On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

doi:10.1016/j.ijnonlinmec.2004.05.010

International Journal of Non-Linear Mechanics

Volume 40, Issue 1, January 2005, Pages 49-58

https://doi.org/10.1016/j.ijnonlinmec.2004.05.010 Get rights and content

Abstract

In this work we develop a mathematical model to predict the velocity profile for an unidirectional, incompressible and steady flow of an Oldroyd 6-constant fluid. The fluid is electrically conducting by a transverse magnetic field. The developed governing equation is non-linear. This equation is solved analytically to obtain the general solution. The governing non-linear equation is also solved numerically subject to appropriate boundary conditions (three cases of typical plane shearing flows) by an iterative technique with the finite-difference discretizations. A parametric study of the physical parameters involved in the problems such as the applied magnetic field and the material constants is conducted. The obtained results are illustrated graphically to show salient features of the solutions. Numerical results show that the applied magnetic field tends to reduce the flow velocity. Depending on the choice of the material parameters, the fluid exhibits shear-thickening or shear-thinning behaviours.

Introduction

Many magnetohydrodynamic problems of practical interest involving fluids as a working medium have attracted engineers, physicists and mathematicians alike. These problems pose challenges to cope with non-linearity of the governing equations, field coupling, and complex boundary conditions. Further, using Newtonian fluid models to analyse, predict and simulate the behaviour of viscoelastic fluids has been widely adopted in industries. However, the flow characteristics of viscoelastic fluids are quite different from those of Newtonian fluids. This suggests that in practical applications the behaviour of viscoelastic fluids cannot be represented by that of Newtonian fluids. Hence, it is necessary to study the flow behaviour of viscoelastic fluids in order to obtain a thorough cognition and improve the utilization in various manufactures. Amongst many models which have been used to describe the viscoelastic behaviour exhibited by these fluids, the fluids of differential type [1] and those of rate type [2] have received a great deal of attention. Various authors [3], [4], [5], [6], [7], [8] considered an Oldroyd 3-constant model, which contains as special cases some of the previous fluid models. Recently, Baris [9] examined the hydrodynamic flow of an Oldroyd 6-constant fluid in the absence of magnetic field. The problem dealing with the steady and slow flow in the wedge between intersecting planes, one of which is fixed and the other one moving, was analysed. Using truncated series expansions and a polar coordinate system, the governing equations of the problem were solved analytically subject to the relevant boundary conditions. Strictly speaking, such approximate solutions are valid only in the domain far from the corner formed by the two planes. In the present paper, the magnetohydrodynamic flow of an Oldroyd 6-constant fluid is tackled. It is concerned with steady Couette, Poiseuille and generalized Couette flows of an Oldroyd 6-constant fluid. The steady behaviour of such flows is motivated by both the fundamental interest and its practical importance. The fluid is conducting and the effect of an applied magnetic field is considered. The presented analysis is of interest because the theoretical study of magnetohydrodynamic (MHD) channel flows has widespread applications in designing cooling systems with liquid metals, MHD generators, accelerators, pumps and flow meters. Owing to the coupling of the equations of fluid mechanics and electrodynamics, the equations governing MHD non-Newtonian flows are rather cumbersome, and analytic solutions are rare. Thus, here we address the daunting task of quantifying steady flows of an Oldroyd 6-constant fluid. The important aspect of the study is that a general analytic solution is obtained for the modelled non-linear differential equation. The method of transformation is used to construct the general solution of the non-linear equation. Additionally, the numerical solutions are also carried out for steady Couette, Poiseuille and generalized Couette flows. The given analysis is general and several limiting situations with their implications can be extracted from the present problems.

Section snippets

Governing equations

The Cauchy stress $T$ in an incompressible Oldroyd 6-constant type fluid is related to the fluid motion in the following manner [10], [11]: $T = - p I + S,$ where $- p I$ is the indeterminate part of the stress due to the constraint of incompressibility. The extra stress tensor $S$ is defined by $S + λ_{1} \frac{D S}{Dt} + \frac{λ_{3}}{2} ({SA}_{1} + A_{1} S) + \frac{λ_{5}}{2} (tr S) A_{1} = μ (A_{1} + λ_{2} \frac{D A_{1}}{Dt} + λ_{4} A_{1}^{2}),$ where $μ$ , $λ_{1}$ , $λ_{2}$ , $λ_{3}$ , $λ_{4}$ , $λ_{5}$ are six material constants. $A_{1}$ is the first Rivlin–Ericksen tensor defined by $A_{1} = L + L^{T},$ where $L$ is the spatial velocity gradient, $L = grad v$ , and $v$

