International Journal of Heat and Mass Transfer 53 (2010) 1290–1297

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Steady flow and heat transfer of a Sisko fluid in annular pipe

M. Khan a,*, S. Munawar a, S. Abbasbandy b

a Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistanb Department of Mathematics, Imam Khomeini University, Ghazvin 34149-16818, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 December 2008Received in revised form 1 July 2009Accepted 10 November 2009Available online 8 January 2010

Keywords:Sisko fluidHeat transferAnnular pipe

0017-9310/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2009.12.037

* Corresponding author.E-mail addresses: mkhan@qau.edu.pk, mkhan_21@

The flow and heat transfer problem of a Sisko fluid in an annular pipe is considered. The governing non-linear equation of an incompressible Sisko fluid is modelled. Both analytical and numerical solutions ofthe governing nonlinear problem are presented. The analytical solutions are developed using homotopyanalysis method (HAM) and for the numerical solutions the finite difference method in combination withan iterative scheme is used. A comparison between the analytical and the numerical solutions is pre-sented. Moreover, the shear-thinning and shear-thickening behaviors of the non-Newtonian Sisko fluidare discussed through several graphs and a comparison is also made with the Newtonian fluid.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The rheological behavior of many of the fluids used in industrialand engineering applications do not obey the Newtonian postulate,e.g. the fluids formed during plastic manufacture, the liquid filmsin bearing lubrication system, the glue used in biological chemistryand so forth. Such fluids are often referred to as non-Newtonianfluids. Non-Newtonian fluid is a broad class of fluids in which therelation connecting the shear stress and the shear rate is nonlinearand hence no single constitutive relation possesses the potential topredict all kinds of non-Newtonian behavior of the fluids flowing invarious situations. Despite this fact several investigators [1–10]have recently studied many interesting flow situations under vary-ing conditions. Although a good number of fluid rheologies are al-ready in existence, particularly for most of those fluids used aslubricant and having non-Newtonian behavior, the flow can beanalyzed with the help of a power-law model whereby, now inaddition to viscosity, another parameter, namely the power-law in-dex (or exponent) is used to characterize the fluid. Further, power-law models characterize both pseudo-plastic and dilatant fluidsdepending on whether they possess shear-thinning or shear-thick-ening characteristics. Due to the practical applications in lubri-cants, a power-law model [11] has been preferred.

The interest in heat transfer problems involving power-lawnon-Newtonian fluids has grown in the past half century since heattransfer process plays an important role in industrial and techno-logical applications. This is due to the fact that the rate of coolinginfluences a lot to the quality of the final product with desired

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characteristics such as metal extrusion, glass fiber production hotrolling, manufacturing of plastic and rubber sheets and so forth.Consequently, the results for the flows and heat transfer of non-Newtonian fluids are needed. Some recent studies dealing withheat transfer of non-Newtonian fluids may be mentioned in thereferences [12–15].

In this paper, we present the analysis for steady flow in annularpipe of a power-law model known as a Sisko fluid. An analysis ofheat transfer is also carried out. The current work has been carriedout to develop the modelling and to construct solutions. Employingthe finite difference method in combination with an iterativescheme the numerical solutions are established. The analyticalsolutions are given using the homotopy analysis method (HAM)[16]. This technique has already been successfully used for thesolution of various problems [17–25].

2. Mathematical formulation

Consider an incompressible Sisko fluid flowing in annular pipe.The motion starts suddenly due to the motion of outer cylinder anda constant pressure gradient in the z-direction which is taken asthe axis of flow. The Cauchy stress tensor T in a Sisko fluid [21] is

T ¼ �pIþ S; ð1Þ

where p is the pressure, I the identity tensor and the extra stresstensor S satisfies the following expression:

S ¼ aþ b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

tr A21

� �r����������n�1

24

35A1; ð2Þ

A1 ¼ L þ LT ; L ¼ gradV: ð3Þ

M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297 1291

In above equations, V is the velocity, A1 the first Rivlin–Ericksontensor and n, a and b are the material parameters defined differentlyfor different fluids. Note that for b = 0 the Newtonian fluid model isrecovered and for a = 0 the generalized power-law model can beobtained.

