Analytical Solutions for Different Motions of Differential and Rate Type Fluids with Fractional Derivatives.

Abstract

In this dissertation, we present the analytical studies of some uid ow models. We analyze the fractional models for the ow of non-Newtonian uids via classical computational techniques to obtain analytical solutions. This study includes the investigation of the unsteady natural convection ow of Maxwell uid with fractional derivative over an exponentially accelerated in nite vertical plate. Slip condition, chemical reaction, transverse magnetic eld and Newtonian heating e ects are also considered using a modern de nition of fractional derivative. Moreover, the unsteady ow of Maxwell uid with noninteger order derivatives through a circular cylinder of in nite length in a rotating frame is studied. The motion of Maxwell uid is generated by a time dependent torsion applied to the surface of the cylinder. As novelty, the dimensionless governing equation related to the non-trivial shear stress is used and the rst exact solution analogous to a ramped shear stress on the surface is obtained. The rotational ow of an Oldroyd-B uid with fractional derivative induced by an in nite circular cylinder that applies a constant couple stress to the uid is investigated. It is worth mentioning that the considered problem of Oldroyd-B uid in the settings of fractional derivatives has not been found in the literature. Some unsteady Couette ows of an Oldroyd-B uid with non-integer derivative in an annular region of two in nite co-axial circular cylinders are investigated. Flows are due to the motion of the outer cylinder, that rotates about its axis with an arbitrary time dependent velocity while the inner cylinder is held xed. Finally, the analysis of the second grade uid with fractional derivative is made. The uid lls the annulus region between two coaxial cylinders in which one cylinder is at rest while the other experiences time dependent shear stress. In all the ow models, we obtained the exact or semi analytical solutions for the motions with technical relevance. These solutions correspond to some ows in which either velocity or the shear stress is given on the boundary are established for di erent kinds of rate and di erential type

uids. The obtained solutions presented in all the uid ow models satisfy the imposed initial and boundary conditions. Further, the ow properties and comparison of models with respect to derivative (fractional or ordinary) are highlighted by graphical illustrations.