Analysis of enhanced diffusion in Taylor dispersion via a model problem

TL;DR

This paper mathematically explains the phenomenon of enhanced diffusion in Taylor dispersion, showing that low modes decay algebraically at an increased rate governed by a finite-dimensional center manifold, using Fourier analysis and center manifold theory.

Contribution

It provides a rigorous mathematical explanation for enhanced diffusion in Taylor dispersion, linking low mode decay to a finite-dimensional center manifold dynamics.

Findings

Low modes decay algebraically at an enhanced rate

High modes decay exponentially

Behavior governed by a finite-dimensional center manifold

Abstract

We consider a simple model of the evolution of the concentration of a tracer, subject to a background shear flow by a fluid with viscosity $\nu \ll 1$ in an infinite channel. Taylor observed in the 1950's that, in such a setting, the tracer diffuses at a rate proportional to $1/\nu$, rather than the expected rate proportional to $\nu$. We provide a mathematical explanation for this enhanced diffusion using a combination of Fourier analysis and center manifold theory. More precisely, we show that, while the high modes of the concentration decay exponentially, the low modes decay algebraically, but at an enhanced rate. Moreover, the behavior of the low modes is governed by finite-dimensional dynamics on an appropriate center manifold, which corresponds exactly to diffusion by a fluid with viscosity proportional to $1/\nu$.