17 - Homogenization for advection-diffusion in a perforated domain

Summary

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.

AMS subject classification (MSC2010) 60G60, 60G65, 35B27, 65C05

Introduction

We consider the problem of the existence and characterization of a homogenized limit for advection-diffusion in a perforated domain. This problem was initially motivated for us as a model for the transport of water vapour in the atmosphere, subject to molecular diffusion and turbulent advection, where the vapour is also lost by condensation on suspended ice crystals. It is of interest to determine the long-time rate of loss and in particular whether this is strongly affected by the advection. In this article we address a simple version of this set-up, where the advection is periodic in space and constant in time and where the ice crystals remain fixed in space.