FKPP fronts in cellular flows: the large-P\'eclet regime

TL;DR

This paper analyzes the speed of chemical fronts in cellular flows under high Péclet number conditions, deriving asymptotic expressions across different reaction regimes and confirming results with numerical simulations.

Contribution

It provides a comprehensive asymptotic analysis of FKPP front speeds in cellular flows for large Péclet numbers, covering multiple reaction regimes with explicit formulas.

Findings

Derived asymptotic formulas for front speed in three distinct regimes.

Identified power-law and logarithmic dependencies on Pe and Da.

Validated asymptotic results with numerical eigenvalue solutions.

Abstract

We investigate the propagation of chemical fronts arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the presence of a steady cellular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large P\'eclet number, $\text{Pe} \gg 1$) and arbitrary reaction rate (arbitrary Damk\"ohler number $\text{Da}$). We identify three regimes corresponding to the distinguished limits $\text{Da} = O(\text{Pe}^{-1})$, $\text{Da}=O\left((\log \text{Pe})^{-1}\right)$ and $\text{Da} = O(\text{Pe})$ and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of $\text{Pe}$ and $\text{Da}$. Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on $\text{Da}$ and $\text{Pe}$ and valid in intermediate regimes, are deduced as limiting cases. Taken together, our asymptotic results provide a complete description of the complex dependence of the front speed on $\text{Da}$ for $\text{Pe} \gg 1$. They are confirmed by numerical solutions of the eigenvalue problem determining the front speed, and illustrated by a number of numerical simulations of the advection--diffusion--reaction equation.