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Communications in Nonlinear Science

and Numerical Simulation 11 (2006) 297–305

Non-Newtonian flow between concentric cylinders

T. Hayat a, M. Khan a, Y. Wang b,*

a Department of Mathematics, Quaid-i-Azam University, 45320 Islamabad, Pakistanb Institute of Mechanics, Darmstadt University of Technology, Hochschulstr. 1, 64289 Darmstadt, Germany

Received 12 October 2004; received in revised form 24 November 2004; accepted 28 November 2004

Available online 28 December 2004

Abstract

The nonlinear rheological effects of Oldroyd 6-constant fluid between concentric cylinders is addressed.Numerical solution of nonlinear differential equation is given. The nonlinear effects on the velocity is shown

and discussed. This reveal that characteristics for shear thickening/shear thinning behavior of a fluid is

dependent upon the rheological properties.

� 2004 Elsevier B.V. All rights reserved.

PACS: 47.50.+d

Keywords: Oldroyd 6-constant fluid; Concentric cylinders; Modeling

1. Introduction

Recently the interest in problems on non-Newtonian fluid has grown considerably because ofwide use of these fluids in chemical process industries, food and construction engineering, inpetroleum production, in power engineering and commercial applications. However, there isnot a single governing equation which exhibits all the properties of non-Newtonian fluids andthese fluids cannot be described simply as Newtonian fluids. Due to this, many constitutive equa-tions for non-Newtonian fluids have been proposed. Important theoretical studies of such fluids

1007-5704/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2004.11.007

* Corresponding author. Tel.: +49 6151 163196; fax: +49 6151 164120.

E-mail address: wang@mechanik.tu-darmstadt.de (Y. Wang).

298 T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305

were made by Oldroyd [1], Rajagopal [2–5], Erdogan [6,7], Baris [8], Hayat et al. [9–12], and Liao[13].

Our objective in this communication is to venture further in the regime of non-Newtonian flu-ids. We deal with numerical solution of differential equation arising in Oldroyd 6-constant fluidflow in concentric cylinders. In Section 2, we shall consider the governing equations, in Section3, we shall consider mathematical formulation, in Section 4, we shall present the numerical methodand in Section 5, we shall obtain the numerical solution of the problem. Concluding remarks aregiven in Section 6.

2. Governing equations

Here, we consider the flow of an incompressible and electrically conducting fluid. The govern-ing laws are:

qoV

otþ ðV � $ÞV

� �¼ $ � Tþ J� B; ð1Þ

divV ¼ 0; ð2Þ

$ � B ¼ 0; $� B ¼ lmJ; $� E ¼ � oB

ot; ð3Þ

J ¼ rðEþ V� BÞ; ð4Þ

where V is the velocity vector, q the density, J the current density, B the total magnetic field sothat B = B0 + b, b is the induced magnetic field, lm the magnetic permeability, E the electric fieldand r the electric conductivity.

Making the following assumptions:

• The quantities q, lm and r are all constant throughout the flow field.• The magnetic field B is perpendicular to the velocity field V and the induced magnetic field is

negligible compared with the imposed magnetic field so that the magnetic Reynolds number issmall [14].

• The electric field is assumed to be zero.

The electromagnetic body force involved in Eq. (1) becomes

J� B ¼ �rB20V: ð5Þ

The Cauchy stress tensor T of an Oldroyd 6-constant fluid is [15,16]

T ¼ �pIþ S; ð6Þ

Sþ k1DS

Dtþ k3

2ðSA1 þ A1SÞ þ

k52ðtrSÞA1 ¼ l A1 þ k2

DA1

Dtþ k4A

21

� �; ð7Þ

T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305 299

A1 ¼ Lþ LT ; L ¼ gradV ð8Þ

in which �pI is the indeterminate part of the stress due to the constraint of incompressibility, S isthe extra stress tensor, A1 is the first Rivlin–Ericksen tensor, l, k1, k2, k3, k4 and k5 are the materialconstants and the contravariant convected derivative D/Dt is defined as

DS

Dt¼ dS

dt� LS� SLT ; ð9Þ

where d/dt is the material derivative.The extra stress tensor and velocity are

Sðr; tÞ ¼Srr Srh Srz

Shr Shh Shz

Szr Szh Szz

0B@1CA; Vðr; tÞ ¼

0

0

u

0B@1CA: ð10Þ

Making use of Eq. (10), continuity equation is identically satisfied and the rest field equationsgive the following equations

