Homogenization of a two-phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two-phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid-fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two-phase fluid in the free domain, a coupled Darcy law connecting two-phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.