Radiation effects on MHD flow in a porous space

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International Journal of Heat and Mass Transfer 51 (2008) 1024–1033

Radiation effects on MHD flow in a porous space

Zaheer Abbas *, Tasawar Hayat

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

Received 8 February 2007; received in revised form 25 May 2007Available online 12 September 2007

Abstract

This paper deals with the study of the radiation effects on the magnetohydrodynamic (MHD) flow of an incompressible viscous fluidin a porous space. The flow is induced due to a non-linear stretching sheet. Two cases of heat transfer analysis are discussed. These are: (i)the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case). By means ofsimilarity transformation, the governing partial differential equations are reduced into highly non-linear ordinary differential equations.The resulting non-linear system has been solved analytically using a very efficient technique namely homotopy analysis method (HAM).Expressions for velocity and temperature fields are developed in series form. Convergence of the series solution is shown explicitly. Theinfluence of various pertinent parameters is also seen on the velocity and temperature fields. The tabulated values of the wall shear stressand the Nusselt number show good agreement with the existing results.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Heat transfer; Non-linear stretching; Porous space; HAM solution; Electrically conducting fluid

1. Introduction

The flow of a fluid past a stretching surface is encoun-tered in many technical and industrial applications. Someof these applications include aerodynamic extrusion ofplastic sheets, the boundary layer along a material handlingconveyers, cooling of an infinite metallic plate in a coolingbath, the boundary layer along a liquid film in a condensa-tion process and heat treated material that travel betweenfeed and wind-up rolls. After the pioneering work of Saki-adis [1], extensive literature is available on this topic for alinear stretching sheet. Some very recent contributions inthis direction are made by Sadeghy et al. [2], Ariel et al.[3], Liao [4], Xu [5] and Cortell [6–8].

In all the above mentioned studies, linear stretching hasbeen taken into account. The literature on the non-linearstretching is fairly scarce. Most recently, Cortell [9] dis-cussed the viscous flow and heat transfer over a non-line-arly stretching sheet.

0017-9310/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2007.05.031

* Corresponding author. Tel.: +92 51 2275341.E-mail address: [email protected] (Z. Abbas).

The objective of the present paper is to extend the anal-ysis of Ref. [9] in four directions (i) to consider a MHDflow (ii) to analyze the flow in a porous medium (iii) toinclude the radiation effects and (iv) to provide analyticsolution to highly non-linear problem. A new developedpowerful technique namely HAM [10,11] has beenemployed for the analytic solution. This technique hasalready been successfully applied to various problems[12–32]. Very recently, Allan [33] showed that the Adomiandecomposition method is a special case of the HAM.Finally, the graphs of velocity and temperature fields aresketched and variations of sundry parameters arediscussed.

2. Formulation of the problem

Let us consider the steady two-dimensional, incompress-ible flow of a viscous fluid with heat transfer past a flat sur-face coinciding with the plane y = 0. The wall is stretchedhorizontally by pulling on both sides with equal and oppo-site forces parallel to the wall keeping the origin fixed. Thefluid is electrically conducting in the presence of a constant

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Z. Abbas, T. Hayat / International Journal of Heat and Mass Transfer 51 (2008) 1024–1033 1025

applied magnetic field B0 in the y-direction. The inducedmagnetic field is neglected using small magnetic Reynoldsnumber assumption [34,35]. The continuity, momentumand energy equations under boundary layer approxima-tions become

ouoxþ ov

oy¼ 0; ð1Þ

uouoxþ m

ouoy¼ m

o2uoy2� rB2

0

qu� m/

ku; ð2Þ

qcp uoToxþ v

oToy

� �¼ k1

o2T

oy2� oqr

oyþ l

ouoy

� �2

; ð3Þ

in which u and v are the velocity components in the x- andy-directions, respectively, q is the fluid density, m is thekinematic viscosity, r is the electrical conductivity, / isthe porosity, k is the permeability of the porous medium,T is the temperature, cp is the specific heat, k1 is the thermalconductivity of the fluid and qr is the radiative heat flux.

