Sh = minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radicSc120587119905

(27)

4 Graphical Results and Discussion

In order to reveal some relevant physical aspects of theobtained solutions the numerical results for velocity tem-perature and concentration are computed and shown graph-ically in Figures 2ndash16 to illustrate the influence of embeddedflow parameters such as time 119905 radiation parameter 119877Prandtl number Pr Grashof number Gr modified Grashof


Table 1 Skin friction variation

119905 119877 Pr Gr Gm Sc 120574 120596 120591

001 1 071 5 2 022 1 1205872 54263002 1 071 5 2 022 1 1205872 36395001 2 071 5 2 022 1 1205872 54049001 1 10 5 2 022 1 1205872 54401001 1 071 10 2 022 1 1205872 53663001 1 071 5 4 022 1 1205872 52727001 1 071 5 2 062 1 1205872 54537001 1 071 5 2 022 2 1205872 53473001 1 071 5 2 022 1 120587 54208

Table 2 Nusselt number variation

119905 119877 Pr 120574 Nu02 2 071 1 1311904 2 071 1 1105402 4 071 1 1147102 2 10 1 1465902 2 071 2 20347

Table 3 Sherwood number variation

119905 Sc Sh02 022 0591704 022 0418402 062 09938

numberGm Schmidt number Sc Newtonian heating param-eter 120574 and phase angle 120596119905 The numerical values for skinfriction Nusselt number and Sherwood number for theseparameters are also presented in Tables 1ndash3

The velocity profiles for different values of time 119905 areshown in Figure 2 It is observed that the velocity increaseswith increasing values of time 119905 The velocity profiles in caseof radiation and pure convection are shown in Figure 3 Itis found from this figure that the radiation parameter 119877 hasan accelerating effect on velocity Physically it is due to thefact that an increase in the radiation parameter 119877 for fixedvalues of other parameters decreases the rate of radiativeheat transfer to the fluid and consequently the fluid velocityincreasesThis behavior of119877 is quite identical with that foundin Figure 6 of Mohamed et al [28]

The effects of Prandtl number Pr on the velocity profilesare shown in Figure 4 for Pr = 071 (air) Pr = 10 (electrolyticsolution) Pr = 70 (water) and Pr = 100 (engine oil)It is seen from this figure that an increase in the values ofPrandtl number Pr results in the decrease of velocity Inheat transfer analysis the role of Prandtl number Pr is tocontrol the relative thickness of the momentum and thermalboundary layers For small value of Pr the heat diffuses veryquickly compared to the velocity This means that for liquidmetals the thickness of the thermal boundary layer is muchbigger than the velocity boundary layerThe effects ofGrashofnumberGr andmodifiedGrashof numberGmon velocity areshown in Figures 5 and 6 It is found that the effects ofGrashof

0 1 2 3 4 50

05

1

15

2

25

Velo

city

y

= 071 1 7 100Pr

Figure 4 Velocity profiles for different values of Pr when 119905 =

05 119877 = 2Gr = 2Gm = 1 Sc = 078 120574 = 1 and 120596 = 1205876

0 05 1 15 2 25 3 350

05

1

15

2

Velo

city

y

= 0 2 5 10Gr

Figure 5 Velocity profiles for different values of Gr when 119905 =

02 119877 = 3Pr = 071Gm = 2 Sc = 078 120574 = 1 and 120596 = 1205873

numberGr andmodifiedGrashof numberGmon velocity aresimilar Velocity increases with increasing values of Gr andGm Physically it is possible because an increase in the valuesof Grashof number Gr and modified Grashof number Gmhas the tendency to increase the thermal and mass buoyancyeffects This gives rise to an increase in the induced flowFurther from these figures it is noticed that Grashof numberand modified Grashof number do not have any influence asthe fluid move away from the bounding surface

The effects of Sc on the velocity profiles are shown inFigure 7 Four different values of Schmidt number Sc =

022 062 078 and 094 are chosen They physically corre-spond to hydrogen water vapour ammonia and carbondioxide respectively It is clear that the velocity decreases asthe Schmidt number Sc increases Further it is clear from thisfigure that the velocity for hydrogen is themaximum and car-bon dioxide carries the minimum velocity Figure 8 displaysthe effect of Newtonian heating parameter 120574 on the dimen-sionless velocity It is found that as the Newtonian heatingparameter increases the density of the fluid decreases andthe momentum boundary layer thickness increases and as aresult and the velocity increases within the boundary layer

The graphical results for the phase angle 120596119905 are shownin Figure 9 It is interesting to note that when the phase


0 05 1 15 2 250

02

04

06

08

1

Velo

city

y

= 0 1 2 3Gm

Figure 6 Velocity profiles for different values of Gm when 119905 =

02 119877 = 1Pr = 071Gr = 5 Sc = 078 120574 = 1 and 120596 = 1205873

0 1 2 3 40

02

04

06

08

Velo

city

y

= 022 062 078 094Sc

Figure 7 Velocity profiles for different values of Sc when 119905 =

04 119877 = 2Pr = 071Gr = 5Gm = 2 120574 = 01 and 120596 = 1205873

angle 120596119905 is zero which physically corresponds to no oscilla-tion then the fluid approaches to its maximum velocity ofmagnitude 1 meter per second whereas for the phase angle120596119905 = 1205872 the velocity gains its minimum value of magnitude0 meter per second The oscillations near the plate are ofgreat significance however these oscillations reduce for largevalues of the independent variable 119910 and approach to zeroas 119910 tends to infinity The velocity profiles are plotted inFigure 10 for different values of Sc when 120596 = 0 (impulsivemotion of the plate) It is found from this figure that thebehavior of Sc on the velocity profiles quite identical with thatfound in Figure 6 of Narahari and Nayan [20] Further allthese graphical results discussed above are in good agreementwith the imposed boundary conditions given by (12) and (13)Hence this ensures the accuracy of our results

