Homogenization of Non-homogeneous Incompressible Navier-Stokes System in Critically Perforated Domains

TL;DR

This paper investigates the homogenization of the 3D non-homogeneous incompressible Navier-Stokes equations in perforated domains with critically sized holes, revealing convergence to a modified system with a Brinkman-type friction term as the holes become densely distributed.

Contribution

It introduces a new homogenization result for non-homogeneous Navier-Stokes in critical perforated domains, showing the emergence of a Brinkman-type friction term in the limit.

Findings

Velocity and density converge to a solution with a Brinkman friction term

Critical hole size of ^ ext{3} is essential for the homogenization result

The limit system differs from the classical Navier-Stokes due to the perforations

Abstract

In this paper, we study the homogenization of 3D non-homogeneous incompressible Navier-Stokes system in perforated domains with holes of critical size. The diameter of the holes is of size \epsilon^3, where \epsilon is a small parameter measuring the mutual distance between the holes. We show that when \epsilon tends to 0, the velocity and density converge to a solution of the non-homogeneous incompressible Navier-Stokes system with a friction term of Brinkman type.