Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes II: Non-critical sizes of the holes for a volume distribution and a surface distribution of holes

Abstract

This paper is devoted to the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. For obstacles of critical size it was established in Part I that the limit problem is described by a law of Brinkman type. Here we prove that for smaller obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law. We also apply the abstract framework of Part I to the case of a domain containing tiny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size e2, located at the nodes of a regular plane mesh of period e.) This provides a mathematical model for fluid flows through mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.