New Results in the Theory of Ordinary and Generalized non-Newtonian Fluids

Abstract

This thesis concerns with the results regarding the flow behavior of some non- Newtonian fluids under different circumstances. First of all, some concepts regarding Newtonian and non-Newtonian fluids, constitutive equations, equations of motion, and integral transforms have been discussed. Then the exact solutions for the ve- locity field and the shear stress corresponding to some flows with technical relevance have been established for second grade, Maxwell, and Oldroyd-B fluids with fractional derivatives model. In Chapter 2, the velocity field and the adequate shear stress, corresponding to the flow of a second grade fluid with fractional derivatives in an annular region, due to a constant/time-dependent shear stress, are determined by means of the Laplace and the finite Hankel transforms. The corresponding solutions for a second grade and Newtonian fluids, performing the same motion, are obtained from our general solutions. Chapter 3 deals with the motion of a Maxwell fluid with fractional derivatives, and we studied the flow starting from rest due to the sliding of the cylinder along its axis with a constant acceleration. The velocity and the adequate shear stress, obtained by means of the finite Hankel and Laplace transforms, are presented under series form in terms of the generalized G functions. The similar solutions for the ordinary Maxwell fluid, performing the same motion, are obtained as special cases of our general solution. Chapter 4 concerns with the unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives, induced by a constantly accelerating plate between two viiiix side walls perpendicular to the plate. The solutions have been studied using Fourier sine and Laplace transforms. The expressions for the velocity field and the shear stresses, written in terms of the generalized G and R functions, are presented as sum of the similar Newtonian solutions and the corresponding non-Newtonian con- tributions. Furthermore, the solutions for Maxwell fluid with fractional derivatives, ordinary Oldroyd-B, Maxwell and Newtonian fluids, performing the same motion, are also obtained as limiting cases of our general solutions. In the absence of the side walls, namely when the distance between the two walls tends to infinity, the solutions corresponding to the motion over an infinite constantly accelerating plate are recovered. Finally, the effect of the material parameters on the velocity profile is spotlighted by means of the graphical illustrations. Chapter 5 intends to establish exact and approximative expressions for dissipa- tion, the power due to the shear stress at the wall and the boundary layer thickness corresponding to the unsteady motion of a second grade fluid, induced by an infinite plate subject to a shear stress. As a limiting case of our general solutions, the similar results for Newtonian fluids performing the same motion, are obtained. The results that have been here obtained are different of those corresponding to the Rayleigh- Stokes problem. A series solution for the velocity field is also determined. Its form, as it was to be expected, is identical to that resulting from the general solution by asymptotic approximations.