Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 682795 10 pageshttpdxdoiorg1011552013682795

Research ArticleAnalytical Solution of Flow and Heat Transfer over a PermeableStretching Wall in a Porous Medium

M Dayyan1 S M Seyyedi2 G G Domairry2 and M Gorji Bandpy2

1 Department of Mechanical and Aerospace Engineering Science and Research Branch Islamic Azad UniversityTehran Iran2Department of Mechanical Engineering Babol University of Technology Babol Iran

Correspondence should be addressed to M Dayyan mortezadayyangmailcom

Received 6 April 2013 Revised 18 July 2013 Accepted 18 July 2013

Academic Editor Waqar Khan

Copyright copy 2013 M Dayyan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Boundary layer flow through a porous medium over a stretching porous wall has seen solved with analytical solution It has beenconsidered twowall boundary conditionswhich are power-law distribution of either wall temperature or heat fluxThese are generalenough to cover the isothermal and isoflux cases In addition tomomentum both first and second laws of thermodynamics analysesof the problem are investigated The governing equations are transformed into a system of ordinary differential equations Thetransformed ordinary equations are solved analytically using homotopy analysis method A comprehensive parametric study ispresented and it is shown that the rate of heat transfer increases with Reynolds number Prandtl number and suction to the surface

1 Introduction

Boundary layer flows over a stretching surface haveapplication in engineering processes such as liquid compositemolding extrusion of plastic sheets paper production glassblowing metal spinning wire drawing and hot rolling[1ndash3] More importantly the quality of the products in theabovementioned processes depends on the kinematics ofstretching and the simultaneous heat and mass transfer ratesduring the fabrication process Sakiadis [4 5] and Crane[6] were the pioneers in the investigations of boundarylayer flow over continuously moving surfaces that are quitedifferent from the free stream flow over stationary flat platesPop and Na [7] studied free convection heat transfer ofnon-Newtonian fluids along a vertical wavy surface in aporous medium

The flow field of a stretching surface with a power-lawvelocity variationwas discussed byBanks [8] Elbashbeshy [9]investigated heat transfer over a stretching surface with vari-able surface heat flux Elbashbeshy and Bazid [10] analyzedheat transfer in a porous medium over a stretching surfacewith internal heat generation and suction or injection Thiswork was extended by Cortell [11] to include power-lawtemperature distribution Steady flow and heat transfer of

a viscous incompressible fluid flow through porous mediumover a stretching sheet were studied by Sriramalu et al [12]Ali [13] investigated thermal boundary layer on a power-law stretched surface with suction or injection for uniformand variable surface temperatures Recently Pantokratoras[14] published analytical solution for velocity distributioninside a Darcy-Brinkman porous medium with a stretch-ing boundary Elbashbeshy [15] included thermal radiationeffects in a stretching surface problem Tamayol and Bahrami[16] understood that porousmaterials can be used to enhancethe heat transfer rate from stretching surfaces to improveprocesses such as hot rolling and composite fabrication

In view of the above an analytical solution is developed inthe present study to solve fluid flow heat transfer in a porousmedium over a porous plate with linear velocity subjectedto different power-law thermal boundary conditions Theanalytical solution is successfully validated in comparisonwith numerical analysis There are many effective methodsfor obtaining the solutions of nonlinear equation such asvariational iteration method [17] Adomian method [18]and homotopy perturbation method [19 20] and we useone of a powerful technique namely the homotopy analysismethod (HAM) which was expressed by Liao [21ndash24] Thismethod has been successfully applied to solve many types

2 Mathematical Problems in Engineering

of nonlinear problems [25ndash29] and provides us with greatfreedom to express solutions of a given nonlinear problemby means of different base functions Secondly unlike allprevious analytic techniques the homotopy analysis methodalways provides us with a family of solution expressions inthe auxiliary parameter ℎ even if a nonlinear problem hasa unique solution Thirdly unlike perturbation techniquesthe homotopy analysis method is independent of any smallor large quantities So the homotopy analysis method canbe applied no matter if governing equations and bound-aryinitial conditions of a given nonlinear problem containsmall or large quantities or not

Above all there are no rigorous theories to direct us tochoose the initial approximations auxiliary linear operatorsauxiliary functions and auxiliary parameter ℎ From thepractical viewpoints there are some fundamental rules suchas the rule of solution expression the rule of coefficientergodicity and the rule of solution existence which playimportant roles within the homotopy analysis methodUnfortunately the rule of solution expression implies suchan assumption that we should have more or less someknowledge about a given nonlinear problem a prior Sotheoretically this assumption impairs the homotopy analysismethod althoughwe can always attempt some base functionseven if a given nonlinear problem is completely new for us

2 Governing Equation

Consider a steady constant property two-dimensional flowthrough a homogenous porous medium of permeability 119870over a stretching surface with linear velocity distribution thatis 119906119908

= 1199060119909119871 (Figure 1) The transport properties of the

medium can be assumed independent of temperature whenthe difference between wall and ambient temperatures is notsignificant [4] The origin is kept fixed while the wall isstretching and the 119910-axis is perpendicular to the surfaceThegoverning equations are [4 5]

120597119906

120597119909+120597120592

120597119910= 0

120588 (119906120597119906

120597119909+ 120592

120597119906

120597119910) = 120583eff

1205972

119906

1205971199102minus

120583

1198701015840119906

119906120597119879

120597119909+ 120592

120597119879

120597119910= 120572eff

1205972

119879

1205971199102

(1)

where 119906 and 120592 are velocity components in the 119909 and 119910

directions respectively 120583eff is the effective viscosity which forsimplicity in the present study is considered to be identicalto the dynamic viscosity 120583 The transport properties ofthe porous medium such as permeability depend on theirmicrostructure and can be calculated either using existingcorrelations in the literature or through experimental mea-surements 120572eff is the effective thermal diffusivity of themediumThe hydrodynamic boundary conditions are

119906 (119909lowast

0) = 1199060119909lowast

120592 (119909lowast

0) = 120592119908 119906 (119909

lowast

infin) = 0

(2)

Porous medium

y 120578

O

vw

uwmax = u0uw = u0xlowast

x xlowast u

Figure 1 Schematic diagram of problem

where 119909lowast

= 119909119871 is the nondimensional 119909-coordinate and119871 is the length of the porous plate The following thermalboundary conditions are considered

119879 (119909lowast

0) = 119879infin

+ 1198790(119909lowast

)119899

119879 (119909lowast

infin) = 119879infin

minus120581120597119879

120597119910

10038161003816100381610038161003816100381610038161003816(119909lowast 0)

= 1199020(119909lowast

)119899

119879 (119909lowast

infin) = 119879infin

(3)

where 120581 is the effective thermal conductivity of the mediumand is a function of thermal conductivities of the fluid andsolid phases and the porous medium microstructure

Using dimensionless parameters

120578 =119910

radic119870

119906 = 1199060119909lowast

1198911015840

(120578)

120592 = minus1199060

119871radic119870119891 (120578)

120595 = 1199060119909lowastradic119870119891 (120578)

(4)

where 1198911015840 is 119889119891119889120578The transformed nonlinear ordinary differential equa-

tions are

119891101584010158401015840

+ Re (11989111989110158401015840 minus 11989110158402

) minus 1198911015840

= 0

12057910158401015840

+ Re sdotPr (1198911205791015840) minus 119899 sdot Re sdotPr (1198911015840120579) = 0

(5)

where Re = 1205881199060119870119871120583 is the Reynolds number Equation

(5) should be solved subject to the following boundaryconditions

1198911015840

(0) = 1 119891 (0) = minus120592119908119871

1199060radic119870

= 119891119908 119891

1015840

(infin) = 0

(6)

120579 (0) = 1 120579 (infin) = 0 Power-law temperature (7a)

1205791015840

(0) = minus1 1205791015840

(infin) = 0 Power-law heat flux (7b)

where 119891119908is the injection parameter Positivenegative values

of 119891119908show suctioninjection intofrom the porous surface

respectively The wall shear stress term can be calculated as

120591119871=

minus120583119906011987111989110158401015840

(0)

2radic119870 (8)

Mathematical Problems in Engineering 3

For power-law fluid wit constant temperature and heatflux boundary conditions respectively Employing the defini-tion of convective heat transfer coefficient the local Nusseltnumbers become

Nu119909=

ℎ119909

119896=

minus1205791015840

(0) 119909

radic119870 Power-law temperature

119902119908119909

119896 (119879119908minus 119879infin)=

119909

120579 (0)radic119870 Power-law heat flux

(9)

3 Solution of Problem by HomotopyAnalysis Method (HAM)

As mentioned by Liao a solution may be expressed withdifferent base functions among which some converge to theexact solution of the problem faster than others Such basefunctions are obviously better suited for the final solution tobe expressed in terms of Noting these facts we have decidedto express 119892(120578) by a set of base functions of the followingform

119891 (120578) =

infin

sum

119898=0

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

120579 (120578) =

infin

sum

119898=0

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

(10)