We seek velocity and temperature fields of the form

V ¼ wðrÞez; h ¼ hðrÞ: ð4Þ

In the absence of body forces, the continuity and the balance of lin-ear momentum

divV ¼ 0; ð5Þ

qdVdt¼ divT; ð6Þ

reduce to

opor¼ op

o/¼ 0; ð7Þ

dpdz¼ 1

rddr

r aþ bdwdr

��������n�1

!dwdr

" #; ð8Þ

under condition (4). Eq. (7) shows that p is independent of r and hand z differential of pressure in Eq. (8) is constant since the flowis due to prescribed pressure gradient. To obtain the energy equa-tion appropriate for this problem we assume that Eq. (4) holds.Thus, the balance of energy is

qCpdhdt¼ T � L � divq; ð9Þ

where q is the density, Cp the specific heat and the heat flux q isrepresented by Fourier’s law with a constant conductivity:

q ¼ �kgradh: ð10Þ

With the help of Eqs. (4) and (10), the balance of energy (9) results

kr

ddr

rdhdr

� �þ aþ b

dwdr

��������n�1

!dwdr

� �2

¼ 0: ð11Þ

The relevant boundary conditions of the problem are

wðrÞ ¼ 0; hðrÞ ¼ T0 at r ¼ R0; ð12Þ

wðrÞ ¼W1; hðrÞ ¼ T1 at r ¼ R1; ð13Þ

where R0 and T0 are the radius and temperature of the inner cylin-der and W1; R1 and T1 are the velocity, radius and temperature ofthe outer cylinder, respectively.

hðrÞ ¼ 1576

Brdpdz 9 dp

dz ð1� r4Þ � 32bð1� r3Þn o

� 18Br 4� 4bð1� dÞ þ dd

n

�2 �4bð1� dÞ þ dp

dz ð1� d2Þn o

logðrÞðlogðdÞÞ2

� 2logðdÞ

�4bð1�

þ dpdz ð1�

8<:

þ 4� 4bð1� dÞ þ dpdz ð1� d2Þ

n oðlogðrÞÞ2

ðlogðdÞÞ2

2666664

þ9Br 32þ 16ð1� d2Þ dpdz þ ð1� 4d2 þ 3d4Þ dp

dz

� �2�

logðrÞlogðdÞ

þ16Brbð1� dÞ �36þ dpdz ð�7þ 2dþ 11d2Þ

n ologðrÞlogðdÞ

þ576 logðrÞlogðdÞ þ 288Brb

2ð1� dÞ2 logðrÞlogðdÞ

26666666666666666666666664

Introducing the new dimensionless variables:

r� ¼ rR0; z� ¼ z

R0; w� ¼ w

W1; b� ¼ b

aW1

R0

��������n�1

; p� ¼ pðaW1=R0Þ

;

h� ¼ h� T0

T1 � T0; Ec ¼

W21

CpðT0 � T1Þ; Pr ¼

aCp

k: ð14Þ

The non-dimensional form of the problem is

ddr

1þ bdwdr

��������

n�1 !

dwdr

" #þ 1

r1þ b

dwdr

��������

n�1 !

dwdr¼ dp

dz; ð15Þ

1r

ddr

rdhdr

� �þ Br 1þ b

dwdr

��������

n�1 !

dwdr

� �2

¼ 0; ð16Þ

wðrÞ ¼ hðrÞ ¼ 0 at r ¼ 1; ð17ÞwðrÞ ¼ hðrÞ ¼ 1 at r ¼ d; ð18Þ

where d ¼ R1=R0, Br ¼ PrEc is the Brinkman number and asteriskshave been suppressed for simplicity.

3. Analytic solutions

Physically the power index n is a non-negative real number.First consider the case when n is a non-negative integer. For n = 0and n = 1, Eqs. (15) and (16) take the forms

d2w

dr2 þ1r

dwdrþ b

r¼ dp

dz; n ¼ 0; ð19Þ

d2w

dr2 þ1r

dwdr¼ 1

1þ bdpdz; n ¼ 1; ð20Þ

and

1r

ddr

rdhdr

� �þ Br

dwdrþ b

� �dwdr¼ 0; n ¼ 0; ð21Þ

1r

ddr

rdhdr

� �þ Br 1þ bð Þ dw

dr

� �2

¼ 0; n ¼ 1: ð22Þ

Solving Eqs. (19)–(22) subject to the boundary conditions (17) and(18), we get exact analytic solutions of the forms:

wðrÞ ¼ logðrÞlogðdÞ þ b ð1� rÞ � ð1� dÞ logðrÞ

logðdÞ

�

� 14

dpdzð1� r2Þ � ð1� d2Þ logðrÞ

logðdÞ

� ; ð23Þ

pz ð1� d2Þ

orÞ

r2Þ

9=;3777775

37777777777777777777777775

; ð24Þ

1292 M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297

for n = 0 and

wðrÞ ¼ 14ð1þ bÞ 4ð1þ bÞ logðrÞ

logðdÞ �dpdzð1� r2Þ � ð1� d2Þ logðrÞ

logðdÞ

� �;

