Résumé
Analysis & PDE 15 (2022) 1131-1167 We study the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size$a$separated by distances$\tilde d$and the fluid fills the exterior. We analyse the asymptotic behavior of the fluid when$(a,\tilde d) \to (0,0)$ . If the inclusions are distributed on the unit square, this issue is studied recently when$\frac{\tilde d}a$tends to zero or infinity, leaving aside the critical case where the volume fraction of the porous medium is below its possible maximal value but non-zero. In this paper, we provide the first result in this regime. In contrast with former results, we obtain an Euler type equation where a homogenized term appears in the elliptic problem relating the velocity and the vorticity. Our analysis is based on the so-called method of reflections whose convergence provides novel estimates on the solutions to the div-curl problem which is involved in the 2D-Euler equations.