The effects of various parameters on the temperatureand concentration profiles are shown in Figures 11ndash16 Inthese figures Figure 11 exhibits the influence of dimensionlesstime 119905 on the temperature It is found that the temperatureprofiles increase with increasing time From Figure 12 it isnoted that an increase in the radiation parameter 119877 leads toan increase in the temperature due to the fact that thermalboundary layer thickness of fluid increases The influenceof Prandtl number Pr on temperature profiles is shown in

0 1 2 3 40

02

04

06

08

1

12

Velo

city

y

120574 = 01 02 03 04


04 119877 = 3Pr = 071Gr = 5Gm = 2 Sc = 078 and 120596 = 1205873

0 1 2 3 4 5

0

02

04

06

08

1

120596t = 0120587

3120587

22120587

3Ve

loci

ty

y

minus02

minus04

Figure 9 Velocity profiles for different values of 120596119905when 119905 = 1 119877 =05Pr = 100Gr = 2Gm = 02 Sc = 094 and 120574 = 001

Figure 13 It is found that the temperature decreases as thePrandtl number Pr increases Physically the increase of Prmeans the decrease of thermal conductivity of fluid FromFigure 14 it is observed that an increase in the Newtonianheating parameter increases the thermal boundary layerthickness and as a result the surface temperature of the plateincreases On the other hand it is found from Figure 15that the influence of time 119905 on concentration profiles issimilar to the velocity and temperature profiles given inFigures 2 and 11 The effects of Schmidt number Sc on theconcentration profiles are shown in Figure 16 It is seen fromthis figure that an increase in the value of Schmidt numbermakes the concentration boundary layer thin and hence theconcentration profiles decrease

The numerical results for skin friction Nusselt numberand Sherwood number are shown in Tables 1 2 and 3 forvarious parameters of interest It is depicted from Table 1that skin friction decreases with increasing 119905 119877GrGm 120574and 120596119905 while it increases as Pr and Sc are increased Table 2reveals that the Nusselt number increases as Pr and 120574 areincreased and decreases when 119905 and 119877 are increased FromTable 3 it is observed that the Sherwood number increaseswith increasing Sc while reverse effect is observed for 119905


Table 4 Inverse Laplace transform formulae119865(119902) = 119871119891(119905) 119891(119905)

1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905

minus 119886radic119905)minus erf c(119910

2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905

+ radic119886119905) + 119890minus119910radic119886 erf c(

119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08

Figure 11 Temperature profiles for different values of 119905 when 119877 =

1Pr = 071 and 120574 = 1

5 Conclusions

In this paper exact solutions of unsteady free convectionflow of an incompressible viscous fluid past an oscillatingvertical plate with Newtonian heating and constant massdiffusion are obtained using Laplace transform techniqueThe results obtained show that the velocity and temperatureare increased with increasing Newtonian heating parameterFurther the effect of Newtonian heating parameter increasesthe Nusselt number but reduces the skin friction Howeverthe Nusselt number is decreased when the radiation param-eter is increased Also the skin friction is decreased when

0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3

Figure 12 Temperature profiles for different values of 119877 when 119905 =02Pr = 071 and 120574 = 1

0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr

Figure 13 Temperature profiles for different values of Pr when 119905 =02 119877 = 5 and 120574 = 001

the radiation parameter phase angle and Grashof numberare increased The exact solutions obtained in this studyare significant not only because they are solutions of somefundamental flows but also they serve as accuracy standardsfor approximate methods whether numerical asymptotic orexperimental

Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04

Figure 14 Temperature profiles for different values of 120574 when 119905 =02 119877 = 1 and Pr = 071

0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08

Figure 15 Concentration profiles for different values of 119905when Sc =022

Nomenclature

1198621015840 Species concentration in the fluid

119862119908 Species concentration near the plate

119862infin Species concentration in the fluid far awayfrom the plate

119862 Dimensionless concentration119888119901 Heat capacity at a constant pressure

119863 Mass diffusivity119892 Acceleration due to gravityℎ119904 Heat transfer parameter for Newtonian

heatingGr Thermal Grashof numberGm Modified Grashof number119896 Thermal conductivity of the fluidPr Prandtl number119902119903 Radiative heat flux in the 1199101015840-direction

119877 Radiation parameterSc Schmidt number1198791015840 Temperature of the fluid

119879infin Ambient temperature

1199051015840 Time119905 Dimensionless time1199061015840 Velocity of the fluid in the 1199091015840-direction119906 Dimensionless velocity

0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc

Figure 16 Concentration profiles for different values of Sc when119905 = 02

1199101015840 Coordinate axis normal to the plate119910 Dimensionless coordinate axis normal

to the plate119896 Thermal conductivity119896lowast Mean absorption coefficient120573 Volumetric coefficient of thermal

expansion120573lowast Volumetric coefficient of mass

expansion120583 Coefficient of viscosity] Kinematic viscosity120588 Fluid density120590 Stefan-Boltzmann constant120591 Skin friction1205911015840 Dimensionless skin friction120579 Dimensionless temperature1205961015840 Frequency of oscillation

120596119905 Phase angleerf 119888 Complementary error function

Acknowledgments

The authors would like to acknowledge MOHE and ResearchManagement Centre UTM for the financial support throughVote nos 4F109 and 04H27 for this research

