The influence of thermal radiation on MHD flow of a second grade fluid

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International Journal of Heat and Mass Transfer 50 (2007) 931–941

The influence of thermal radiation on MHD flowof a second grade fluid

T. Hayat a,*, Z. Abbas a, M. Sajid a,b, S. Asghar c

a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistanb Physics Research Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistanc COMSATS Institute of Information Technology, H-8, Islamabad 44000, Pakistan

Received 8 July 2006; received in revised form 22 July 2006Available online 17 October 2006

Abstract

The present analysis deals with the steady magnetohydrodynamic (MHD) flow of a second grade fluid in the presence of radiation. Bymeans of similarity transformation, the arising non-linear partial differential equations are reduced to a system of four coupled ordinarydifferential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopyanalysis method (HAM). Convergence of the obtained series solution is discussed. The effects of various involved interesting parameterson the velocity and temperature fields are shown and discussed.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Boundary layer flow; Heat transfer analysis; Analytical solution

1. Introduction

Due to their application in industry and technology fewproblems in fluid mechanics have enjoyed the attentionthat has been accorded to the flow which involves non-Newtonian fluids. It is well known that mechanics ofnon-Newtonian fluids present a special challenge to engi-neers, physicists and mathematicians. The non-linearitycan manifest itself in a variety of ways in many fields, suchas food, drilling operations and bio-engineering. TheNavier–Stokes theory is inadequate for such fluids andno single constitutive equation is available in the literaturewhich exhibits the properties of all fluids. Because of com-plex behavior many fluid models have been suggested.Amongst these, the fluids of viscoelastic type have receivedmuch attention. In fact interest in viscoelastic fluids goesback almost to 65 years, triggered by the discovery ofMysels [1] and Toms [2] who found that the addition of

0017-9310/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2006.08.014

* Corresponding author. Tel.: +92512275341.E-mail addresses: [email protected], [email protected]

(T. Hayat).

small amounts of a high molecular weight polymer to aNewtonian fluid in turbulent pipe flows resulted in a dra-matic decrease in pressure drop. The second grade fluidmodel is the simplest subclass of viscoelastic fluids forwhich one can reasonably hope to obtain the analytic solu-tion. Some typical works on the topic are given in the ref-erences [3–12]. Even though considerable progress has beenmade in our understanding of the flow phenomena, moreworks are needed to understand the effects of the variousparameters involved in the non-Newtonian models andthe formulation of an accurate method of analysis forany body shapes of engineering significance. Also, theboundary layer concept for such fluids is of special impor-tance because of its application to many practical prob-lems, among which we cite the possibility of reducingfrictional drag on the hulls of ships and submarines. Fur-ther, thermal radiation effects and MHD flow problemshave assumed an increasing importance at a fundamentalfabrication level. Specifically, such flows occurs in electricalpower generation, astrophysical flows, solar power technol-ogy, space vehicle re-entry and other industrial areas[13,14]. Related studies regarding the thermal radiation of

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932 T. Hayat et al. / International Journal of Heat and Mass Transfer 50 (2007) 931–941

a gray fluid have been made in the references [15–20]. Morerecently, Raptis et al. [21] discussed the thermal radiationeffects on the MHD flow of a viscous fluid.

The purpose of the present study is to examine the influ-ence of thermal radiation on the MHD flow of a secondgrade fluid. The homotopy analysis method proposed byLiao [22,23] has been used for the analytic solution.HAM is recently developed powerful technique and hasbeen successfully applied to several non-linear problems[25–44]. The organization of the paper is as follows:

In Section 2 the problem of MHD second grade fluidwith radiation effects is formulated. Sections 3 and 4 com-prise the series solutions for the flow and heat transferanalysis, respectively. The convergence of the solution isdiscussed in Section 5. The graphical results are presentedand discussed in Section 6. Section 7 contains the conclud-ing remarks.

2. Problem statement

Let us consider MHD flow of an incompressible secondgrade fluid past a semi-infinite fixed plate. We choosex-axis parallel and y-axis normal to the plate. A transversemagnetic field of strength B0 is imposed. MHD equationsare the usual electromagnetic and hydrodynamic equa-tions, but modified to take account of the interactionbetween the motion and magnetic field. For small magneticReynolds number, the induced magnetic field is neglected.Moreover, the radiative heat flux in the x-direction is neg-ligible when compared with the y-direction. For the presentproblem the conservation of mass and momentum equa-tions can be expressed as [24]

ouoxþ ov

oy¼ 0; ð1Þ

uouoxþ v

ouoy¼ m

o2uoy2þ U

dUdxþ rB2

0

qðU � uÞ

þ a1

qu

o3u

oxoy2þ ou

oxo

2uoy2þ ou

oyo

2voy2þ v

o3u

oy3

� �:

ð2Þ

In the above equations V = (u,v), U is the free streamvelocity, a1 is the material constant of second grade fluid,q and m are the respective density and kinematic viscosityof fluid and r is the electrical conductivity.

