Homogenization of the Stokes equations with slender bodies

In this paper we study the convergence of solutions of the Stokes equations in a domain perforated by slender bodies placed on a fixed periodic grid of size $2\epsilon$ as the size of the bodies goes to zero and the number of bodies goes to infinity. We assume that the width $b_{\epsilon }$ of the bodies vanishes faster than the length $a_{\epsilon }$ of the particles. We identify the critical scaling regimes for $a_{\epsilon }$ and $b_{\epsilon }$ and prove that in this case the system is effectively described by the Brinkman equations.