The periodic solution of Stokes' second problem for viscoelastic fluids as characterized by a fractional constitutive equation

Stokes' second problem is about the steady-state oscillatory flow of a viscous fluid due to an oscillating plate. We consider Stokes' second problem for a class of viscoelastic fluids that are characterized by a fractional constitutive equation. The exact analytical solution as parametrized by the order of the fractional derivative is obtained. We provide detailed analyses and discussions for effects of the model parameters on the wave length and the amplitude in the flow field. We show that, as the order varies from 0 to 1, the flow displays a transition from elastic to viscous behavior. Finally, we consider the case of the constitutive equation for a fractional element or a spring-pot in series with a dashpot.