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

Christophe Lacave, Nader Masmoudi

Research output: Contribution to journalArticle

Abstract

We 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 (Formula presented.) separated by distances (Formula presented.) and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of (Formula presented.) when (Formula presented.) goes to zero. If (Formula presented.), then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, (Formula presented.), 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 (Formula presented.) where (Formula presented.) is related to the geometry of the lateral boundaries of the obstacles. If (Formula presented.), 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 (Formula presented.) for balls.

Original languageEnglish (US)
Pages (from-to)1-44
Number of pages44
JournalArchive for Rational Mechanics and Analysis
DOIs
StateAccepted/In press - Mar 11 2016

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Perforated Domains
Euler equations
Euler Equations
Inclusion
Fluids
Porous materials
Geometry
Porous Media
Asymptotic Behavior
Distance formula
Fluid
Unit
Ideal Fluid
Motion
Zero
Incompressible Fluid
Euler
Lateral
Ball
Verify

ASJC Scopus subject areas

  • Analysis
  • Mechanical Engineering
  • Mathematics (miscellaneous)

Cite this

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abstract = "We 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 (Formula presented.) separated by distances (Formula presented.) and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of (Formula presented.) when (Formula presented.) goes to zero. If (Formula presented.), then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, (Formula presented.), 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 (Formula presented.) where (Formula presented.) is related to the geometry of the lateral boundaries of the obstacles. If (Formula presented.), 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 (Formula presented.) for balls.",
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