Periodic Homogenization for Inertial Particles

TL;DR

This paper investigates the homogenization of inertial particles in periodic velocity fields with diffusion, deriving an effective diffusion equation and analyzing the diffusivity tensor, including effects of particle inertia and numerical validation.

Contribution

It introduces a homogenization framework for inertial particles using multiple scale expansions, extending passive tracer results to include particle inertia effects.

Findings

Effective diffusion equation for inertial particles derived

Expression and properties of the diffusivity tensor established

Numerical simulations support theoretical results

Abstract

We study the problem of homogenization for inertial particles moving in a periodic velocity field, and subject to molecular diffusion. We show that, under appropriate assumptions on the velocity field, the large scale, long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone. To achieve this we use a formal multiple scale expansion in the scale parameter. This expansion relies on the hypo-ellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties studied. In particular, an expansion in terms of the non-dimensional particle relaxation time $\tau$ (the Stokes number) is shown to co-incide with the known result for passive (non-inertial) tracers in the singular limit $\tau \to 0$. This requires the solution of a singular perturbation problem, achieved by means of a formal multiple scales expansion in $\tau.$ Incompressible and potential fields are studied, as well as fields which are neither, and theoretical findings are supported by numerical simulations.