References

[1] P Chandrakala ldquoRadiation effects on flow past an impulsivelystarted vertical oscillating plate with uniform heat fluxrdquo Inter-national Journal of Dynamics of Fluids vol 6 pp 209ndash215 2010

[2] R K Deka and S K Das ldquoRadiation effects on free convectionflow near a vertical plate with ramped wall temperaturerdquoEngineering vol 3 pp 1197ndash1206 2011

[3] P Chandran N C Sacheti and A K Singh ldquoNatural convec-tion near a vertical plate with ramped wall temperaturerdquo Heatand Mass Transfer vol 41 no 5 pp 459ndash464 2005

[4] VM Soundalgekar RM Lahurikar S G Pohanerkar andN SBirajdar ldquoEffects of mass transfer on the flow past an oscillatinginfinite vertical plate with constant heat fluxrdquo Thermophysicsand AeroMechanics vol 1 pp 119ndash124 1994


[5] VM Soundalgekar RM Lahurikar S G Pohanerkar andN SBirajdar ldquoMass transfer effects on flow past a vertical oscillatingplate with variable temperaturerdquo Heat and Mass Transfer vol30 no 5 pp 309ndash312 1995

[6] G Pathak C H Maheshwari and S P Gupta ldquoEffects ofradiation onunsteady free convection flowbounded by an oscil-lating plate with variable temperaturerdquo International Journal ofApplied Mechanics and Engineering vol 11 pp 371ndash382 2006

[7] R Muthucumaraswamy ldquoNatural convection on flow past animpulsively started vertical plate with variable surface heat fluxrdquoFar East Journal of Applied Mathematics vol 14 no 1 pp 99ndash119 2004

[8] J H Merkin ldquoNatural-convection boundary-layer flow on avertical surface with Newtonian heatingrdquo International Journalof Heat and Fluid Flow vol 15 no 5 pp 392ndash398 1994

[9] D Lesnic D B Ingham and I Pop ldquoFree convection boundary-layer flow along a vertical surface in a porous medium withNewtonian heatingrdquo International Journal of Heat and MassTransfer vol 42 no 14 pp 2621ndash2627 1999

[10] D Lesnic D B Ingham and I Pop ldquoFree convection froma horizontal surface in a porous medium with newtonianheatingrdquo Journal of Porous Media vol 3 no 3 pp 227ndash2352000

[11] D Lesnic D B Ingham I Pop and C Storr ldquoFree convectionboundary-layer flow above a nearly horizontal surface in aporous medium with newtonian heatingrdquo Heat and MassTransfer vol 40 no 9 pp 665ndash672 2004

[12] M Z Salleh R Nazar and I Pop ldquoBoundary layer flow andheat transfer over a stretching sheet with Newtonian heatingrdquoJournal of the Taiwan Institute of Chemical Engineers vol 41 no6 pp 651ndash655 2010

[13] M Z Salleh R Nazar and I Pop ldquoNumerical solutions offree convection boundary layer flow on a solid sphere withNewtonian heating in a micropolar fluidrdquo Meccanica vol 47pp 1261ndash1269 2012

[14] M Z Salleh R Nazar N M Arifin I Pop and J H MerkinldquoForced-convection heat transfer over a circular cylinder withNewtonian heatingrdquo Journal of Engineering Mathematics vol69 no 1 pp 101ndash110 2011

[15] S Das C Mandal and R N Jana ldquoRadiation effects onunsteady free convection flow past a vertical plate with New-tonian heatingrdquo International Journal of Computer Applicationsvol 41 pp 36ndash41 2012

[16] A RM Kasim N FMohammad Aurangzaib and S SharidanldquoNatural convection boundary layer flow of a viscoelastic fluidon solid sphere with Newtonian heatingrdquo World Academy ofScience Engineering and Technology vol 64 pp 628ndash633 2012

[17] R C Chaudhary and P Jain ldquoUnsteady free convection bound-ary layer flow past an impulsively started vertical surface withNewtonian heatingrdquo Romanian Journal of Physics vol 51 pp911ndash925 2006

[18] P Mebine and E M Adigio ldquoUnsteady free convection flowwith thermal radiation past a vertical porous plate with newto-nian heatingrdquo Turkish Journal of Physics vol 33 no 2 pp 109ndash119 2009

[19] M Narahari and A Ishak ldquoRadiation effects on free convectionflow near a moving vertical plate with newtonian heatingrdquoJournal of Applied Sciences vol 11 no 7 pp 1096ndash1104 2011

[20] MNarahari andMYunusNayan ldquoFree convection flowpast animpulsively started infinite vertical plate with Newtonian heat-ing in the presence of thermal radiation and mass diffusionrdquo

Turkish Journal of Engineering and Environmental Sciences vol35 no 3 pp 187ndash198 2011

[21] V Rajesh ldquoEffects of mass transfer on flow past an impulsivelystarted infinite vertical platewithNewtonian heating and chem-ical reactionrdquo Journal of Engineering Physics andThermophysicsvol 85 no 1 pp 221ndash228 2012

[22] P Chandrakala and P N Bhaskar ldquoRadiation effects on oscil-lating vertical plate with uniform heat flux and mass diffusionrdquoInternational Journal of Fluids Engineering vol 4 pp 1ndash11 2012

[23] R Muthucumaraswamy P Chandrakala and S A Raj ldquoRadia-tive heat andmass transfer effects onmoving isothermal verticalplate in the presence of chemical reactionrdquo International Journalof AppliedMechanics and Engineering vol 11 no 3 pp 639ndash6462006

[24] R Muthucumaraswamy and A Vijayalakshmi ldquoEffects of heatand mass transfer on flow past an oscillating vertical platewith variable temperaturerdquo International Journal of AppliedMathematics and Mechanics vol 4 no 1 pp 59ndash65 2008

