A note on the regularity of the holes for permeability property through a perforated domain for the 2D Euler equations

TL;DR

This paper investigates how the boundary regularity of holes in a perforated domain affects the permeability of 2D Euler flows, showing that even with corners, irregular obstacles do not impede fluid flow when certain conditions are met.

Contribution

It extends previous results by relaxing boundary regularity conditions, demonstrating that irregular obstacles with corners do not hinder fluid permeability under specific asymptotic regimes.

Findings

Irregular obstacles with corners do not block fluid flow when  _{} o 0.

Permeability depends on the geometry of the obstacle boundaries.

Relaxed regularity conditions still ensure non-negligible permeability.

Abstract

For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much larger than the size of the holes $\varepsilon$. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when $(\varepsilon,d_{\varepsilon})\to (0,0)$. Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -- or few -- the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distance much smaller than the size of the holes. For this result, the regularity of holes boundary plays a crucial role, and the permeability criterium depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such that $\varepsilon \ln d_{\varepsilon} \to 0$.