0 ¼ � opor

þ 1

ro

orðrSrrÞ �

Shh

r; ð11Þ

0 ¼ � 1

ropoh

þ 1

r2o

orðr2SrhÞ; ð12Þ

qouot

¼ � opoz

þ 1

ro

orðrSrzÞ � rB2

0u; ð13Þ

1þ k1o

ot

� �Srr þ k3Srz

ouor

¼ lk4ouor

� �2

; ð14Þ

1þ k1o

ot

� �Srh þ

k32Szh

ouor

¼ 0; ð15Þ

1þ k1o

ot

� �Srz þ

ðk3 þ k5Þ2

ðSrr þ SzzÞouor

þ k52Shh

ouor

� k1Srrouor

¼ l 1þ k1o

ot

� �ouor

; ð16Þ

1þ k1o

ot

� �Shh ¼ 0; ð17Þ

1þ k1o

ot

� �Shz þ

ðk3 � 2k1Þ2

Shrouor

¼ 0; ð18Þ

1þ k1o

ot

� �Szz þ ðk3 � 2k1ÞSrz

ouor

¼ ðk4 � 2k2Þouor

� �2

: ð19Þ

300 T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305

For steady case the above equations become

opor

¼ 1

rd

drðrSrrÞ �

Shh

r; ð20Þ

1

ropoh

¼ 1

r2d

drðr2SrhÞ; ð21Þ

opoz

¼ 1

rd

drðrSrzÞ � rB2

0u; ð22Þ

Srr þ k3Srzdudr

¼ lk4dudr

� �2

; ð23Þ

Srh þk32Szh

dudr

¼ 0; ð24Þ

Srz þðk3 þ k5Þ

2ðSrr þ SzzÞ

dudr

þ k52Shh

dudr

� k1Srrdudr

¼ ldudr

; ð25Þ

Shh ¼ 0; ð26Þ

Shz þðk3 � 2k1Þ

2Shr

dudr

¼ 0; ð27Þ

Szz þ ðk3 � 2k1ÞSrzdudr

¼ ðk4 � 2k2Þdudr

� �2

ð28Þ

and consequently

Srh ¼ Shz ¼ 0; ð29Þ

Srr ¼ �k3Srzdudr

þ lk4dudr

� �2

; ð30Þ

Srr þ Szz ¼ �2ðk3 � k1ÞSrzdudr

þ 2lðk4 � k2Þdudr

� �2

; ð31Þ

Srz ¼l du

dr þ la1 dudr

� �31þ a2 du

dr

� �2 ð32Þ

in which

a1 ¼ k1k4 � ðk3 þ k5Þðk4 � k2Þ;

a2 ¼ k1k3 � ðk3 þ k5Þðk3 � k1Þ:ð33Þ

T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305 301

3. Mathematical model

Considering two infinite coaxial cylinders full of incompressible Oldroyd 6-constant fluid, char-acterized by Eq. (6). The fluid starts suddenly due to a constant pressure gradient and by themotion of the inner cylinder parallel to it�s length, while the outer cylinder keeps stationary. UsingEqs. (26) and (29), Eqs. (20)–(22) reduce to the following

� opor

þ 1

rd

drðrSrrÞ ¼ 0; ð34Þ

� 1

ropoh

¼ 0; ð35Þ

� opoz

þ 1

rd

drðrSrzÞ � rB2

0u ¼ 0: ð36Þ

In the derivation of the above equations, the transverse magnetic field is taken to be as bodyforce. Moreover, the z-differential of pressure is constant since the flow is due to a prescribed pres-sure gradient and the motion of the inner cylinder. From Eq. (36) the velocity field is determined.Next, the pressure field is determined from Eq. (34).

Using value of Srz from Eq. (32) into Eq. (36), we arrive at

1

rd

drr

l dudr þ la1 du

dr

� �31þ a2 du

dr

� �2( )" #

� rB20u ¼ dp

dz: ð37Þ

Eq. (37) can be rewritten as

1þ ð3a1 � a2Þ dudr

� �2 þ a1a2 dudr

� �41þ a2 du

dr

� �2� �2

264375 d2udr2

þ 1

r

"1þ a1 du

dr

� �21þ a2 du

dr

� �2#dudr

� rB20

lu ¼ 1

ldpdz

: ð38Þ

The equation which is to be solved is Eq. (38) subject to the following boundary conditions

uðrÞ ¼ U 0 at r ¼ R0;

uðrÞ ¼ 0 at r ¼ R1;ð39Þ

where R0 is the radius of the inner cylinder, R1 is that of the outer cylinder and U0 is the velocity ofthe inner cylinder.

