Meccanica (2010) 45: 857–868DOI 10.1007/s11012-010-9295-y

Homotopy Analysis Method to Walter’s B fluid in a verticalchannel with porous wall

A.A. Joneidi · G. Domairry · M. Babaelahi

Received: 13 October 2008 / Accepted: 16 March 2010 / Published online: 9 June 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract In this research the steady three-dimen-sional flow of a Walter’s B fluid in a vertical chan-nel with porous wall, through which the fluid is in-jected uniformly into the channel through one side ofthe channel, is studied analytically using HomotopyAnalysis Method (HAM). The channel is assumed tobe infinite and uniform. The effects of the elasticity ofthe fluid on the flow and heat transfer on the walls ofthe channel are discussed.

Keywords Homotopy Analysis Method (HAM) ·Velocity equation · Walter’s B fluid · Verticalchannel · Porous wall · Viscoelastic fluid

Nomenclaturecp Specific heat at constant pressuree Rate of strain tensorg Gravitational acceleration vectorI Identity tensor

A.A. JoneidiDepartment of Mechanical Engineering, EindhovenUniversity of Technology, Eindhoven, Netherlands

G. Domairry (�)Department of Mechanical Engineering, Babol Universityof Technology, Babol, Iran, P.O. Box 484e-mail: amirganga111@yahoo.com

M. BabaelahiDepartment of Mechanical Engineering, K.N. ToosiUniversity of Technology, Tehran, Iran

k Thermal conductivityk0 Short memory coefficientL, B, D Dimensions of the channelPe Peclet numberP PressurePx , Py Pressure variations in the x and y directionsp Embedding parameterRe Cross-flow Reynolds numberS Elastic numberT TemperatureT Cauchy stress tensorT0, T1 Temperatures of the wallst TimeU Uniform injection velocityu, v, w Components of the velocity vectorV Velocity vectorθ Dimensionless temperatureτ Relaxation timeϕ0 Arbitrary functionρ Density

1 Introduction

Non-Newtonian fluids in a porous surface channelhave always attracted the interest of many investiga-tors because of many applications of them in differ-ent scientific fields. Examples of these applicationsare the boundary layer control, transpiration cooling,

858 Meccanica (2010) 45: 857–868

gaseous diffusion, prevent corrosion, reactants addi-tion and drag reduction. From a technological pointof view, flows of this type are always important, espe-cially in transpiration cooling, which is a very effectiveprocess to protect certain structural elements in turbo-jet and rocket engines, like combustion chamber walls,or gas turbine blades, from the influence of hot gases.

The flow of Newtonian and non-Newtonian fluidsthrough porous channels has been investigated by nu-merous authors. The case of a two-dimensional, in-compressible, steady, laminar suction flow of a New-tonian fluid in a parallel-walled porous channel wasstudied by Berman [1]. He has solved the Navier-Stokes equations by using a perturbation method forvery low cross-flow Reynolds numbers. After his pio-neering work, this problem has been studied by manyresearchers considering various variations in the prob-lem [2, 3].

Wang and Skalak were the first persons who presentthe solution for a three-dimensional problem of fluidinjection through one side of a long vertical chan-nel for Newtonian fluid [4]. They have obtained aseries solution valid for small values of the cross-flow Reynolds numbers and a numerical solutionfor both small and large cross-flow Reynolds num-bers. Huang re-examined Wang and Skalak’s prob-lem using a method based upon quasilinearization [5].The same flow problem was solved for large cross-flow Reynolds numbers by Ascher using a spline-collocation method. Sharma and Chaudhary reconsid-ered the above-mentioned problem by introducing asecond order viscoelastic fluid [6, 7]. They obtainedthe second order perturbation solution by assumingthat the cross-flow Reynolds number is small. How-ever, their results seem to be in error and their ex-pression for pressure does not satisfy the equations ofmotion. In the recent years, Baris continued the lastmentioned research by substituting thermodynami-cally compatible fluid of second grade instead of New-tonian fluid [8]. The used analytical method by Bariswas traditional perturbation solution which was one ofthe old analytical methods.

