Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

Abstract

International audienceWe study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $\varepsilon$ separated by distances $d_\varepsilon$ and the fluid fills the exterior.If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $\frac{d_{\varepsilon}}\varepsilon$ when $\varepsilon$ goes to zero. If $\frac{d_{\varepsilon}}\varepsilon\to \infty$, then the limit motion is not perturbed by the porous medium, namely we recover the Euler solution in the whole space. On the contrary, if $\frac{d_{\varepsilon}}\varepsilon\to 0$, then the fluid cannot penetrate the porous region, namely the limit velocity verifies the Euler equations in the exterior of an impermeable square.If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}$ where $\gamma\in (0,\infty]$ is related to the geometry of the lateral boundaries of the obstacles. If $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty$, then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions. In particular it is equal to $\varepsilon^3$ for balls

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

“…For any β > 0, we note that the case d ε = ε β satisfies |ε ln d ε | → 0. Even if this result is a natural extension of the previous works [5,15], the proof is not obvious. Indeed, it was listed in [15] as an open question because the conformal mapping is not regular if the boundary admits a corner.…”

“…Even if this result is a natural extension of the previous works [5,15], the proof is not obvious. Indeed, it was listed in [15] as an open question because the conformal mapping is not regular if the boundary admits a corner. The behavior of solutions of elliptic problems with the respect to the boundary regularity is an important research focus in analysis of PDE's-as testified by the huge literature (see e.g.…”

“…in the regime d ε = ε when the holes are distributed on a surface. The Euler equations were treated in [5,15], where the cases of inter-holes distances smaller than the hole sizes were finally achieved. Here, the geometry of the lateral boundaries of the holes plays a role in the criterium, so we define precisely the domain properties.…”

“…In the case of holes distributed in one direction, the main results of [5,15] are as follows: if dε ε 2+ 1 γ → ∞ then the limit motion is not perturbed by the porous medium, whereas, if dε ε 2+ 1 γ → 0 then the unit segment becomes impermeable at the limit. We note here that the regimes considered correspond to close inclusions d ε ≪ ε, and it is physically natural that a fluid passes easier between disks than between flat solids.…”

“…In both results, it was crucial that γ > 0, and the case of a corner-which corresponds to γ = 0-was one of the open problems listed in [15]. More precisely, with a corner, the following questions are unsolved: 1 or any regular compact set whose the curvature is non zero and finite, see (H2) in [15] for the extension of the tangency exponent to any boundary.…”