Using the Rosseland approximation for radiation for anoptically thick layer [36] one can obtain

qr ¼ �4r�

3k�oT 4

oy; ð4Þ

where r* is the Stefan–Boltzmann constant and k* is themean absorption coefficient. We express the term T4 as alinear function of temperature in a Taylor series aboutT1 and neglecting higher terms, therefore we have

T 4 ffi 4T 31T � 3T 4

1: ð5ÞFrom Eqs. (3)–(5) we can write

qcp uoToxþ v

oToy

� �¼ o

oy16r�

3k�þ k1

� �oToy

� �þ l

ouoy

� �2

: ð6Þ

The appropriate boundary conditions are

uwðxÞ ¼ Cxn; v ¼ 0 at y ¼ 0; ð7Þu! 0 as y !1; ð8Þ

where C and n are parameters related to the surface stretch-ing speed.

For temperature we have the following two sets ofboundary conditions.

Case a: Constant surface temperature (CST)

T ¼ T w at y ¼ 0;

T ! T1 as y !1:ð9Þ

Case b: Prescribed surface temperature (PST)

T ¼ T w ¼ T1 þ AxK� �

at y ¼ 0;

T ! T1 as y !1;ð10Þ

where K is the surface temperature parameter.

Defining the following non-dimensionlized quantities:

g ¼ y

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCðnþ 1Þ

2m

r; u ¼ Cxnf 0ðgÞ;

v ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCmðnþ 1Þ

2

rx

n�12 f ðgÞ þ n� 1

nþ 1gf 0ðgÞ

� �;

hðgÞ ¼ T � T1T w � T1

;

ð11Þ

continuity Eq. (1) is satisfied automatically and Eqs. (2)and (7) become

f 000 þ ff 00 � 2nnþ 1

f 02 �M2f 0 � kf 0 ¼ 0; ð12Þ

f ð0Þ ¼ 0; f 0ð0Þ ¼ 1; f 0ð1Þ ¼ 0: ð13Þ

The shear stress at the stretched surface is

sw ¼ louoy

� �w

; ð14Þ

which in non-dimensional form becomes

sw ¼ Cl


2m

rx

3n�12 f 00ð0Þ: ð15Þ

For temperature with the help of Eqs. (6) and (9)–(11)we get:

Case a: Constant surface temperature (CST)

1þ 4

3Rd

� �h00 þ Prf h0 þ PrEcf 002 ¼ 0; ð16Þ

hð0Þ ¼ 1; hð1Þ ¼ 0 ð17Þ

and the rate of heat transfer of the surface is

�kTdTdy

� �y¼0

¼ �kT T w � T1ð Þ 1þ 4

3Rd

� �h0ð0Þxn�1

2

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCðnþ 1Þ

2m

r:

ð18ÞCase b: Prescribed surface temperature (PST)

1þ 4

3Rd

� �h00 þ Prf h0 � 2K

nþ 1

� �Prf 0h

þ PrEc0x2n�Kf 002 ¼ 0: ð19Þ

The above equation for K = 2n reduces to

1þ 4

3Rd

� �h00 þ Prf h0 � 4n

nþ 1

� �Prf 0hþ PrEc0f 002 ¼ 0;

ð20Þhð0Þ ¼ 1; hð1Þ ¼ 0: ð21Þ

and the local surface heat flux is

qw ¼ �kTdTdy

� �w

¼ �kT Ax2Kþn�1

2 1þ 4

3Rd

� �h0ð0Þ


2m

r:

ð22Þ

1026 Z. Abbas, T. Hayat / International Journal of Heat and Mass Transfer 51 (2008) 1024–1033

Here the local Hartmen number M, the local porosityparameter k, the Prandtl number Pr, the Eckert numberEc, the radiation parameter Rd and Ec0 are respectively

M2 ¼ 2xn�1rB20

Cqðnþ 1Þ ; k ¼ 2xn�1m/Ckðnþ 1Þ ; Pr ¼ lcp

k1

;

Ec ¼ u2w

cpðT w � T1Þ; Rd ¼

4r�T 31

k�k1

; Ec0 ¼ C2

Acp:

In the next two sections, we will solve the non-linear sys-tems consisting of Eqs. (12), (13), (16), (17), (20) and (21)by HAM.

