Homogenization and Low Mach Number Limit of Compressible Navier-Stokes Equations in Critically Perforated Domains

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