General solution of the non-linear equation

It is fairly difficult to get the general solution of a non-linear equation, even for a linear Newtonian fluid. We attempt to construct the general solution of the non-linear differential equation (29) by means of the method of transformation. Let $P = \frac{μ (d u / d y) + μ α_{1} (d u / d y)^{3}}{1 + α_{2} (d u / d y)^{2}} .$ Then, (29) becomes $\frac{d P}{d y} - Nu = \frac{d \hat{p}}{d x} .$ Eq. (33) can be solved for $d u / d y$ in terms of P. In order to do this we assume the transformation $\frac{\bar{d u}}{d y} = α_{1} μ \frac{d u}{d y} - \frac{α_{2}}{3} P .$ This transformation effectively gets rid of the quadratic first

Boundary value problems and numerical method

We now consider some simple shearing flows between two parallel plates at a distance H so as to acquire an idea of the effect of the material constants of the Oldroyd 6-constant fluid. All the flows considered are steady, magnetohydrodynamic, fully developed plane flows. It is assumed that the top plate at $y = H$ moves with a constant velocity $U_{0}$ in the positive x direction and the bottom plate at $y = 0$ is fixed. The appropriate non-slip boundary conditions are $\begin{matrix} u = 0, & at & y = 0, \\ u = U_{0}, & at & y = H . \end{matrix}$ We shall now write

Numerical results for some simple plane flows

In Figs. 1 and 2, the velocity profiles along the transverse y-direction are plotted for various values of the non-Newtonian material parameters $α_{1}$ and $α_{2}$ , respectively, with several given values of the pressure gradient $d \hat{p} / d x$ . The dimensionless parameter of the magnetic strength is fixed as $N = 5$ . It is easily seen from the governing differential equation (46) that if $α_{1} = α_{2}$ the Oldroyd fluid for the simple plane shearing flow behaves as a Newtonian fluid. Therefore, the solid curves in Figs. 1

Concluding remarks

In this paper, the steady MHD flows of an Oldroyd 6-constant fluid have been studied in the presence of an external uniform magnetic field. The variation of the material constants of the fluid has a significant influence on the velocity field. Likewise, the variation in the applied magnetic field has also a pronounced effect on the velocity profile. The findings are summarized as follows:

1.
The non-linear differential equation for the simple magnetohydrodynamic shearing flow of an Oldroyd

References (18)

J.E. Dunn et al.
Fluids of differential typecritical review and thermodynamic analysis
Int. J. Eng. Sci.
(1995)
T. Hayat et al.
Some simple flows of an Oldroyd-B fluid
Int. J. Eng. Sci.
(2001)
T. Hayat et al.
MHD flows of an Oldroyd-B fluid
Math. Comput. Modell.
(2002)
K. Vajravelu et al.
Hydromagnetic flow at an oscillating plate
Int. J. Non-Linear Mech.
(2003)
T. Hayat et al.
Magnetohydrodynamic flow due to non-coaxial rotations of a porous oscillating disk and a fluid at infinity
Int. J. Eng. Sci.
(2003)
S. Abel et al.
Non-Newtonian magnetohydrodynamic flow over a stretching surface with heat and mass transfer
Int. J. Non-Linear Mech.
(2004)
K.R. Rajagopal
Mechanics of non-Newtonian fluids
K.R. Rajagopal et al.
Exact solutions for some simple flows of an Oldroyd-B fluid
Acta Mech.
(1995)
K.R. Rajagopal
On an exact solution for the flow of an Oldroyd-B fluid
Bull. Tech. Univ. Istanbul.
(1996)

There are more references available in the full text version of this article.

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On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

Abstract

Introduction

Section snippets

Governing equations

General solution of the non-linear equation

Boundary value problems and numerical method

Numerical results for some simple plane flows

Concluding remarks

Int. J. Eng. Sci.

Int. J. Eng. Sci.

Math. Comput. Modell.

Int. J. Non-Linear Mech.

Int. J. Eng. Sci.

Int. J. Non-Linear Mech.

Mechanics of non-Newtonian fluids

Exact solutions for some simple flows of an Oldroyd-B fluid

Acta Mech.

On an exact solution for the flow of an Oldroyd-B fluid

Bull. Tech. Univ. Istanbul.