These scientific problems are modeled by ordinaryor partial differential equations and should be solvedusing special techniques, because in most cases, an-alytical solutions can’t be applied to these problems.In recent years, much attention has been devoted tothe newly developed methods to construct an analyticsolution of these equations such as Homotopy Per-turbation Method [9–11], Differential Transformation

Fig. 1 Sketch of the problem under discussion

Method [12–14], and Optimal Homotopy AsymptoticMethod [15, 16] and so on. One of these techniquesis Homotopy Analysis Method (HAM), which was in-troduced by Liao [17–20] and has been successfullyapplied to solve many types of nonlinear problems[21–31].

In this paper, HAM is employed to find velocity,heat transfer and pressure variation profiles of steadythree-dimensional Walter’s B fluid in a vertical chan-nel with porous wall. The major purpose of presentresearch is to obtain new analytical technique calledHAM for better analytical relations for velocity, tem-perature and pressure variations and to examine qual-itatively the effect of elasticity of the fluid, Reynoldsand Peclet number on these values. Sketch of the prob-lem is depicted in Fig. 1.

2 Governing equations

The steady flow of a Walter’s B fluid in a vertical chan-nel with porous wall is considered. Figure 1 shows thephysical model and coordinate system. A fluid is in-jected through a vertical porous plate at y = D withuniform velocity U . The fluid strikes another verticalimpermeable plate at y = 0. It flows out through the

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Fig. 2 Walter’s B fluid flow in a vertical channel with porouswall

opening of the plates, due to the action of gravity alongthe z-axis. The distance between the walls is assumedD, is small compared to the dimensions of the plates,i.e. L � B � D. Due to this assumption the edge ef-fects can be ignored and the isobars are parallel to thez-axis.

The model of Walter’s B fluid is chosen for ourstudy as it involves only one non-Newtonian parame-ter. The Cauchy stress tensor T in such a fluid is re-lated to the motion in the following manner [27]:

T = −P I + 2η0e − 2k0δe

δt. (1)

In this equation, P is the pressure, I is the identitytensor, and the rate of strain tensor e is defined by:

2e = ∇v + (∇v)T , (2)

where v is the velocity vector, ∇ is the gradient op-erator and δ/δt denotes the convected differentiationof a tensor quantity in relation to the material in mo-tion. The convected differentiation of the rate of straintensor is given by:

δe

δt= ∂e

∂t+ v.∇e − e.∇v − (∇v)T .e. (3)

Finally η0 and k0 are, respectively, the limiting viscos-ity at small rate of shear and the short memory coeffi-cient which are defined through

η0 =∫ ∞

0N(τ)dτ,

(4)

k0 =∫ ∞

0τN(τ)dτ,

where N(τ) is the distribution function with relaxationtime τ . This idealized model is a valid approximationof Walter’s B fluid taking very short memory into ac-count so that terms involving∫ ∞

0τnN(τ)dτ, n ≥ 2, (5)

have been neglected.In addition to (1), the basic equations of the prob-

lem are in the following form:

∇.v = 0, (6)

ρ(v.∇v) = ∇.T + ρg, (7)

ρcp(v.∇T ) = kT . (8)

Equations (6), (7) and (8) are continuity, momen-tum and energy equations, respectively. Where ρ isthe density and g is the gravitational acceleration vec-tor. The fluid flow is assumed to be steady and lami-nar. The fluid is incompressible and the body force perunit mass is taken to be equal to the gravitational ac-celeration. Also the effects of viscous dissipation areassumed to be negligible.

Substituting Cauchy stress tensor from (1) into (7),with the aid of (2) and (3) yields:

ρ(v.∇v) = −∇P + ρg + η0∇2v

− 2k0v.∇∇2v + k0∇2(v.∇v). (9)

The velocity components corresponding to the x,y and z directions are respectively denoted by u, v

and w. Following [4], we look for a solution, com-patible with the continuity equation, of the form:

u = U

Dxf ′(η), v = −Uf (η),

(10)

w = D2gρ

η0h(η),

where η = y/D and the prime denotes the differentia-tion with respect to η. The boundary conditions for thevelocity field are:

f (0) = 0, f (1) = 1, f ′(0) = 0,(11)

f ′(1) = 0, h(0) = 0, h(1) = 0.

