Time-dependent Stokes flow through a randomly perforated porous medium. (English) Zbl 0973.76087

Summary: 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 \(\mathbb{R}^n\) that is perforated for each \(\varepsilon> 0\) by \(\varepsilon\)-dilations of a subset of \(\mathbb{R}^n\) 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 \(\varepsilon\to 0\) 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.


76S05 Flows in porous media; filtration; seepage
76M35 Stochastic analysis applied to problems in fluid mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76D07 Stokes and related (Oseen, etc.) flows
35R60 PDEs with randomness, stochastic partial differential equations