The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects

TL;DR

This paper develops a time-fractional transport model for 2D heterogeneous media with memory effects, extending homogenisation theory to compute effective diffusivity tensors, validated through numerical and real-world applications.

Contribution

It introduces a novel time-fractional transport model for homogenisation of media with memory effects, extending classical theory to include fractional dynamics and steady-state diffusivity computation.

Findings

Effective diffusivity tensor can be computed at steady state.

Memory effects significantly alter diffusivity compared to classical models.

Model validated on both synthetic and real cellular structures.

Abstract

In this work, a two-dimensional time-fractional subdiffusion model is developed to investigate the underlying transport phenomena evolving in a binary medium comprised of two sub-domains occupied by homogeneous material. We utilise an unstructured mesh control volume method to validate the model against a derived semi-analytical solution for a class of two-layered problems. This generalised transport model is then used to perform computational homogenisation on various two-dimensional heterogenous porous media. A key contribution of our work is to extend the classical homogenisation theory to accommodate the new framework and show that the effective diffusivity tensor can be computed once the cell problems reach steady state at the microscopic scale. We verify the theory for binary media via a series of well-known test problems and then investigate media having inclusions that exhibit a molecular relaxation (memory) effect. Finally, we apply the generalised transport model to estimate the bound water diffusivity tensor on cellular structures obtained from environmental scanning electron microscope (ESEM) images for Spruce wood and Australian hardwood. A highlight of our work is that the computed diffusivity for the heterogeneous media with molecular relaxation is quite different from the classical diffusion cases, being dominated at steady-state by the material with memory effects.