Homogenization of the Navier-Stokes equations in perforated domains in the inviscid limit

Abstract

We study the solution $u_{\epsilon }$ to the Navier-Stokes equations in ${\mathbb{R}}^3$ perforated by small particles centered at $(\epsilon \mathbb{Z}{)}^3$ with no-slip boundary conditions at the particles. We study the behavior of $u_{\epsilon }$ for small $\epsilon$, depending on the diameter ${\epsilon }^{\alpha }$, $\alpha >1$, of the particles and the viscosity ${\epsilon }^{\gamma }$, $\gamma >0$, of the fluid. We prove quantitative convergence results for $u_{\epsilon }$ in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain a) the Euler-Brinkman equations in the critical regime, b) the Euler equations in the subcritical regime and c) Darcy's law in the supercritical regime.

Published version (in Nonlinearity)