The vanishing viscosity limit in the presence of a porous medium

Abstract

International audienceWe consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. The domain is the exterior of a regular lattice of rigid particles. We study the simultaneous limit of vanishing particle size and distance, and of vanishing viscosity. Under suitable conditions on the particle size, particle distance, and viscosity, we prove that solutions of the Navier-Stokes system in the perforated domain converges to solutions of the Euler system, modeling inviscid, incompressible flow, in the full plane. That is, the flow is not disturbed by the porous medium and becomes inviscid in the limit. Convergence is obtained in the energy norm with explicit rates of convergence

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

“…By a dilatation argument and standard estimates for such a problem, we can prove the existence of h ε which satisfies the same estimate than ϕ ε i,j , namely Lemma 2.1 in [22] gives…”

“…Nevertheless, this is not a sufficient justification: Allaire showed in [3] that the scaling is the same for Navier-Stokes equations for the two types of boundary conditions (Dirichlet/Robin). We finish this introduction by referring to a recent result of Lacave and Mazzucato [22], where one considers the limit of the porous medium together with the vanishing viscosity problem.…”

“…The use of the Biot Savart law in the exterior of one inclusion allowed us to get the smallness of the stream function ψ ε on the support of ∇ϕ ε i,j . Nevertheless, there is another classical way to construct a compatible correction, as it used in [1,2,34] and more recently in [22]. The general idea is to utilize a direct cut-off function on the velocity and a correction to restore the divergence-free condition, based on Bogovskiȋ operator.…”

“…For distances d ε ε, we would need to understand the behavior of the estimates of h ε on domains of the form (−1, 1) 2 \ρK as ρ → 1 − , which is a complicated question. This explains why [2,22] need to assume that size is still smaller than the inter-hole distance (see [2,Equation (3.4.2)] and [22, Remark 2.1]).…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…In [86], the limit was established under the condition that d ǫ > ǫ and ǫ ≤ A ν; see Theorem 3.10 below. In this regime, one expects that the limit Euler flow, defined in the whole plane, does not feel the presence of the porous medium.…”