Flow of a fractional Oldroyd-B fluid over a plane wall that applies a time-dependent shear to the fluid

Abstract

The unsteady flow of an incompressible Oldroyd-B fluid with fractional derivatives induced by a plane wall that applies a time-dependent shear stress fta to the fluid is studied using Fourier sine and Laplace transforms. Exact solutions for velocity and shear stress distributions are found in integral and series form in terms of generalized G functions. They are presented as a sum between the corresponding Newtonian solutions and non-Newtonian contributions and reduce to Newtonian solutions if relaxation and retardation times tend to zero. The solutions for fractional second grade and Maxwell fluids, as well as those for ordinary fluids, are obtained as limiting cases of general solutions. Finally, some special cases are considered and known solutions from the literature are recovered. An important relation with the first problem of Stokes is brought to light. The influence of fractional parameters on the fluid motion, as well as a comparison between models, is graphically illustrated.