The rule of solution expression provides us with a startingpoint It is under the rule of solution expression that initialapproximations auxiliary linear operators and the auxiliaryfunctions are determined So according to the rule of solutionexpression we choose the initial guess and auxiliary linearoperator in the following form

1198910(120578) = 119891

119908+ 1 minus exp (minus120578) 120579

0(120578) = exp (minus120578) (11)

We choose linear operator as follows

L1(119891) = 119891

101584010158401015840

+ 11989110158401015840

L2(120579) = 120579

10158401015840

+ 1205791015840

L1(1198881+ 1198882120578 + 1198883exp (minus120578)) = 0

L2(1198884+ 1198885exp (minus120578)) = 0

(12)

where 119888119894(119894 = 1ndash5) are constants Let 119875 isin [0 1] denote the

embedding parameter and let ℎ indicate nonzero auxiliaryparameters Then we construct the following equations

31 Consider ZerothndashOrder Deformation Equations

(1 minus 119875)L1[119891 (120578 119901) minus 119891

0(120578)] = 119901ℎ

1119867(120578)119873

1[119891 (120578 119901)]

119891 (0 119901) = 119891119908

1198911015840

(0 119901) = 1 1198911015840

(infin 119901) = 0

(1 minus 119875)L2[120579 (120578 119901) minus 120579

0(120578)] = 119901ℎ

2119867(120578)119873

2[120579 (120578 119901)]

120579 (0 119901) = 1 120579 (infin 119901) = 0 Power-law temperature

1205791015840

(0 119901) = minus1 120579 (infin 119901) = 0 Power-law heat flux

1198731[119891 (120578 119901) 120579 (120578 119901)]

=1205973

119891 (120578 119901)

1205971205783

+ Re(119891 (120578 119901)1205972

119891 (120578 119901)

1205971205782minus (

120597119891 (120578 119901)

120597120578)

2

)

minus120597119891 (120578 119901)

120597120578

1198732[119891 (120578 119901) 120579 (120578 119901)]

=1205972

120579 (120578 119901)

1205971205782+ Re sdotPr(119891 (120578 119901)

120597120579 (120578 119901)

120597120578)

minus 119899 sdot Re sdotPr(120579 (120578 119901)120597119891 (120578 119901)

120597120578)

(13)

For 119901 = 0 and 119901 = 1 we have

119891 (120578 0) = 1198910(120578) 119891 (120578 1) = 119891 (120578)

120579 (120578 0) = 1205790(120578) 120579 (120578 1) = 120579 (120578)

(14)

When 119901 increases from 0 to 1 then 119891(120578 119901) and 120579(120578 119901) varyfrom 119891

0(120578) and 120579

0(120578) to 119891(120578) and 120579(120578) By Taylorrsquos theorem

and using (14) we can write the following

119891 (120578 119901) = 1198910(120578) +

infin

sum

119898minus1

119891119898(120578) 119901119898

119891119898(120578) =

1

119898

120597119898

(119891 (120578 119901))

120597119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898minus1

120579119898(120578) 119901119898

120579119898(120578) =

1

119898

120597119898

(120579 (120578 119901))

120597119901119898

(15)

In which ℎ1and ℎ

2are chosen in such a way that these two

series are convergent at119901 = 1 therefore we have the followingthrough (15)

119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

(16)

32 Consider Mth-Order Deformation Equations

L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ1119867(120578) 119877

119891

119898(120578) (17)

4 Mathematical Problems in Engineering

119891119898(0) = 0 119891

1015840

119898(0) = 0 119891

1015840

119898(infin) = 0 (18)

119877119891

119898(120578) = 119891

101584010158401015840

119898minus1+

119898minus1

sum

119896=0

11989111989611989110158401015840

119898minus1minus119896

minus

119898minus1

sum

119896=0

1198911015840

1198961198911015840

119898minus1minus119896minus 1198701198911015840

119898minus1

(19)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ2119867(120578) 119877 120579

119898(120578) (20)

120579119898(0) = 0 120579

119898(infin) = 0 (21)

1205791015840

119898(0) = 0 120579

119898(infin) = 0 (22)

119877120579

119898(120578) = 120579

10158401015840

119898minus1+ Re sdotPr(

119898minus1

sum

119896=0

1198911198961205791015840

119898minus1minus119896)

minus 119899 sdot Re sdotPr(119898minus1

sum

119896=0

1205791198961198911015840

119898minus1minus119896)

(23)

120594119898=

0 119898 le 1

1 119898 gt 1(24)

The general solutions of (17)ndash(23) are

119891119898(120578) minus 120594

119898119891119898minus1

(120578) = 119891lowast

119898(120578) + 119862

119898

1+ 119862119898

2120578 + 119862119898

3exp (minus120578)

120579119898(120578) minus 120594

119898120579119898minus1

(120578) = 120579lowast

119898(120578) + 119862

119898

4+ 119862119898

5exp (minus120578)

(25)

where 119862119898

1to 119862119898

5are constants that can be obtained by

applying the boundary condition in (18) (21) and (22)As discussed by Liao the rule of coefficient ergodicity and

the rule of solution existence play important roles in deter-mining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutionsIn many cases by means of the rule of solution expressionand the rule of coefficient ergodicity auxiliary functions canbe uniquely determined So we define the auxiliary function119867(120578) in the following form

119867(120578) = exp (minus120578) (26)

4 Convergence of the HAM Solution

As was mentioned in introduction the convergence and therate of approximation of the HAM solution strongly dependon the values of auxiliary parameter ℎ By means of the so-called ℎ curves it is easy to find out the so-called valid regionsof ℎ to gain a convergent solution series According to Figures2 and 3 the acceptable range of auxiliary parameter for Pr =1 119899 = 0Re = 1 and 119891

119908= 0 is minus03 lt ℎ

1lt minus01 and minus18 lt

ℎ2lt minus03Figures 4 and 5 show how auxiliary parameters ℎ

1and

ℎ2vary with changing 119891

119908 If 119891

119908increases the range of

convergency of solution is restricted and then decreased

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus2

f998400998400(0)

minus03 minus02 minus01 0

13th-order app14th-order app15th-order app

ℏ1

Figure 2 The ℎ1validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

13th-order app14th-order app15th-order app

ℏ2

Figure 3 The ℎ2validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

5 Results and Discussion

In the present study the Homotopy analysis method hasbeen used to solve the nonlinear equations of heat transferand fluid flow over a permeable stretching wall in a porousmedium The nondimensional numbers introduced in thepresent analysis are Reynolds number (Re) Prandtl number

Mathematical Problems in Engineering 5

Table 1 The results of HAM and NS for 119891(120578) 1198911015840(120578) and 120579(120578) when 119891119908= 0 Pr = 1 119899 = 0 and Re = 1

120578119891(120578) 119891

1015840

(120578) 120579(120578) (temperature) 120579(120578) (heat flux)HAM NS HAM NS HAM NS HAM NS

00 000000 000000 099999 100000 0999999 100000 19565 1961502 017042 017420 075376 075363 089633 089996 17677 1762704 030564 030549 056813 075363 080205 080327 15658 1570606 040571 040443 042820 042804 071301 071225 13856 1389808 047505 047900 032272 032259 062792 062826 12210 1222910 053802 053519 024322 024311 055226 055191 10690 1071212 058010 057754 018330 018322 048134 048329 09313 0934014 061161 060946 013814 013808 042056 042214 08094 0813416 064096 063352 010410 010406 036595 036800 06943 0705818 065374 065165 007845 007842 031817 032029 06095 0611020 066952 066531 005912 005910 027657 027841 05193 0527822 067812 067560 004455 004454 023934 024175 04480 0454924 068567 068336 003357 003357 020813 020971 03861 0391326 069054 068921 002530 002529 017972 018177 03323 0335828 069675 069362 001907 001906 015519 015743 02858 0287430 069842 069694 001437 001436 013407 013623 02456 02453

05

0

minus05

minus1

minus15

minus05 minus04 minus03 minus02 minus01 0 01

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

f998400998400(0)

ℏ1

Figure 4 The ℎ1validity for various 119891

119908 Pr = 1 119899 = 0 and Re = 1

(Pr) and the injection number (119891119908) Another important

parameter is the power of the surface temperatureheat fluxdistribution (119899) which is considered in the present study

In order to ensure the convergence of the solution seriesthe variation of 119891 has been plotted at different ordersof approximation in Figure 6 The comparison betweenresults of HAM and numerical solution (Runge-Kutta) hasbeen displayed in Table 1 It can be observed that there isa good agreement between HAM method and numericalsolution Figures 7 8 and 9 illustrate effect of variationof wall injectionsuction parameter (119891

119908) on velocity and

minus02

minus04

minus06

minus08

minus12

minus1

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

ℏ2

Figure 5 The ℎ2validity for various 119891

119908 Pr = 1 119899 = 0 Re = 1 and

119891119908= 0

temperature distribution It can be observed that all boundarylayer thickness decreased by increasing 119891

119908from negative to

positive (from injection to suction) Figures 10 11 and 12show the effect of Reynolds number on the velocity profilewhen Pr = 1 119899 = 0 and 119891

