Homogenization approximations for unidirectional transport past randomly distributed sinks

TL;DR

This paper develops homogenization approximations for solute transport past randomly distributed sinks, capturing spatial correlations and fluctuations, and refines classical methods for better accuracy in complex biological transport models.

Contribution

It introduces a non-standard integral approximation approach that improves upon classical two-scale methods for random sink distributions, providing explicit predictions for fluctuations and disorder effects.

Findings

Refined higher-order corrections for periodic sink arrays.

Explicit predictions for concentration fluctuations.

Validated models across diverse parameters.

Abstract

Transport in biological systems often occurs in complex spatial environments involving random structures. Motivated by such applications, we investigate an idealised model for solute transport past an array of point sinks, randomly distributed along a line, which remove solute via first-order kinetics. Random sink locations give rise to long-range spatial correlations in the solute field and influence the mean concentration. We present a non-standard approach to evaluating these features based on rationally approximating integrals of a suitable Green's function, which accommodates contributions varying on short and long lengthscales and has deterministic and stochastic components. We refine the results of classical two-scale methods for a periodic sink array (giving more accurate higher-order corrections with non-local contributions) and find explicit predictions for the fluctuations in concentration and disorder-induced corrections to the mean for both weakly and strongly disordered sink locations. Our predictions are validated across a large region of parameter space.