Advancements in the Theory of Generalized non-Newtonian Fluids

Abstract

In this work I have presented the exact solution of some non-Newtonian fluids in dif- ferent 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 deriva- tives 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 deter- mined using Laplace and finite Hankel transforms, are presented under integral and series form in terms of the generalized G a, 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 veloc- ity 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 α → 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 viiiix 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 im- posed initial and boundary conditions.Finally, the influence of the material constants on the velocity and shear stress is underlined by graphical illustrations.