Reaction-Diffusion Front Speed Enhancement by Flows

TL;DR

This paper investigates how fluid flows can enhance the speed of reaction-diffusion fronts and quench reactions, establishing a proportional relationship in 2D flows and exploring limitations in higher dimensions.

Contribution

It proves the proportionality between front speed and effective diffusivity in 2D flows and confirms conjectures on front speed-up and quenching thresholds for cellular flows.

Findings

Front speed scales with square root of effective diffusivity in 2D.

The proportionality does not hold in three or more dimensions.

Confirmed conjectures on speed-up rates and quenching thresholds for cellular flows.

Abstract

We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction-diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from [1,3,11] for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a fixed (large) size