Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 135424, 14 pages

http://dx.doi.org/10.1155/2015/135424

## Self-Similar Unsteady Flow of a Sisko Fluid in a Cylindrical Tube Undergoing Translation

^{1}Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan^{2}School of Computational and Applied Mathematics, DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa^{3}School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

Received 2 August 2014; Accepted 14 September 2014

Academic Editor: Sandile Motsa

Copyright © 2015 M. Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The governing nonlinear equation for unidirectional flow of a Sisko fluid in a cylindrical tube due to translation of the tube wall is modelled in cylindrical polar coordinates. The exact steady-state solution for the nonlinear problem is obtained. The reduction of the nonlinear initial value problem is carried out by using a similarity transformation. The partial differential equation is transformed into an ordinary differential equation, which is integrated numerically taking into account the influence of the exponent *n* and the material parameter *b* of the Sisko fluid. The initial approximation for the fluid velocity on the axis of the cylinder is obtained by matching inner and outer expansions for the fluid velocity. A comparison of the velocity, vorticity, and shear stress of Newtonian and Sisko fluids is presented.

#### 1. Introduction

Over the past few decades, use of the Newtonian fluid model to analyze and predict the behaviour of many real fluids has been extensively adopted in industry. However, the flow characteristics of many real fluids have been found to be quite different from those of the Newtonian fluid and hence researchers have proposed many non-Newtonian fluid models to explain the deviation in the behaviour of real fluids from that of the Newtonian fluid. Several rheological models of non-Newtonian fluids have been proposed to represent the viscosity function of these fluids. Amongst these models is the Sisko model [1], which is the most suitable for the flow of greases. The appropriateness of the Sisko model has been successfully extended to the shear thinning rheological behaviour of concentrated non-Newtonian slurries [2]. Some polymeric suspensions such as waterbone coating are known to be non-Newtonian in nature and follow the Sisko model [3]. The viscosity of such coatings depends on the shear rate and the strain history. Many rheological fluids such as drilling fluids and cement slurries without yield stress also obey this model.

The properties of Sisko fluids have been investigated by the study of a range of problems. In [4] the problem of a Sisko fluid in Taylor’s scraping problem has been considered and magnetohydrodynamic [MHD] peristaltic motion of a Sisko fluid in symmetric and asymmetric channels has been considered in [5]. In [6] a Lie group analysis of the boundary layer equations for a Sisko fluid has been performed, and thin film flow of non-Newtonian and second grade fluids on a moving belt has been analysed in [7, 8]. Flow of a Sisko fluid in a porous medium has also been investigated. Solutions for MHD flow have been obtained in [9] and an analysis of heat transfer of an MHD flow in a porous medium has been performed in [10]. Stokes’ first problem for a rotating Sisko fluid with porous space has been studied in [11]. The Rayleigh problem has been investigated for a rotating Sisko fluid in [12] and for an MHD Sisko fluid in [13], while Stokes’ first problem for a Sisko fluid over a porous wall has been considered in [14].

There have been several investigations of non-Newtonian and Sisko fluids in cylindrical geometry. Steady flow and heat transfer of a Sisko fluid in an annular pipe have been investigated in [15]. Exact solutions for rotating flows of a generalised Burgers’ fluid have been derived in cylindrical geometry in [16] and exact solutions for the helical flow of a generalised Oldroyd-B fluid in a circular cylinder have also been obtained in [17]. The unsteady flow of an MHD Sisko fluid between two concentric tubes due to a prescribed pressure gradient along the tube, with the tube walls at rest, has been investigated in [18]. The problem was solved numerically by using the fourth-order Runge-Kutta method. More recently the steady flow and heat transfer of an MHD fluid in the porous space between concentric tubes have been considered in [19]. The flow was due to the motion of the outer cylinder and a constant pressure gradient along the tube. An analytical solution was derived using homotopy analysis and a numerical solution was obtained by an iterative method.

In this paper we consider the flow of a Sisko fluid in a cylindrical tube due to the translation of the tube wall parallel to the axis of the tube. The velocity of the wall is not prescribed but is determined from a similarity solution of the partial differential equation. There is no inner tube on which a no-slip boundary condition could be imposed as in [16, 17]. The flow is unsteady and the fluid velocity on the axis of the tube is estimated by matching inner and outer expansions for the fluid velocity. Using this initial estimate a numerical solution for the fluid velocity is obtained by a shooting method. This numerical method for a Sisko fluid is a new feature of the paper.