860 Meccanica (2010) 45: 857–868

It follows from (10) and the equation of motion that:

∂P

∂x= Ux

D2

(−Uρf ′2 + Uρff ′′ + Uk0

D2f ′′2 + η0

Df ′′′

− 2Uk0

D2f ′f ′′′ + Uk0

D2ff ′′′′

), (12)

∂P

∂η= −U2ρff ′ − Uη0

Df ′′ + 3U2k0

D2f ′f ′′

− U2k0

D2ff ′′′, (13)

h′′ + Ref h′ + ReS(f h′′′ − f ′′h′ − 2f ′h′′)

+ 1 = 0, (14)

where the cross-flow Reynolds number, Re, and theelastic number, S, are defined through, respectively

Re = UDρ

η0, S = k0

ρD2. (15)

Integrating (13) with respect to η, obtained equationis:

P(x,η) = −1

2ρU2f 2 − Uη0

Df ′ + 2U2k0

D2f ′2

− U2k0

D2ff ′′ + ϕ(x), (16)

where ϕ(x) is an arbitrary function of x. Differentia-tion of the above equation with respect to x yields:

∂P

∂x= ∂ϕ

∂x. (17)

Combining of (12) and (17) delivers:

dϕ

dx= Uxη0

D3{f ′′′ + R(ff ′′ − f ′2)

+ RS(ff ′′′′ + f ′′2 − 2f ′f ′′′)}. (18)

It is apparent that the quantity in parentheses in (18)must be independent of η. Hence, the following equa-tion for f is:

f ′′′ + Re (ff ′′ − f ′2)

+ ReS(ff ′′′′ + f ′′2 − 2f ′f ′′′) = C, (19)

where C is an arbitrary constant which takes the value

C = f ′′′′(0) + ReSf ′′2(0). (20)

Now differentiating (19) with respect to η to elimi-nate the constant C, gives:

f ′′′′ + Re (f ′′′f − f ′f ′′) + ReS(ff ′′′′′ − f ′f ′′′′) = 0.

(21)

By using (19), ϕ(x) can now be written as

ϕ(x) = Uη0C

2D3x2 + P0, (22)

where P0 is the constant of integration. Inserting ϕ(x)

from (22) into (16):

P(x,η) = −1

2ρU2f 2 − Uη0

Df ′ + 2U2k0

D2f ′2

− U2k0

D2ff ′′ + Uη0C

2D3x2 + P0. (23)

Solving (21), leads to obtain the pressure at anypoint. From (23), the pressure variation in the x andy direction can be written in non-dimensional form asfollows:

P(x) = P(0, η) − P(x,η)

ρU2

= −f ′′′(0) + ReSf ′′2(0)

2Re

(x

d

)2, (24)

P(y) = P(x,0) − P(x,η)

ρU2

= f 2

2+ f ′

Re+ S(ff ′′ − 2f ′2). (25)

3 Governing equations for temperature

In this section, temperature field is introduced as bel-low

T = T0 + (T1 − T0)θ(η), (26)

where T0 and T1 are temperatures of the imperme-able and porous plates, respectively and with constantvalue. Substituting (10) and (26) into (8) leads to theordinary differential equation

θ ′′ + Pef θ ′ = 0, (27)

where Pe = ρUDcp/k is the Peclet number. Equation(27) is solved subject to the boundary conditions

θ(0) = 0, θ(1) = 1. (28)

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4 Solution using Homotopy Analysis Method

In this section, HAM is applied to solve (14), (21) and(27) subject to boundary conditions (11) and (28). Theinitial guesses and auxiliary linear operators are cho-sen in the following form:

f0(η) = −2η3 + 3η2,

θ0(η) = η,

h0(η) = 0.

(29)

As the initial guess approximation for f (η), θ(η) andh(η)

L1(f ) = f IV , L2(θ) = θ ′′, L3(h) = h′′.