[25] V R Prasad N B Reddy and R Muthucumaraswamy ldquoRadia-tion and mass transfer effects on two-dimensional flow past animpulsively started infinite vertical platerdquo International Journalof Thermal Sciences vol 46 no 12 pp 1251ndash1258 2007

[26] R Siegel and J R Howell Thermal Radiation Heat TransferTaylor amp Francis New York NY USA 4th edition 2002

[27] E Magyari and A Pantokratoras ldquoNote on the effect of thermalradiation in the linearized Rosseland approximation on theheat transfer characteristics of various boundary layer flowsrdquoInternational Communications in Heat and Mass Transfer vol38 no 5 pp 554ndash556 2011

[28] R A Mohamed S M Abo-Dahab and T A Nofal ldquoThermalradiation and MHD effects on free convective flow of a polarfluid through a porous medium in the presence of internal heatgeneration and chemical reactionrdquo Mathematical Problems inEngineering vol 2010 Article ID 804719 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Momentum boundary layer

Thermal boundary layer

Concentration boundary layer

0

T998400 = Tinfin C998400 = Cinfin

g

y998400

x998400u998400=U0

cos(120596998400 t998400 )120597T998400

120597y998400=minushsT

998400 C998400=C120596

Figure 1 Physical model and coordinate system

and concentration Moreover expressions for skin frictionNusselt number and Sherwood number are obtained andpresented in tabular forms Finally the obtained results areplotted graphically and discussed for the pertinent flowparameters

2 Mathematical Formulation

Consider the unsteady free convection flow of a viscousincompressible fluid past a vertical plate The 1199091015840-axis is takenalong the vertical plate and the 1199101015840-axis is taken normal tothe plate Initially for time 1199051015840 le 0 both the plate and fluidare at stationary condition with the constant temperature 119879

infin

and concentration 119862infin At time 1199051015840 = 0

+ the plate started anoscillatory motion in its plane with the velocity 119880

0cos (12059610158401199051015840)

against the gravitational field where1198800is the amplitude of the

plate oscillations At the same time the heat transfer from theplate to the fluid is directly proportional to the local surfacetemperature 1198791015840 and the concentration level near the plate israised from 119862

infinto 119862119908 As the plate is considered infinite in

the 1199091015840-axis therefore all physical variables are independent of1199091015840 and are functions of 1199101015840 and 1199051015840 onlyThe physical model and

coordinate system are presented in Figure 1Under the above assumptions the governing equations of

the free convective flow with Boussinesqrsquos approximation areas follows

1205971199061015840

1205971199051015840= ]

12059721199061015840

12059711991010158402+ 119892120573 (119879

1015840minus 119879infin) + 119892120573

lowast(1198621015840minus 119862infin) (1)

120588119888119901

1205971198791015840

1205971199051015840= 119896

12059721198791015840

12059711991010158402minus

120597119902119903

1205971199101015840 (2)

1205971198621015840

1205971199051015840= 119863

12059721198621015840

12059711991010158402 (3)

The initial and boundary conditions are1199051015840le 0 119906

1015840= 0 119879

1015840= 119879infin 1198621015840= 119862infin

forall1199101015840ge 0

1199051015840gt 0 119906

1015840= 1198800cos (12059610158401199051015840)

1205971198791015840

1205971199101015840= minusℎ1199041198791015840 1198621015840= 119862119908

at 1199101015840 = 0

1199061015840997888rarr 0 119879

1015840997888rarr 119879infin 1198621015840997888rarr 119862

infinas 1199101015840 997888rarr infin

(4)

The radiation heat flux under Rosseland approximation[26] is expressed by

119902119903= minus

4120590

3119896lowast

12059711987910158404

1205971199101015840 (5)

It should be noted that by using the Rosseland approx-imation we limit our analysis to optically thick fluids It isassumed that the temperature difference within the flow aresufficiently small and then (5) can be linearized by expanding11987910158404 into the Taylor series about 119879

infin which after neglecting

higher order terms takes the form

11987910158404cong 41198793

infin1198791015840minus 31198794

infin (6)

In view of (5) and (6) (2) reduces to

120588119888119901

1205971198791015840

1205971199051015840= 119896(1 +

161205901198793

infin

3119896119896lowast)1198791015840 (7)

To reduce the above equations into their nondimensionalforms we introduce the following nondimensional quanti-ties

119910 =

11991010158401198800

] 119905 =

11990510158401198802

0

] 119906 =

1199061015840

1198800

120579 =

1198791015840minus 119879infin

119879infin

119862 =

1198621015840minus 119862infin

119862119908minus 119862infin

120596 =

1205961015840]

1198802

0

(8)

Substituting (8) into (1) (7) and (3) we obtain thefollowing nondimensional Partial Differential Equations

120597119906

120597119905

=

1205972119906

1205971199102+ Gr 120579 + Gm119862

Pr120597120579120597119905

= (1 + 119877)

1205972120579

1205971199102

Sc120597119862120597119905

=

1205972119862

1205971199102

(9)

where

Gr =]119892120573119879infin

1198803

0

Gm =

]119892120573lowast (119862119908minus 119862infin)

1198803

0

Pr =120583119888119901

119896

119877 =

161205901198793

infin

3119896119896lowast Sc = ]

119863

(10)

are the Grashof number modified Grashof number Prandtlnumber radiation parameter and Schmidt number respec-tively The corresponding initial and boundary conditions innondimensional form are

119905 le 0 119906 = 0 120579 = 0 119862 = 0 forall119910 ge 0 (11)

