Permeability through a perforated domain for the incompressible 2D Euler equations

TL;DR

This paper studies how small perforations in a domain affect the behavior of 2D incompressible Euler equations, showing that under certain conditions, the perforations do not influence the fluid flow as their size diminishes.

Contribution

It provides a rigorous analysis of the asymptotic behavior of 2D Euler flows in perforated domains, identifying conditions under which perforations become negligible.

Findings

Perforations do not affect the fluid flow at leading order when sufficiently small.

The influence depends on the separation distance exponent $eta$ relative to the size $oldsymbol{ ext{epsilon}}$.

The results differ for perforations arranged on a segment versus a square.

Abstract

We investigate the influence of a perforated domain on the 2D Euler equations. Small inclusions of size $\varepsilon$ are uniformly distributed on the unit segment or a rectangle, and the fluid fills the exterior. These inclusions are at least separated by a distance $\varepsilon^\alpha$ and we prove that for $\alpha$ small enough (namely, less than 2 in the case of the segment, and less than 1 in the case of the square), the limit behavior of the ideal fluid does not feel the effect of the perforated domain at leading order when $\varepsilon\to 0$.