Stokes’ first problem for an Oldroyd-B fluid in a porous half space

TLDR

The study investigates the effect of viscoelasticity on unsteady flow in porous media. Using a modified Darcy’s law and Fourier sine transform, an exact solution for Stokes’ first problem in an Oldroyd‑B fluid within a porous half‑space was derived. Unlike the classical Stokes’ first problem for a clear fluid, the Oldroyd‑B fluid in a porous half‑space exhibits a y‑dependent steady‑state solution that decays exponentially with distance from the plate, a boundary‑layer thickness that approaches a finite limit, and oscillatory velocity behavior when α>¼[(αt/Re)+Re]², with the results reducing to known Maxwell or Newtonian cases in appropriate limits.

Abstract

Based on a modified Darcy’s law for a viscoelastic fluid, Stokes’ first problem was extended to that for an Oldroyd-B fluid in a porous half space. By using Fourier sine transform, an exact solution was obtained. In contrast to the classical Stokes’ first problem for a clear fluid, there is a y-dependent steady state solution for an Oldroyd-B fluid in the porous half space, which is a damping exponential function with respect to the distance from the flat plate. The thickness of the boundary layer, which tends to be a limited value, is also different from that of a clear fluid. The effect of viscoelasticity on the unsteady flow in porous media is investigated. It was found if α>1∕4[(αt∕Re)+Re]2, oscillations in velocity occur obviously and the system exhibits viscoelastic behaviors, where α and αt are nondimensional relaxation and retardation times, respectively, Re is Reynold number in porous media. Some previous solutions of Stokes’ first problem corresponding to Maxwell fluid and Newtonian fluid in porous or nonporous half space can be easily obtained from our results in different limiting cases.