Homogenization of Stokes System using Bloch Waves

TL;DR

This paper advances the understanding of fluid flow in periodic media by applying Bloch wave homogenization to the Stokes system, revealing spectral properties of the homogenized tensor despite challenges from incompressibility.

Contribution

It provides the first spectral interpretation of the homogenized tensor for the Stokes system with periodic viscosity, addressing issues caused by the incompressibility constraint.

Findings

Spectral interpretation of the homogenized tensor obtained.

Resolved issues related to non-smoothness at the bottom of the Bloch spectrum.

Completed the homogenization process for the Stokes system using Bloch waves.

Abstract

In this work, we study the Bloch wave homogenization for the Stokes system with periodic viscosity coefficient. In particular, we obtain the spectral interpretation of the homogenized tensor. The presence of the incompressibility constraint in the model raises new issues linking the homogenized tensor and the Bloch spectral data. The main difficulty is a lack of smoothness for the bottom of the Bloch spectrum, a phenomenon which is not present in the case of the elasticity system. This issue is solved in the present work, completing the homogenization process of the Stokes system via the Bloch wave method.