Abstract:
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.