Darcy’s Law for Flow in a Periodic Thin Porous Medium Confined Between Two Parallel Plates

We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance between the plates is δ , and the perforation consists of ε -periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters ε , δ are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) ε δ , (2) δ /ε ~ constant and (3) ε δ . For each case, a permeability tensor is obtained by solving local problems. In the intermediate case, the cell problems are 3D, whereas they are 2D in the other cases, which is a considerable simplification. The dimensional reduction can be used for a wide range of ε and δ with maintained accuracy. This is illustrated by some numerical examples.