119908= 0 This figure shows that

the boundary layer thickness and thermal boundary layerthickness are quite opposite to that of Reynolds number Theinfluence of 119899 on temperature field for both types of thethermal boundary conditions considered has been studiedin Figures 13 and 14 We notice that increasing 119899 reduces

6 Mathematical Problems in Engineering

1

05

0

minus05

f(120578)

0 2 4 6 8 10(120578)

n = 11

n = 12n = 13

n = 14

n = 15

142

14

138

136

134

13292 94 96 98

Figure 6 The variation of 119891 at different orders of approximationsPr = 1 119899 = 0 Re = 1 and 119891

119908= 0

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

1

08

06

04

02

0 1 2 3 4 5 6

f998400 (120578)

120578

Figure 7 Velocity profile 1198911015840 for various 119891119908when Pr = 1 119899 = 0 and

Re = 1

the thermal boundary layer thickness regardless of theboundary condition type leading to a heat transfer aug-mentation Effect of Prandtl number (isothermal) on thetemperature field is plotted in Figure 15 This figure showsthat thermal boundary layer thickness directly depends onPrandtl number

Tables 2 and 3 compare the results ofHAMand numericalsolution when the Reynolds number varies for 119891

119908= 0

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 8 Temperature profile (isothermal) 120579 for various 119891119908when

Pr = 1 119899 = 0 and Re = 1

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 9 Temperature profile (isoflux) 120579 for various 119891119908when Pr =

1 119899 = 0 and Re = 1

Pr = 1 and 119899 = 0 for isothermal and heat flux boundaryconditions respectively It can be seen that for isothermalboundary condition with increasing Reynolds number thewall shear stress for both boundary conditions consideredhere (isothermal and heat flux) and Nusselt number increasewith Reynolds number for isothermal state and independentof Reynolds number for heat flux boundary condition

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of nonlinear problems [25ndash29] and provides us with greatfreedom to express solutions of a given nonlinear problemby means of different base functions Secondly unlike allprevious analytic techniques the homotopy analysis methodalways provides us with a family of solution expressions inthe auxiliary parameter ℎ even if a nonlinear problem hasa unique solution Thirdly unlike perturbation techniquesthe homotopy analysis method is independent of any smallor large quantities So the homotopy analysis method canbe applied no matter if governing equations and bound-aryinitial conditions of a given nonlinear problem containsmall or large quantities or not

Above all there are no rigorous theories to direct us tochoose the initial approximations auxiliary linear operatorsauxiliary functions and auxiliary parameter ℎ From thepractical viewpoints there are some fundamental rules suchas the rule of solution expression the rule of coefficientergodicity and the rule of solution existence which playimportant roles within the homotopy analysis methodUnfortunately the rule of solution expression implies suchan assumption that we should have more or less someknowledge about a given nonlinear problem a prior Sotheoretically this assumption impairs the homotopy analysismethod althoughwe can always attempt some base functionseven if a given nonlinear problem is completely new for us

2 Governing Equation

Consider a steady constant property two-dimensional flowthrough a homogenous porous medium of permeability 119870over a stretching surface with linear velocity distribution thatis 119906119908

= 1199060119909119871 (Figure 1) The transport properties of the

medium can be assumed independent of temperature whenthe difference between wall and ambient temperatures is notsignificant [4] The origin is kept fixed while the wall isstretching and the 119910-axis is perpendicular to the surfaceThegoverning equations are [4 5]

120597119906

120597119909+120597120592

120597119910= 0

120588 (119906120597119906

120597119909+ 120592

120597119906

120597119910) = 120583eff

1205972

119906

1205971199102minus

120583

1198701015840119906

119906120597119879

120597119909+ 120592

120597119879

120597119910= 120572eff

1205972

119879

1205971199102

(1)

where 119906 and 120592 are velocity components in the 119909 and 119910

directions respectively 120583eff is the effective viscosity which forsimplicity in the present study is considered to be identicalto the dynamic viscosity 120583 The transport properties ofthe porous medium such as permeability depend on theirmicrostructure and can be calculated either using existingcorrelations in the literature or through experimental mea-surements 120572eff is the effective thermal diffusivity of themediumThe hydrodynamic boundary conditions are

119906 (119909lowast

0) = 1199060119909lowast

120592 (119909lowast

0) = 120592119908 119906 (119909

lowast

infin) = 0

(2)

Porous medium

y 120578

O

vw

uwmax = u0uw = u0xlowast

x xlowast u

Figure 1 Schematic diagram of problem

where 119909lowast

= 119909119871 is the nondimensional 119909-coordinate and119871 is the length of the porous plate The following thermalboundary conditions are considered

119879 (119909lowast

0) = 119879infin

+ 1198790(119909lowast

)119899

119879 (119909lowast

infin) = 119879infin

minus120581120597119879

120597119910

10038161003816100381610038161003816100381610038161003816(119909lowast 0)

= 1199020(119909lowast

)119899

119879 (119909lowast

infin) = 119879infin

(3)

where 120581 is the effective thermal conductivity of the mediumand is a function of thermal conductivities of the fluid andsolid phases and the porous medium microstructure

Using dimensionless parameters

120578 =119910

radic119870

119906 = 1199060119909lowast

1198911015840

(120578)

120592 = minus1199060

119871radic119870119891 (120578)

120595 = 1199060119909lowastradic119870119891 (120578)

(4)

where 1198911015840 is 119889119891119889120578The transformed nonlinear ordinary differential equa-

tions are

119891101584010158401015840

+ Re (11989111989110158401015840 minus 11989110158402

) minus 1198911015840

= 0

12057910158401015840

+ Re sdotPr (1198911205791015840) minus 119899 sdot Re sdotPr (1198911015840120579) = 0

(5)

where Re = 1205881199060119870119871120583 is the Reynolds number Equation

(5) should be solved subject to the following boundaryconditions

1198911015840

(0) = 1 119891 (0) = minus120592119908119871

1199060radic119870

= 119891119908 119891

1015840

(infin) = 0

(6)

120579 (0) = 1 120579 (infin) = 0 Power-law temperature (7a)

1205791015840

(0) = minus1 1205791015840

(infin) = 0 Power-law heat flux (7b)

where 119891119908is the injection parameter Positivenegative values

of 119891119908show suctioninjection intofrom the porous surface

respectively The wall shear stress term can be calculated as

120591119871=

minus120583119906011987111989110158401015840

(0)

2radic119870 (8)

Mathematical Problems in Engineering 3

For power-law fluid wit constant temperature and heatflux boundary conditions respectively Employing the defini-tion of convective heat transfer coefficient the local Nusseltnumbers become

Nu119909=

ℎ119909

119896=

minus1205791015840

(0) 119909

radic119870 Power-law temperature

119902119908119909

119896 (119879119908minus 119879infin)=

119909

120579 (0)radic119870 Power-law heat flux

(9)

3 Solution of Problem by HomotopyAnalysis Method (HAM)

As mentioned by Liao a solution may be expressed withdifferent base functions among which some converge to theexact solution of the problem faster than others Such basefunctions are obviously better suited for the final solution tobe expressed in terms of Noting these facts we have decidedto express 119892(120578) by a set of base functions of the followingform

119891 (120578) =

infin

sum

119898=0

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

120579 (120578) =

infin

sum

119898=0

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

(10)

The rule of solution expression provides us with a startingpoint It is under the rule of solution expression that initialapproximations auxiliary linear operators and the auxiliaryfunctions are determined So according to the rule of solutionexpression we choose the initial guess and auxiliary linearoperator in the following form

1198910(120578) = 119891

119908+ 1 minus exp (minus120578) 120579

0(120578) = exp (minus120578) (11)

We choose linear operator as follows

L1(119891) = 119891

101584010158401015840

+ 11989110158401015840

L2(120579) = 120579

10158401015840

+ 1205791015840

L1(1198881+ 1198882120578 + 1198883exp (minus120578)) = 0

L2(1198884+ 1198885exp (minus120578)) = 0

(12)

where 119888119894(119894 = 1ndash5) are constants Let 119875 isin [0 1] denote the

embedding parameter and let ℎ indicate nonzero auxiliaryparameters Then we construct the following equations

31 Consider ZerothndashOrder Deformation Equations

(1 minus 119875)L1[119891 (120578 119901) minus 119891

0(120578)] = 119901ℎ

1119867(120578)119873

1[119891 (120578 119901)]

119891 (0 119901) = 119891119908

1198911015840

(0 119901) = 1 1198911015840

(infin 119901) = 0

(1 minus 119875)L2[120579 (120578 119901) minus 120579

0(120578)] = 119901ℎ

2119867(120578)119873

2[120579 (120578 119901)]

120579 (0 119901) = 1 120579 (infin 119901) = 0 Power-law temperature

1205791015840

(0 119901) = minus1 120579 (infin 119901) = 0 Power-law heat flux

1198731[119891 (120578 119901) 120579 (120578 119901)]

=1205973

119891 (120578 119901)

