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

Abstract

In this note, we consider the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in $${{\,\mathrm{{\mathbb {R}}}\,}}^3$$ R 3 . Assuming that the particle size scales like $$\varepsilon ^3$$ ε 3 , where $$\varepsilon >0$$ ε > 0 is their mutual distance, and that the Mach number decreases fast enough, we show that in the limit $$\varepsilon \rightarrow 0$$ ε → 0 , the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of Höfer, Kowalczyk and Schwarzacher [https://doi.org/10.1142/S0218202521500391], where they proved convergence to Darcy’s law for the particle size scaling like $$\varepsilon ^\alpha $$ ε α with $$\alpha \in (1,3)$$ α ∈ ( 1 , 3 ) .