A Perturbation Expansion for Diffusion in a Random Velocity Field

Abstract

We present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity, $V(x)$, and constant nonrandom diffusion matrix, a, on a small scale. If $\mu $ denotes the mean of $V(x)$, and we express $V(x)$ as $\mu + \varepsilon U(x)$, then the expansion is in powers of $\varepsilon $ with $\mu $, U and a fixed. We obtain explicit expressions up to second order in $\varepsilon $ which generalize the standard formulae to the case $a \ne 0$.