The energy equation is

uoToxþ v

oToy¼ k

qcp

o2Toy2þ l

qcp

ouoy

� �2

þ a1

qcp

ouoy

� �o

oyuouoxþ v

ouoy

� �� 1

qcp

oqr

oy;

ð3Þ

in which T is the fluid temperature, k is the thermal con-ductivity, cp is the specific heat of the fluid under constantpressure and qr is the radiative heat flux.

The appropriate boundary conditions are

u ¼ 0; v ¼ 0; T ¼ T w at y ¼ 0;

u! UðxÞ; T ! T1 as y !1:ð4Þ

In above conditions Tw is the temperature at the plate, T1is the temperature of the fluid far away from the plate andthe free stream velocity is

UðxÞ ¼ axþ cx2; ð5Þ

in which a and c are constants.Making use of the Rosseland approximation for radia-

tion for an optically thick layer [15] one obtains

qr ¼ �4r�

3k�oT 4

oy; ð6Þ

where k* is the mean absorption coefficient and r* is theStefan–Boltzmann constant. We express the term T4 as alinear function of temperature. It is recognized by expand-ing T4 in a Taylor series about T1 and neglecting higherterms, thus

T 4 ffi 4T 31T � 3T 4

1: ð7Þ

With the help of Eqs. (3), (6) and (7) we can write

uoToxþ v

oToy¼ k

qcp

o2T

oy2þ l

qcp

ouoy

� �2

þ a1

qcp

ouoy

� �o

oyuouoxþ v

ouoy

� �þ 16r�T 3

13k�qcp

o2Toy2

:

ð8Þ

Employing the following transformations

g ¼ffiffiffiam

ry; u ¼ axf 0ðgÞ þ cx2g0ðgÞ;

v ¼ �ffiffiffiffiffiamp

f ðgÞ � 2cxffiffiffiffiffiffiffia=m

p gðgÞ;

T ¼ T w þ ðT1 � T wÞ T 0ðgÞ þ2ca

xT 1ðgÞ� �

:

ð9Þ

Eq. (1) is automatically satisfied and Eqs. (2), (4) and (8)yield

f 000 �M2f 0 � f 02 þ ff 00 þ a �f 002 þ 2f 0f 000 � ff0000

� �þ ðM2 þ 1Þ ¼ 0; ð10Þ

g000 �M2g0 � 3g0f 0 þ 2gf 00 þ g00f þ ðM2 þ 3Þþ að3g000f 0 � 3g00f 00 þ 3g0f 000 � 2gf

0000 � fg0000 Þ ¼ 0; ð11Þ

ð3K þ 4ÞT 000 þ 3KPfT 00 ¼ 0; ð12Þð3K þ 4ÞT 001 þ 3KP �T 1f 0 þ T 00g þ T 01f

¼ 0; ð13Þ

f ð0Þ ¼ 0; f 0ð0Þ ¼ 0; f 0ð1Þ ¼ 1;

gð0Þ ¼ 0; g0ð0Þ ¼ 0; g0ð1Þ ¼ 1;

T 0ð0Þ ¼ 0; T 0ð1Þ ¼ 1;

T 1ð0Þ ¼ 0; T 1ð1Þ ¼ 0:

ð14Þ

T. Hayat et al. / International Journal of Heat and Mass Transfer 50 (2007) 931–941 933

In the above, the Prandtl number Pr, the radiation numberK, the Hartman number M and the dimensionless materialparameter a are defined, respectively, as:

Pr ¼ mqcp

k; K ¼ k�k

4r�T 31; M2 ¼ rB2

0

aq; a ¼ aa1

l:

Note that the energy equations (12) and (13) in the presentproblem becomes similar to that of a viscous fluid case. It isfurther interesting to note that for M ¼ 1=

ffiffiffiap

Eq. (10) hasthe following exact solution

f ¼ gþffiffiffiap

e�g=ffiffiap� 1

� �:

3. HAM solution for f(g) and g(g)

In this section we employ the homotopy analysismethod to solve Eqs. (10)–(14). For that we select

f0 gð Þ ¼ g� 1þ e�g; ð15Þg0 gð Þ ¼ g� 1þ e�g; ð16Þ

as the initial guess approximations for f(g) and g(g), respec-tively and

L1 fð Þ ¼ f 000 þ f 00; ð17Þ

as the auxiliary linear operator which has the property

L1 C1gþ C2 þ C3e�g½ � ¼ 0; ð18Þ

where C1, C2 and C3 are arbitrary constants.We construct the zeroth order deformation problems as

1� pð ÞL1 f g; pð Þ � f0 gð Þh i

¼ p�h1N1 f g; pð Þh i

; ð19Þ

1� pð ÞL1 g g; pð Þ � g0 gð Þ½ � ¼ p�h2N2 f g; pð Þ; g g; pð Þh i

; ð20Þ

f 0; pð Þ ¼ 0; f 0 0; pð Þ ¼ 0; f 0 1; pð Þ ¼ 1; ð21Þg 0; pð Þ ¼ 0; g0 0; pð Þ ¼ 0; g0 1; pð Þ ¼ 1; ð22Þ

where �h1 and �h2 are non-zero auxiliary parameters,p 2 [0, 1] is the embedding parameter and the non-lineardifferential operators N1 and N2 are defined by

