Homogenization and Dimension Reduction of the Stokes Problem with Navier-Slip Condition in Thin Perforated Layers
John Fabricius, Markus Gahn · Multiscale Modeling and Simulation · 2023
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 [Formula: see text], which is related to the thickness of the layer as well as the size of the periodic structure. By letting [Formula: see text] 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.