Arrested shear dispersion and other models of anomalous diffusion

Abstract

The macroscopic dispersion of tracer in microscopically disordered fluid flow can ultimately, at large times, be described by an advection-diffusion equation. But before this asymptotic regime is reached there is an intermediate regime in which first and second spatial moments of the distribution are proportional to tν. Conventional advection-diffusion (which applies at large times) has ν = 1 but in the intermediate regime ν < 1. This phenomenon is referred to as ‘anomalous diffusion’ and this article discusses the special case ν = ½ in detail. This particular value of ν results from tracer dispersion in a central pipe with many stagnant side branches leading away from it. The tracer is “held up” or ‘arrested’ when it wanders into the side branches and so the dispersion in the central duct is more gradual than in conventional advection-diffusion (i.e. ν = ½ < 1).This particular example serves as an entry point into a more general class of models which describe tracer arrest in closed pockets of recirculation, permeable particles, etc. with an integro-differential equation. In this view tracer is arrested and detained at a particular site for a random period. A quantity of fundamental importance in formulating a continuum model of this interrupted random walk is the distribution of stopping times at a site. Distributions with slowly decaying tails (long sojourns) produce anomalous diffusion while the conventional model results from distributions with short tails.