Chemical front propagation in periodic flows: FKPP vs G

TL;DR

This study compares FKPP and G equation models for chemical front propagation in periodic flows, revealing conditions where FKPP fronts can move against flows while G fronts cannot, using a variational approach and numerical analysis.

Contribution

Introduces a variational formulation to compare FKPP and G front speeds in periodic flows, highlighting differences and conditions for opposing flow propagation.

Findings

FKPP front speed is always greater than or equal to G front speed.

Large differences occur when a strong opposing mean flow exists.

FKPP fronts can propagate against flow while G fronts cannot under certain parameters.

Abstract

We investigate the influence of steady periodic flows on the propagation of chemical fronts in an infinite channel domain. We focus on the sharp front arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large P\'eclet and Damk\"ohler numbers, $\mathrm{Pe}$ and $\mathrm{Da}$) and on its heuristic approximation by the G equation. We introduce a variational formulation that expresses the two front speeds in terms of periodic trajectories minimizing the time of travel across the period of the flow, under a constraint that differs between the FKPP and G equations. This formulation shows that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows used in experiments. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot. We verify our computations against closed-form expressions derived for $\mathrm{Da}\ll \mathrm{Pe}$ and for $\mathrm{Da}\gg \mathrm{Pe}$.