Homogenization and Dimension Reduction of the Stokes Problem with Navier-Slip Condition in Thin Perforated Layers

Abstract

We study a Stokes system posed in a thin perforated layer with a Navier-slip condition on the internal oscillating boundary from two viewpoints: (1) dimensional reduction of the layer and (2) homogenization of the perforated structure. Assuming the perforations are periodic, both aspects can be described through a small parameter , which is related to the thickness of the layer as well as the size of the periodic structure. By letting tend to zero, we prove that the sequence of solutions converges to a limit which satisfies a well-defined macroscopic problem. More precisely, the limit velocity and limit pressure satisfy a two pressure Stokes model, from which a Darcy law for thin layers can be derived. Due to nonstandard boundary conditions, some additional terms appear in Darcy’s law.