Coupled Transport and Adsorption in Graded Filters: A Multi-Scale Analysis of Non-Solenoidal Effects

Abstract

We investigate the transport and adsorption of solutes within graded porous filters characterised by a spatially varying microstructure. While classical homogenisation theory typically assumes periodic media, we employ the method of multiple scales to derive an effective macroscopic model for ``near-periodic'' geometries where the porosity varies slowly over the longitudinal coordinate. A key novelty of this work is the departure from the standard solenoidal constraint; instead, we introduce a modified incompressibility condition derived from non-equilibrium thermodynamics that accounts for the coupling between the solute concentration and the solvent velocity. This leads to a generalised Darcy-scale description where the fluid velocity field is non-solenoidal within the porous domain. Through asymptotic analysis, we determine the leading-order concentration profiles and quantify first-order corrections that capture the interplay between the porosity gradient and the mixture composition. We evaluate filter performance across several metrics, including outflux concentration and total adsorption rate, under both fixed-flow and fixed-pressure-drop operating conditions. Our results demonstrate that the porosity gradient and the coupling parameter significantly influence the filtration efficiency, particularly as the medium approaches the clogging limit. The analysis reveals that the optimal filter design is highly sensitive to the chosen performance metric, highlighting the necessity of physically consistent boundary conditions and mixture dynamics in the design of high-efficiency graded filters.