Homogenization for advection-diffusion in a perforated domain

TL;DR

This paper establishes a homogenized limit for advection-diffusion in perforated domains with obstacles, analyzing the asymptotic growth rate of a Wiener sausage and introducing a Monte Carlo method for numerical computation.

Contribution

It provides a rigorous homogenization result for advection-diffusion with obstacles and introduces a Monte Carlo algorithm for estimating the volume growth rate.

Findings

The Wiener sausage volume growth rate is non-random and positive in higher dimensions.

A homogenized limit exists for diffusion with obstacles and periodic velocity fields.

Numerical results demonstrate the impact of obstacle configuration and velocity field on the growth rate.

Abstract

The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to establish the existence of a homogenized limit for such a diffusion when subject to Dirichlet conditions on the boundaries of a sparse and independent array of obstacles. There is a constant effective long-time loss rate at the obstacles. The dependence of this rate on the form and intensity of the obstacles and on the velocity field is investigated. A Monte Carlo algorithm for the computation of the volume growth rate of the sausage is introduced and some numerical results are presented for the Taylor--Green velocity field.