A macroscopic model for inertial flows through thin permeable membranes

Porous membranes are thin solids that allow fluid to flow through their pores. In many natural and industrial situations, inertial effects at the pore scale play a relevant role, and the fluid and solute flow cannot be described by usual inertia-less, linear, models. In the present contribution, we develop a macroscopic predictive model describing the solvent and solute flow fields via a homogenization-based methodology in the presence of inertial effects. We homogenize the Navier-Stokes and advection-diffusion equations to obtain effective stress and flux jump conditions across the membrane, modelled as an interface separating two fluid sub-domains. The jump conditions rely on several coefficients, which stem from the solution of non-linear problems in the microscopic periodic cell, the elementary brick of the membrane structure. Because of the above-mentioned non-linearity, the microscopic and the macroscopic problems are coupled and a strategy to pass information from the macro- to the microscopic world is implemented using an iterative fixed-point scheme. The accuracy of the present method is assessed by comparison with full-scale direct numerical simulations of the solvent and solute flow, showing a substantial improvement with respect to the classic linear theory.