Rigorous derivation of the Darcy boundary condition on the porous wall

Abstract

An effective boundary condition on a porous wall is derived, starting from basic principles of mechanics. Stokes system, governing the viscous flow through a reservoir with an array of small pores on the boundary, was studied, and the corresponding macroscopic model via rigorous asymptotic analysis is found. Under the assumption of periodicity of the pores, the effective boundary condition of the Darcy type is derived, using homogenization and boundary layer techniques. Further asymptotic analysis with respect to the porosity yields a recursive sequence of boundary value problems showing that the large pressure jump occurs on the boundary.