119905 gt 0 119906 = cos (120596119905) 120597120579120597119910

= minus120574 (1 + 120579) 119862 = 1 at 119910 = 0

(12)

119906 997888rarr 0 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin (13)Here 120574 = ℎ

119904]1198800is theNewtonian heating parameterWe

note that (12) gives 120579 = 0when 120574 = 0 which physically meansthat no heating from the plate exists [12 16]




119902119906 (119910 119902) minus 119906 (119910 0) =

1198892119906 (119910 119902)

1198891199102

+ Gr 120579 (119910 119902) + Gm119862 (119910 119902)

Pr [119902120579 (119910 119902) minus 120579 (119910 0)] = (1 + 119877)1198892120579 (119910 119902)

1198891199102

Sc [119902119862 (119910 119902) minus 119862 (119910 0)] =1198892119862 (119910 119902)

1198891199102

(14)

Here 119906(119910 119902) = int

infin

0119890minus119902119905119906(119910 119905)119889119905 120579(119910 119902) = int

infin

0119890minus119902119905120579(119910 119905)119889119905



1198892119906 (119910 119902)

1198891199102

minus 119902119906 (119910 119902) + Gr 120579 (119910 119902) + Gm119862 (119910 119902) = 0

1198892120579 (119910 119902)

1198891199102

minus 119902 (

Pr1 + 119877

) 120579 (119910 119902) = 0

1198892119862 (119910 119902)

1198891199102

minus 119902 Sc119862 (119910 119902) = 0

(15)


119906 (119910 119902) =

119902

1199022+ 1205962

119889120579 (119910 119902)

119889119910

= minus120574 [

1

119902

+ 120579 (119910 119902)]

119862 (119910 119902) =

1

119902

at 119910 = 0

119906 (119910 119902) 997888rarr 0 120579 (119910 119902) 997888rarr0 119862 (119910 119902) 997888rarr 0


(16)


119906 (119910 119902) =

1

2 (119902 + 119894120596)

119890minus119910radic119902

+

1

2 (119902 minus 119894120596)

119890minus119910radic119902

+

119886119888

1199022(radic119902 minus 119888)

119890minus119910radic119902

minus

119886119888

1199022(radic119902 minus 119888)


+

119887

1199022119890minus119910radic119902

minus

119887

1199022119890minus119910radic119902Sc

120579 (119910 119902) =

119888

119902 (radic119902 minus 119888)


119862 (119910 119902) =

1

119902


(17)


120579 (119910 119905) = 1198654(119910radicPreff 119905 119888) (18)

119862 (119910 119905) = 1198651(119910radicSc 119905) (19)

119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

minus

119886


+

119886

119888




3(119910 119905)]

(20)



119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

minus

119886


+

119886

119888



119910Gm2

1198652(119910 119905)

(21)



119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]


3(119910 119905)] +

119910Gr2120574

[1198654(119910 119905 120574)]

(22)


119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

+

119910Gr2120574

[1198654(119910 119905 120574)] minus

119910Gr2

times [1198652(119910 119905)] +

119910Gm2

[1198652(119910 119905)]

(23)

Here

1198651(V 119905) = erf 119888 ( V

2radic119905

)

1198652(V 119905) = 2radic

119905

120587

119890minusV24119905


2radic119905

)

1198653(V 119905) = (

V2

2

+ 119905) erf 119888 ( V

2radic119905

)

minus Vradic119905

120587

119890minusV24119905

1198654(V 119905 120572) = 119890(120572

2119905minus120572V) erf 119888 ( V

2radic119905



2radic119905

)

1198655(V 119905 120572) =

1

2


2radic119905


+119890Vradic120572 erf 119888 ( V

2radic119905

+ radic120572119905)]

(24)


1 1198652 1198653 1198654 and 119865

5are dummy



120591 =

1205911015840

1205881198802

0

= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=

1

2

[119890minus119894120596119905


+119890119894120596119905radic119894120596 erf(radic119894120596119905)]

+

119886

119888

[1 minus 1198901198882119905(1 + erf(119888radic119905))]


radic120587119905



120587


120587

(radicSc minus 1)

(25)

0 1 2 3 40

05

1

15

2

25

3

Velo

city

y

t = 01 02 03 04


0 05 1 15 2 25 30

02

04

06

08

1

Velo

city

y

R = 0 1 2 3


02Pr = 071Gr = 2Gm = 3 Sc = 094 120574 = 1 and 120596 = 1205874


Nu = minus

]

1198800(1198791015840minus 119879infin)

1205971198791015840

1205971199101015840

1003816100381610038161003816100381610038161003816100381610038161199101015840=0

=

1

120579 (0 119905)

+ 1


1198901198882119905[1 + erf (119888radic119905)] minus 1

)

(26)


Sh = minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radicSc120587119905

(27)





119905 119877 Pr Gr Gm Sc 120574 120596 120591

001 1 071 5 2 022 1 1205872 54263002 1 071 5 2 022 1 1205872 36395001 2 071 5 2 022 1 1205872 54049001 1 10 5 2 022 1 1205872 54401001 1 071 10 2 022 1 1205872 53663001 1 071 5 4 022 1 1205872 52727001 1 071 5 2 062 1 1205872 54537001 1 071 5 2 022 2 1205872 53473001 1 071 5 2 022 1 120587 54208


119905 119877 Pr 120574 Nu02 2 071 1 1311904 2 071 1 1105402 4 071 1 1147102 2 10 1 1465902 2 071 2 20347