We take the following dimensionless parameters

u� ¼ uU 0

; r� ¼ rR0

; z� ¼ zR0

; p� ¼ plU 0=R0

; a�1 ¼a1

ðR0=U 0Þ2; a�2 ¼

a2ðR0=U 0Þ2

: ð40Þ

The dimensionless form of equation of motion (38) and boundary conditions (39) after drop-ping asterisks are

1þ ð3a1 � a2Þ dudr

� �2 þ a1a2 dudr

� �41þ a2 du

dr

� �2� �2

264375 d2udr2

þ 1

r

1þ a1 dudr

� �21þ a2 du

dr

� �2" #

dudr

�M2u ¼ dpdz

; ð41Þ

302 T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305

uðrÞ ¼ 1 at r ¼ 1;

uðrÞ ¼ 0 at r ¼ b;ð42Þ

where M2 ¼ rB20=ðl=R2

0Þ and b = R1/R0.

4. Numerical method

In this section, we have an interest to give direct numerical solution of the nonlinear problemconsisting of differential equation (41) and the boundary conditions (42) by iterative method.

For that we construct an iterative procedure as

1þ ð3a1 � a2Þ duðnÞ

dr

� �2

þ a1a2 duðnÞ

dr

� �4

1þ a2 duðnÞdr

� �2� �2

264375 d2uðnþ1Þ

dr2þ 1

r

1þ a1 duðnÞ

dr

� �2

1þ a2 duðnÞdr

� �2264

375 duðnþ1Þ

dr�M2uðnþ1Þ ¼ dp

dz

ð43Þ

in which the index (n) is used for the iterative step. If the indices (n) and (n + 1) are withdrawn,Eq. (43) is consistent with Eq. (41). The following boundary conditions

uðnþ1ÞðrÞ ¼ 1 at r ¼ 1;

uðnþ1ÞðrÞ ¼ 0 at r ¼ bð44Þ

and Eq. (43) is a linear differential boundary value problem for u(n+1). Using finite-difference meth-od we deduce a linear algebraic equation system which is solved for each iterative step. Thus, asequence of functions u(0)(r), u(1)(r), u(2)(r), . . . is obtained in the following manner: if u(0)(r) isgiven, then u(1)(r), u(2)(r), . . .are calculated successively as the solutions of the boundary valueproblem (43) and (44).

For a better convergence, the so-called method of under-relaxation is used. We solve theboundary value problem (43) and (44) for the iterative step n + 1 to obtain an estimated valueof uðnþ1Þ : euðnþ1Þ, then u(n+1) is defined by the formula

uðnþ1Þ ¼ uðnÞ þ s euðnþ1Þ � uðnÞ� �

; s 2 ð0; 1�; ð45Þ

where s 2 (0,1] is an under-relaxation parameter. We should choose s so small that convergentiteration is reached. The iteration should be carried out until the relative difference of the com-puted u(n+1) and u(n) between two iterative steps are smaller than a given error chosen to be 10�16.

5. Numerical results and discussion

We compute and compare the profiles of the dimensionless velocities for two kinds of fluids: aNewtonian fluid, for which a1 = a2 and an Oldroyd 6-constant fluid for different values of pressuregradient dp/dz, non-Newtonian parameters a1 and a2, and the magnetic parameter M. It is notedthat the differential equation (41) for a1 = a2 reduces to that of Newtonian fluid.

T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305 303

In Figs. 1–3 the numerical results are given for Newtonian and Oldroyd fluids. The variationsof pressure gradient on Newtonian and Oldroyd fluids are shown in Fig. 1a and b respectively. Itis worth noting from Fig. 1 that dp/dz < 0(>0) causes the velocity profiles to curve toward thepositive (negative) z-direction for both the Newtonian fluid and Oldroyd fluid. Their amplitudesdepend on the magnitude of the pressure gradient and the flow directions are against the directionof the pressure gradient. Also the velocities in Fig. 1b (for a1 = 2, a2 = 8) is larger than the veloc-ities in Fig. 1a. This observation is not general to an Oldroyd 6-constant fluid. Interestingly, fromEq. (41), it is noted that for a1 < a2(a1 > a2) the flow velocity of an Oldroyd fluid is larger (smaller)than that of a Newtonian fluid.

