We study the motion of an ideal incompressible fluid in a perforated
domain. The porous medium is composed of inclusions of size
separated by
distances
,
and the fluid fills the exterior. We analyze the asymptotic behavior of the fluid when
.
If the inclusions are distributed on the unit square, this issue was studied recently
when
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 nonzero. 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 2-dimensional Euler equations.
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