3. Analytic solution for velocity f(g)

For HAM solution of Eqs. (12) and (13), it is straight-forward to choose the initial approximation

f0ðgÞ ¼ 1� expð�gÞ ð23Þand the auxiliary linear operator

L1ðf Þ ¼ f 000 � f 0; ð24Þwith the following property

L1½C1 þ C2eg þ C3e�g� ¼ 0; ð25Þin which Ci, i = 1 � 3 are arbitrary constants. If p (2 [01])and �h1 indicate the embedding and non-zero auxiliaryparameters, respectively then we have:

The zeroth-order deformation problem

ð1� pÞL1bf ðg; pÞ � f0ðgÞh i

¼ p�h1N1bf ðg; pÞh i

; ð26Þ

bf ð0; pÞ ¼ 0; bf 0 ð0; pÞ ¼ 1; bf 0 ð1; pÞ ¼ 0: ð27ÞThe mth-order deformation problem

L1½fmðgÞ � vmfm�1ðgÞ� ¼ �h1R1mðgÞ; ð28Þfmð0Þ ¼ f 0mð0Þ ¼ f 0mð1Þ ¼ 0; ð29Þ

where

N1bf ðg;pÞh i

¼ o3bf ðg;pÞog3

þ bf ðg;pÞo2bf ðg;pÞog2

� 2nnþ1

obf ðg;pÞog

!2

�M2 obf ðg;pÞog

�kobf ðg;pÞ

og;

ð30Þ

R1mðgÞ ¼ f 000m�1ðgÞ �M2f 0m�1 � kf 0m�1

þXm�1

k¼0

fm�1�kf 00k� 2n

nþ 1f 0m�1�kf 0k

� �; ð31Þ

vm ¼0; m 6 1;

1; m > 1:

ð32Þ

Now using Mathematics one can solve Eqs. (28) and(29) up to first few order of approximations. The formsof Eqs. (28) and (29) give us the following type of solutionof fm:

fmðgÞ ¼Xmþ1

n¼0

Xmþ1�n

q¼0

aqm;ng

qe�ng; m P 0; ð33Þ

where the recurrence formulas for the coefficients akm;n of

fm(g) are obtained for m P 1, 0 6 n 6 m + 1 and 0 6 k 6

m + l � n as

a0m;0¼ vmvmþ2a0

m�1;0�Xm

q¼0

Dqm;1l

q1;1

�Xmþ1

n¼2

Xmþ1�n

q¼1

Dqm;nfl

qn;1�ðn�1Þlq

n;0g;

akm;0¼ vmvmþ2�kak

m�1;0; 06 k6mþ1;

a0m;1¼ vmvmþ1a0

m�1;1þXm

q¼0

Dqm;1l

q1;1

�Xmþ1

n¼2

nD0m;nl

0n;0þ

Xmþ1�n

q¼1

Dqm;n nlq

n;0�lqn;1

�" #;

akm;1¼ vmvmþ1�kak

m�1;1�Xm

q¼k�1

Dqm;1l

q1;k ;16 k6 2mþ1;

akm;n¼ vmvmþ2�n�kak

m�1;n�Xmþ1�n

q¼k

Dqm;nl

qn;k ;

26 n6mþ1; 06 k6mþ1�n;

lq1;k ¼

Xqþ1�k

p¼0

q!

k!2qþ1�k�p ; q P 0; 16 k6 qþ1; ð34Þ

lqn;k ¼

Xq�k

r¼0

Xq�k�r

p¼0

q!

k!ðn�1Þqþ1�k�r�pnrþ1ðnþ1Þpþ1; q P 0; 16 k6 q; n P 2;

ð35Þ

Dqm;n¼ �h1 vm�n�qþ2 eq

m�1;n�ðM2þkÞcqm�1;n

�hþ aq

m;n�2n

nþ1bq

m;n

� ��:

ð36Þ

The coefficients aqm;n and bq

m;n, where m P 1, 0 6 n 6

m + 1, 0 6 q 6 m + 1 � n are

aqm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg

Xminfq;kþ1�jg

i¼maxf0;q�mþkþn�jgdi

k;jaq�im�1�k;n�j;

bqm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg

Xminfq;kþ1�jg

i¼maxf0;q�mþkþn�jgci

k;jcq�im�1�k;n�j;

ckm;n ¼ ðk þ 1Þakþ1

m;n � nakm;n;

dkm;n ¼ ðk þ 1Þckþ1

m;n � nckm;n;

ekm;n ¼ ðk þ 1Þdkþ1

m;n � ndkm;n:

For the detailed procedure of deriving the above rela-tions the reader is referred to [11]. Using the above recur-rence formulas, we can calculate all coefficients ak

m;n usingonly the first few

a00;0 ¼ 1; a1

0;0 ¼ 0; a00;1 ¼ �1; ð37Þ

given by the initial guess approximation in Eq. (23).


The explicit, totally analytic solution is

f ðgÞ ¼X1m¼0

fmðgÞ

¼ limM!1

XM

m¼0

a0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

Xmþ1�n

k¼0

akm;ng

k

!" #:

ð38Þ

4. Analytic solutions for temperature h(g)

Case a: Constant surface temperature (CST)To seek the analytic solution of Eqs. (16) and(17) using HAM, we take the initial guessapproximation of h(g) and auxiliary linear oper-ator as

h0ðgÞ ¼ expð�gÞ; ð39ÞL2ðf Þ ¼ f 00 � f ; ð40Þ

with

L2½C4eg þ C5e�g� ¼ 0; ð41Þ

in which C4 and C5 are the arbitrary constants. If�h2 indicates the non-zero auxiliary parameterthen we get:
The zeroth-order deformation problem
;
ð1� pÞL2bhðg; pÞ � h0ðgÞh i
¼ p�h2N2bhðg; pÞ; bf ðg; pÞh i

ð42Þ

bhð0; pÞ ¼ 1; hð1; pÞ ¼ 0; ð43Þ

in which the non-linear operator N2 is

N2bhðg; pÞ; bf ðg; pÞh i

¼ 1þ 4

3Rd

� �o

2bhðg; pÞog2

þ Prbf ðg; pÞ obhðg; pÞog

þ PrEco

2bf ðg; pÞog2

!2

;

ð44Þ

andThe mth-order deformation problem

L2 hmðgÞ � vmhm�1ðgÞ½ � ¼ �h2R2mðgÞ; ð45Þhmð0Þ ¼ hmð1Þ ¼ 0; ð46Þ

R2mðgÞ ¼ 1þ 4

3Rd

� �h00m�1ðgÞ þ Pr

Xm�1

k¼0

h0m�1�kfk þ Ecf 00m�1�kf 00k�

: ð47Þ

Using the software Mathematica one can solve Eqs.(45) and (46) up to first few order of approxima-tions. We get the following type of solution of hm:

hmðgÞ ¼Xmþ1

n¼0

Xmþ1�n

q¼0

bqm;ng

qe�ng; m P 0; ð48Þ

where the recurrence formulas for the coefficientsbk

m;n of hm(g) are obtained for m P 1,0 6 n 6 m + 1 and 0 6 k 6 m + 1 � n as

bkm;0 ¼ vmvmþ2�kbk

m�1;0 þXmþ1

q¼0

Cqm;0l1q

1;0;

b0m;1 ¼ vmvmþ1b0

m�1;1 �Xmþ1

q¼0

Cqm;0l1q

0;0

�Xmþ1

n¼2

Xmþ1�n

q¼0

Cqm;nl1q

n;0;

bkm;1 ¼ vmvmþ1�kbk

m�1;1 þXm

q¼k�1

Cqm;1l1q

1;k ; 1 6 k 6 mþ 1;

bkm;n ¼ vmvmþ2�n�kbk

m�1;n þXmþ1�n

q¼k

Cqm;nl1q

n;k ; 2 6 n 6 mþ 1;

0 6 k 6 mþ 1� n;

l1q1;k ¼

ð�1Þ2qþ1�2k�pq!