(30)

As the auxiliary linear operator which has the prop-erty:

L(c1 + c2η + c3η2 + c4η

3) = 0,

L(c5 + c6η) = 0, (31)

L(c7 + c8η) = 0,

and ci (i = 1–8) are constants. Let p ∈ [0,1] denotesthe embedding parameter and � indicates non-zeroauxiliary parameters. Then, the following equationsare constructed:

Zeroth-order deformation equations

(1 − P)L1[f (η;p) − f0(η)] = p�1N1[f (η;p)], (32)

(1 − P)L2[θ(η;p) − θ0(η)] = p�2N2[θ(η;p)], (33)

(1 − p)L3[h(η;p) − h0(η)] = p�3N3[h(η;p)], (34)

f (0;p) = 0; f ′(0;p) = 0;(35)

f (1;p) = 1; f ′(1;p) = 1;θ(0;p) = 0; θ(1;p) = 1; (36)

h(0;p) = 0; h(1;p) = 0; (37)

N1[f (η;p)] = d4f (η;p)

dη4

+ Re

(f (η;p)

d3f (η;p)

dη3

− df (η;p)

dη

d2f (η;p)

dη2

)

+ ReS

(f (η;p)

d5f (η;p)

dη5

− df (η;p)

dη

d4f (η;p)

dη4

)= 0, (38)

N2[θ(η;p)] = d2θ(η;p)

dη2

+ Pef (η;p)dθ(η;p)

dη= 0, (39)

N3[h(η;p)]

= d2f (η;p)

dη2+ Re

(f (η;p)

dh(η;p)

dη

)

+ ReS

(f (η;p)

d3h(η;p)

dη3

− d2f (η;p)

dη2

dh(η;p)

dη

− 2df (η;p)

dη

d2h(η;p)

dη2

)= 0. (40)

For p = 0 and p = 1:

f (η;0) = f0(η); f (η;1) = f (η);θ(η;0) = θ0(η); θ(η;1) = θ(η);h(η;0) = h0(η); h(η;1) = h(η);

(41)

when p increases from 0 to 1 then f (η;p) vary fromf0(η) to f (η), θ(η;p) vary from θ0(η) to θ(η) andh(η;p) vary from h0(η) to h(η). By Taylor’s theoremand using (41):

f (η;p) = f0(η) +∞∑

m−1

fm(η)pm,

(42)

fm(η) = 1

m!∂m(f (η;p))

∂pm,

θ(η;p) = θ0(η) +∞∑

m−1

θm(η)pm,

(43)

θm(η) = 1

m!∂m(θ(η;p))

∂pm,

862 Meccanica (2010) 45: 857–868

h(η;p) = h0(η) +∞∑

m−1

hm(η)pm,

(44)

hm(η) = 1

m!∂m(h(η;p))

∂pm.

For simplicity, �1 = �2 = �3 = � is supposed, which �

is chosen in such a way that these three series are con-vergent at p = 1. Therefore, through (42)–(44) belowequations are obtained:

f (η) = f0(η) +∞∑

m−1

fm(η), (45)

θ(η) = θ0(η) +∞∑

m−1

θm(η), (46)

h(η) = h0(η) +∞∑

m−1

hm(η). (47)

mth-order deformation equations

L[fm(η) − χmfm−1(η)] = �Rfm(η), (48)

fm(0) = f ′m(0) = fm(1) = f ′

m(1) = 0, (49)

Rfm(η) = f IV

m−1 + Rem−1∑n=0

fm−1−nf′′′n − f ′

m−1−nf′′n

+ ReS

m−1∑n=0

fm−1−nfVn − f ′

m−1−nfIVn , (50)

L[θm(η) − χmθm−1(η)

] = �Rθm(η), (51)

θm(0) = θm(1) = 0, (52)

Rθm(η) = θ ′′

m−1 +m−1∑n=0

Pefm−1−nθ′n, (53)

L[hm(η) − χmhm−1(η)

] = �Rhm(η), (54)

hm(0) = hm(1) = 0, (55)

