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

Abstract

We consider the time-dependent compressible Navier-Stokes equations in the low Mach number regime inside a family of domains $({\Omega }_{\epsilon }{)}_{\epsilon >0}$ in ${\mathbb{R}}^3$. Assuming that ${\lim }_{\epsilon \to 0}{\Omega }_{\epsilon }=\Omega \subset {\mathbb{R}}^3$ 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.