Brinkman's law as $\Gamma$-limit of compressible low Mach Navier-Stokes equations and application to randomly perforated domains

TL;DR

This paper demonstrates that as the perforated domains become dense, the compressible Navier-Stokes equations converge to the incompressible Navier-Stokes-Brinkman equations, with applications to stochastic homogenization in random perforations.

Contribution

It establishes the $ ext{Gamma}$-limit of compressible low Mach number Navier-Stokes equations to Brinkman's law in perforated domains, including stochastic homogenization results.

Findings

Convergence of compressible Navier-Stokes to Brinkman's law in perforated domains.

Validation of the limit under stochastic homogenization.

Application to randomly perforated domain models.

Abstract

We consider the time-dependent compressible Navier-Stokes equations in the low Mach number regime inside a family of domains $(\Omega_\varepsilon)_{\varepsilon > 0}$ in $\mathbb{R}^3$. Assuming that $\lim_{\varepsilon \to 0} \Omega_\varepsilon = \Omega \subset \mathbb{R}^3$ in a suitable sense, we show that in the limit the fluid flow inside $\Omega$ is governed by the incompressible Navier-Stokes-Brinkman equations, provided the latter one admits a strong solution. The abstract convergence result is complemented with a stochastic homogenization result for randomly perforated domains in the critical regime.