Homogenization of the Navier-Stokes equations in a randomly perforated domain in the inviscid limit

TL;DR

This paper analyzes the limit behavior of Navier-Stokes solutions in a randomly perforated domain with vanishing viscosity, showing convergence to Euler or Euler-Brinkman equations depending on particle size and viscosity scaling.

Contribution

It extends homogenization results for Navier-Stokes equations from periodic to random perforations, identifying the limiting equations in different regimes.

Findings

In the subcritical regime, solutions converge to Euler equations.

In the critical regime, solutions converge to Euler-Brinkman equations.

Results depend on the local Reynolds number being small.

Abstract

We study the behaviour of the solution $u_\varepsilon$ to the Navier-Stokes equations with vanishing viscosity and a non-slip condition in a randomly perforated domain. We consider the space $\mathbb{R}^3$ where we remove $N$ holes that are i.i.d. distributed. The behaviour depends on the particle size $\varepsilon^\alpha=N^{-\alpha/3}$ and the viscosity $\varepsilon^\gamma=N^{-\gamma/3}$ of the fluid. We prove quantitative convergence results to a function $u$, provided that the local Reynolds number is small, in the subcritical ($\alpha+\gamma>3$) and critical ($\alpha+\gamma=3$) regime. In the first case, $u$ solves the Euler equations, whereas in the second case $u$ solves the Euler-Brinkman equations. This extends the results of https://doi.org/10.1088/1361-6544/acfe56 from the periodic to the random setting. We only treat the case $\alpha>2$ so that the particles do not overlap with overwhelming probability.