Axial Couette flow of an Oldroyd-B fluid in an annulus

Muhammad Jamil1,2 a) and Najeeb Alam Khan3

Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

2) Department of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan

3) Department of Mathematics, University of Karachi, Karachi-75270, Pakistan

(Received 18 September 2011; accepted 27 September 2011; published online 10 January 2012)

Abstract This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated. © 2012 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1201201]

Keywords Oldroyd-B fluid, velocity field, time-dependent shear stress, Hankel transform

In recent years, the interest for non-Newtonian fluid flows has considerably increased and many exact solutions have been obtained.1-5 Obtaining the exact solutions for the equations of motion of non-Newtonian fluids, as well as for the Navier-Stokes fluids, is very important for many reasons. They provide a standard for checking the accuracies of many approximate methods which can be numerical or empirical. Although the computer techniques make the complete integration of the equations of motion of these fluids feasible, the accuracy of the results can be established only by comparison with an exact solution.6 The exact solutions can also be used as tests to verify numerical schemes which are developed to study more complex unsteady flow problems.

The inadequacy of the Navier-Stokes theory in describing rheologically complex fluids used in industrial processing, such as polymer solutions, melts and paints, has led to the formulation of other mathematical models, which are able to predict the flow of such materials. However there are many rheologically complicated fluids which do not show the relaxation and retardation phenomena. For this reason, many models have been proposed in which fluids are usually classified as of differential, rate and integral type.7 The differential and rate type models are used to describe the response of the fluids which have slight memory such as dilute polymeric solutions whereas the integral models are used to describe the response of the fluids which have considerable memory such as polymeric melts. A large number of non-Newtonian fluid models are concerned with the fluids of grade two and three, but these fluids do not predict stress relaxation and retardation. Models of the Maxwell, Oldroyd-B and Burgers' fluid type can predict these phenomena, and have therefore become more popular. The Oldroyd-B fluid model,8 which takes into account elastic and memory effects exhibited by most

a) Corresponding author. Email: jqrza26@yahoo.com.

polymeric and biological liquids, has been used quite widely in many applications and the results of simulations fit to experimental data in a wide range.9 The Oldroyd-B fluids belong to the class of rate type fluid models.10-18

Generally, there exist three kinds of boundary value problems: (1) when the velocity is given on the boundary; (2) when the shear stress is given on the boundary and (3) mixed boundary value problems. For the applications point of view, all three types of boundary value problems are of interest. Unfortunately, there are few exact solutions corresponding to the cases 2 and 3. In this paper, we tried to solve a motion problem in which the velocity is given on a part of the boundary and the shear stress on the other part. The first exact solutions for motions of non-Newtonian fluids due to a given shear stress on the boundary are those of Refs. 19-21 over an infinite plate and Ref. 22 in cylindrical domains.

Axial flow in an annulus between a rotating inner cylinder and a fixed outer cylinder has several important engineering applications including journal bearings, biological separation devices, rotating machinery, desalination to magnetohydrodynamics and also in viscosimetric analysis.23 This paper studies exact analytical solutions corresponding to the longitudinal flow of an Oldroyd-B fluid between two infinite coaxial circular cylinders. In order to produce the flow of fluid, the boundary of inner cylinder is applied a time-dependent longitudinal shear stress. The general solutions, obtained by means of finite Hankel transforms and presented in series form in terms of Bessel functions Jo(-), JL(), Yi(-) and Y1 (•), satisfy all the imposed initial and boundary conditions. Moreover, they can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, the influence of material parameters on the velocity profile as well as shear stress is graphically underlined and discussed by graphical illustrations.

The Cauchy stress T in an incompressible Oldroyd-

B fluid is given by

T = -pI + S, S + A(S - LS - SLT) =

+ Ar(A - LA - ALT)],

where —pi denotes the indeterminate spherical stress due to the constraint of incompressibility, S is the extrastress tensor, L is the velocity gradient, A = L + LT is the first Rivlin Ericsen tensor, ^ is the dynamic viscosity of the fluid, A and Ar are relaxation and retardation times respectively, the superscript T indicates the transpose operation and the superposed dot indicates the material time derivative. The model characterized by the constitutive equation (1) contains as special cases the upper-convected Maxwell model for Ar ^ 0 and the Newtonian fluid model for Ar ^ 0 and A ^ 0. In some special flows, as those to be considered here, the governing equations for an Oldroyd-B fluid resemble those for a fluid of second grade. For the problem under consideration we shall assume a velocity field and an extra-stress of the form

V = V(r, t) = v(r,t)ez, S = S(r,t),

where ez is the unit vector in the z-direction of the system of cylindrical coordinates r, 0 and z. For such flows the constraint of incompressibility is automatically satisfied. If the fluid is at rest up to the moment t = 0, then

V(r, 0) = 0, S(r, 0) = 0, (3)

and Eqs. (1) and (2) imply Srr = Srg = Sgz = See = 0.