N1½f g; pð Þ� ¼ o3f g; pð Þog3

�M2 of g; pð Þog

þ ð1� vmÞðM2 þ 1Þ

� of g; pð Þog

!2

þ f g; pð Þ o2f g; pð Þog2

þ a � o2f g; pð Þog2

!2

þ 2of g; pð Þ

ogo

3f g; pð Þog3

(

�f g; pð Þ o4f g; pð Þog4

); ð23Þ

N2½f g; pð Þ; g g; pð Þ�

¼ o3g g; pð Þog3

�M2 og g; pð Þog

þ ð1� vmÞðM2 þ 3Þ

� 3ogðg; pÞ

ogof ðg; pÞ

ogþ 2gðg; pÞ o

2f ðg; pÞog2

þ f ðg; pÞ o2gðg; pÞog2

þ a 3o3gðg; pÞ

og3

of ðg; pÞog

� 3o2gðg; pÞ

og2

o2f ðg; pÞog2

(

þ3ogðg; pÞ

ogo3f ðg; pÞ

og3� 2gðg; pÞ o

4f ðg; pÞog4

� o4gðg; pÞog24

f ðg; pÞ):

ð24Þ

Obviously for p = 0 and p = 1 we have

f ðg; 0Þ ¼ f0ðgÞ; f ðg; 1Þ ¼ f ðgÞ; ð25Þgðg; 0Þ ¼ g0ðgÞ; gðg; 1Þ ¼ gðgÞ: ð26Þ

As p increases from 0 to 1, f ðg; pÞ and gðg; pÞ vary fromf0(g) and g0(g) to the exact solutions f(g) and g(g). Dueto Taylor’s theorem and Eqs. (25) and (26), we can expressthat

f ðg; pÞ ¼ f0ðgÞ þX1m¼1

fmðgÞpm; ð27Þ

gðg; pÞ ¼ g0ðgÞ þX1m¼1

gmðgÞpm; ð28Þ

fmðgÞ ¼1

m!

omf ðg; pÞopm

��p¼0

; gmðgÞ ¼1

m!

omgðg; pÞopm

��p¼0

; ð29Þ

where the convergence of the series in Eqs. (27) and (28) isdependent upon �h1 and �h2. Assume that �h1 and �h2 are se-lected such that the series in Eqs. (27) and (28) are conver-gent at p = 1, then due to Eqs. (25) and (26) one can write

f ðgÞ ¼ f0ðgÞ þX1m¼1

fmðgÞ; ð30Þ

gðgÞ ¼ g0ðgÞ þX1m¼1

gmðgÞ: ð31Þ

Differentiating the zeroth order deformation equations (19)and (20) m times with respect to p, then dividing by m!, andfinally setting p = 0 we get the following mth-order defor-mation problems

L1½fmðgÞ � vmfm�1ðgÞ� ¼ �h1R1mðgÞ; ð32ÞL1½gmðgÞ � vmgm�1ðgÞ� ¼ �h2R2mðgÞ; ð33Þfmð0Þ ¼ f 0mð0Þ ¼ f 0mð1Þ ¼ 0; ð34Þgmð0Þ ¼ g0mð0Þ ¼ g0mð1Þ ¼ 0; ð35ÞR1mðgÞ ¼ f 000m�1ðgÞ �M2f 0m�1 þ ð1� vmÞðM2 þ 1Þ

þXm�1

k¼0

�f 0m�1�kf 0k þ fm�1�kf 00k�

þa �f 00m�1�kf 00k þ 2f 0m�1�kf 000k � fm�1�kf0000

k

; ð36Þ


R2mðgÞ ¼ g000m�1ðgÞ �M2g0m�1 þ ð1� vmÞðM2 þ 3Þ

þXm�1

k¼0

h�3g0m�1�kf 0k þ 2gm�1�kf 00k þ g00m�1�kfk:

þa 3g000m�1�kf 0k � 3g00m�1�kf 00k þ 3g0m�1�kf 000k

�2gm�1�kf

0000

k � g0000

m�1�kfk

i; ð37Þ

where

vm ¼0; m 6 1;

1; m > 1:

�ð38Þ

MATHEMATICA is used to solve the linear equations(32)–(35) up to first few order of approximations and it isfound that f and g can be expressed as

fmðgÞ ¼Xmþ1

n¼0

X2ðmþ1�nÞ

q¼0

aqm;ng

qe�ng; m P 0; ð39Þ

gmðgÞ ¼Xmþ1

n¼0

X2ðmþ1�nÞ

q¼0

bqm;ng

qe�ng; m P 0: ð40Þ

Substituting Eqs. (39) and (40) into Eqs. (32) and (33) therecurrence formulae for the coefficients aq

m;n and bqm;n of

fm(g) and gm(g) are obtained, respectively, for m P 1,0 6 n 6 m + 1 and 0 6 q 6 2(m + 1 � n) as

a0m;0 ¼ vmv2mþ2a0

m�1;0 �X2m

q¼0

D1qm;1l

q1;1

�Xmþ1

n¼2

ðn� 1ÞD10m;nl

0n;0 þ

P2ðmþ1�nÞ

q¼1

D1qm;n ðn� 1Þlq

n;0 � lqn;1

� �" #;