Rhm(η) = h′′

m−1 + Rem−1∑n=0

fm−1−nh′n

+m−1∑n=0

ReS(fm−1−nh′′′n − h′

m−1−nf′′n

− 2f ′m−1−nh

′′n) + 1, (56)

χm ={

0, m ≤ 1,

1, m > 1.(57)

5 Convergence of HAM

HAM provides us with a great freedom in choos-ing the solution of a nonlinear problem by differentbase functions. This has a great effect on the con-vergence region because the convergence region andthe rate of a series are chiefly determined by the basefunctions used to express the solution. Therefore, anonlinear problem could be approximated more effi-ciently by ensuring its convergency. The convergenceand rate of approximation for the HAM solution de-pends on the value of auxiliary parameter � strongly.By means of the so-called �-curves, it is easy to findout the so-called valid regions of auxiliary parame-ters to gain a convergent solution series. According toFigs. 3 to 6, the convergence ranges of f ′′(0), θ ′(0)

and h′(0) are variable for different values of parame-ters.

6 Results and discussion

In this section, based on the Homotopy AnalysisMethod (HAM) solution which presented in Sect. 4,the tangential, normal and axial velocities of the steadyflow of a Walter’s B fluid in a vertical channel are ob-tained.

In Figs. 7–9, the functions which correspond tothe velocity components have been plotted versus η

for two different values of the cross-flow Reynoldsnumber (Re), with the elastic number S as a pa-rameter. For low values of the cross-flow Reynoldsnumber, e.g. Re = 1, the velocity profiles in the vis-coelastic fluid case, are indistinguishable from thosein the Newtonian fluid cases and so they are not pre-sented.

Figure 7 depicts the normal velocity componentfor various values of the elastic number when Re isfixed at 10 and 20, respectively. As evident, by increas-ing of the elastic number, the normal velocity intensi-fies.

Figure 8 illustrates the effect of elastic numberon tangential velocity profiles for the same values ofcross-flow Reynolds number and the elastic number.It is obvious from this figure that the tangential ve-locity profile behavior is different from approximately

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Fig. 3 The �-validity for f (η), when Re = 10, S = 0

Fig. 4 The �-validity for θ(η), when Re = 10, S = 0

η = 0.4 to the η = 1. According to this figure, the in-creasing of the elastic number enhances the maximumof tangential velocity and moves it away from porousplate. Also increasing the cross flow Reynolds num-ber switch the maximum velocity point to further dis-tance from the porous surface. This is due to incrementof shear stresses on impermeable surface of the chan-nel.

Axial velocity which is due to gravity force hasbeen shown in Fig. 9 when Re is keeping fixed in 10. Itis obvious that the increasing of elastic parameter en-

Fig. 5 The �-validity for h(η), when Re = 10, S = 0.012

Fig. 6 The �-validity for θ(η), when Re = 20, S = 0

hances the maximum of axial velocity. Therefore, thevalue of axial velocity for viscoelastic fluids is morethan that for Newtonian fluids.

In Fig. 10, the effects of suction and injectionhave been investigated. As it is shown by contours inFig. 10, velocity contour concentration in Re = 10 oc-curs approximately in η = 0.4 and for Re = −10 isapproximately in η = 0.8. Regarding to this figure, theposition of maximum tangential velocity gets closer tothe porous surface when Reynolds number has a neg-ative value. Vectors of tangential velocity are depictedin Fig. 11. For Re = 10, in η = 0.4 size of vectors are

864 Meccanica (2010) 45: 857–868

Fig. 7 Normal velocity profile when (a) Re = 10, (b) Re = 20

Fig. 8 Tangential velocity profile for (a) Re = 10, (b) Re = 20

in maximum values. This is corresponding to velocitycontours presented in Fig. 10. The same results can beobtained for Re = −10.

Figure 12 depicts the profiles of temperature dif-ference in viscoelastic fluid in a vertical channel withporous wall. In this figure, variation of temperature isshown when S = 0.012 and Reynolds number is keep-

ing fixed in 10 and 20 that the effect of Peclet numberon temperature profile is shown. According to defin-ition of Peclet number, increasing of Peclet is equalto increasing of uniform injection velocity. Due to theconvection effects, increasing of Peclet number leadsto intensify the temperature distribution in the chan-nel.

