Homogenization and dimension reduction of the Stokes-problem with Navier-Slip condition in thin perforated layers

TL;DR

This paper analyzes the behavior of a Stokes system in thin perforated layers with Navier-slip boundary conditions, deriving a macroscopic model through homogenization and dimension reduction as the layer thickness and perforation size tend to zero.

Contribution

It introduces a novel homogenization and dimension reduction approach for Stokes problems with Navier-slip conditions in thin perforated layers, leading to a new macroscopic Darcy law with additional boundary terms.

Findings

Solutions converge to a two-pressure Stokes model as ε→0

Derived a Darcy law with extra terms due to boundary conditions

Established the limit behavior of velocity and pressure in thin 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 $\epsilon>0,$ which is related to the thickness of the layer as well as the size of the periodic structure. By letting $\epsilon$ 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 non-standard boundary conditions, some additional terms appear in Darcy's law.