1205971205783

+ Re(119891 (120578 119901)1205972

119891 (120578 119901)

1205971205782minus (

120597119891 (120578 119901)

120597120578)

2

)

minus120597119891 (120578 119901)

120597120578

1198732[119891 (120578 119901) 120579 (120578 119901)]

=1205972

120579 (120578 119901)

1205971205782+ Re sdotPr(119891 (120578 119901)

120597120579 (120578 119901)

120597120578)

minus 119899 sdot Re sdotPr(120579 (120578 119901)120597119891 (120578 119901)

120597120578)

(13)

For 119901 = 0 and 119901 = 1 we have

119891 (120578 0) = 1198910(120578) 119891 (120578 1) = 119891 (120578)

120579 (120578 0) = 1205790(120578) 120579 (120578 1) = 120579 (120578)

(14)

When 119901 increases from 0 to 1 then 119891(120578 119901) and 120579(120578 119901) varyfrom 119891

0(120578) and 120579

0(120578) to 119891(120578) and 120579(120578) By Taylorrsquos theorem

and using (14) we can write the following

119891 (120578 119901) = 1198910(120578) +

infin

sum

119898minus1

119891119898(120578) 119901119898

119891119898(120578) =

1

119898

120597119898

(119891 (120578 119901))

120597119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898minus1

120579119898(120578) 119901119898

120579119898(120578) =

1

119898

120597119898

(120579 (120578 119901))

120597119901119898

(15)

In which ℎ1and ℎ

2are chosen in such a way that these two

series are convergent at119901 = 1 therefore we have the followingthrough (15)

119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

(16)

32 Consider Mth-Order Deformation Equations

L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ1119867(120578) 119877

119891

119898(120578) (17)

4 Mathematical Problems in Engineering

119891119898(0) = 0 119891

1015840

119898(0) = 0 119891

1015840

119898(infin) = 0 (18)

119877119891

119898(120578) = 119891

101584010158401015840

119898minus1+

119898minus1

sum

119896=0

11989111989611989110158401015840

119898minus1minus119896

minus

119898minus1

sum

119896=0

1198911015840

1198961198911015840

119898minus1minus119896minus 1198701198911015840

119898minus1

(19)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ2119867(120578) 119877 120579

119898(120578) (20)

120579119898(0) = 0 120579

119898(infin) = 0 (21)

1205791015840

119898(0) = 0 120579

119898(infin) = 0 (22)

119877120579

119898(120578) = 120579

10158401015840

119898minus1+ Re sdotPr(

119898minus1

sum

119896=0

1198911198961205791015840

119898minus1minus119896)

minus 119899 sdot Re sdotPr(119898minus1

sum

119896=0

1205791198961198911015840

119898minus1minus119896)

(23)

120594119898=

0 119898 le 1

1 119898 gt 1(24)

The general solutions of (17)ndash(23) are

119891119898(120578) minus 120594

119898119891119898minus1

(120578) = 119891lowast

119898(120578) + 119862

119898

1+ 119862119898

2120578 + 119862119898

3exp (minus120578)

120579119898(120578) minus 120594

119898120579119898minus1

(120578) = 120579lowast

119898(120578) + 119862

119898

4+ 119862119898

5exp (minus120578)

(25)

where 119862119898

1to 119862119898

5are constants that can be obtained by

applying the boundary condition in (18) (21) and (22)As discussed by Liao the rule of coefficient ergodicity and

the rule of solution existence play important roles in deter-mining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutionsIn many cases by means of the rule of solution expressionand the rule of coefficient ergodicity auxiliary functions canbe uniquely determined So we define the auxiliary function119867(120578) in the following form

119867(120578) = exp (minus120578) (26)

4 Convergence of the HAM Solution

As was mentioned in introduction the convergence and therate of approximation of the HAM solution strongly dependon the values of auxiliary parameter ℎ By means of the so-called ℎ curves it is easy to find out the so-called valid regionsof ℎ to gain a convergent solution series According to Figures2 and 3 the acceptable range of auxiliary parameter for Pr =1 119899 = 0Re = 1 and 119891

119908= 0 is minus03 lt ℎ

1lt minus01 and minus18 lt

ℎ2lt minus03Figures 4 and 5 show how auxiliary parameters ℎ

1and

ℎ2vary with changing 119891

119908 If 119891

119908increases the range of

convergency of solution is restricted and then decreased

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus2

f998400998400(0)

minus03 minus02 minus01 0

13th-order app14th-order app15th-order app

ℏ1

Figure 2 The ℎ1validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

13th-order app14th-order app15th-order app

ℏ2

Figure 3 The ℎ2validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

5 Results and Discussion

In the present study the Homotopy analysis method hasbeen used to solve the nonlinear equations of heat transferand fluid flow over a permeable stretching wall in a porousmedium The nondimensional numbers introduced in thepresent analysis are Reynolds number (Re) Prandtl number

Mathematical Problems in Engineering 5

Table 1 The results of HAM and NS for 119891(120578) 1198911015840(120578) and 120579(120578) when 119891119908= 0 Pr = 1 119899 = 0 and Re = 1

120578119891(120578) 119891

1015840

(120578) 120579(120578) (temperature) 120579(120578) (heat flux)HAM NS HAM NS HAM NS HAM NS

00 000000 000000 099999 100000 0999999 100000 19565 1961502 017042 017420 075376 075363 089633 089996 17677 1762704 030564 030549 056813 075363 080205 080327 15658 1570606 040571 040443 042820 042804 071301 071225 13856 1389808 047505 047900 032272 032259 062792 062826 12210 1222910 053802 053519 024322 024311 055226 055191 10690 1071212 058010 057754 018330 018322 048134 048329 09313 0934014 061161 060946 013814 013808 042056 042214 08094 0813416 064096 063352 010410 010406 036595 036800 06943 0705818 065374 065165 007845 007842 031817 032029 06095 0611020 066952 066531 005912 005910 027657 027841 05193 0527822 067812 067560 004455 004454 023934 024175 04480 0454924 068567 068336 003357 003357 020813 020971 03861 0391326 069054 068921 002530 002529 017972 018177 03323 0335828 069675 069362 001907 001906 015519 015743 02858 0287430 069842 069694 001437 001436 013407 013623 02456 02453

05

0

minus05

minus1

minus15

minus05 minus04 minus03 minus02 minus01 0 01

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

f998400998400(0)

ℏ1

Figure 4 The ℎ1validity for various 119891

119908 Pr = 1 119899 = 0 and Re = 1

(Pr) and the injection number (119891119908) Another important

parameter is the power of the surface temperatureheat fluxdistribution (119899) which is considered in the present study

In order to ensure the convergence of the solution seriesthe variation of 119891 has been plotted at different ordersof approximation in Figure 6 The comparison betweenresults of HAM and numerical solution (Runge-Kutta) hasbeen displayed in Table 1 It can be observed that there isa good agreement between HAM method and numericalsolution Figures 7 8 and 9 illustrate effect of variationof wall injectionsuction parameter (119891

119908) on velocity and

minus02

minus04

minus06

minus08

minus12

minus1

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

ℏ2

Figure 5 The ℎ2validity for various 119891

119908 Pr = 1 119899 = 0 Re = 1 and

119891119908= 0

temperature distribution It can be observed that all boundarylayer thickness decreased by increasing 119891

119908from negative to

positive (from injection to suction) Figures 10 11 and 12show the effect of Reynolds number on the velocity profilewhen Pr = 1 119899 = 0 and 119891

119908= 0 This figure shows that

the boundary layer thickness and thermal boundary layerthickness are quite opposite to that of Reynolds number Theinfluence of 119899 on temperature field for both types of thethermal boundary conditions considered has been studiedin Figures 13 and 14 We notice that increasing 119899 reduces

6 Mathematical Problems in Engineering

1

05

0

minus05

f(120578)

0 2 4 6 8 10(120578)

n = 11

n = 12n = 13

n = 14

n = 15

142

14

138

136

134

13292 94 96 98

Figure 6 The variation of 119891 at different orders of approximationsPr = 1 119899 = 0 Re = 1 and 119891

119908= 0

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

1

08

06

04

02

0 1 2 3 4 5 6

f998400 (120578)

120578

Figure 7 Velocity profile 1198911015840 for various 119891119908when Pr = 1 119899 = 0 and

Re = 1

the thermal boundary layer thickness regardless of theboundary condition type leading to a heat transfer aug-mentation Effect of Prandtl number (isothermal) on thetemperature field is plotted in Figure 15 This figure showsthat thermal boundary layer thickness directly depends onPrandtl number

Tables 2 and 3 compare the results ofHAMand numericalsolution when the Reynolds number varies for 119891

119908= 0

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 8 Temperature profile (isothermal) 120579 for various 119891119908when

Pr = 1 119899 = 0 and Re = 1

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 9 Temperature profile (isoflux) 120579 for various 119891119908when Pr =