2qþ2�kk!; q P 0; 0 6 k 6 qþ 1; ð49Þ

l1qn;k ¼

Xqþ1�k

p¼0

q!

k!ðnþ 1Þpþ1ðn� 1Þqþ1�k�p ;

q P 0; 1 6 k 6 q; n P 2; ð50Þ

Cqm;n ¼ �h2 vm�n�qþ1 1þ 4

3Rd

� �gq

m�1;n þ Pr cqm;n þ Ecdq

m;n

�� :

ð51Þ

The coefficients cqm;n and dq

m;n, where m P 1,0 6 n 6 m + 1, 0 6 q 6 m + 1 � n are

cqm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkgXminfq;kþ1�jg

i¼maxf0;q�mþkþn�jgai

k;jfq�im�1�k;n�j;

dqm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkgXminfq;kþ1�jg

i¼maxf0;q�mþkþn�jgdi

k;jdq�im�1�k;n�j;

f km;n ¼ ðk þ 1Þbkþ1

m;n � nbkm;n;

gkm;n ¼ ðk þ 1Þf kþ1

m;n � nf km;n;

Employing the above recurrence formulas, we cancalculate all coefficients bk

m;n using only

b00;0 ¼ b1

0;0 ¼ 0; b00;1 ¼ 1; ð52Þ

given by the initial guess approximations in Eq.(39) and thus the temperature field is


hðgÞ¼X1m¼0

hmðnÞ

¼ limM!1

XM

m¼0

b0m;0þ

XMþ1

n¼1

e�ngXM

m¼n�1

Xmþ1�n

k¼0

bkm;ng

k

!" #:

ð53Þ

Case b: Prescribed surface temperature (PST)

Employing the same methodology as in case (a), thesolution here is

hðgÞ ¼X1m¼0

hmðnÞ

¼ limM!1

XM

m¼0

b0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

Xmþ1�n

k¼0

bkm;ng

k

!" #;

ð54Þ

where

N2bhðg; pÞ; bf ðg; pÞh i¼ 1þ 4

3Rd

� �o2bhðg; pÞ

og2þ Prbf ðg; pÞ o

bhðg; pÞog

� 2Knþ 1

� �Prbhðg; pÞ o

bf ðg; pÞog

þ PrEc0o2bf ðg; pÞ

og2

!2

;

ð55Þ

R3mðgÞ¼ 1þ4

3Rd

� �h000m�1ðgÞ

þPrXm�1

k¼0

h0m�1�kfk�2K

nþ1

� �hm�1�kf 0kþEc0f 000m�1�kf 0k

� �ð56Þ

Pqm;n ¼ �h2 vm�n�qþ2 1þ 4

3Rd

� �gq

m�1;n

�þPr cq

m;n �2K

nþ 1

� �xq

m;n þ Ec0dqm;n

� ��; ð57Þ

Fig. 1. �h-curves for 25th-o

xqm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg

Xminfq;kþ1�jg

i¼maxf0;q�mþkþn�jgci

k;jbq�im�1�k;n�j:

Putting Pqm;n instead of Cq

m;n in case (a), we can get the

formulas for the coefficients bkm;n for case (b).

5. Convergence of the HAM solution

As pointed out by Liao [10], the convergence and rate ofapproximation for the HAM solutions, i.e., the series (38),(53) and (54) are strongly dependent upon �h1 and �h2. Inorder to find the admissible values of �h1 and �h2, �h-curvesare plotted for 25th-order of approximations. It is obviousfrom Fig. 1 that the range for the admissible values of �h1

and �h2 are �1.05 6 �h1 6 �0.1 and �1.5 6 �h2 6 �0.2. Ourcomputations show that the series (38), (53) and (54) con-verge in the whole region of g when �h1 = �0.6 and�h2 = �0.8.