In the absence of body forces and a pressure gradient in the axial direction, the balance of linear momentum and the constitutive equation (1) leads to the relevant equations

(1 + )t (,,<) = ,(1+ Ar |) ^,

dv(r, t) f d 1\

( dr + r ) ( , ),

where p is the constant density of the fluid and t is the shear stress which is different from zero.

Eliminating t between two equations in Eq. (4), we obtain the governing equation

, d2 v(r, t) dv(r, t)

d\ ( d2

v + UT2 + rdr'v(r,t),

where a = vAr and v = ^,/p is the kinematic viscosity of the fluid.

Suppose that an incompressible Oldroyd-B fluid at rest is situated between two infinite coaxial circular cylinders of radii Ri and R2(> Ri). At time t = 0+, a time-dependent longitudinal shear stress is applied along the inner cylinder boundary and has the form

t(R1,t) = f [(t — A)2 + A2(1 — 2e-*)!, t> 0, (6)

where f is a constant, while the outer one is held fixed. Due to the shear, the fluid between cylinders is gradually moved, its velocity is of the form of the first equation in Eq. (2). The governing equations are given by first equation in Eqs. (4) and (5), the appropriate initial and boundary conditions are

v(r, 0) t(r, 0)

dv(r, 0) dt

0; r G [Ri,R2]

1 + Adt)T (r,t)

(, + A d\dv(r,t)

^1 + Ar

v(R2,t)=0; t > 0 .

ft2; t > 0, (8a)

Of course, t(R1,t) given by Eq. (6) is just the solution of the first differential equation in Eq. (8a). In order to solve this problem, we shall use the finite Hankel transforms.

Let vH (t) be the finite Hankel transform of the function v(r,t) defined by Refs. 2, 24, 25

VH(rn,t)= / rv(r,t)B(r, rn)dr, Ri

n = 1, 2, 3, ••• where

B(r,r„) = Jo(rrn)Yi(Rir„) — Ji(Rir„)Yo(rr„),

with rn being the positive roots of the transcendental equation B(R2,r) = 0, Jp(-) and Yp(^) are Bessel functions of the first and second kind of order p . Furthermore, the inverse Hankel transform is2,24,25

;(r t) = ^ j rn J02(R2r„)B(r,r„)

2 n= J°2(Rir„) - J2(R2rn)

VH(r„,t),

Multiplying Eq. (5) by rB(r, rn), integrating the result with respect to r from Ri to R2 and using the boundary condition (second equation in Eq. (8a)) and the known results (Eq. (18) in Ref. 2), we find that

A%(rn,t) + (1 + arj; )vH(r„,t) + 2

vrn VH(r„,t)

t > 0.

npr„

From Eq. (7) we obtain

VH(r„, 0) = v H(r„, 0) = 0.

The solution of the linear ordinary differential equation (11), with the initial conditions Eq. (12), is given by

4f f t2 1 + arl

eq2nt _e^int- "

(1 + arn )2 A

g2»eqint - qiweg2nt q2n — qin

qin, q2n

— (1 + ar2) (1 + arn)2 — 4vArn

Finally, applying the inverse Hankel transform formula and using Eq. (27) in Ref. 2, we find for the velocity field v(r, t) the simplified expression

v(r, t) = f M

(t — Ar )2 + A

Riln( R1 —

Jo2(R2rn )B(r,rn)

Mv n=Î r3[J?(Rirn) — Jo2(R2rn)] t + A — 2Ar —

1 L eqint — q?neq2ni

-4 1-a 2

q2n — qin

Solving the first equation in Eq. (4) with respect to t (r, t) and taking into account the third equation in Eq. (7), we find that

t(r, t) = Me a A

t T * ( d\ dv(r, s)

1 Jo e*(1 + Ads)-^-'.

Substituting Eq. (14) into Eq. (15) and using the identities

qin?2n = q3n?4n =

2 1 + Ar qin qin?3n = — vrn-T—

v (A — Ar )r

r )' n

?2n?4n = —vrn

2 1 + Ar q2n _ vrn + qin

qin?4n =

vrn + g2n A

q2n?3n

where Aq3n = 1 + Aq1n and Aq4n = 1 + Aq2n, we obtain, after lengthy but straightforward manipulations, the suitable form for the shear stress,

t (r,t) = /fR

(t — A)2 + A2(1 — 2e- a ) J02(R2rn )B(r,rn)

v n=i rn[Ji2(Rirn) — J2(R2rn)] t — A — — ( 1 + Aq°neg1nt — g2neq2nt . r vrn V q2n— qin

B(r,r„) = Ji(rrn)Yi(Rirn) —

Ji(Rirn)Yi(rrn).

