Upscaled equations for the Fokker-Planck diffusion through arrays of permeable and of impermeable inclusions

TL;DR

This paper derives upscaled Fokker-Planck equations for diffusion in periodic arrays of inclusions, revealing different limiting behaviors based on the order of limits and inclusion scale.

Contribution

It introduces new homogenized models for diffusion in arrays of permeable and impermeable inclusions considering combined degeneration and homogenization limits.

Findings

Different limiting behaviors identified: pure diffusion, diffusion with mass deposition, no diffusion.

The order of limits and the ratio of inclusion size to periodicity scale determine the diffusion behavior.

Upscaled equations provide a unified framework for understanding diffusion in complex media.

Abstract

We study the Fokker-Planck diffusion equation with diffusion coefficient depending periodically on the space variable. Inside a periodic array of inclusions the diffusion coefficient is reduced by a factor called the diffusion magnitude. We find the upscaled equations obtained by taking both the degeneration and the homogenization limits in which the diffusion magnitude and the scale of the periodicity tends, respectively, to zero. Different behaviors, classified as pure diffusion, diffusion with mass deposition, and absence of diffusion, are found depending on the order in which the two limits are taken and on the ratio between the size of the inclusions and the scale of the periodicity.