In the reduction of a partial differential equation subject to initial and boundary conditions to an ordinary differential equation with boundary conditions by a similarity transformation, the initial and boundary conditions cannot be arbitrary because they must be expressible in terms of the similarity variable. The value of such a transformation is the great simplification achieved by the reduction of a partial differential equation to an ordinary differential equation. The present investigation is an extension to cylindrical geometry of the two-dimensional problem of the flow induced by an infinite sliding solid plate on a half-space of viscous fluid [20]. When the plate is impulsively set in motion with constant speed , the flow is referred to as Stokes’ first problem or the Rayleigh problem. The analytical solution was used by Rayleigh as a model to study the diffusion of vorticity in a boundary layer on a flat plate. In the two-dimensional flow the plate velocity cannot be arbitrary for a similarity solution to exist but must be a power law of time . If the velocity of the plate is proportional to then the applied stress on the plate which induces the flow is constant, while if it is proportional to , the acceleration of the plate is constant. Other power laws can be considered leading to numerical solutions. We will find that, for a similarity solution to exist for a Sisko fluid in a cylindrical tube undergoing translation, the velocity of the tube wall must depend on time in a determined way. The initial velocity of the Sisko fluid across the tube cannot be arbitrary but must have a -shaped profile. Although these conditions would be difficult to realise in practice, the physical relevance of the similarity transformation is that it does yield a model to investigate the evolution with time of a Sisko fluid in a tube undergoing translation and to study the diffusion of its vorticity and shear stress from the translating wall to the axis of the tube.

There are several ways to derive similarity solutions of partial differential equations. We will derive the similarity solution by first obtaining the Lie point symmetries of the partial differential equation. This is a powerful systematic method which does not assume a form for the solution. Only one Lie point symmetry of the partial differential equation will be used which presents the possibility of other forms of solution for different boundary conditions. Other methods could be applied which do not require a knowledge of Lie group analysis of differential equations, for example, the approach of Dresner [21].

Although it is difficult to obtain exact solutions of the equations of motion of a non-Newtonian fluid, travelling wave and similarity solutions of nonlinear equations are desirable as such solutions play a very important role in the study of nonlinear wave and fluid flow phenomena. The analytical solutions, if available, facilitate the verification of numerical solvers and are also helpful in the stability analysis of solutions. In the literature, there are very few analytical solutions for non-Newtonian fluids. It is due to the fact that the governing equations of such fluids are much more complicated and of higher order than the Navier-Stokes equations. Unlike the Navier-Stokes equations which have nonlinear terms only in the inertia term, the equations for a non-Newtonian fluid have higher order nonlinear terms in the viscous term. Although an analytical solution may not be derived, it may be possible to reduce the partial differential equations to ordinary differential equations. With this background, the investigation of unsteady flow of a Sisko fluid in a cylindrical tube subject to initial and boundary conditions is carried out in the present study.

In this paper we concentrate on the reduction and numerical solution of the partial differential equation for the unsteady flow of a Sisko fluid in a cylindrical tube. This partial differential equation is transformed into an ordinary differential equation by using one of the Lie point symmetries of the partial differential equation. Numerical solutions of the ordinary differential equation are derived for values of the exponent corresponding to a shear thinning, Newtonian, and shear thickening fluid and for a fixed value of the material parameter .

The underlying physical process that the problem seeks to clarify is diffusion in a Sisko fluid. It is an ideal problem for investigation of this process. The diffusion of velocity, vorticity, and shear stress from the wall to the axis of the tube due to the translation of the wall will be studied. The process will be illustrated by computer generated graphs.

A gap in the literature for Sisko fluids which this investigation attempts to fill is the extension of Stokes’ first problem (Rayleigh problem), from a flat plate set in motion to the wall of a cylindrical tube set in motion. For a Newtonian fluid Stokes’ first problem yielded important insights into boundary layers.

There have been investigations of steady and unsteady flow of a Sisko fluid between concentric tubes [15, 18] with the no-slip boundary condition on the inner tube. A novel feature of the present study is the absence of the inner tube for unsteady flow driven by the translation of the tube wall. A matching procedure used to obtain an estimate for the boundary condition on the axis of the cylinder and the shooting method take the place of the no-slip boundary condition. The solution is possible because a similarity transformation is found that not only reduces the partial differential equation to an ordinary differential equation, but also determines the initial condition in the form of a -shaped velocity profile that has to be imposed.

The remainder of the paper is organized as follows. Section 2 deals with the formulation of the nonlinear initial boundary value problem. In Section 3 steady-state solutions are investigated, while in Section 4 the reduction of the partial differential equation and the formulation of the problem in terms of similarity variables are given. In Section 5 the problem is reformulated as a boundary value problem suitable for numerical computation. In Section 6 the numerical results are presented and discussed. Finally concluding remarks are made in Section 7.

#### 2. Problem Formulation

Consider the unsteady unidirectional flow of an incompressible Sisko fluid in a circular cylinder parallel to the axis of the cylinder (see Figure 1). Cylindrical polar coordinates are chosen with the -axis along the axis of the cylinder. We assume the velocity, the stress fields, and pressure are of the form The fluid flow is generated by the translation of the wall of the cylindrical tube and not by a pressure gradient along the tube. The incompressibility condition is identically satisfied. The fluid flow is illustrated in Figure 1.