1 119899 = 0 and Re = 1

Pr = 1 and 119899 = 0 for isothermal and heat flux boundaryconditions respectively It can be seen that for isothermalboundary condition with increasing Reynolds number thewall shear stress for both boundary conditions consideredhere (isothermal and heat flux) and Nusselt number increasewith Reynolds number for isothermal state and independentof Reynolds number for heat flux boundary condition

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

For power-law fluid wit constant temperature and heatflux boundary conditions respectively Employing the defini-tion of convective heat transfer coefficient the local Nusseltnumbers become

Nu119909=

ℎ119909

119896=

minus1205791015840

(0) 119909

radic119870 Power-law temperature

119902119908119909

119896 (119879119908minus 119879infin)=

119909

120579 (0)radic119870 Power-law heat flux

(9)

3 Solution of Problem by HomotopyAnalysis Method (HAM)

As mentioned by Liao a solution may be expressed withdifferent base functions among which some converge to theexact solution of the problem faster than others Such basefunctions are obviously better suited for the final solution tobe expressed in terms of Noting these facts we have decidedto express 119892(120578) by a set of base functions of the followingform

119891 (120578) =

infin

sum

119898=0

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

120579 (120578) =

infin

sum

119898=0

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

(10)

The rule of solution expression provides us with a startingpoint It is under the rule of solution expression that initialapproximations auxiliary linear operators and the auxiliaryfunctions are determined So according to the rule of solutionexpression we choose the initial guess and auxiliary linearoperator in the following form

1198910(120578) = 119891

119908+ 1 minus exp (minus120578) 120579

0(120578) = exp (minus120578) (11)

We choose linear operator as follows

L1(119891) = 119891

101584010158401015840

+ 11989110158401015840

L2(120579) = 120579

10158401015840

+ 1205791015840

L1(1198881+ 1198882120578 + 1198883exp (minus120578)) = 0

L2(1198884+ 1198885exp (minus120578)) = 0

(12)

where 119888119894(119894 = 1ndash5) are constants Let 119875 isin [0 1] denote the

embedding parameter and let ℎ indicate nonzero auxiliaryparameters Then we construct the following equations

31 Consider ZerothndashOrder Deformation Equations

(1 minus 119875)L1[119891 (120578 119901) minus 119891

0(120578)] = 119901ℎ

1119867(120578)119873

1[119891 (120578 119901)]

119891 (0 119901) = 119891119908

1198911015840

(0 119901) = 1 1198911015840

(infin 119901) = 0

(1 minus 119875)L2[120579 (120578 119901) minus 120579

0(120578)] = 119901ℎ

2119867(120578)119873

2[120579 (120578 119901)]

120579 (0 119901) = 1 120579 (infin 119901) = 0 Power-law temperature

1205791015840

(0 119901) = minus1 120579 (infin 119901) = 0 Power-law heat flux

1198731[119891 (120578 119901) 120579 (120578 119901)]

=1205973

119891 (120578 119901)

1205971205783

+ Re(119891 (120578 119901)1205972

119891 (120578 119901)

1205971205782minus (

120597119891 (120578 119901)

120597120578)

2

)

minus120597119891 (120578 119901)

120597120578

1198732[119891 (120578 119901) 120579 (120578 119901)]

=1205972

120579 (120578 119901)

1205971205782+ Re sdotPr(119891 (120578 119901)

120597120579 (120578 119901)

120597120578)

minus 119899 sdot Re sdotPr(120579 (120578 119901)120597119891 (120578 119901)

120597120578)

(13)

For 119901 = 0 and 119901 = 1 we have

119891 (120578 0) = 1198910(120578) 119891 (120578 1) = 119891 (120578)

120579 (120578 0) = 1205790(120578) 120579 (120578 1) = 120579 (120578)

(14)

When 119901 increases from 0 to 1 then 119891(120578 119901) and 120579(120578 119901) varyfrom 119891

0(120578) and 120579

0(120578) to 119891(120578) and 120579(120578) By Taylorrsquos theorem

and using (14) we can write the following

119891 (120578 119901) = 1198910(120578) +

infin

sum

119898minus1

119891119898(120578) 119901119898

119891119898(120578) =

1

119898

120597119898

(119891 (120578 119901))

120597119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898minus1

120579119898(120578) 119901119898

120579119898(120578) =

1

119898

120597119898

(120579 (120578 119901))

120597119901119898

(15)

In which ℎ1and ℎ

2are chosen in such a way that these two

series are convergent at119901 = 1 therefore we have the followingthrough (15)

119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

(16)

32 Consider Mth-Order Deformation Equations

L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ1119867(120578) 119877

119891

119898(120578) (17)

4 Mathematical Problems in Engineering

119891119898(0) = 0 119891

1015840

119898(0) = 0 119891

1015840

119898(infin) = 0 (18)

119877119891

119898(120578) = 119891

101584010158401015840

119898minus1+

119898minus1

sum

119896=0

11989111989611989110158401015840

119898minus1minus119896

minus

119898minus1

sum

119896=0

1198911015840

1198961198911015840

119898minus1minus119896minus 1198701198911015840

119898minus1

(19)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ2119867(120578) 119877 120579

119898(120578) (20)

120579119898(0) = 0 120579

119898(infin) = 0 (21)

1205791015840

119898(0) = 0 120579

119898(infin) = 0 (22)

119877120579

119898(120578) = 120579

10158401015840

119898minus1+ Re sdotPr(

119898minus1

sum

119896=0

1198911198961205791015840

119898minus1minus119896)

minus 119899 sdot Re sdotPr(119898minus1

sum

119896=0

1205791198961198911015840

119898minus1minus119896)

(23)

120594119898=

0 119898 le 1

1 119898 gt 1(24)

The general solutions of (17)ndash(23) are

119891119898(120578) minus 120594

119898119891119898minus1

(120578) = 119891lowast

119898(120578) + 119862

119898

1+ 119862119898

2120578 + 119862119898

3exp (minus120578)

120579119898(120578) minus 120594

119898120579119898minus1

(120578) = 120579lowast

119898(120578) + 119862

119898

4+ 119862119898

5exp (minus120578)

(25)

where 119862119898

1to 119862119898

5are constants that can be obtained by

applying the boundary condition in (18) (21) and (22)As discussed by Liao the rule of coefficient ergodicity and

the rule of solution existence play important roles in deter-mining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutionsIn many cases by means of the rule of solution expressionand the rule of coefficient ergodicity auxiliary functions canbe uniquely determined So we define the auxiliary function119867(120578) in the following form

119867(120578) = exp (minus120578) (26)

4 Convergence of the HAM Solution

As was mentioned in introduction the convergence and therate of approximation of the HAM solution strongly dependon the values of auxiliary parameter ℎ By means of the so-called ℎ curves it is easy to find out the so-called valid regionsof ℎ to gain a convergent solution series According to Figures2 and 3 the acceptable range of auxiliary parameter for Pr =1 119899 = 0Re = 1 and 119891

119908= 0 is minus03 lt ℎ

1lt minus01 and minus18 lt

ℎ2lt minus03Figures 4 and 5 show how auxiliary parameters ℎ

1and

ℎ2vary with changing 119891

119908 If 119891

119908increases the range of

convergency of solution is restricted and then decreased

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus2

f998400998400(0)

minus03 minus02 minus01 0

13th-order app14th-order app15th-order app

ℏ1

Figure 2 The ℎ1validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

13th-order app14th-order app15th-order app

ℏ2

Figure 3 The ℎ2validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

5 Results and Discussion

In the present study the Homotopy analysis method hasbeen used to solve the nonlinear equations of heat transferand fluid flow over a permeable stretching wall in a porousmedium The nondimensional numbers introduced in thepresent analysis are Reynolds number (Re) Prandtl number

Mathematical Problems in Engineering 5

Table 1 The results of HAM and NS for 119891(120578) 1198911015840(120578) and 120579(120578) when 119891119908= 0 Pr = 1 119899 = 0 and Re = 1

120578119891(120578) 119891

1015840

(120578) 120579(120578) (temperature) 120579(120578) (heat flux)HAM NS HAM NS HAM NS HAM NS

00 000000 000000 099999 100000 0999999 100000 19565 1961502 017042 017420 075376 075363 089633 089996 17677 1762704 030564 030549 056813 075363 080205 080327 15658 1570606 040571 040443 042820 042804 071301 071225 13856 1389808 047505 047900 032272 032259 062792 062826 12210 1222910 053802 053519 024322 024311 055226 055191 10690 1071212 058010 057754 018330 018322 048134 048329 09313 0934014 061161 060946 013814 013808 042056 042214 08094 0813416 064096 063352 010410 010406 036595 036800 06943 0705818 065374 065165 007845 007842 031817 032029 06095 0611020 066952 066531 005912 005910 027657 027841 05193 0527822 067812 067560 004455 004454 023934 024175 04480 0454924 068567 068336 003357 003357 020813 020971 03861 0391326 069054 068921 002530 002529 017972 018177 03323 0335828 069675 069362 001907 001906 015519 015743 02858 0287430 069842 069694 001437 001436 013407 013623 02456 02453