ð41Þak

m;0 ¼ vmv2mþ2�kakm�1;0; 1 6 k 6 2mþ 2; ð42Þ

a0m;1 ¼ vmv2ma0

m�1;1 þX2m

q¼0

D1qm;1l

q1;1

þXmþ1

n¼2

nD10m;nl

0n;0 þ

X2ðmþ1�nÞ

q¼1

D1qm;nðnlq

n;0 � lqn;1Þ

( );

ð43Þ

akm;1 ¼ vmv2m�kak

m�1;1 þX2m

q¼k�1

D1qm;1l

q1;k ; 1 6 k 6 2m; ð44Þ

akm;n ¼ vmv2ðm�nÞ�kþ2ak

m�1;n þX2ðmþ1�nÞ

q¼k

D1qm;nl

qn;k ;

2 6 n 6 mþ 1; 0 6 k 6 2ðmþ 1� nÞ; ð45Þ

b0m;0 ¼ vmv2mþ2b0

m�1;0 �X2m

q¼0

D2qm;1l

q1;1

�Xmþ1

n¼2

�ðn� 1ÞD20

m;nl0n;0

þX2ðmþ1�nÞ

q¼1

D2qm;nððn� 1Þlq

n;0 � lqn;1Þ�; ð46Þ

bkm;0 ¼ vmv2mþ2�kbk

m�1;0; 1 6 k 6 2mþ 2; ð47Þ

b0m;1 ¼ vmv2mb0

m�1;1 þX2m

q¼0

D2qm;1l

q1;1

þXmþ1

n¼2

nD20m;nl

0n;0 þ

X2ðmþ1�nÞ

q¼1

D2qm;n nlq

n;0 � lqn;1

� �( );

ð48Þ

bkm;1 ¼ vmv2m�kbk

m�1;1 þX2m

q¼k�1

D2qm;1l

q1;k ; 1 6 k 6 2m; ð49Þ

bkm;n ¼ vmv2ðm�nÞ�kþ2bk

m�1;n þX2ðmþ1�nÞ

q¼k

D2qm;nl

qn;k ;

2 6 n 6 mþ 1; 0 6 k 6 2ðmþ 1� nÞ; ð50Þ

where

lq1;k ¼

q� k þ 2ð Þq!

k!; 0 6 k 6 qþ 1; q P 0; ð51Þ

lqn;k ¼

Xq�k

p¼0

ðq� k � p þ 1Þq!

k!nq�k�pþ2ðn� 1Þpþ1; 0 6 k 6 q; q P 0; n P 2;

ð52Þ

D1qm;n ¼ �h1 v2ðm�nÞ�qþ2ða3q

m�1;n �M2a1qm�1;nÞ

h�ðd1q

m;n � d2qm;nÞ þ að�d3q

m;n þ 2d4qm;n � d5q

m;nÞi; ð53Þ

D2qm;n ¼ �h2 v2ðm�nÞ�qþ2ðb3q

m�1;n �M2b1qm�1;nÞ

h�ð3d6q

m;n � 2d7qm;n � d8q

m;nÞ

það3d9qm;n � 3d10q

m;n þ 3d11qm;n � 2d12q

m;n � d13qm;nÞi;

ð54Þ

and d1qm;n � d13q

m;n, where m P 1, 0 6 n 6 3m + 3, 0 6 q 6

3m + 3 � n are

d1qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg

Xminfq;2ðkþ1�jÞg

i¼maxf0;q�2ðm�k�nþjÞg

� a1ik;ja1q�i

m�1�k;n�j;

d2qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a2ik;ja

q�im�1�k;n�j;

d3qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a2ik;ja2q�i

m�1�k;n�j;

d4qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a3ik;ja1q�i

m�1�k;n�j;

d5qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a4ik;ja



d6qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a1ik;jb1q�i

m�1�k;n�j;

d7qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a2ik;jb


d8qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a1ik;jb2q�i

m�1�k;n�j;

d9qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a1ik;jb3q�i

m�1�k;n�j;

d10qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a2ik;jb2q�i

m�1�k;n�j;

d11qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a3ik;jb1q�i

m�1�k;n�j;

d12qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� a4ik;jb


d13qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� aik;jb4q�i

m�1�k;n�j:

The values a1km;n; a2k

m;n, a3km;n; a4k

m;n; b1km;n; b2k

m;n; b3km;n and

b4km;n are

a1km;n ¼ k þ 1ð Þakþ1

m;n � nakm;n;

a2km;n ¼ k þ 1ð Þa1kþ1

m;n � na1km;n;


m;n � na2km;n;


m;n � na3km;n;

