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

This paper is devoted to the homogenizat ion of the Stokes or Navier-Stokes equations with a Dirichlet boundary condit ion 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 f ramework of Par t I to the case of a domain containing t iny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size e 2, located at the nodes of a regular plane mesh of period e.) This provides a mathematical model for fluid flows th rough mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.