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 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 -> 0, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. We strongly follow the methods of Hofer, Kowalczyk and Schwarzacher [https://doi.org/10.1142/S0218202521500391], where they proved convergence to Darcy's law for the particle size scaling like epsilon(alpha) with alpha is an element of (1,3).

Automated summary

In this note, the homogenization of the compressible Navier-Stokes equations in a periodically perforated domain in R-3 is investigated. By assuming that the particle size scales with their mutual distance and that the Mach number decreases sufficiently fast, it is shown that in the limit as the size tends to zero, the velocity and density converge to a solution of the incompressible Navier-Stokes equations with Brinkman term. The methods used in the paper closely follow those of Hofer, Kowalczyk, and Schwarzacher [https://doi.org/10.1142/S0218202521500391], where the convergence to Darcy's law was proven for particle size scaling with epsilon(alpha) and alpha ∈ (1,3).