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Also Figs. 13(a) and 13(b) present the pressurevariations in the x and y directions, respectively whenRe = 10. From the evident, the pressure variationswill decrease with increasing of elastic number. Thisis due to the pressure increment along the x axisof the channel which leads to decreasing of pres-sure difference between origin and farther points alongx axis.

Stream functions of fluid flow for two cases of suc-tion and injection are displayed in Fig. 14. Injection

Fig. 9 Axial velocity profile for Re = 10

has more concentration stream functions than suctionand due to injection streamlines are closer to the originin comparison with suction.

7 Conclusion

In this paper, the steady three-dimensional flow of aWalter’s B fluid in a vertical channel with porous wallhas been studied using Homotopy Analysis Method(HAM) to present an analytical solution for the flowand temperature field. It is supposed that the fluid isinjected into the channel through one side of the chan-nel. The governing equations of problem are reducedto a set of nonlinear ordinary differential equations us-ing similarity transformation and are solved by meansof HAM.

Most important results of this research can be out-lined as below:

– The elastic element of the fluid increases the normalvelocity at any point.

– Tangential velocity increases with an increase in theelastic number S, up to approximately η = 0.4 andthereafter decreases with increasing S.

– With an increase in the value of the cross-flowReynolds number, the point at which maximum ve-

Fig. 10 Tangential velocity contours for (a) Re = 10, (b) Re = −10

866 Meccanica (2010) 45: 857–868

Fig. 11 Tangential velocity vectors for (a) Re = 10, (b) Re = −10

Fig. 12 Temperature variation when (a) Re = 10, (b) Re = 20

locity occurs moves away from the porous plate.Moreover, the elastic elements of the fluid make thispoint closer to the impermeable wall.

– When Re = 10, that the axial velocity for a vis-coelastic fluid is more than that for a Newtonianfluid.

– In Re = 10 increasing of Peclet number is cause of

increment of temperature distribution of the fluid.

When Re = 20, the same result is obtained.

– The elasticity of the fluid decreases the pressure

variations in both directions.

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Fig. 13 Pressure variation when Re = 10 (a) in x direction, (b) in y direction

Fig. 14 Stream function of fluid flow (a) Re = 10, (b) Re = −10

– Streamlines in injection is closer to the origin of the

coordinate than suction.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

References

1. Berman AS (1953) Laminar flow in channels with porouswalls. J Appl Phys 24:1232–1235

2. Cox SM (1991) Two dimensional flow of a viscous fluid ina channel with porous walls. J Fluid Mech 227:1–33

3. Choi JJ, Rusak Z, Tichy JA (1999) Maxwell fluid suc-tion flow in a channel. J Non-Newtonian Fluid Mech85:165–187

868 Meccanica (2010) 45: 857–868

4. Wang CY, Skalak F (1974) Fluid injection through one sideof a long vertical channel. AIChE J 20:603–605

5. Huang CL (1978) Application of quasilinearization tech-nique to the vertical channel flow and heat convection. Int JNon-Linear Mech 13:55–60

6. Ascher U (1980) Solving boundary value problems with aspline-collocation code. J Comput Phys 34:401–413

7. Sharma PR, Chaudhary RC (1982) Fluid injection of aRivlin-Ericksen fluid through one side of a long verticalchannel. Bull Tech Univ Istanbul 35:175–185

8. Baris S (2001) Injection of a non-Newtonian fluidthrough one side of a long vertical channel. Acta Mech151:163–170

9. He JH (2005) Homotopy perturbation method for bifurca-tion on nonlinear problems. Int J Non-Linear Sci NumerSimul 8:207–208

10. Ganji DD, Rajabi A (2006) Assessment of homotopy-perturbation and perturbation methods in heat radiationequations. Int Commun Heat Mass Transf 33:391–400

