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

TL;DR

This paper analyzes the asymptotic behavior of solutions to the Navier-Stokes equations in perforated domains with small particles, deriving effective macroscopic equations depending on particle size and viscosity regimes.

Contribution

It provides a rigorous derivation of homogenized equations for fluid flow in perforated domains, revealing different regimes like Euler, Euler-Brinkman, and Darcy laws based on particle parameters.

Findings

In the negligible local Reynolds number regime, solutions converge to effective equations.

Particles exert a linear friction force on the fluid in the limit.

Different macroscopic models emerge depending on particle size and viscosity regimes.

Abstract

We study the solution $u_\varepsilon$ to the Navier-Stokes equations in $\mathbb R^3$ perforated by small particles centered at $(\varepsilon \mathbb Z)^3$ with no-slip boundary conditions at the particles. We study the behavior of $u_\varepsilon$ for small $\varepsilon$, depending on the diameter $\varepsilon^\alpha$, $\alpha > 1$, of the particles and the viscosity $\varepsilon^\gamma$, $\gamma > 0$, of the fluid. We prove quantitative convergence results for $u_\varepsilon$ 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.