The vanishing viscosity limit in the presence of a porous medium

TL;DR

This paper investigates the behavior of viscous fluid flow around a lattice of particles as both the particles and viscosity vanish, showing that the flow approaches an inviscid, free-flow state under certain conditions.

Contribution

It provides a rigorous analysis of the vanishing viscosity limit in perforated domains, establishing convergence to Euler solutions with explicit rates.

Findings

Flow converges to Euler solutions as viscosity and particle size vanish.

Explicit convergence rates are derived in the energy norm.

Porous medium effects become negligible in the limit.

Abstract

We 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.