05

0

minus05

minus1

minus15

minus05 minus04 minus03 minus02 minus01 0 01

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

f998400998400(0)

ℏ1

Figure 4 The ℎ1validity for various 119891

119908 Pr = 1 119899 = 0 and Re = 1

(Pr) and the injection number (119891119908) Another important

parameter is the power of the surface temperatureheat fluxdistribution (119899) which is considered in the present study

In order to ensure the convergence of the solution seriesthe variation of 119891 has been plotted at different ordersof approximation in Figure 6 The comparison betweenresults of HAM and numerical solution (Runge-Kutta) hasbeen displayed in Table 1 It can be observed that there isa good agreement between HAM method and numericalsolution Figures 7 8 and 9 illustrate effect of variationof wall injectionsuction parameter (119891

119908) on velocity and

minus02

minus04

minus06

minus08

minus12

minus1

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

ℏ2

Figure 5 The ℎ2validity for various 119891

119908 Pr = 1 119899 = 0 Re = 1 and

119891119908= 0

temperature distribution It can be observed that all boundarylayer thickness decreased by increasing 119891

119908from negative to

positive (from injection to suction) Figures 10 11 and 12show the effect of Reynolds number on the velocity profilewhen Pr = 1 119899 = 0 and 119891

119908= 0 This figure shows that

the boundary layer thickness and thermal boundary layerthickness are quite opposite to that of Reynolds number Theinfluence of 119899 on temperature field for both types of thethermal boundary conditions considered has been studiedin Figures 13 and 14 We notice that increasing 119899 reduces

6 Mathematical Problems in Engineering

1

05

0

minus05

f(120578)

0 2 4 6 8 10(120578)

n = 11

n = 12n = 13

n = 14

n = 15

142

14

138

136

134

13292 94 96 98

Figure 6 The variation of 119891 at different orders of approximationsPr = 1 119899 = 0 Re = 1 and 119891

119908= 0

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

1

08

06

04

02

0 1 2 3 4 5 6

f998400 (120578)

120578

Figure 7 Velocity profile 1198911015840 for various 119891119908when Pr = 1 119899 = 0 and

Re = 1

the thermal boundary layer thickness regardless of theboundary condition type leading to a heat transfer aug-mentation Effect of Prandtl number (isothermal) on thetemperature field is plotted in Figure 15 This figure showsthat thermal boundary layer thickness directly depends onPrandtl number

Tables 2 and 3 compare the results ofHAMand numericalsolution when the Reynolds number varies for 119891

119908= 0

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 8 Temperature profile (isothermal) 120579 for various 119891119908when

Pr = 1 119899 = 0 and Re = 1

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 9 Temperature profile (isoflux) 120579 for various 119891119908when Pr =

1 119899 = 0 and Re = 1

Pr = 1 and 119899 = 0 for isothermal and heat flux boundaryconditions respectively It can be seen that for isothermalboundary condition with increasing Reynolds number thewall shear stress for both boundary conditions consideredhere (isothermal and heat flux) and Nusselt number increasewith Reynolds number for isothermal state and independentof Reynolds number for heat flux boundary condition

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

119891119898(0) = 0 119891

1015840

119898(0) = 0 119891

1015840

119898(infin) = 0 (18)

119877119891

119898(120578) = 119891

101584010158401015840

119898minus1+

119898minus1

sum

119896=0

11989111989611989110158401015840

119898minus1minus119896

minus

119898minus1

sum

119896=0

1198911015840

1198961198911015840

119898minus1minus119896minus 1198701198911015840

119898minus1

(19)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ2119867(120578) 119877 120579

119898(120578) (20)

120579119898(0) = 0 120579

119898(infin) = 0 (21)

1205791015840

119898(0) = 0 120579

119898(infin) = 0 (22)

119877120579

119898(120578) = 120579

10158401015840

119898minus1+ Re sdotPr(

119898minus1

sum

119896=0

1198911198961205791015840

119898minus1minus119896)

minus 119899 sdot Re sdotPr(119898minus1

sum

119896=0

1205791198961198911015840

119898minus1minus119896)

(23)

120594119898=

0 119898 le 1

1 119898 gt 1(24)

The general solutions of (17)ndash(23) are

119891119898(120578) minus 120594

119898119891119898minus1

(120578) = 119891lowast

119898(120578) + 119862

119898

1+ 119862119898

2120578 + 119862119898

3exp (minus120578)

120579119898(120578) minus 120594

119898120579119898minus1

(120578) = 120579lowast

119898(120578) + 119862

119898

4+ 119862119898

5exp (minus120578)

(25)

where 119862119898

1to 119862119898

5are constants that can be obtained by

applying the boundary condition in (18) (21) and (22)As discussed by Liao the rule of coefficient ergodicity and

the rule of solution existence play important roles in deter-mining the auxiliary function and ensuring that the high-order deformation equations are closed and have solutionsIn many cases by means of the rule of solution expressionand the rule of coefficient ergodicity auxiliary functions canbe uniquely determined So we define the auxiliary function119867(120578) in the following form

119867(120578) = exp (minus120578) (26)

4 Convergence of the HAM Solution

As was mentioned in introduction the convergence and therate of approximation of the HAM solution strongly dependon the values of auxiliary parameter ℎ By means of the so-called ℎ curves it is easy to find out the so-called valid regionsof ℎ to gain a convergent solution series According to Figures2 and 3 the acceptable range of auxiliary parameter for Pr =1 119899 = 0Re = 1 and 119891

119908= 0 is minus03 lt ℎ

1lt minus01 and minus18 lt

ℎ2lt minus03Figures 4 and 5 show how auxiliary parameters ℎ

1and

ℎ2vary with changing 119891

119908 If 119891

119908increases the range of

convergency of solution is restricted and then decreased

minus06

minus08

minus1

minus12

minus14

minus16

minus18

minus2

f998400998400(0)

minus03 minus02 minus01 0

13th-order app14th-order app15th-order app

ℏ1

Figure 2 The ℎ1validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

0

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

13th-order app14th-order app15th-order app

ℏ2

Figure 3 The ℎ2validity for Pr = 1 119899 = 0 Re = 1 and 119891

119908= 0

5 Results and Discussion

In the present study the Homotopy analysis method hasbeen used to solve the nonlinear equations of heat transferand fluid flow over a permeable stretching wall in a porousmedium The nondimensional numbers introduced in thepresent analysis are Reynolds number (Re) Prandtl number

Mathematical Problems in Engineering 5

Table 1 The results of HAM and NS for 119891(120578) 1198911015840(120578) and 120579(120578) when 119891119908= 0 Pr = 1 119899 = 0 and Re = 1

120578119891(120578) 119891

1015840

(120578) 120579(120578) (temperature) 120579(120578) (heat flux)HAM NS HAM NS HAM NS HAM NS

00 000000 000000 099999 100000 0999999 100000 19565 1961502 017042 017420 075376 075363 089633 089996 17677 1762704 030564 030549 056813 075363 080205 080327 15658 1570606 040571 040443 042820 042804 071301 071225 13856 1389808 047505 047900 032272 032259 062792 062826 12210 1222910 053802 053519 024322 024311 055226 055191 10690 1071212 058010 057754 018330 018322 048134 048329 09313 0934014 061161 060946 013814 013808 042056 042214 08094 0813416 064096 063352 010410 010406 036595 036800 06943 0705818 065374 065165 007845 007842 031817 032029 06095 0611020 066952 066531 005912 005910 027657 027841 05193 0527822 067812 067560 004455 004454 023934 024175 04480 0454924 068567 068336 003357 003357 020813 020971 03861 0391326 069054 068921 002530 002529 017972 018177 03323 0335828 069675 069362 001907 001906 015519 015743 02858 0287430 069842 069694 001437 001436 013407 013623 02456 02453

05

0

minus05

minus1

minus15

minus05 minus04 minus03 minus02 minus01 0 01

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

f998400998400(0)

ℏ1

Figure 4 The ℎ1validity for various 119891

119908 Pr = 1 119899 = 0 and Re = 1

(Pr) and the injection number (119891119908) Another important

parameter is the power of the surface temperatureheat fluxdistribution (119899) which is considered in the present study

In order to ensure the convergence of the solution seriesthe variation of 119891 has been plotted at different ordersof approximation in Figure 6 The comparison betweenresults of HAM and numerical solution (Runge-Kutta) hasbeen displayed in Table 1 It can be observed that there isa good agreement between HAM method and numericalsolution Figures 7 8 and 9 illustrate effect of variationof wall injectionsuction parameter (119891

119908) on velocity and

minus02

minus04

minus06

minus08

minus12

minus1

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

ℏ2

Figure 5 The ℎ2validity for various 119891

119908 Pr = 1 119899 = 0 Re = 1 and

119891119908= 0

temperature distribution It can be observed that all boundarylayer thickness decreased by increasing 119891

119908from negative to

positive (from injection to suction) Figures 10 11 and 12show the effect of Reynolds number on the velocity profilewhen Pr = 1 119899 = 0 and 119891

