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

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