119905 Sc Sh02 022 0591704 022 0418402 062 09938




0 1 2 3 4 50

05

1

15

2

25

Velo

city

y

= 071 1 7 100Pr


05 119877 = 2Gr = 2Gm = 1 Sc = 078 120574 = 1 and 120596 = 1205876

0 05 1 15 2 25 3 350

05

1

15

2

Velo

city

y

= 0 2 5 10Gr


02 119877 = 3Pr = 071Gm = 2 Sc = 078 120574 = 1 and 120596 = 1205873






0 05 1 15 2 250

02

04

06

08

1

Velo

city

y

= 0 1 2 3Gm


02 119877 = 1Pr = 071Gr = 5 Sc = 078 120574 = 1 and 120596 = 1205873

0 1 2 3 40

02

04

06

08

Velo

city

y

= 022 062 078 094Sc


04 119877 = 2Pr = 071Gr = 5Gm = 2 120574 = 01 and 120596 = 1205873



0 1 2 3 40

02

04

06

08

1

12

Velo

city

y

120574 = 01 02 03 04


04 119877 = 3Pr = 071Gr = 5Gm = 2 Sc = 078 and 120596 = 1205873

0 1 2 3 4 5

0

02

04

06

08

1

120596t = 0120587

3120587

22120587

3Ve

loci

ty

y

minus02

minus04






1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905


2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905


119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08


1Pr = 071 and 120574 = 1

5 Conclusions


0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3


0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr



Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119902119906 (119910 119902) minus 119906 (119910 0) =

1198892119906 (119910 119902)

1198891199102

+ Gr 120579 (119910 119902) + Gm119862 (119910 119902)

Pr [119902120579 (119910 119902) minus 120579 (119910 0)] = (1 + 119877)1198892120579 (119910 119902)

1198891199102

Sc [119902119862 (119910 119902) minus 119862 (119910 0)] =1198892119862 (119910 119902)

1198891199102

(14)

Here 119906(119910 119902) = int

infin

0119890minus119902119905119906(119910 119905)119889119905 120579(119910 119902) = int

infin

0119890minus119902119905120579(119910 119905)119889119905



1198892119906 (119910 119902)

1198891199102

minus 119902119906 (119910 119902) + Gr 120579 (119910 119902) + Gm119862 (119910 119902) = 0

1198892120579 (119910 119902)

1198891199102

minus 119902 (

Pr1 + 119877

) 120579 (119910 119902) = 0

1198892119862 (119910 119902)

1198891199102

minus 119902 Sc119862 (119910 119902) = 0

(15)


119906 (119910 119902) =

119902

1199022+ 1205962

119889120579 (119910 119902)

119889119910

= minus120574 [

1

119902

+ 120579 (119910 119902)]

119862 (119910 119902) =

1

119902

at 119910 = 0

119906 (119910 119902) 997888rarr 0 120579 (119910 119902) 997888rarr0 119862 (119910 119902) 997888rarr 0


(16)


119906 (119910 119902) =

1

2 (119902 + 119894120596)

119890minus119910radic119902

+

1

2 (119902 minus 119894120596)

119890minus119910radic119902

+

119886119888

1199022(radic119902 minus 119888)

119890minus119910radic119902

minus

119886119888

1199022(radic119902 minus 119888)


+

119887

1199022119890minus119910radic119902

minus

119887

1199022119890minus119910radic119902Sc

120579 (119910 119902) =

119888

119902 (radic119902 minus 119888)


119862 (119910 119902) =

1

119902


(17)


120579 (119910 119905) = 1198654(119910radicPreff 119905 119888) (18)

119862 (119910 119905) = 1198651(119910radicSc 119905) (19)

119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

minus

119886


+

119886

119888




3(119910 119905)]

(20)



119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

minus

119886


+

119886

119888



119910Gm2

1198652(119910 119905)

(21)



119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]


3(119910 119905)] +

119910Gr2120574

[1198654(119910 119905 120574)]

(22)


119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

+

119910Gr2120574

[1198654(119910 119905 120574)] minus

119910Gr2

times [1198652(119910 119905)] +

119910Gm2

[1198652(119910 119905)]

(23)

Here

1198651(V 119905) = erf 119888 ( V

2radic119905

)

1198652(V 119905) = 2radic

119905

120587

119890minusV24119905


2radic119905

)

1198653(V 119905) = (

V2

2

+ 119905) erf 119888 ( V

2radic119905

)

minus Vradic119905

120587

119890minusV24119905

1198654(V 119905 120572) = 119890(120572

2119905minus120572V) erf 119888 ( V

2radic119905



2radic119905

)

1198655(V 119905 120572) =

1

2


2radic119905


+119890Vradic120572 erf 119888 ( V

2radic119905

+ radic120572119905)]

(24)


1 1198652 1198653 1198654 and 119865

5are dummy



120591 =

1205911015840

1205881198802

0

= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=

1

2

[119890minus119894120596119905


+119890119894120596119905radic119894120596 erf(radic119894120596119905)]

+

119886

119888

[1 minus 1198901198882119905(1 + erf(119888radic119905))]


radic120587119905



120587


120587

(radicSc minus 1)

(25)

0 1 2 3 40

05

1

15

2

25

3

Velo

city

y

t = 01 02 03 04


0 05 1 15 2 25 30

02

04

06

08

1

Velo

city

y

R = 0 1 2 3


02Pr = 071Gr = 2Gm = 3 Sc = 094 120574 = 1 and 120596 = 1205874


Nu = minus

]

1198800(1198791015840minus 119879infin)

1205971198791015840

1205971199101015840

1003816100381610038161003816100381610038161003816100381610038161199101015840=0

=

1

120579 (0 119905)

+ 1


1198901198882119905[1 + erf (119888radic119905)] minus 1

)

(26)


Sh = minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radicSc120587119905

(27)