Taking the limit of Eqs. (14) and (17) as Ar ^ 0, we obtain the solutions corresponding to a Maxwell fluid performing the same motion.

Letting now A ^ 0 in Eqs. (14) and (17), we recover the similar solutions25

vSG(r,t) = ^ (t — Ar)2 + a: M

Ri ln ( — I —

__Jo (—2rn)B(r, rn)

M^ rn[JQ(Rirn) — J0(Rirn)]

t — 2Ar —

1 — (1 + arn )2

1 + arn2

tsg (r, t) = ft2( Ri) +

_ J0(Rirn)^(r,rn)

v é^rn[JQ(Rirn) — J0(Rirn)]

t — Ar —

1 — (1 + arn )

1 + arn2

which correspond to a second grade fluid.

Finally, letting A and Ar ^ 0 in Eqs. (14) and (17) or Ar ^ 0 in Eqs. (19) and (20), we can obtain the solutions for a Newtonian fluid.

In this paper we have provided exact analytical solutions for the velocity field and the shear stress corresponding to the longitudinal flow of an Oldroyd-B fluid between two infinite coaxial circular cylinders. The motion of the fluid was produced by a time-dependent longitudinal shear stress applied on the inner cylinder boundary, while the outer cylinder was kept at rest. The solutions, obtained by means of finite Han-kel transforms, were presented in series form in terms of the Bessel functions J0(•), JL(^),Y0(•) and Yl(•). Direct computations showed that they satisfy all initial and boundary conditions. Furthermore, for Ar ^ 0 or A ^ 0 the general solutions given by Eqs. (14) and (17) reduce to the corresponding solutions for Maxwell fluids and second grade fluids, respectively. The general solutions as well as the solutions for Maxwell fluids and second grade fluids can be easily specialized to give similar solutions for Newtonian fluids performing the same motion.

In order to reveal some relevant physical aspects of the obtained results, the velocity v(r, t) and the shear stress profiles t(r, t) given by Eqs. (14) and (17) have been illustrated graphically for different values of the time t and the material parameters. As was expected, the velocity and the shear stress in absolute value are increasing functions of t as presented in Fig. 1.

Fig. 2. Profiles of v(r,t) and t(r,t), for Ri = 0.5, R2 = 0.9, f = -1, v = 0.111 2, ^ = 34.806, Ar = 2, t = 5 s, and different values of A.

Fig. 3. Profiles of v(r,t) and t(r,t), for Ri = 0.5, R2 = 0.9, f = -1, v = 0.111 2, ^ = 34.806, A = 5, t = 5 s, and different values of Ar.

Fig. 6. Profiles of v(r, t) and t(r, t), for Oldroyd-B, Maxwell and Newtonian fluids, for Ri = 0.5, R2 = 0.9, f = -2, v = 0.111 2, ^ = 34.806, A = 3, Ar = 2 and t = 63 s.

The influence of the relaxation time on the fluid motion is shown in Fig. 2. It is clear that the effects of the relaxation time A on the velocity and shear stress profiles are opposite to those depicted in Fig. 1. Figures 3 and 4 show the behavior of retardation time Ar and kinematic viscosity v on the fluid motion. It is observed that these two parameters have, as expected, similar qualitative effects on the velocity and shear stress profiles. The velocity v(r, t) is a decreasing function with respect to Ar and v while the absolute value of shear stress is an increasing one.

Finally, for comparison, the profiles of the velocity and the shear stress corresponding to the four models (Oldroyd-B, Maxwell, second grade and Newtonian) are together depicted in Figs. 5 and 6 for the same values of A, Ar and v. In all these cases, the velocity is a decreasing function of r while the shear stress is an increasing function of r. From these figures it is clearly seen that the Newtonian fluid is the swiftest and the Oldroyd-B fluid is the slowest. Figure 6 highlights the fact that for large time t the velocity as well as the shear stress profiles corresponding to the Oldroyd-B fluids, are tending, as expected, to those of a Newtonian fluid. Consequently, the non-Newtonian effects disappear in time. The units of the material parameters in Figs. 1-6 are SI units and the roots rn have been approximated by

(2n - 1)n/[2(R2 - Ri)].

The author Muhammad Jamil highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan and also Higher Education Commission of Pakistan for generous support and facilitating this research work.

The author Najeeb Alam Khan is highly thankful and, grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi-75270, Pakistan for supporting and facilitating this research work.

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