Full asymptotic expansion of the permeability matrix of a dilute periodic porous medium

We compute full asymptotic expansions of the permeability matrix of a laminar fluid flowing through a periodic array of small solid particles. The derivation considers obstacles with arbitrary shape in arbitrary space dimension. In the first step, we use hydrodynamics layer potential theory to obtain the asymptotic expansion of the velocity and pressure fields across the periodic array. The terms of these expansions can be computed through a procedure involving a cascade of exterior and interior problems. In the second step, we deduce the asymptotic expansion of the permeability matrix. The derivation requires evaluating Hadamard finite part integrals and tensors depending on the values of the fundamental solution or its derivatives on the faces of the unit cell. We verify that our expansions agree to the leading order with the expressions found by Hasimoto [24] in the case of spherical obstacles in two and three dimensions.