Unsteady Rotating Flows of Oldroyd-B Fluids with Fractional Derivatives

Abstract

Exact solutions corresponding to the unsteady flows of an Oldroyd-B fluid with fractional derivatives, between two infinite coaxial circular cylinders are obtained by means of Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t=0+, is applied a time dependent rotational shear stress to the fluid. The expressions of the velocity field and the shear stress are presented in series form in term of generalized G_{a,b,c}(•,t) and R_{a,b}(•,t) 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. Moreover, as a check of our calculi, our present solutions for ordinary second grade and Oldroyd-B fluids are compared with known solutions form the literature. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between the models is underlined by graphical illustrations.