Homogenization for Inertial Particles in a Random Flow

TL;DR

This paper analyzes how inertial particles in a random flow behave over large scales and long times, showing they effectively diffuse with a calculable diffusivity tensor, supported by rigorous analysis and numerical simulations.

Contribution

It provides a rigorous homogenization framework for inertial particles in time-dependent random flows, deriving an explicit effective diffusion equation and diffusivity tensor.

Findings

Effective diffusion equation for inertial particles derived

Explicit expression and properties of the diffusivity tensor obtained

Numerical results show parametric dependence of diffusivity

Abstract

We study the problem of homogenization for inertial particles moving in a time dependent random 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. This is achieved by the use of a formal multiple scales expansion in the scale parameter. The expansion relies on the hypoellipticity of the underlying diffusion. An expression for the diffusivity tensor is found and various of its properties are studied. The results of the formal multiscale analysis are justified rigorously by the use of the martingale central limit theorem. Our theoretical findings are supported by numerical investigations where we study the parametric dependence of the effective diffusivity on the various non--dimensional parameters of the problem.