ð55Þ

b1km;n ¼ k þ 1ð Þbkþ1

m;n � nbkm;n;

b2km;n ¼ k þ 1ð Þb1kþ1

m;n � nb1km;n;


m;n � nb2km;n;


m;n � nb3km;n:

ð56Þ

In order to see the detailed procedure of deriving the aboverelations the reader may consult [25]. Using the aboverecurrence formulae, we can calculate all coefficients ak

m;n

and bkm;n using only the first four

a00;0 ¼ �1; a1

0;0 ¼ 1; a00;1 ¼ 1; a2

0;0 ¼ 0;

b00;0 ¼ �1; b1

0;0 ¼ 1; b00;1 ¼ 1; b2

0;0 ¼ 0ð57Þ

given by the initial guess approximation in Eqs. (15) and(16). The corresponding Mth-order approximation ofEqs. (10), (11) and (14) are

XM

m¼0

fmðgÞ ¼XM

m¼0

a0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

X2ðmþ1�nÞ

k¼0

akm;ng

k

!;

ð58ÞXM

m¼0

gm gð Þ ¼XM

m¼0

b0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

X2ðmþ1�nÞ

k¼0

bkm;ng

k

!:

ð59Þ

Therefore the following explicit, totally analytic solution ofthe present flow is

f ðgÞ ¼X1m¼0

fmðgÞ

¼ limM!1

XM

m¼0

a0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

X2ðmþ1�nÞ

k¼0

akm;ng

k

!" #;

ð60Þ

gðgÞ ¼X1m¼0

gmðgÞ

¼ limM!1

XM

m¼0

b0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

X2ðmþ1�nÞ

k¼0

bkm;ng

k

!" #:

ð61Þ

4. HAM solution for T0(g) and T1(g)

The initial guess approximations for T0(g) and T1(g) are

T 00ðgÞ ¼ 1� e�g; ð62Þ

T 01ðgÞ ¼ ge�g; ð63Þ

and the auxiliary operators are

L2 ¼ f 00 þ f 0; ð64ÞL3 ¼ f 00 þ 2f 0 þ f ; ð65Þ

which satisfy the properties

L2½C5 þ C4e�g� ¼ 0; ð66ÞL3½ðC7 þ C6gÞe�g� ¼ 0; ð67Þ

in which C4, C5, C6 and C7 are arbitrary constants. Thezeroth order deformation problems are

ð1� pÞL2½bT 0ðg; pÞ � T 00ðgÞ�

¼ p�h3N3½bT 0ðg; pÞ; f ðg; pÞ�; ð68Þð1� pÞL3½bT 1ðg; pÞ � T 0

1ðgÞ�¼ p�h4N4½bT 0ðg; pÞ; bT 1ðg; pÞ; f ðg; pÞ; gðg; pÞ�; ð69Þ


bT 0ð0; pÞ ¼ 0; bT 0ð1; pÞ ¼ 1;bT 1ð0; pÞ ¼ 0; bT 1ð1; pÞ ¼ 0; ð70Þ

N3bT 0ðg; pÞ; f ðg; pÞh i¼ o

2bT 0ðg; pÞog2

þ 3KPð3K þ 4Þ f ðg; pÞ

obT 0ðg; pÞog

; ð71Þ

N4bT 0ðg; pÞ; bT 1ðg; pÞ; f ðg; pÞ; gðg; pÞh i¼ o2bT 1 g; pð Þ

og2þ 3KPð3K þ 4Þ �

of g; pð Þog

bT 1ðg; pÞ

þgðg; pÞ obT 0 g; pð Þ

ogþ f g; pð Þ o

bT 1 g; pð Þog

!: ð72Þ

In Eqs. (68) and (69), �h3 and �h4 are the auxiliary nonzeroparameters. For p = 0 and p = 1 we have

bT 0 g; 0ð Þ ¼ T 00 gð Þ; bT 0 g; 1ð Þ ¼ T 0 gð Þ; ð73ÞbT 1 g; 0ð Þ ¼ T 01 gð Þ; bT 1 g; 1ð Þ ¼ T 1 gð Þ: ð74Þ

As p increases from 0 to 1, bT 0ðg; pÞ and bT 1ðg; pÞ vary fromthe initial guesses T 0

0ðgÞ and T 01ðgÞ to the exact solutions

T0(g) and T1(g) respectively. By Taylor’s theorem andEqs. (68) and (69), one obtains

bT 0 g; pð Þ ¼ T 00 gð Þ þ

X1m¼1

T m0 gð Þpm; ð75Þ

bT 1 g; pð Þ ¼ T 01 gð Þ þ

X1m¼1

T m1 gð Þpm; ð76Þ

T m0 gð Þ ¼ 1

m!

ombT 0 g; pð Þopm

��p¼0

;

T m1 gð Þ ¼ 1

m!