ð25Þ

hðrÞ ¼ 164ð1þ bÞ

�4Brdpdz 4ð1þ bÞ þ dp

dz ð1� d2Þ� �

ð1� r2Þ 1logðdÞ � ð1� d2Þ logðrÞ

ðlogðdÞÞ2

n oþBr

dpdz

� �2ð1� r4Þ � 2Br 4ð1þ bÞ þ dp

dz ð1� d2Þ� �2 ðlogðrÞÞ2

ðlogðdÞÞ2

þBr

32ð1þ b2Þ þ 16ð1� d2Þ dpdz

þð1� 4d2 þ 3d4Þ dpdz

� �2

8<:

9=; logðrÞ

logðdÞ

þ64 logðrÞlogðdÞ þ 16b 4þ Br 4þ ð1� d2Þ dp

dz

� �n ologðrÞlogðdÞ

2666666666664

3777777777775; ð26Þ

for n = 1. For b = 0, the above solutions reduce to those for a Newto-nian fluid.

To find out the solutions for the other values of viscoelastic in-dex n we use the homotopy analysis method.

4. HAM solutions

4.1. Zero-order deformation problems

For HAM solutions, wðrÞ and hðrÞ can be expressed by the set ofbase functions

rljl P 0�

; ð27Þ

in the form

wmðrÞ ¼X2mþ1

l¼0

al;mrl; ð28Þ

hmðrÞ ¼X2mþ1

l¼0

a�l;mrl; ð29Þ

where al;m and a�l;m are the coefficients. According to the rule ofsolutions expressions (27)–(29) and the boundary conditions (17)and (18), the initial approximation of wðrÞ and hðrÞ are

w0ðrÞ ¼ r � 1; ð30Þh0ðrÞ ¼ r � 1; ð31Þ

and the auxiliary linear operator is

L ¼ d2

dr2 ; ð32Þ

satisfying

L½C1 þ C2r� ¼ 0; ð33Þ

where C1 and C2 are constants.From Eqs. (15) and (16), we define the nonlinear operators as

Nw wðr; pÞ½ � ¼ o

or1þ b

owðr; pÞor

��������

n�1 !

owðr; pÞor

" #

þ 1r

1þ bowðr; pÞ

or

��������n�1

!owðr; pÞ

or� dp

dz; ð34Þ

Nh hðr; pÞ; wðr; pÞh i

¼ 1r

o

orrohðr; pÞ

or

!

þ Br 1þ bowðr; pÞ

or

��������n�1

!owðr; pÞ

or

� �2

: ð35Þ

Taking �h as a non-zero auxiliary parameter one can construct thezero-order deformation problems as follows:

ð1� pÞL wðr; pÞ �w0ðrÞ½ � ¼ p�hNw wðr; pÞ½ �; ð36Þ

ð1� pÞL hðr; pÞ � h0ðrÞh i

¼ p�hNh hðr; pÞ; wðr; pÞh i

; ð37Þ

wðr; pÞ ¼ hðr; pÞ ¼ 0 at r ¼ 1; ð38Þ

wðr; pÞ ¼ hðr; pÞ ¼ 1 at r ¼ d; ð39Þ

in which p 2 ½0;1� is the embedding parameter. For p = 0 and p = 1,we, respectively, have

wðr; 0Þ ¼ w0ðrÞ; hðr; 0Þ ¼ h0ðrÞ; ð40Þ

wðr; 1Þ ¼ wðrÞ; hðr; 1Þ ¼ hðrÞ: ð41Þ

When p increases from 0 to 1, wðr; pÞ and hðr; pÞ varies from the ini-tial guess to final solutions. Due to Taylor’s theorem we can write

