Quasi-linear homogenization for large-inertia laminar transport across permeable membranes

Porous membranes are thin solid structures that allow the flow to pass through their tiny openings, called pores. Flow inertia may play a significant role in several filtration flows of natural and engineering interest. Here, we develop a predictive macroscopic model to describe solvent and solute flows past thin membranes for non-negligible inertia. We leverage homogenization theory to link the solvent velocity and solute concentration to the jumps of solvent stress and solute flux across the membrane. Within this framework, the membrane acts as a boundary separating two distinct fluid regions. These jump conditions rely on several coefficients, stemming from closure problems at the microscopic pore scale. Two approximations for the advective terms of Navier–Stokes and advection–diffusion equations are introduced to include inertia in the microscopic problem. The approximate inertial terms couple the micro- and macroscopic fields. Here, this coupling is solved numerically using an iterative fixed-point procedure. We compare the resulting models against full-scale simulations, with a good agreement both in terms of averaged values across the membrane and far-field values. Eventually, we develop a strategy based on unsupervised machine learning to improve the computational efficiency of the iterative procedure. The extension of homogenization towards weak-inertia flow configurations as well as the performed data-driven approximation may find application in preliminary analyses as well as optimization procedures towards the design of filtration systems, where inertia effects can be instrumental in broadening the spectrum of permeability and selectivity properties of these filters.