Advection-dominated transport past isolated disordered sinks: stepping beyond homogenization

TL;DR

This paper examines how advection-dominated solute transport past isolated disordered sinks deviates from homogenization predictions, revealing non-local effects, singularities, and the importance of sink distribution and size.

Contribution

It introduces a moment-expansion method to approximate corrections to homogenization in high Pe regimes with disordered sinks, validated by simulations.

Findings

Corrections can be non-local, non-smooth, and non-Gaussian.

Ensemble averaging smooths moments despite singularities at sinks.

Homogenization can be modified to account for disorder effects.

Abstract

We investigate the transport of a solute past isolated sinks in a bounded domain when advection is dominant over diffusion, evaluating the effectiveness of homogenization approximations when sinks are distributed uniformly randomly in space. Corrections to such approximations can be non-local, non-smooth and non-Gaussian, depending on the physical parameters (a P\'eclet number Pe, assumed large, and a Damk\"ohler number Da) and the compactness of the sinks. In one spatial dimension, solute distributions develop a staircase structure for large Pe, with corrections being better described with credible intervals than with traditional moments. In two and three dimensions, solute distributions are near-singular at each sink (and regularized by sink size), but their moments can be smooth as a result of ensemble averaging over variable sink locations. We approximate corrections to a homogenization approximation using a moment-expansion method, replacing the Green's function by its free-space form, and test predictions against simulation. We show how, in two or three dimensions, the leading-order impact of disorder can be captured in a homogenization approximation for the ensemble mean concentration through a modification to Da that grows with diminishing sink size.