wðr; pÞ ¼ w0ðrÞ þX1m¼1

wmðrÞpm; ð42Þ

hðr; pÞ ¼ h0ðrÞ þX1m¼1

hmðrÞpm; ð43Þ

wmðrÞ ¼1

m!� owðr; pÞ

opm

����p¼0

; ð44Þ

hmðrÞ ¼1

m!� ohðr; pÞ

opm

�����p¼0

: ð45Þ

The convergence of the series (42) and (43) depend upon auxiliaryparameter �h. Assume that the �h is chosen so properly that series(42) and (43) are convergent at p = 1, then due to Eqs. (40) and(41) one can write,

wðrÞ ¼ w0ðrÞ þX1m¼1

wmðrÞ; ð46Þ

hðrÞ ¼ h0ðrÞ þX1m¼1

hmðrÞ: ð47Þ

4.2. High-order deformation problems

Differentiating the zero-order deformation problems (36)–(39)m-times with respect to p, setting p = 0 and then dividing by m!we get the following mth-order deformation problems:

L½wmðrÞ � vmwm�1ðrÞ� ¼ �hR1mðrÞ; ð48Þ

L½hmðrÞ � vmhm�1ðrÞ� ¼ �hR2mðrÞ; ð49Þ

wmð1Þ ¼ hmð1Þ ¼ 0; ð50Þ

wmðdÞ ¼ hmðdÞ ¼ 0; ð51Þ

M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297 1293

in which R1mðrÞ and R2

mðrÞ depend upon the value of n.Hence

R1mðrÞ ¼ w00m�1 þ

1r

w0m�1 �dpdzð1� vmÞ þ buwðrÞ; ð52Þ

R2mðrÞ ¼ h00m�1 þ

1rh0m�1 þ BruhðrÞ; ð53Þ

whence

uwðrÞ ¼Xm�1

k¼0

w0m�1�k 2w00k þ1r

w0k

�; n ¼ 2

¼Xm�1

k¼0

w0m�1�k

Xk

l¼0

w0k�l 3w00l þ1r

w0l

� " #;

n ¼ 3 ð54Þ

uhðrÞ ¼Xm�1

k¼0

w0m�1�kw0k þ bXm�1

k¼0

w0m�1�k

Xk

l¼0

w0k�lw0l; n ¼ 2

¼Xm�1

k¼0

w0m�1�kw0k þ bXm�1

k¼0

w0m�1�k

Xk

l¼0

w0k�l

Xl

q¼0

w0l�qw0q; n ¼ 3 ð55Þ

vm ¼0; m 6 1;1; m > 1:

�ð56Þ

The linear non-homogeneous problems (48)–(51) can be solvedby using MATHEMATICA in the order m ¼ 1;2;3; . . ..

Fig. 1. The �h-curves of w0ð1Þ for 30th-order of approximation, when b ¼ 0:8and dp=dz ¼ �4:0 are fixed.

Fig. 2. The �h-curves of h0ð1Þ for 30th-order of approximation when b ¼ 0:4;Br ¼ 1:0 and dp=dz ¼ �2:0 are fixed.

5. Numerical method

In this section, we are interested to find the direct numericalsolution of the nonlinear problem consisting of the differentialequation (15) and boundary conditions (17) and (18) by meansof suitable numerical technique. Since, Eq. (15) is a highly nonlin-ear ordinary differential equation so we can solve it by an iterativemethod.

The iterative procedure is set as follows:

1þ bdwdr

ðkÞ�����

�����n�1

0@

1Ad2w

dr2

ðkþ1Þ

þ ddr

1þ bdwdr

ðkÞ�����

�����n�1

0@

1Adw

dr

ðkþ1Þ

þ 1r

1þ bdwdr

ðkÞ�����

�����n�1

0@

1Adw

dr

ðkþ1Þ

¼ dpdz; ð57Þ

wðkþ1ÞðrÞ ¼ 0 at r ¼ 1; ð58Þwðkþ1ÞðrÞ ¼ 1 at r ¼ d; ð59Þ

in which the index (k) is used for the iterative step.Eqs. (57)–(59) define the linear boundary value problem for

wðkþ1Þ. Using finite difference method, a linear algebraic equationssystem is obtained and solved for each step ðkþ 1Þ. Therefore, a se-quence of functions wð0ÞðrÞ;wð1ÞðrÞ;wð2ÞðrÞ; . . . is determined in thefollowing manner: if an initial estimated wð0ÞðrÞ is given thenwð1ÞðrÞ;wð2ÞðrÞ; . . . are calculated successively to the solutions ofthe boundary value problems (15), (17) and (18).