ombT 1 g; pð Þopm

��p¼0

: ð77Þ

Clearly the convergence of the series (75) and (76) stronglydepends upon �h3 and �h4. The values of �h3 and �h4 are se-lected in such a way that the series (75) and (76) are conver-gent at p = 1, then due to Eqs. (73) and (74) we have

T 0 gð Þ ¼ T 00 gð Þ þ

X1m¼1

T m0 gð Þ; ð78Þ

T 1 gð Þ ¼ T 01 gð Þ þ

X1m¼1

T m1 gð Þ: ð79Þ

For mth-order deformation problems, we employ a similarprocedure as in previous section and obtain

L2 T m0 gð Þ � vmT m�1

0 gð Þ�

¼ �h3R3m gð Þ; ð80ÞL3 T m

1 gð Þ � vmT m�11 gð Þ

� ¼ �h4R4m gð Þ; ð81Þ

T m0 0ð Þ ¼ T m

0 1ð Þ ¼ 0; T m1 0ð Þ ¼ T m

1 1ð Þ ¼ 0; ð82Þ

R3m gð Þ ¼ o2T m�1

0 gð Þog2

þ 3KPð3K þ 4Þ

Xm�1

k¼0

oT m�1�k0

ogfk; ð83Þ

R4m gð Þ ¼ o2T m�11 gð Þog2

þ 3KPð3K þ 4Þ

�Xm�1

k¼0

�T m�1�k1 f 0k þ

oT m�1�k1

ogfk þ

oT m�1�k0

oggk

� �:

ð84Þ

The solutions of Eqs. (80)–(82) is

T m0 gð Þ ¼

Xmþ1

n¼0

X2ðmþ1�nÞ

q¼0

cqm;ng

qe�ng; m P 0; ð85Þ

T m1 gð Þ ¼

Xmþ1

n¼0

X2mþ6�2n

q¼0

dqm;ng

qe�ng; m P 0: ð86Þ

Using Eqs. (85) and (86) into Eqs. (80) and (81), the recur-rence formulae for the coefficients cq

m;n of T m0 ðgÞ for m P 1,

0 6 n 6 m + 1, and 0 6 q 6 2m + 2 � 2n and dqm;n of T m

1 ðgÞfor m P 1, 0 6 n 6 m + 1, and 0 6 q 6 2m + 6 � 2n areobtained as

ckm;0 ¼ vmv2mþ2�kck

m�1;0; 0 6 k 6 2mþ 2; ð87Þ

c0m;1 ¼ vmv2mc0

m�1;1 �Xmþ1

n¼2

X2ðmþ1�nÞ

q¼0

D3qm;nl1q

n;0; ð88Þ

ckm;1 ¼ vmv2m�kck

m�1;1 �X2m

q¼k�1

D3qm;1l1q

1;k; 1 6 k 6 2m; ð89Þ

ckm;n ¼ vmv2ðm�nÞþ2�kck

m�1;n �X2ðmþ1�nÞ

q¼k

D3qm;nl1q

n;k;

2 6 n 6 mþ 1; 0 6 k 6 2ðmþ 1� nÞ; ð90Þ

d0m;1 ¼ vmv2mþ4d0

m�1;1 �Xmþ1

n¼2

X2mþ6�2n

q¼0

D4qm;nl2q

n;0; ð91Þ

dkm;1 ¼ vmv2m�kþ4dk

m�1;1 þX2mþ4

q¼k�1

D4qm;1l2q

1;k;

1 6 k 6 2mþ 4; ð92Þ

dkm;n ¼ vmv2m�2n�kþ6dk

m�1;n þX2mþ6�2n

q¼k

D4qm;nl2q

n;k;

2 6 n 6 mþ 1; 0 6 k 6 2mþ 6� 2n; ð93Þ

where

l1q1;k ¼

q!

k!; 0 6 k 6 qþ 1; q P 0; ð94Þ

l1qn;k ¼

Xq�k

p¼0

q!

k!npþ1 n� 1ð Þq�pþ1;

0 6 k 6 q; q P 0; n P 2; ð95Þ

l2q1;k ¼

1

ðqþ 1Þðqþ 2Þ ; q P 0; ð96Þ

Fig. 1. �h1-curve for the 8th-order of approximation.



l2qn;k ¼

ðq� k þ 1Þq!

k! n� 1ð Þq�pþ2; 0 6 k 6 q; q P 0; n P 2; ð97Þ

D3qm;n ¼ �h3 v2ðm�nÞ�qþ2c2q

m�1;n þ3KPð3K þ 4Þ d14q

m;n

� �; ð98Þ

D4qm;n ¼ �h4 v2mþ6�2n�qd2q

m�1;n þ3KPð3K þ 4Þ

�� v2mþ4�2n�qð�d15q

m;n þ d16qm;nÞ þ d17q

m;n

� ��; ð99Þ

and the coefficients d14qm;n; d15q

m;n; d16qm;n; and d17q

m;n, wherem P 1, 0 6 n 6 m + 1, 0 6 q 6 2(m + 1 � n) and 0 6 q 6

2m + 6 � 2n are

d14qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� aik;jc1q�i

m�1�k;n�j;

