Numerical investigation of a homogenized model with effective interface conditions for Stokes flow through porous membranes

Abstract

We investigate numerically fluid flow through a thin porous layer, which is modelled as a lower-dimensional interface separating two bulk domains. At this interface, we impose effective transmission conditions involving homogenized coefficients rigorously derived via two-scale methods in previous works. These coefficients are determined by cell problems encoding the microscale properties of the porous layer. In a first step, we analyze qualitatively how the effective coefficients depend on the pore geometry. Based on these computations, we then present numerical simulations of the macroscopic models incorporating the effective interface conditions. Although replacing the thin porous layer by an effective interface substantially reduces the computational cost, the resulting nonstandard transmission conditions pose additional challenges for the numerical implementation. We employ a Taylor--Hood-type mixed finite element discretization with discontinuous pressure and tangential velocity across the interface and prove stability as well as error estimates for the resulting scheme. Finally, for microscale geometries that are still computationally resolvable, we compare the results of the macroscopic models with those obtained from direct simulations of the microscopic models.