Driven particle flux through a membrane: Two-scale asymptotics of a diffusion equation with polynomial drift

TL;DR

This paper develops a two-scale asymptotic analysis to model particle diffusion through a heterogeneous membrane with periodic microstructures, deriving effective coefficients for different microstructure scales and illustrating applications to CO2 transport.

Contribution

It introduces a novel two-scale asymptotic framework for diffusion with polynomial drift through heterogeneous membranes, accounting for microstructure scale effects.

Findings

Derived effective diffusion and drift tensors for different microstructure regimes.

Identified two scaling regimes based on microstructure size relative to membrane thickness.

Numerically demonstrated the model's application to CO2 transport through paperboard.

Abstract

Diffusion of particles through an heterogenous obstacle line is modeled as a two-dimensional diffusion problem with a one--directional nonlinear convective drift and is examined using two-scale asymptotic analysis. At the scale where the structure of heterogeneities is observable the obstacle line has an inherent thickness. Assuming the heterogeneity to be made of an array of periodically arranged microstructures (e.g. impenetrable solid rectangles), two scaling regimes are identified: the characteristic size of the microstructure is either significantly smaller than the thickness of the obstacle line or it is of the same order of magnitude. We scale up the convection-diffusion model and compute the effective diffusion and drift tensorial coefficients for both scaling regimes. The upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We numerically illustrate the behavior of the upscaled membrane in the finite thickness regime and apply it to describe the transport of {\rm CO}$_2$ through paperboard.