d15qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg

Xminfq;2ðkþ2�2jÞg

i¼maxf0;q�2mþ2k�4þ2n�2jg

� a1ik;jd


d16qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg

Xminfq;2ðkþ2�2jÞg

i¼maxf0;q�2mþ2k�4þ2n�2jg

� aik;jd1q�i

m�1�k;n�j;

d17qm;n ¼

Xm�1

k¼0

Xminfn;kþ1g

j¼maxf0;n�mþkg



� bik;jc1q�i

m�1�k;n�j;

where

c1km;n ¼ k þ 1ð Þckþ1

m;n � nckm;n;

c2km;n ¼ k þ 1ð Þac1kþ1

m;n � ca1km;n;

ð100Þ

d1km;n ¼ k þ 1ð Þdkþ1

m;n � ndkm;n;

d2km;n ¼ k þ 1ð Þd1kþ1

m;n � nd1km;n:

ð101Þ

Using the same procedure as in Section 3, we can calculateall coefficients ck

m;n and dkm;n using only the first few

c00;0 ¼ 1; c0

0;1 ¼ �1; c10;0 ¼ c2

0;0 ¼ 0;

d10;1 ¼ 0; d0

0;0 ¼ b�10;0 ¼ b0

0;1 ¼ b�20;0 ¼ 0;

ð102Þ

given by the initial guess approximation in Eqs. (62) and(63). The corresponding Mth-order approximation ofEqs. (12)–(14) areXM

m¼0

T m0 gð Þ ¼

XM

m¼0

c0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

X2ðmþ1�nÞ

k¼0

ckm;ng

k

!;

ð103ÞXM

m¼0

T m1 gð Þ ¼

XMþ1

n¼1

e�ngXM

m¼n�1

X2mþ6�2n

k¼0

dkm;ng

k

!: ð104Þ

and thus

T 0 gð Þ ¼XM

m¼0

T m0 gð Þ

¼ limM!1

XM

m¼0

c0m;0 þ

XMþ1

n¼1

e�ngXM

m¼n�1

X2ðmþ1�nÞ

k¼0

ckm;ng

k

!" #;

ð105Þ

T 1 gð Þ ¼XM

m¼0

T m1 gð Þ

¼ limM!1

XMþ1

n¼1

e�ngXM

m¼n�1

X2mþ6�2n

k¼0

dkm;ng

k

!" #: ð106Þ

5. Convergence of the HAM solution

As pointed out by Liao [22], the convergence and rate ofapproximation for the HAM solution strongly depends onthe values of auxiliary parameters �h1, �h2, �h3 and �h4. To seethe admissible values of �h1, �h2, �h3 and �h4, �h-curves are plot-ted for two different orders of approximations. Figs. 1–4clearly depict that the range for the admissible values for�h1, �h2, �h3 and �h4 is �0.04 6 �h1 < 0, �0.15 6 �h2 < 0,



Fig. 6. Effects of a on 8th-order approximation for g0 at �h2 = �0.05.

Fig. 7. Effects of M on 8th-order approximation for f 0 at �h1 = �0.01.


�0.6 6 �h3 < 0 and �0.004 6 �h4 < 0. Our calculationsclearly indicate that the series (60), (61), (104) and (106)converge for whole region of g when �h1 = � 0.01,�h1 = � 0.05, �h3 = � 0.3 and �h4 = � 0.002.

6. Results and discussion

Figs. 5–16 have been drawn to see the effects of the sec-ond grade parameter a, Hartman number M, radiation

Fig. 5. Effects of a on 8th-order approximation for f 0 at �h1 = �0.01.

Fig. 8. Effects of M on 8th-order approximation for g0 at �h2 = �0.05.

parameter K and the Prandtl number Pr on the velocityand the temperature fields.

Figs. 5–8 are made in order to see the effects of a and M

on the velocity components f 0 and g0. From Figs. 5 and 6, itis seen that f 0 and g0 are increased as the second gradeparameter a increases but this change is larger in g0 whencompared with f 0. However, the boundary layer thicknessdecreases in both the cases f 0 and g0. Figs. 7 and 8 show

Fig. 9. Effects of a on 8th-order approximation for T0 at �h3 = �0.3.

Fig. 10. Effects of a on 8th-order approximation for T1 at �h4 = �0.002.

Fig. 11. Effects of M on 8th-order approximation for T0 at �h3 = �0.3.

Fig. 12. Effects of M on 8th-order approximation for T1 at �h4 = �0.002.

Fig. 13. Effects of K on 8th-order approximation for T0 at �h3 = �0.3.

Fig. 14. Effects of K on 8th-order approximation for T1 at �h4 = �0.002.


the effects of M on f 0 and g0. The velocity f 0 in Fig. 7 inincreasing function of M and the boundary layer thicknessdecreases in case of f0. From Fig. 8, it is seen that the veloc-ity component g0 is decreased as the Hartman number M

increases. However, this decrement is very larger in g0 onthe small values of M. The boundary layer thicknessincreases in this cases.