11. Ganji DD, Rafei M, Sadighi A, Ganji ZZ (2009) A com-parative comparison of He’s method with perturbation andnumerical methods for nonlinear vibrations equations. Int JNonlinear Dyn Eng Sci 1(1):1–20

12. Zhou JK (1986) Differential transformation and Its appli-cations for electrical circuits. Huarjung University Press,Wuuhahn (in Chinese)

13. Joneidi AA, Ganji DD, Babaelahi M (2009) Differen-tial transformation method to determine fin efficiency ofconvective straight fins with temperature dependent ther-mal conductivity. Int Commun Heat and Mass Transf36:757–762

14. Babaelahi M, Ganji DD, Joneidi AA (2009) Analysis of ve-locity equation of steady flow of a viscous Incompressiblefluid in channel with porous walls. Int J Numer MethodsFluids. doi:10.1002/fld.2114

15. Marinca V, Herisanu N (2008) Application of Optimal Ho-motopy Asymptotic Method for solving nonlinear equa-tions arising in heat transfer. Int Commun Heat Mass Transf35:710–715

16. Joneidi AA, Ganji DD, Babaelahi M (2009) Micropolarflow in a porous channel with high mass transfer. Int Com-mun Heat and Mass Transf 36(10):1082–1088

17. Liao SJ (1992) The proposed homotopy analysis techniquefor the solution of nonlinear problems. PhD thesis, Shang-hai Jiao Tong University

18. Liao SJ (2003) Beyond perturbation: introduction to thehomotopy analysis method. Chapman & Hall, CRC Press,Boca Raton

19. Liao SJ (2003) On the analytic solution of magnetohydro-dynamic flows of non-Newtonian fluids over a stretchingsheet. J Fluid Mech 488:189–212

20. Liao SJ (2004) On the homotopy analysis method for non-linear problems. Appl Math Comput 147:499–513

21. Hayat T, Khan M, Ayub M (2004) On the explicit ana-lytic solutions of an Oldroyd 6-constant fluid. Int J Eng Sci42:123–135

22. Hayat T, Khan M, Ayub M (2004) Couett and Poisevill flowof an oldroyd 6-constant fluid with magnetic field. J MathAnal Appl 298:225–244

23. Joneidi AA, Domairry G, Babaelahi M, Mozaffari M(2009) Analytical treatment on Magnetohydrodynamic(MHD) flow and heat transfer due to a stretch-ing hollow cylinder. Int J Numer Methods Fluids.doi:10.1002/fld.2087

24. Joneidi AA, Domairry G, Babaelahi M (2010) Analyticaltreatment of MHD free convective flow and mass transferover a stretching sheet with chemical reaction. J Taiwan InstChem Eng 41(1), 35–43

25. Farzaneh-Gord Mahmood, Joneidi AA, Haghighi B (2010)Investigating the effects of the important parameters onMHD flow and heat transfer over a stretching sheet. ProcInst Mech Ehg, E J Process Mech Eng 224(1):1–9

26. Joneidi AA, Domairry G, Babaelahi M (2010) Threeanalytical method applied to Jeffery–Hamel flow. Com-mun Nonlinear Sci Numer Simul. doi:10.1016/j.cnsns.2009.12.023

27. Mehmood A, Ali A, Shah T (2008) Heat transferanalysis of unsteady boundary layer flow by homotopyanalysis method. Commun Nonlinear Sci Numer Simul13(5):902–912

28. Tan Y, Abbasbandy S (2008) Homotopy analysis methodfor quadratic Riccati differential equation. Commun Non-linear Sci Numer Simul 13(3):539–546

29. Hayat T, Sajid M, Ayub M (2007) A note on series solu-tion for generalized Couette flow. Commun Nonlinear SciNumer Simul 12(8):1481–1497

30. Abbasbandy S (2007) The application of homotopy analy-sis method to solve a generalized Hirota–Satsuma coupledKdV equation. Phys Lett A 361(6):478–483

31. Abbasbandy S (2008) Soliton solutions for the Fitzhugh–Nagumo equation with the homotopy analysis method.Appl Math Model 32(12):2706–2714