For a better convergence, we usually use the method of succes-sive under-relaxation. We solve the boundary value problems(57)–(59) for the iterative step ðkþ 1Þ to get an estimated valueof wðkþ1Þ : ~wðkþ1Þ. Then wðkþ1Þ is given by

wðkþ1Þ ¼ wðkÞ þ s ~wðkþ1Þ �wðkÞ� �

; s 2 ð0;1�; ð60Þ

with s as an under-relaxation parameter and it is chosen very smallso that convergent iteration is reached. The iteration should be

carried out until the relative difference of the computedwðkþ1Þ and wðkÞ between two iterative steps are smaller than a givenerror chosen to be 10�16.

6. Numerical results and discussion

This section includes the numerical results and discussion forthe velocity and temperature fields. The boundary value problems(15)–(18) are solved analytically to obtain the explicit expressionsin the form of series for various values of integer power index. Forn = 0 and n = 1, the exact analytic solution is also included. More-over, direct numerical solution for the velocity field is obtainedand a comparison is also made with the analytic solution. Further,we also compare the profiles of velocity and temperature for twokinds of fluids: a Newtonian fluid (when b = 0) and a Sisko fluid(when b – 0). We interpret these results with respect to variationof the emerging parameters of interest especially the power indexn, the material parameter b, the Brinkman number Br and the pres-sure gradient dp/dz.

The analytic expressions given by Eqs. (46) and (47) are the ser-ies solutions of the flow and heat transfer problem if one developsthe convergence of these solutions. The convergence of series thesolutions (46) and (47) strongly depends upon the auxiliaryparameter �h. By means of the so-called �h-curve, it is straightfor-ward to choose an appropriate range for �h which ensures the con-vergence of the series solution. As pointed out by Liao [16], theappropriate region of �h is a horizontal line segment. We can inves-tigate the influence of �h on the convergence of w0ð1Þ and h0ð1Þ, byplotting the curve of it versus �h, as shown in Figs. 1 and 2 for bothn = 2 and n = 3. By considering the �h-curve we can obtain the rea-sonable interval for �h.

Fig. 5. Profiles of the velocity, given by Eqs. (23) and (25), for vario

Fig. 6. HAM solutions for various values of material parameter b in the 30th-order of appfixed.

Fig. 3. The norm 2 error of w30ðrÞ versus �h when b ¼ 0:8 and dp=dz ¼ �4:0 are fixed.

Fig. 4. The norm 2 error of h30ðrÞ versus �h when n ¼ 3; b ¼ 0:4; dp=dz ¼ �2:0and Br ¼ 1:0 are fixed.

1294 M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297

Also, by computing the error of norm 2 for two successiveapproximation of wmðrÞ and hmðrÞ, we can obtain the best valuefor �h. Figs. 3 and 4 show this error for w30ðrÞ and h30ðrÞ. One cancompute easily that, in case of the velocity field for n = 2, we have�h = �0.208 and for n = 3, we have �h = �0.060 and in case of thetemperature field, for n = 3, we have �h = �0.181 and these valuesmatch with �h-curves (see Figs. 1 and 2).

To see the effect of the rheology of the fluid, Figs. 5–7 with fourpower index values n = 0 (a shear-thinning fluid), n = 1 (a Newto-nian fluid) and n = 2 and 3 (a shear-thickening fluid) are prepared.These figures also display a comparison between a Newtonian fluid(when b = 0) and a Sisko fluid (when b – 0). From these figures, it isclear that the velocity profile for a Newtonian fluid is smaller whencompared with the Sisko fluid for n = 0 and it is much greater whencompared with the Sisko fluid for n = 1, 2 and 3. Moreover, it can beseen that increasing b increases the velocity for n = 0 and reduces

us values of material parameter b when dp=dz ¼ �4:0 is fixed.

roximation when dp=dz ¼ �4:0 and �h ¼ �0:208 for n = 2 and �h = �0.060 for n = 3 are

Fig. 7. Direct numerical solutions for various values of material parameter b when dp=dz ¼ �4:0 are fixed.

M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297 1295

the velocity for n = 1, 2 and 3. This reduction in the velocity forn = 3 is significant when compared with that of n = 1, 2.

The variation of the pressure gradient on the velocity is shownin Figs. 8 and 9. It is worth noting that dp=dz < 0 (>0) causes thevelocity profiles to curve toward the positive (negative) z-directionfor both n = 2 and n = 3. Their amplitudes depend on the magnitudeof the pressure gradient and the flow directions are against thedirection of the pressure gradient. The effect of different power val-ues n can also be clearly observed from these figures. Here, we bet-ter observe the effect of the power index n and we investigate astrong shear-thickening for n = 3.

