The Navier-Stokes equations are a good model for the flow of a wide class of fluids. The underlying constitutive model is a linear Newtonian relationship between shear stress and shear rate governed by the fluidā€™s (constant) viscosity. Known exact solutions of Navier-Stokes equation are few in number. This is in general due to the non-linearities which occur in the inertial part of these equations. However, many flow situations of interest are such that a number of terms in the equations of motion either disappear automatically or may be neglected, and the resulting equations reduce to a form that can be readily solved. There are many fluids with complex microstructure such as biological fluids, as well as polymeric liquids, suspensions, liquid crystals which are used in current industrial processes and show non-linear viscoelastic behavior that can not be described by the classical linearly Newtonian model. For such fluids the non-linearities occur not only in the inertial part but also in viscosity part of the governing equations. As a result, the number of exact solutions become rare as compared to the analytical solutions of Navier Stokes equations. But the inadequacy of the Navier-Stokes theory to describe rheological complex fluids has led to the development of several theories of non-Newtonian fluids. Amongst these, the fluids of second and third-grade have acquired a special status.
Another important aspect in the study of fluid dynamics is the consideration of magnetohydrodynamic (MHD) situation. Magnetic fields are used in many engineering applications that involve electrically conducting fluids. They are employed, for example, to drive flows, induce stirring, levitation or to suppress turbulence. Keeping the above aspects into account, this thesis consists of five chapters. In chapter zero there is a development of history and introduction of the work given in this thesis. Chapter one consists of some basic ideas and equations which will be used subsequently. The modeling of the equations which govern the MHD flow of third grade is also presented.
Exact solutions of two flow problems, namely unsteady MHD flow due to eccentric rotations of porous disk and an oscillating fluid at infinity and unsteady MHD flow due to eccentric rotations of porous disk and a fluid at infinity oscillating with different frequencies are obtained in chapter 2. It is observed that the boundary layer thickness decreases by increasing magnetic and suction parameters.
The viscous flow analysis of chapter 2 has been extended to the case of second grade fluid in chapter 3. The exact solutions of both problems are obtained using Laplace transform and perturbation techniques. It is found that boundary layer thickness increases with the increase in material parameter of the second grade fluid. The values where the second grade parameter starts influencing the velocity are determined.
Chapter 4 is devoted to the solutions of the problems presented in chapter 3 for the third grade fluid. The governing partial differential equations are non-linear for which closed form solution is impossible. Thus numerical solutions are developed. It is noted that boundary layer thickness increases with the increase in material parameter of third grade fluid.
The unsteady MHD flows due to non-coaxial rotations of a porous disk with variable axes of rotation and a fluid at infinity for the viscous and second grade fluid are presented in chapter 5. It is found that increase in suction and magnetic field causes reduction in the boundary layer thickness and blowing enhances the layer thickness in comparison to that of suction.