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 \(\delta \), and the perforation consists of \(\varepsilon \)-periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters \(\varepsilon \), \(\delta \) are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) \(\varepsilon \ll \delta \), (2) \(\delta /\varepsilon \sim \text {constant}\) and (3) \(\varepsilon \gg \delta \). 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 \(\varepsilon \) and \(\delta \) with maintained accuracy. This is illustrated by some numerical examples.