Translational flows of an Oldroyd-B fluid with fractional derivatives

Abstract

The objective of this paper is to study the unsteady flow of an Oldroyd-B fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t = 0 + , applies a time dependent longitudinal shear stress to the fluid. Velocity field and the adequate shear stress are presented in series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional Maxwell, ordinary Maxwell, fractional second grade, ordinary second grade and Newtonian fluids performing the same motion are obtained as limiting cases of general solutions. In particular, the existing solutions for ordinary Oldroyd-B and second grade fluids are compared with the present solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between models is underlined by graphical illustrations.