Title of Thesis

Advancements in the Theory of Generalized Non-Newtonian Fluids

Author(s)

Muhammad Athar

Abstract
In this work I have presented the exact solution of some non-Newtonian fluids in different situation after some preliminaries regarding continuity equation, constitutive equation, equation of motion and integral transforms, the newly exact solutions for second grade fluid with fractional derivatives, Maxwell fluid with fractional derivatives and ordinary Oldroyd-B fluid have been found in which we have calculated the velocity and shear stress.
In chapter 2, we established exact solutions for the velocity field and shear stress corresponding to the flow of a second grade fluid with fractional derivatives (SGFFD) between two infinite circular cylinders due to an azimuthal constant/time-dependent shear stress on the surface of the inner cylinder. The solutions that have been determined using Laplace and finite Hankel transforms, are presented under integral and series form in terms of the generalized Ga; b; c(: ; :) functions.
In chapter 3, we present exact solutions for the unsteady flow of a Maxwell fluid with fractional derivatives (MFFD) due to a constantly accelerating plate. The velocity field and the adequate shear stress corresponding to the unsteady flow of a MFFD are determined using Fourier sine and Laplace transforms. They are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. Graphical illustrations show that the velocity profiles corresponding to a MFFD are going to that for an ordinary Maxwell fluid if → 1.
In chapter 4, we succeeded to provide exact solutions for the unsteady flow of a MFFD between two side walls perpendicular to a plate. The motion is produced by the infinite plate that at time t = 0+ begins to slide into its plane with a constant acceleration A. The exact solutions for the velocity field and the adequate shear stresses, obtained by means of Fourier sine and Laplace transforms, are presented in terms of the generalized Mittag-Leffler functions. In the absence of the side walls, all solutions that have been obtained reduce to the solutions corresponding to the motion over an infinite constantly accelerating plate.
Chapter 5 deals with the study of unsteady rotational flow of an Oldroyd-B fluid in an annular pipe. The motion of the fluid is produced by the inner cylinder that, at the initial moment, is subject to a time dependent couple per unit length. The exact solutions, obtained by means of Laplace and finite Hankel transforms, satisfy all imposed initial and boundary conditions.Finally, the influence of the material constants on the velocity and shear stress is underlined by graphical illustrations.

1

1.1 Newtonian and non-Newtonian fluids
1.2 The fluids of rate type
1.3 Differential type fluids
1.4 Constitutive equations
1.5 Continuity equation
1.6 Equations of motion
1.7 Some integral transforms

2.1 Introduction
2.2 Governing equations
2.3 Taylor-Couette flow due to a constant shear on the boundary
2.4 The special case → 1 (Second grade fluid)
2.5 Taylor-Couette flow due to a time-dependent shear on the boundary
2.6 The special case → 1 (Second grade fluid)
2.7 Conclusions

3.1 Introduction
3.2 Governing equations
3.3 Formulation and solution of the problem
3.4 Conclusions

4.1 Introduction
4.2 Governing equations
4.3 Statement of the problem
4.4 Exact solutions
4.5 The special case = 1 (Maxwell fluid)
4.6 Newtonian case
4.7 Limiting case h → ∞ (Flow over an infinite plate)
4.8 Conclusions

5.1 Introduction
5.2 Governing equations
5.3 Rotational flow in an annulus due to a time dependent couple
5.4 Limiting cases
5.5 Numerical results and conclusions