Figs. 9–16 are plotted to see the effects of a, M, K and Pr

on the temperature profiles T0 and T1. Figs. 9 and 10 indi-cate the effects of a on T0 and T1. In Fig. 9, the temperatureT0 decreases as a increases but this decrement is made onvery large values of second grade parameter a and inFig. 10, T1 is increasing when a increases. The boundarylayer thickness increases in case of T0 and decreases in case

Fig. 15. Effects of Pr on 8th-order approximation for T0 at �h3 = �0.3.

Fig. 16. Effects of Pr on 8th-order approximation for T1 at �h4 = �0.002.

Table 2Effects of the non-dimensional parameters M, a, K and Pr on T 00ð0Þ andT 01ð0Þa M K Pr T 00ð0Þ T 01ð0Þ1 0 3 1 0.36170565 1.44329

0.1 0.361766 1.443090.5 0.363197 1.43841.0 0.36751389 1.423981.5 0.374217 1.400593.0 0.40228349 1.273574.0 0.42214090 1.09502

1 2 0 1 0.057648010 1.00000000.1 0.155971 1.003820.5 0.31501 1.020451.0 0.38762665 1.038781.5 0.421388 1.056972.0 0.43071212 1.092873.0 0.38268495 1.368615.0 0.10071477 4.68049

1 1 3 0 0.057648010 1.00000000.1 0.152492 1.012450.3 0.268866 1.038230.5 0.342648 1.064970.7 0.396276 1.092491.0 0.36751389 1.135121.2 �0.205072 1.16406

0 1 3 1 0.38138688 1.124660.1 0.379944 1.155570.5 0.374298 1.277041.0 0.36751389 1.423981.5 0.361019 1.565441.7 0.358499 1.620492.0 0.35479890 1.70141


of T1. Figs. 11 and 12 illustrate the effects of M on temper-ature T0 and T1. It is found in Fig. 11 that T0 initiallyincreases and after the value at M P 10 it goes to decreaseand the boundary layer thickness is increased. Fig. 12 givesthat T1 is decreasing function of M. The boundary layerthickness is also increased in the case of T1. Figs. 13 and14 elucidate the effects of K on T0 and T1. It is seen thatT0 and T1 are increasing function of K. Moreover theboundary layer thickness decreases in both the cases T0

and T1. Figs. 15 and 16 show the effects of Pr on T0 andT1. The temperatures T0 and T1 increase for large valuesof Pr. However, this increment is larger in T0 when com-pared with T1 at very small values of Pr. The boundarylayer thickness decreases in both cases of T0 and T1.

Table 1Effects of the non-dimensional parameters M and a on f 00(0) and g00(0)

a M f 00(0) g00(0) a M f 00(0) g00(0)

1 0 0.93245764 0.87630411 0 1 1.1329356 2.08316930.1 0.933154 0.877628 0.1 1.11856 1.910210.5 0.949745 0.908814 0.5 1.06353 1.387041.0 1.0000000 1.0000000 1.0 1.0000000 1.00000001.5 1.07884 1.13584 1.5 0.941841 0.7835863.0 1.4266105 1.5686494 1.7 0.919979 0.7248264.0 1.7023142 �21.505764 2.0 0.88860210 0.65521489

Tables 1 and 2 have been included just to show the var-iation of a, M, K, and Pr. Table 1 indicates the effects of aand M on f00(0) and g00(0). It is found that f 00(0) increaseswith an increase in M but decreases for large values of a[45]. The behavior of a and M on the magnitude of g00(0)is similar to that of f00(0). Table 2 shows the variation ofM, K, Pr and a on T 00ð0Þ and T 01ð0Þ. As expected T 00ð0Þincreases and T 01ð0Þ decreases when values of M areincreased. Further T 01ð0Þ is an increasing function of K

whereas T 00ð0Þ increases for 0 6 K 6 2.0 and then decreaseswhen K P 2. The effects of Pr here indicates that T 00ð0Þ firstincreases and then decreases. However T 01ð0Þ is an increas-ing function of Pr. Moreover, the large values of a areresponsible for increasing T 01ð0Þ and decreasing T 00ð0Þ.

7. Concluding remarks

MHD steady flow of an incompressible second gradefluid in the presence of radiation has been examined in thistreatise. The system is stressed by a uniform transversemagnetic field. The non-linear equations are solved analyt-ically using HAM. The obtained results are quite new andhave never been reported. Even such results for MHD vis-cous fluid have not be reported yet. However, the numeri-cal solution for MHD viscous fluid is given in reference


[21]. The results for MHD viscous fluid in the presence of aradiation can be recovered by taking a = 0. The presentsolution for f00(0) and T 00ð0Þ are qualitatively similar to thatgiven in reference [45]. This provides useful comparison.

Acknowledgement

The authors are grateful to the anonymous reviewers forthe useful suggestions.

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The influence of thermal radiation on MHD flow of a second grade fluid

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