Homogenization and low Mach number limit of compressible Navier-Stokes equations in critically perforated domains

TL;DR

This paper studies the homogenization and low Mach number limit of compressible Navier-Stokes equations in a periodically perforated domain, showing convergence to incompressible Navier-Stokes with Brinkman term under specific scaling assumptions.

Contribution

It extends previous results by analyzing the case where particle size scales as ^3 and the Mach number decreases rapidly, leading to convergence to an incompressible model with Brinkman correction.

Findings

Velocity and density converge to incompressible Navier-Stokes solutions with Brinkman term.

The analysis covers the critical particle size scaling of ^3.

Methodology follows and extends prior homogenization techniques.

Abstract

In this note, we consider the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in $\mathbb{R}^3$. Assuming that the particle size scales like $\varepsilon^3$, where $\varepsilon>0$ is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit $\varepsilon\to 0$, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of H\"ofer, Kowalczik and Schwarzacher [arXiv:2007.09031], where they proved convergence to Darcy's law for the particle size scaling like $\varepsilon^\alpha$ with $\alpha\in (1,3)$.