119905 119877 Pr Gr Gm Sc 120574 120596 120591

001 1 071 5 2 022 1 1205872 54263002 1 071 5 2 022 1 1205872 36395001 2 071 5 2 022 1 1205872 54049001 1 10 5 2 022 1 1205872 54401001 1 071 10 2 022 1 1205872 53663001 1 071 5 4 022 1 1205872 52727001 1 071 5 2 062 1 1205872 54537001 1 071 5 2 022 2 1205872 53473001 1 071 5 2 022 1 120587 54208


119905 119877 Pr 120574 Nu02 2 071 1 1311904 2 071 1 1105402 4 071 1 1147102 2 10 1 1465902 2 071 2 20347


119905 Sc Sh02 022 0591704 022 0418402 062 09938




0 1 2 3 4 50

05

1

15

2

25

Velo

city

y

= 071 1 7 100Pr


05 119877 = 2Gr = 2Gm = 1 Sc = 078 120574 = 1 and 120596 = 1205876

0 05 1 15 2 25 3 350

05

1

15

2

Velo

city

y

= 0 2 5 10Gr


02 119877 = 3Pr = 071Gm = 2 Sc = 078 120574 = 1 and 120596 = 1205873






0 05 1 15 2 250

02

04

06

08

1

Velo

city

y

= 0 1 2 3Gm


02 119877 = 1Pr = 071Gr = 5 Sc = 078 120574 = 1 and 120596 = 1205873

0 1 2 3 40

02

04

06

08

Velo

city

y

= 022 062 078 094Sc


04 119877 = 2Pr = 071Gr = 5Gm = 2 120574 = 01 and 120596 = 1205873



0 1 2 3 40

02

04

06

08

1

12

Velo

city

y

120574 = 01 02 03 04


04 119877 = 3Pr = 071Gr = 5Gm = 2 Sc = 078 and 120596 = 1205873

0 1 2 3 4 5

0

02

04

06

08

1

120596t = 0120587

3120587

22120587

3Ve

loci

ty

y

minus02

minus04






1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905


2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905


119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08


1Pr = 071 and 120574 = 1

5 Conclusions


0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3


0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr



Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]


3(119910 119905)] +

119910Gr2120574

[1198654(119910 119905 120574)]

(22)


119906 (119910 119905) =

1

2

[1198655(119910 119905 minus119894120596) + 119865

5(119910 119905 119894120596)]

+

119910Gr2120574

[1198654(119910 119905 120574)] minus

119910Gr2

times [1198652(119910 119905)] +

119910Gm2

[1198652(119910 119905)]

(23)

Here

1198651(V 119905) = erf 119888 ( V

2radic119905

)

1198652(V 119905) = 2radic

119905

120587

119890minusV24119905


2radic119905

)

1198653(V 119905) = (

V2

2

+ 119905) erf 119888 ( V

2radic119905

)

minus Vradic119905

120587

119890minusV24119905

1198654(V 119905 120572) = 119890(120572

2119905minus120572V) erf 119888 ( V

2radic119905



2radic119905

)

1198655(V 119905 120572) =

1

2


2radic119905


+119890Vradic120572 erf 119888 ( V

2radic119905

+ radic120572119905)]

(24)


1 1198652 1198653 1198654 and 119865

5are dummy



120591 =

1205911015840

1205881198802

0

= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=

1

2

[119890minus119894120596119905


+119890119894120596119905radic119894120596 erf(radic119894120596119905)]

+

119886

119888

[1 minus 1198901198882119905(1 + erf(119888radic119905))]


radic120587119905



120587


120587

(radicSc minus 1)

(25)

0 1 2 3 40

05

1

15

2

25

3

Velo

city

y

t = 01 02 03 04


0 05 1 15 2 25 30

02

04

06

08

1

Velo

city

y

R = 0 1 2 3


02Pr = 071Gr = 2Gm = 3 Sc = 094 120574 = 1 and 120596 = 1205874


Nu = minus

]

1198800(1198791015840minus 119879infin)

1205971198791015840

1205971199101015840

1003816100381610038161003816100381610038161003816100381610038161199101015840=0

=

1

120579 (0 119905)

+ 1


1198901198882119905[1 + erf (119888radic119905)] minus 1

)

(26)


Sh = minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= radicSc120587119905

(27)





119905 119877 Pr Gr Gm Sc 120574 120596 120591

001 1 071 5 2 022 1 1205872 54263002 1 071 5 2 022 1 1205872 36395001 2 071 5 2 022 1 1205872 54049001 1 10 5 2 022 1 1205872 54401001 1 071 10 2 022 1 1205872 53663001 1 071 5 4 022 1 1205872 52727001 1 071 5 2 062 1 1205872 54537001 1 071 5 2 022 2 1205872 53473001 1 071 5 2 022 1 120587 54208


119905 119877 Pr 120574 Nu02 2 071 1 1311904 2 071 1 1105402 4 071 1 1147102 2 10 1 1465902 2 071 2 20347