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

r

u

dp/dz=5

dp/dz=2

dp/dz=−2

dp/dz=−5

dp/dz=−5

dp/dz=−2

dp/dz=2

dp/dz=2

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

1.0 1.2 1.4 1.6 1.8 2.0

1.0 1.2 1.4 1.6 1.8 2.0

r

u

(a) (b)

Newtonian fluid 2,1=(α 2)2=α Oldroyd fluid 2,1=(α 8)2=α

Fig. 1. Profiles of the dimensionless velocity u(r) for various values of the pressure gradient dp/dz with fixed value of

M = 1.

0.01

0.2

0.4

0.6

0.8

1.0

1.2

1.2 1.4 1.6 1.8 2 1.0 1.2 1.4 1.6 1.8 2.0r

u

21=α

22=α 21=α

41=α

61=α81=α

0

0.5

1

1.5

2

r

u 22=α

42=α

62=α

82=α

(a) (b)

Fig. 2. Profiles of the dimensionless velocity u(r) for various values of the non-Newtonian material parameters a1 anda2, respectively, with fixed values of dp/dz = � 5 and M = 1.

r

u

1=M

2=M

5=M

10=M

r

u

1=M

2=M

5=M

10=M

0.01.0

0.2

0.4

0.6

0.8

1.0

1.2

1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.00.0

0.5

1.0

1.5

2.0

(a) (b)

Newtonian fluid 2,1=(α 2)2=α Oldroyd fluid 2,1=(α 8)2=α

Fig. 3. Profiles of the dimensionless velocity u(r) for various values of magnetic parameter M with fixed value of

dp/dz = � 5.

304 T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305

The influence of a1 and a2 on the velocity can be found in Fig. 2. As can be seen from Fig. 2athat for an Oldroyd 6-constant fluid, when the material parameter a1 increases from a1 = 2 toa1 = 8 while a2 maintains a constant value a2 = 2, the flow profiles tend to approach the linear dis-tribution; thus, the shearing can unattenuately extend to the whole flow domain from the bound-aries, corresponding to a shear-thickening phenomenon. Further, the variation of a2 is given inFig. 2b when a1 is fixed. It can be found that for a1 = 2 and a2 = 2–8, a shear thinning effect ofthe examined Oldroyd fluid is noted which is opposite phenomenon to that shown in Fig. 2a.

Fig. 3a and b are prepared to discuss the effects of the magnetic field M on the flow for New-tonian and Oldroyd 6-constant fluids, respectively. As expected, by increasing the magnitude ofmagnetic fieldM reduces the velocity monotonically due to the effect of the magnetic force againstthe flow direction. It can be seen that with the increase of magnetic field M the velocity profilesdecrease more rapidly for Oldroyd fluid when compared with Newtonian fluid.

6. Concluding remarks

The steady and magnetohydrodynamic flow of an incompressible Oldroyd 6-constant fluid be-tween concentric cylinders has been modeled and solved. The nonlinear problem is solved numer-ically to examine the dependence of the flow on the pressure gradient, material parametersemerging in the constitutive model and the applied magnetic field. The numerical results of theOldroyd 6-constant fluid are compared with those of Newtonian fluids. The major findings ofthe present study can be summarized as follows:

• The Oldroyd 6-constant fluid is the general case of the Newtonian, Maxwell, second grade andOldroyd 3-constant fluids. When ki = 0 (i = 1–5), it reduces to Newtonian fluid. For k150,kj = 0 (j = 2–5), it corresponds to Maxwell fluid. When k1 = 0, lk2 ¼ ea1, kk = 0 (k = 3–5), itreduces to second grade fluid. For k1505k2, kk = 0 (k = 3–5), it becomes Oldroyd 3-constantfluid.

T. Hayat et al. / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 297–305 305

• The velocity in case of an Oldroyd 6-constant fluid is larger (smaller) than that of Newtonianfluid case for a1 < a2 (a1 > a2).

• The transverse magnetic field decreases the fluid motion.

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