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

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 ${\epsilon }^3$, where $\epsilon >0$ is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit $\epsilon \to 0$, 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 ${\epsilon }^{\alpha }$ with $\alpha \in (1,3)$.