# 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.

## Key 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 $\varepsilon \ln d_{\varepsilon} \to 0$.

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Source: https://tomesphere.com/paper/1904.05645