119908= 0 This figure shows that

the boundary layer thickness and thermal boundary layerthickness are quite opposite to that of Reynolds number Theinfluence of 119899 on temperature field for both types of thethermal boundary conditions considered has been studiedin Figures 13 and 14 We notice that increasing 119899 reduces

6 Mathematical Problems in Engineering

1

05

0

minus05

f(120578)

0 2 4 6 8 10(120578)

n = 11

n = 12n = 13

n = 14

n = 15

142

14

138

136

134

13292 94 96 98

Figure 6 The variation of 119891 at different orders of approximationsPr = 1 119899 = 0 Re = 1 and 119891

119908= 0

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

1

08

06

04

02

0 1 2 3 4 5 6

f998400 (120578)

120578

Figure 7 Velocity profile 1198911015840 for various 119891119908when Pr = 1 119899 = 0 and

Re = 1

the thermal boundary layer thickness regardless of theboundary condition type leading to a heat transfer aug-mentation Effect of Prandtl number (isothermal) on thetemperature field is plotted in Figure 15 This figure showsthat thermal boundary layer thickness directly depends onPrandtl number

Tables 2 and 3 compare the results ofHAMand numericalsolution when the Reynolds number varies for 119891

119908= 0

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 8 Temperature profile (isothermal) 120579 for various 119891119908when

Pr = 1 119899 = 0 and Re = 1

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 9 Temperature profile (isoflux) 120579 for various 119891119908when Pr =

1 119899 = 0 and Re = 1

Pr = 1 and 119899 = 0 for isothermal and heat flux boundaryconditions respectively It can be seen that for isothermalboundary condition with increasing Reynolds number thewall shear stress for both boundary conditions consideredhere (isothermal and heat flux) and Nusselt number increasewith Reynolds number for isothermal state and independentof Reynolds number for heat flux boundary condition

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 The results of HAM and NS for 119891(120578) 1198911015840(120578) and 120579(120578) when 119891119908= 0 Pr = 1 119899 = 0 and Re = 1

120578119891(120578) 119891

1015840

(120578) 120579(120578) (temperature) 120579(120578) (heat flux)HAM NS HAM NS HAM NS HAM NS

00 000000 000000 099999 100000 0999999 100000 19565 1961502 017042 017420 075376 075363 089633 089996 17677 1762704 030564 030549 056813 075363 080205 080327 15658 1570606 040571 040443 042820 042804 071301 071225 13856 1389808 047505 047900 032272 032259 062792 062826 12210 1222910 053802 053519 024322 024311 055226 055191 10690 1071212 058010 057754 018330 018322 048134 048329 09313 0934014 061161 060946 013814 013808 042056 042214 08094 0813416 064096 063352 010410 010406 036595 036800 06943 0705818 065374 065165 007845 007842 031817 032029 06095 0611020 066952 066531 005912 005910 027657 027841 05193 0527822 067812 067560 004455 004454 023934 024175 04480 0454924 068567 068336 003357 003357 020813 020971 03861 0391326 069054 068921 002530 002529 017972 018177 03323 0335828 069675 069362 001907 001906 015519 015743 02858 0287430 069842 069694 001437 001436 013407 013623 02456 02453

05

0

minus05

minus1

minus15

minus05 minus04 minus03 minus02 minus01 0 01

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

f998400998400(0)

ℏ1

Figure 4 The ℎ1validity for various 119891

119908 Pr = 1 119899 = 0 and Re = 1

(Pr) and the injection number (119891119908) Another important

parameter is the power of the surface temperatureheat fluxdistribution (119899) which is considered in the present study

In order to ensure the convergence of the solution seriesthe variation of 119891 has been plotted at different ordersof approximation in Figure 6 The comparison betweenresults of HAM and numerical solution (Runge-Kutta) hasbeen displayed in Table 1 It can be observed that there isa good agreement between HAM method and numericalsolution Figures 7 8 and 9 illustrate effect of variationof wall injectionsuction parameter (119891

119908) on velocity and

minus02

minus04

minus06

minus08

minus12

minus1

minus14

minus16

minus2 minus15 minus1 minus05 0

120579998400 (0)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

ℏ2

Figure 5 The ℎ2validity for various 119891

119908 Pr = 1 119899 = 0 Re = 1 and

119891119908= 0

temperature distribution It can be observed that all boundarylayer thickness decreased by increasing 119891

119908from negative to

positive (from injection to suction) Figures 10 11 and 12show the effect of Reynolds number on the velocity profilewhen Pr = 1 119899 = 0 and 119891

119908= 0 This figure shows that

the boundary layer thickness and thermal boundary layerthickness are quite opposite to that of Reynolds number Theinfluence of 119899 on temperature field for both types of thethermal boundary conditions considered has been studiedin Figures 13 and 14 We notice that increasing 119899 reduces

6 Mathematical Problems in Engineering

1

05

0

minus05

f(120578)

0 2 4 6 8 10(120578)

n = 11

n = 12n = 13

n = 14

n = 15

142

14

138

136

134

13292 94 96 98

Figure 6 The variation of 119891 at different orders of approximationsPr = 1 119899 = 0 Re = 1 and 119891

119908= 0

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

1

08

06

04

02

0 1 2 3 4 5 6

f998400 (120578)

120578

Figure 7 Velocity profile 1198911015840 for various 119891119908when Pr = 1 119899 = 0 and

Re = 1

the thermal boundary layer thickness regardless of theboundary condition type leading to a heat transfer aug-mentation Effect of Prandtl number (isothermal) on thetemperature field is plotted in Figure 15 This figure showsthat thermal boundary layer thickness directly depends onPrandtl number

Tables 2 and 3 compare the results ofHAMand numericalsolution when the Reynolds number varies for 119891

119908= 0

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 8 Temperature profile (isothermal) 120579 for various 119891119908when

Pr = 1 119899 = 0 and Re = 1

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 9 Temperature profile (isoflux) 120579 for various 119891119908when Pr =

1 119899 = 0 and Re = 1

Pr = 1 and 119899 = 0 for isothermal and heat flux boundaryconditions respectively It can be seen that for isothermalboundary condition with increasing Reynolds number thewall shear stress for both boundary conditions consideredhere (isothermal and heat flux) and Nusselt number increasewith Reynolds number for isothermal state and independentof Reynolds number for heat flux boundary condition

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

1

05

0

minus05

f(120578)

0 2 4 6 8 10(120578)

n = 11

n = 12n = 13

n = 14

n = 15

142

14

138

136

134

13292 94 96 98

Figure 6 The variation of 119891 at different orders of approximationsPr = 1 119899 = 0 Re = 1 and 119891

119908= 0

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

1

08

06

04

02

0 1 2 3 4 5 6

f998400 (120578)

120578

Figure 7 Velocity profile 1198911015840 for various 119891119908when Pr = 1 119899 = 0 and

Re = 1

the thermal boundary layer thickness regardless of theboundary condition type leading to a heat transfer aug-mentation Effect of Prandtl number (isothermal) on thetemperature field is plotted in Figure 15 This figure showsthat thermal boundary layer thickness directly depends onPrandtl number

Tables 2 and 3 compare the results ofHAMand numericalsolution when the Reynolds number varies for 119891

119908= 0

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 8 Temperature profile (isothermal) 120579 for various 119891119908when

Pr = 1 119899 = 0 and Re = 1

0 2 4 6 8120578

1

08

06

04

02

120579(120578)

fw = minus03

fw = minus01

fw = 0

fw = 01

fw = 03

Figure 9 Temperature profile (isoflux) 120579 for various 119891119908when Pr =

1 119899 = 0 and Re = 1

Pr = 1 and 119899 = 0 for isothermal and heat flux boundaryconditions respectively It can be seen that for isothermalboundary condition with increasing Reynolds number thewall shear stress for both boundary conditions consideredhere (isothermal and heat flux) and Nusselt number increasewith Reynolds number for isothermal state and independentof Reynolds number for heat flux boundary condition

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

1

08

06

04

02

00

1 2 3 4 5120578

f998400 (120578)

Re = 1

Re = 15

Re = 2

Re = 5

Figure 10 Velocity profile 1198911015840 for various Re when Pr = 1 119899 = 0and 119891

119908= 0

1

08

06

04

02

Re = 1

Re = 15

Re = 2

Re = 5

120579(120578)

0 2 4 6 8120578

Figure 11 Temperature profile (isothermal) 120579 for various Re whenPr = 1 119899 = 0 and 119891

119908= 0

Comparison of the results of HAM and numerical solu-tion has been shown in Tables 4 and 5 for various Prandtlnumbers when Re = 1 119899 = 0 and 119891

119908= minus03 119891

119908=

03 respectively It can be observed that the Prandtl numberhas no effect on the wall shear stress for both boundaryconditions and Nusselt number for heat flux boundarycondition

2

15

1

05

Re = 1

Re = 15

Re = 2

Re = 5

0 2 4 6 8120578

120579(120578)

Figure 12 Temperature profile (isoflux) 120579 for various Re when Pr =1 119899 = 0 and 119891