Fig. 10 displays a comparison of direct numerical solution andthe series solution obtained using HAM. From this figure, we cansee a good agreement between the direct numerical and HAMsolutions.

Fig. 8. HAM solutions for various values of pressure gradient dp/dz in the 30th-order offixed.

Fig. 9. Direct numerical solutions for various values

In order to observe the effects of various physical parameters onthe temperature, hðrÞ is plotted against r in Figs. 11–14. The influ-ence of the power index can also be seen from these figures. Figs.11 and 13 show the effect of the material parameter b and theBrinkman number Br on the temperature for various power indexn. We see that an increase in b and Br causes an increase in temper-ature for n = 0, 1, 2 and 3. From these figures, it is obvious that theeffect of the Brinkman number in the temperature is much stron-ger than the material parameter b. Further, the temperature for Sis-ko fluids is much greater than the Newtonian fluid in all cases.

The pattern of the temperature profile based on the value anddirection of the pressure gradient dp/dz is given in Fig. 14. For ad-verse pressure gradient ðdp=dz > 0Þ the temperature is smaller inthe narrow part of the channel ðwhen r 2 ½1;1:5�Þ than that forfavorable pressure gradient ðdp=dz < 0Þ, whereas for wider part

approximation when b ¼ 0:5 and �h ¼ �0:208 for n = 2 and �h = �0.060 for n = 3 are

of pressure gradient dp/dz when b = 0.5 is fixed.

Fig. 10. A comparison of direct numerical and HAM solutions in the 30th-order of approximation when b ¼ 0:5 and dp=dz ¼ �4:0 are fixed.

Fig. 11. Profiles of temperature, given by Eqs. (24) and (26), for various values of material parameter b when dp=dz ¼ �2:0 and Br ¼ 1:0 are fixed.

Fig. 12. HAM solutions for various values of material parameter b in the 30th-order of approximation when Br ¼ 1:0; dp=dz ¼ �2:0 and �h ¼ �0:310 for n = 2 and �h = �0.181for n = 3 are fixed.

Fig. 13. HAM solutions for various values of Brinkman number Br in the 30th-order of approximation when b ¼ 0:4; dp=dz ¼ �2:0 and �h ¼ �0:310 for n = 2 and �h = �0.181 forn = 3 are fixed.

1296 M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297

Fig. 14. HAM solutions for various values of pressure gradient dp/dz in the 30th-order of approximation when b ¼ 0:4; Br ¼ 1:0 and �h ¼ �0:310 for n = 2 and �h = �0.181 forn = 3 are fixed.

M. Khan et al. / International Journal of Heat and Mass Transfer 53 (2010) 1290–1297 1297

of the channel ðwhen r 2 ½1:5;2�Þ the behavior of the temperatureis quite opposite.

7. Concluding remarks

In this paper, the steady flow and heat transfer characteristics ofa Sisko fluid in annular pipe are analyzed. The governing nonlinearequations of an incompressible Sisko fluid are modelled usingcylindrical polar coordinates. The highly nonlinear problems aresolved analytically using homotopy analysis method and numeri-cally using finite difference method in combination with an itera-tive scheme. Moreover, the analytical solutions for n = 0 and n = 1are also given. A comparison between analytical and numericalsolutions is made. The influence of various parameters of intereston the velocity and temperature fields is graphically presented.The results categorically indicate the following findings:

� It is noted that the velocity in case of the Newtonian fluid issmaller than the Sisko fluid for n = 0 and it is much greater thanthe Sisko fluid for n = 1, 2 and 3.

� It is observed that the velocity decreases monotonically byincreasing material parameter b for n = 1, 2 and 3 and itincreases for n = 0.

� We observe strong shear-thickening effects for n = 3 when com-pared with n = 1 and 2.

� An increase in material parameter b increases the temperaturefor n = 0, 1, 2 and 3.

� The temperature profiles for Sisko fluid are much greater thanthe Newtonian fluid in all cases.

� The effects of the Brinkman number Br and the material param-eter b on the temperature are similar.

Acknowledgements

The authors are grateful to the Higher Education Commission(HEC) for the financial support. The authors are also grateful tothe referees for providing many useful suggestions to improvethe paper in its present form.

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