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It is known now that the Navier-Stokes equations cannot describe the behaviour of fluids having high molecular weights. Due to the variety of such fluids it is very difficult to suggest a single constitutive equation which can describe the properties of all non-Newtonian fluids. Therefore many models of non-Newtonian fluids have been proposed. The flow of non-Newtonian fluids offer special challenges to the engineers, modellers, mathematicians, numerical simulists, computer scientists and physicists alike. In general the equations of non-Newtonian fluids are of higher order and much more complicated than the Newtonian fluids. The adherence boundary conditions are insufficient and one requires additional conditions for a unique solution. Also the flow characteristics of non-Newtonian fluids are quite different from those of the Newtonian fluids. Therefore, in practical applications, one cannot replace the behaviour of non-Newtonian fluids with Newtonian fluids and it is necessary to examine the flow behaviour of non-Newtonian fluids in order to obtain a thorough understanding and improve the utilization in various manufactures. Although the non-Newtonian behaviour of many fluids has been recognized for a long time, the science of rheology is, in many respects, still in its infancy, and new phenomena are constantly being discovered and new theories proposed. Analysis of fluid flow operations is typically performed by examining local conservation relations, conservation of mass, momentum and energy. This analysis gives rise to highly non-linear relationships given in terms of differential equations, which are solved using special non-linear techniques. Advancements in computational techniques are making easier the derivation of solutions to linear problems. However, it is still difficult to solve non-linear problems analytically. Engineers, chemists, physicists, and mathematicians are actively developing non-linear analytical techniques, and one such method which is known for systematically searching for exact solutions of differential equations is the Lie symmetry approach for differential equations. Lie theory of differential equations originated in the 1870s and was introduced by the Norwegian mathematician Marius Sophus Lie (1842 - 1899). However it was the Russian scientist Ovsyannikov by his work of 1958 who awakened interest in modern group analysis. Today, the Lie group approach to differential equations is widely applied in various fields of mathematics, mechanics, and theoretical physics and many results published in these area demonstrates that Lie’s theory is an efficient tool for solving intricate problems formulated in terms of differential equations. The conditional symmetry approach or what is called the non-classical symmetry approach is an extension of the Lie approach. It was proposed by Bluman and Cole 1969. Many equations arising in applications have a paucity of Lie symmetries but have conditional symmetries. Thus this method is powerful in obtaining exact solutions of such equations. Numerical methods for the solutions of non-linear differential equations are important and nowadays there several software packages to obtain such solutions. Some of the common ones are included in Maple, Mathematica and Matlab. This thesis is divided into six chapters and an introduction and conclusion. The first chapter deals with basic concepts of fluids dynamics and an introduction to symmetry approaches to differential equations. In Chapter 2 we investigate the influence of a time-dependentmagnetic field on the flow of an incompressible third grade fluid bounded by a rigid plate. Chapter 3 describes the modelling of a fourth grade flow caused by a rigid plate moving in its own plane. The resulting fifth order partial differential equation is reduced using symmetries and conditional symmetries. In Chapter 4 we present a Lie group analysis of the third oder PDE obtained by investigating the unsteady flow of third grade fluid using the modified Darcy’s law. Chapter 5 looks at the magnetohydrodynamic (MHD) flow of a Sisko fluid over a moving plate. The flow of a fourth grade fluid in a porous medium is analyzed in Chapter 6. The flow is induced by a moving plate. Several graphs are included in the ensuing discussions. Chapters 2 to 6 have been published or submitted for publication. Details are given in the references at the end of the thesis.

The motion of an incompressible fluid caused by the oscillation of a plane at plate of in nite length is termed Stokes' second problem. We assume zero velocity normal to the plate and thus simpli ed Navier-Stokes equations. For the unsteady Stokes' second problem, solutions may be obtained by using Laplace transforms, perturbation techniques, homotopy, di erential transform method or Adomian decomposition method. Stokes' second problem is discussed for second-grade and Oldroyd-B non-Newtonian fluids. This dissertation summarizes previously published work.

Graduation date: 1990

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The purpose of this thesis is to study the problem of a low Reynolds number swimmer that is in very close proximity to a wall or solid boundary in a non- Newtonian fluid. We assume that it moves by propagating waves down its length in one direction, creating a thrust and therefore propelling it in the opposite direction. We model the swimmer as an infinite, inextensible waving sheet. We consider two main cases of this swimming sheet problem. In the first case, the type of wave being propagated down the length of the swimmer is specified. We compare the swimming speeds of viscoelastic shear thinning, shear thickening and Newtonian fluids for a fixed propagating wave speed. We then compare the swimming speeds of these same fluids for a fixed rate of work per wavelength. In the latter situation, we find that a shear thinning fluid always yields the fastest swimming speed regardless of the amplitude of the propagating waves. We conclude that a shear thinning fluid is optimal for the swimmer. Analytical results are obtained for various limiting cases. Next, we consider the problem with a Bingham fluid. Yield surfaces and flow profiles are obtained. In the second case, the forcing along the length of the swimmer is specified, but the shape of the swimmer is unknown. First, we solve this problem for a Newtonian fluid. Large amplitude forcing yields a swimmer shape that has a plateau region following by a large spike region. It is found that there exists an optimal forcing that will yield a maximum swimming speed. Next, we solve the problem for moderate forcing amplitudes for viscoelastic shear thickening and shear thinning fluids. For a given forcing, it is found that a shear thinning fluid yields the fastest swimming speed when compared to a shear thickening fluid and a Newtonian fluid. The difference in swimming speeds decreases as the bending stiffness of the swimmer increases.

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