119908= 0

1

08

06

04

02

0 2 4 6 8120578

120579(120578)

n = 0

n = 1n = 10

Figure 13 Temperature profile (isothermal) 120579 for various 119899 whenPr = 1 119899 = 0 and 119891

119908= 0

6 Summary and Conclusions

Homotopy analysis method (HAM) is applied to computewall driven flow through a porous medium over a stretch-ing permeable surface subjected to power-law temperatureand heat flux boundary conditions The proper range of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 2 4 6 8120578

n = 0

n = 1n = 10

2

15

1

05

120579(120578)

Figure 14 Temperature profile (isoflux) 120579 for various 119899when Pr = 1119899 = 0 and 119891

119908= 0

Table 2 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for temperature

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus02 minus04 14198 14242 05030 0503315 minus025 minus04 15799 15811 06456 064222 minus02 minus02 17234 17320 07518 075925 minus01 minus01 24394 24494 12636 12576

Table 3 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) when 119891119908= 0

Pr = 1 and 119899 = 0 for heat flux

Re ℎ1

ℎ2

minus11989110158401015840

(0) minus1205791015840

(0)

HAM NS HAM NS1 minus025 minus02 14191 14242 09997 115 minus02 minus04 15791 15811 10000 12 minus015 minus03 17298 17320 09985 125 minus01 minus01 24445 24494 09939 1

the auxiliary parameter ℎ to ensure the convergency of thesolution series was obtained through the so-called ℎ curvesWhen comparedwith other analyticalmethods it is clear thatHAM provides highly accurate analytical solutions for non-linear problems Moreover second-law (of thermodynamics)aspects of the problem are investigatedThe highlights of thisstudy are the following

(i) The nondimensional viscous boundary layer thick-ness has a direct relationship with Reynolds numberthus Nusselt number rate increases with Re

1

09

08

07

06

05

04

03

02

01

Pr = 05

Pr = 1

Pr = 5

120578

120579(120578)

0 1 2 3 4 5

Figure 15 Temperature profile 120579 (isothermal) for various Pr whenPr = 1 119899 = 0 and 119891

119908= 0

Table 4 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= minus03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= minus03 119891

119908= minus03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 12699 12721 12711 12721 0216 0219 1000 09991 12699 12721 12711 12721 03128 0313 1000 10005 12699 12721 12711 12721 0636 0623 1000 0999

Table 5 The results of HAM and NS for 11989110158401015840(0) 1205791015840(0) for various Prwhen Re = 1 119899 = 0 and 119891

119908= 03

Pr

minus11989110158401015840

(0) minus1205791015840

(0)

119891119908= 03 119891

119908= 03

Temperature Heat flux Temperature Heat fluxHAM NS HAM NS HAM NS HAM NS

05 15699 15721 15695 15721 04145 04100 10002 100001 15699 15721 15695 15721 07258 07216 10002 100005 15699 15721 15695 15721 25821 25826 10002 10000

(ii) Nusselt number wall shear stress have a reverserelationship with andmass transfer from the wall 119891

119908

(iii) Increasing the Prandtl number results in reduction ofthermal boundary layer thickness

Consequently Nusselt number increase with Pr

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Nomenclature

119891 Similarity function for velocity119891119908 Injection parameter 119891

119908= minus1205921199081198711199060radic119870

ℎ Auxiliary parameterHAM Homotopy analysis method119867 Auxiliary functionL Linear operator of HAM119870 Permeability of the porous medium m2119873 Non-linear operator119899 Power of temperatureheat flux distributionNu Local Nusselt numberNu119871 Averaged Nusselt number

Pr Prandtl number Pr = V120572eff1199020 Wall heat flux coefficient Wm2

Re Reynolds number Re = 1205881199060119870119871

119879 Temperature1198790 Wall temperature coefficient 119870

119906 Velocity in 119909 direction1199060 Wall velocity coefficient ms

120592 Velocity in 119910 direction120592119908 Injection velocity ms

119861 Positive constant120579 Similarity function for temperature119909 Coordinate system m119910 Coordinate system m119876 Volumetric rate of heat generation120582 Heat generation119873 Radiation parameter120588 Density of the fluidΨ Stream function120583 Dynamic viscosity1205901 Effective viscosity

1205811 Absorption coefficient

120578 Dimensionless similarity variableV Kinematic viscosity

References

[1] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 pp 289ndash299 2006

[2] B Yu H T Chiu Z Ding and L J Lee ldquoAnalysis of flowand heat transfer in liquid composite moldingrdquo InternationalPolymer Processing vol 15 no 3 pp 273ndash283 2000

[3] R Nazar A Ishak and I Pop ldquoUnsteady boundary layer flowover a stretching sheet in a micropolar fluidrdquo InternationalJournal of Mathematical Physical and Engineering Sciences vol2 pp 161ndash165 2008

[4] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces I boundary layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II boundary layer behaviour on con-tinuous flatsurfacesrdquo AIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[7] I Pop and T Y Na ldquoFree convection heat transfer of non-Newtonian fluids along a vertical wavy surface in a porousmediumrdquo in Proceedings of the 4th International Symposium onHeat Transfer (ISHT rsquo96) pp 452ndash457 Beijing China October1996

[8] W H H Banks ldquoSimilarity solutions of the boundary layerequations for a stretching wallrdquo Journal deMecaniqueTheoriqueet Appliquee vol 2 no 3 pp 375ndash392 1983

[9] E M A Elbashbeshy ldquoHeat transfer over a stretching surfacewith variable surface heat fluxrdquo Journal of Physics D vol 31 no16 pp 1951ndash1954 1998

[10] E M A Elbashbeshy and M A A Bazid ldquoHeat transfer overa stretching surface with internal heat generationrdquo CanadianJournal of Physics vol 81 no 4 pp 699ndash703 2003

[11] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[12] A Sriramalu N Kishan and R J Anand ldquoSteady flow and heattransfer of a viscous incompressible fluid flow through porousmedium over a stretching sheetrdquo Journal of Energy Heat andMass Transfer vol 23 pp 483ndash495 2001

[13] M E Ali ldquoOn thermal boundary layer on a power-law stretchedsurface with suction or injectionrdquo International Journal of Heatand Fluid Flow vol 16 no 4 pp 280ndash290 1995

[14] A Pantokratoras ldquoFlow adjacent to a stretching permeablesheet in a Darcy-Brinkman porous mediumrdquo Transport inPorous Media vol 80 no 2 pp 223ndash227 2009

[15] E M A Elbashbeshy ldquoRadiation effect on heat transfer over astretching surfacerdquo Canadian Journal of Physics vol 78 no 12pp 1107ndash1112 2000

[16] A Tamayol and M Bahrami ldquoAnalytical determination ofviscous permeability of fibrous porous mediardquo InternationalJournal of Heat and Mass Transfer vol 52 no 9-10 pp 2407ndash2414 2009

[17] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[18] G Adomian ldquoA review of the decomposition method andsome recent results for nonlinear equationsrdquoMathematical andComputer Modelling vol 13 no 7 pp 17ndash43 1990

[19] AHNayfehPerturbationMethodsWileyNewYorkNYUSA2000

[20] D D Ganji ldquoThe application of Hersquos homotopy perturbationmethod to nonlinear equations arising in heat transferrdquo PhysicsLetters A vol 355 no 4-5 pp 337ndash341 2006

[21] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[22] S J Liao ldquoBoundary element method for general nonlineardifferential operatorsrdquo Engineering Analysis with BoundaryElements vol 202 pp 91ndash99 1997

[23] S J Liao ldquoAn approximate solution technique not dependingon small parameters a special examplerdquo International Journalof Non-Linear Mechanics vol 30 no 3 pp 371ndash380 1995

[24] S Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method ChapmanampHallCRC Boca Raton Fla USA2003

[25] T Hayat and M Khan ldquoHomotopy solutions for a generalizedsecond-grade fluid past a porous platerdquo Nonlinear Dynamicsvol 42 no 4 pp 395ndash405 2005

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[26] A Fakhari G Domairry and E Ebrahimpour ldquoApproximateexplicit solutions of nonlinear BBMB equations by homotopyanalysis method and comparison with the exact solutionrdquoPhysics Letters A vol 368 no 1-2 pp 64ndash68 2007

[27] M Dayyan D D Ganji and S M Seyyedi ldquoApplicationof homotopy analysis method for velocity and temperaturedistribution of viscose stagnation pointrdquo International Journalof Nonlinear Dynamics in Engineering and Sciences vol 2 no 2pp 189ndash205 2010

[28] G Domairry and M Fazeli ldquoHomotopy analysis methodto determine the fin efficiency of convective straight finswith temperature-dependent thermal conductivityrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 14no 2 pp 489ndash499 2009

[29] MDayyanDDGanji A Imam and SM Seyyedi ldquoAnalyticalsolution of heat transfer over a flat plate with radiation for bing-ham non-newtonian fluidrdquo International Journal of NonlinearDynamics in Engineering and Sciences vol 4 no 1 pp 155ndash1672012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of