6. Results and discussion

This section describes the influence of some interestingparameters on the velocity and temperature fields. In par-ticular, attention has been focused to the variations of n,the Hartman number M, porosity parameter k, Prandtlnumber Pr, Eckert number Ec, Ec0, and the radiationparameter Rd on the velocity and temperature fields,respectively. Moreover, the values of the wall shear stress�f

00(0) and the Nusselt number �h

0(0) are computed in

the Tables 1–4.In order to see the effects of n, the Hartman number M

and the porosity parameter k on the velocity componentf0, we prepared Figs. 2–4. Fig. 2 depicts the effects of n onf0. It shows that the velocity is a decreasing function of n.Figs. 3 and 4 describe the effects of M and k on f0, respec-tively. It is noted that velocity f0 is an increasing functionof M and k. But this increment, is larger in a porousmedium.

rder approximations.

Table 1Values of the wall shear stress �f00(0) when M = k = 0

n Cortell [9] HAM solution

0.0 0.627547 0.6275470.2 0.766758 0.7668370.5 0.889477 0.8895440.75 0.953786 0.9539561.0 1.0 1.01.5 1.061587 1.0616013.0 1.148588 1.1485937.0 1.216847 1.216851

10.0 1.234875 1.23487420.0 1.257418 1.257423

100.0 1.276768 1.276773

Table 2Values of the wall shear stress �f00(0) in the presence of M and k

n M k �f00(0) n M k �f00(0)

0.0 0.5 0.5 1.052407 1.5 0.0 0.5 1.2764540.2 1.148901 0.2 1.2920710.5 1.238663 0.5 1.3711130.75 1.287402 0.7 1.4561541.0 1.322875 1.0 1.6220411.5 1.371113 2.0 2.3732473.0 1.440650 0.5 0.0 1.1740687.0 1.496283 0.2 1.256655

10.0 1.511120 0.5 1.37111320.0 1.529766 0.7 1.442332

100.0 1.545838 1.0 1.542966

Table 3Heat transfer characteristics at the wall �h0(0) for CST case when M = k = R

Cortell [9]

Ec n Pr = 1

0.0 0.2 0.6102620.5 0.5952771.5 0.5745373.0 0.564472

10.0 0.554960

0.1 0.2 0.5749850.5 0.5566231.5 0.5309663.0 0.517977

10.0 0.505121

Table 4Heat transfer characteristics at the wall �h0(0) for PST case when K = 2n at M

Cortell [9]

Ec0 n Pr = 1

0.0 0.75 1.2526721.5 1.4393937.0 1.699298

10.0 1.728934

0.1 0.75 1.2199851.5 1.4050787.0 1.662506

10.0 1.691822


Figs. 5–10 are made for the effects of n, M, k, Pr, Ec andRd on the temperature field h in CST (Constant surfacetemperature) case. Fig. 5 shows that h increases for largevalues of n. Figs. 6 and 7 illustrate the effects of M and kon h. The temperature profile h increases as M and kincrease, respectively. But this increment in case of M islarger when compared with k. Fig. 8 elucidates the varia-tion of Pr on temperature h. It is observed that h decreasesas Pr increases. Fig. 9 gives the behavior of Ec on h. It isnoted that h has opposite results when compared withFig. 8. Fig. 10 shows the effects of radiation parameterRd on h. It is obvious from Fig. 10 that the temperatureh is an increasing function of Rd.

Figs. 11–16 are plotted for the effects of n, M, k, Pr, Ec0

and Rd, on the temperature field h in PST (Prescribed sur-face temperature) case. Fig. 11 depicts the effects of n on h.It has the similar results when compared with Fig. 5 in(CST) case. Figs. 12 and 13 show the influence of M andk on h, respectively. It is found that these Figs. have thesimilar results as in Figs. 6 and 7 for (CST) case. Fig. 14indicates that h is a decreasing function of Pr. But this dec-rement is larger when compared with Fig. 8. Fig. 15 showsthat h increases as Ec0 increases. Fig. 16 elucidates theeffects of Rd on h. It is noted that h is increased when Rd

increases. The change in Fig. 16 is larger when comparedwith Fig. 10.

