Time‐dependent Stokes flow through a randomly perforated porous medium

Abstract

An incompressible fluid is assumed to satisfy the time‐dependent Stokes equations in a porous medium. The porous medium is modeled by a bounded domain in $R^n$ that is perforated for each ε > 0 by ε‐dilations of a subset of $R^n$ arising from a family of stochastic processes which generalize the homogeneous random fields. The solution of the Stokes equations on these perforated domains is homogenized as ε → 0 by means of stochastic two‐scale convergence in the mean, and the homogenized limit is shown to satisfy a two‐pressure Stokes system containing both deterministic and stochastic derivatives and a Darcy‐type law with memory which generalizes the Darcy law obtained for fluid flow in periodically perforated porous media.