# FKPP fronts in cellular flows: The Large-Péclet Regime

Alexandra Tzella, Jacques Vanneste

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

## Abstract

We investigate the propagation of chemical fronts arising in Fisher--Kolmogorov--Petrovskii--Piskunov-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 Péclet number, $\text{Pe} \gg 1$) and arbitrary reaction rate (arbitrary Damköhler number $\text{Da}$). We identify three regimes corresponding to the distinguished limits $\text{Da} = O(\text{Pe}^{-1})$, $\text{Da}=O((\log \text{Pe})^{-1})$, and $\text{Da} = O(\text{Pe})$ and, in each regime, obtain the front speed in terms of a different nontrivial function of the relevant combination of $\text{Pe}$ and $\text{Da}$. Closed-form expressions for the speed, characterized 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.

Original language English 1789–1816 28 SIAM Journal on Applied Mathematics 75 4 20 Aug 2015 https://doi.org/10.1137/15M1006714 Published - 2015

## Keywords

• front propagation
• large deviations
• cellular
• homogenization
• Hamilton-Jacobi
• boundary layer
• WKB

## Fingerprint

Dive into the research topics of 'FKPP fronts in cellular flows: The Large-Péclet Regime'. Together they form a unique fingerprint.