Aspects of diffusive-relaxation dynamics with a non-uniform, partially absorbing boundary in general porous media

TL;DR

This paper investigates how inhomogeneous absorption at boundaries affects the spectral properties of diffusive-relaxation processes in porous media, revealing limitations in geometric inference methods.

Contribution

It provides theoretical, perturbative, and numerical analysis of the impact of non-uniform boundary absorption on spectral properties in porous media.

Findings

Inhomogeneous absorption significantly alters spectral characteristics in certain parameter regimes.

The study identifies conditions where boundary inhomogeneity affects the diffusion spectrum.

Results highlight limitations in using spectral data to infer boundary geometry.

Abstract

We consider the Helmholtz problem in the context of the evolution of uniform initial distribution of a physical attribute in general porous media subject to a partially absorbing boundary condition. Its spectral property as a reflection of the boundary geometry has been widely exploited, such as in biological and geophysical applications. We consider the situation where the critical assumptions which enable such applications break down. Specifically, what are the consequences of an inhomogeneous absorption strength? By means of perturbation theory, exact theoretical results, and numerical simulations on random sphere packs, we identify the regions of parameter space in which such inhomogeneity is important and those in which it is not. Our findings shed light on the issue that limits the mapping between the diffusion/relaxation spectrum and the underlying boundary geometry.