119905 Sc Sh02 022 0591704 022 0418402 062 09938




0 1 2 3 4 50

05

1

15

2

25

Velo

city

y

= 071 1 7 100Pr


05 119877 = 2Gr = 2Gm = 1 Sc = 078 120574 = 1 and 120596 = 1205876

0 05 1 15 2 25 3 350

05

1

15

2

Velo

city

y

= 0 2 5 10Gr


02 119877 = 3Pr = 071Gm = 2 Sc = 078 120574 = 1 and 120596 = 1205873






0 05 1 15 2 250

02

04

06

08

1

Velo

city

y

= 0 1 2 3Gm


02 119877 = 1Pr = 071Gr = 5 Sc = 078 120574 = 1 and 120596 = 1205873

0 1 2 3 40

02

04

06

08

Velo

city

y

= 022 062 078 094Sc


04 119877 = 2Pr = 071Gr = 5Gm = 2 120574 = 01 and 120596 = 1205873



0 1 2 3 40

02

04

06

08

1

12

Velo

city

y

120574 = 01 02 03 04


04 119877 = 3Pr = 071Gr = 5Gm = 2 Sc = 078 and 120596 = 1205873

0 1 2 3 4 5

0

02

04

06

08

1

120596t = 0120587

3120587

22120587

3Ve

loci

ty

y

minus02

minus04






1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905


2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905


119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08


1Pr = 071 and 120574 = 1

5 Conclusions


0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3


0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr



Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905 119877 Pr Gr Gm Sc 120574 120596 120591

001 1 071 5 2 022 1 1205872 54263002 1 071 5 2 022 1 1205872 36395001 2 071 5 2 022 1 1205872 54049001 1 10 5 2 022 1 1205872 54401001 1 071 10 2 022 1 1205872 53663001 1 071 5 4 022 1 1205872 52727001 1 071 5 2 062 1 1205872 54537001 1 071 5 2 022 2 1205872 53473001 1 071 5 2 022 1 120587 54208


119905 119877 Pr 120574 Nu02 2 071 1 1311904 2 071 1 1105402 4 071 1 1147102 2 10 1 1465902 2 071 2 20347


119905 Sc Sh02 022 0591704 022 0418402 062 09938




0 1 2 3 4 50

05

1

15

2

25

Velo

city

y

= 071 1 7 100Pr


05 119877 = 2Gr = 2Gm = 1 Sc = 078 120574 = 1 and 120596 = 1205876

0 05 1 15 2 25 3 350

05

1

15

2

Velo

city

y

= 0 2 5 10Gr


02 119877 = 3Pr = 071Gm = 2 Sc = 078 120574 = 1 and 120596 = 1205873






0 05 1 15 2 250

02

04

06

08

1

Velo

city

y

= 0 1 2 3Gm


02 119877 = 1Pr = 071Gr = 5 Sc = 078 120574 = 1 and 120596 = 1205873

0 1 2 3 40

02

04

06

08

Velo

city

y

= 022 062 078 094Sc


04 119877 = 2Pr = 071Gr = 5Gm = 2 120574 = 01 and 120596 = 1205873



0 1 2 3 40

02

04

06

08

1

12

Velo

city

y

120574 = 01 02 03 04


04 119877 = 3Pr = 071Gr = 5Gm = 2 Sc = 078 and 120596 = 1205873

0 1 2 3 4 5

0

02

04

06

08

1

120596t = 0120587

3120587

22120587

3Ve

loci

ty

y

minus02

minus04






1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905


2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905


119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08


1Pr = 071 and 120574 = 1

5 Conclusions


0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3


0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr



Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 250

02

04

06

08

1

Velo

city

y

= 0 1 2 3Gm


02 119877 = 1Pr = 071Gr = 5 Sc = 078 120574 = 1 and 120596 = 1205873

0 1 2 3 40

02

04

06

08

Velo

city

y

= 022 062 078 094Sc


04 119877 = 2Pr = 071Gr = 5Gm = 2 120574 = 01 and 120596 = 1205873



0 1 2 3 40

02

04

06

08

1

12

Velo

city

y

120574 = 01 02 03 04


04 119877 = 3Pr = 071Gr = 5Gm = 2 Sc = 078 and 120596 = 1205873

0 1 2 3 4 5

0

02

04

06

08

1

120596t = 0120587

3120587

22120587

3Ve

loci

ty

y

minus02

minus04






1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905


2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905


119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08


1Pr = 071 and 120574 = 1

5 Conclusions


0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3


0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr



Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 1

119902

119890minus119910radic119902 erf 119888 (

119910

2radic119905

)

2 1

1199022119890minus119910radic119902

(

1199102

2

+ 119905) erf c(119910

2radic119905

) minus 119910radic

119905

120587

119890minus11991024119905

3 1

119902radic119902 minus 119886

119890minus119910radic119902

119890(1198862119905minus119886119910) erf c(

119910

2radic119905


2radic119905

)

4 1

1199022radic119902 minus 119886

119890minus119910radic119902

1

4radic120587119905

119890minus11991024119905+ 119886119890(1198862119905minus119886119910) erf c(

119910

2radic119905


5 1

119902 + 119886

119890minus119910radic119902

119890119886119905

2

119890119910radic119886 erf c(

119910

2radic119905


119910

2radic119905


0 1 2 3 40

02

04

06

08

1

Velo

city

y

= 016 06 201 500Sc


03 119877 = 05Pr = 071Gr = 3Gm = 2 120574 = 1 and 120596 = 0

0 1 2 3 40

25

5

75

10

125

15

175

Tem

pera

ture

y

t = 02 04 06 08


1Pr = 071 and 120574 = 1

5 Conclusions


0 05 1 15 2 25 3 350

1

2

3

4

Tem

pera

ture

y

R = 0 1 2 3


0 02 04 06 08 1 120

0002

0004

0006

0008

001

0012

0014

Tem

pera

ture

y

= 071 1 7 100Pr



Appendix

See Table 4


0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 20

01

02

03

04

Tem

pera

ture

y

120574 = 01 02 03 04


0 2 4 60

02

04

06

08

1

Con

cent

ratio

n

y

t = 02 04 06 08


Nomenclature










0 1 2 3 40

02

04

06

08

1

Con

cent

ratio

n

y

= 022 062 078 094Sc







Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Exact Analysis of Heat and Mass ...

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