d = 0

HAM solution Cortell [9] HAM solution

Pr = 1 Pr = 5 Pr = 5

0.610217 1.607175 1.6079250.595201 1.586744 1.5868330.574729 1.557463 1.5576720.564661 1.542337 1.5421450.554878 1.528573 1.528857

0.574955 1.474764 1.4742030.556775 1.436789 1.4372420.530962 1.381861 1.3820030.518043 1.352768 1.3525480.505127 1.324772 1.324943

= k = Rd = 0

HAM solution Cortell [9] HAM solution

Pr = 1 Pr = 5 Pr = 5

1.252700 3.124975 3.1243471.439375 3.567737 3.5679441.699318 4.185373 4.1853781.728952 4.255972 4.255935

1.219940 3.016983 3.0169341.405184 3.455721 3.4558751.662599 4.065722 4.0657911.691812 4.135296 4.135299

Fig. 2. Effects of n on f0 at �h1 = �0.6.

Fig. 3. Effects of M on f0 at �h1 = �0.6.

Fig. 4. Effects of k on f0 at �h1 = �0.6.

Fig. 5. Effects of n on h at �h2 = �0.8 for CST case.

Fig. 6. Effects of M on h at �h2 = �0.8 for CST case.

Fig. 7. Effects of k on h at �h2 = �0.8 for CST case.


Tables 1–4 have been made in order to show the varia-tions of wall shear stress �f00(0) and the heat transfer char-acteristics at the wall �h0(0) for different values of involvingparameters. Table 1 gives the variations of n on the wallshear stress �f00(0) when (M = k = 0). The magnitude of

shear stress increases as n increases and the HAM solutionhas good agreement with the numerical solution [9]. Table2 shows the values of �f00(0) for different values of n, M andk. The magnitude of the shear stress is increased for largevalues of n, M and k. Tables 3 and 4 illustrate the heat

Fig. 8. Effects of Pr on h at �h2 = �0.8 for CST case.

Fig. 9. Effects of Ec on h at �h2 = �0.8 for CST case.

Fig. 10. Effects of Rd on h at �h2 = �0.8 for CST case.

Fig. 11. Effects of n on h at �h2 = �0.8 for PST case.

Fig. 12. Effects of M on h at �h2 = �0.8 for PST case.

Fig. 13. Effects of k on h at �h2 = �0.8 for PST case.


transfer characteristics at the wall �h0(0) when (M = k =Rd = 0) in CST and PST cases, respectively. Table 3 showsthat the magnitude of �h0(0) decreases for large values of nand increases as Pr increases for (Ec = 0 and Ec 6¼ 0).The magnitude of �h0(0) increases for large values of n

and Pr for (Ec0 = 0 and Ec0 6¼ 0) in Table 4. From thesetables one can see that the HAM solution has good agree-ment with the numerical solution [9] for both cases of CSTand PST.

Fig. 15. Effects of Ed on h at �h2 = �0.8 for PST case.

Fig. 14. Effects of Pr on h at �h2 = �0.8 for PST case.

Fig. 16. Effects of Rd on h at �h2 = �0.8 for PST case.


7. Conclusions

In this analysis the effects of radiation on MHD flow ofa viscous fluid with heat transfer is investigated. Seriessolutions for velocity and temperature fields are first devel-oped and then discussed for various emerging parameters.The values of wall shear stress and heat transfer at the wallare also tabulated. The following observations have beenmade from the present analysis.

� The velocity f0 and temperature h increase for large val-ues of n, M, k, Ec, Ec0 and Rd.� The temperature h decreases as Pr increases.� The magnitude of the wall shear stress �f00(0) increases

as n, M and k increases.� The magnitude of �h0(0) increases for large values of n

and Pr and decreases for large vales of n in GST case.� Tables 1, 3 and 4 show that HAM solution has good

agreement with the numerical solution [9].

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Radiation effects on MHD flow in a porous space

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