## 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.

Read More: http://epubs.siam.org/doi/abs/10.1137/15M1006714

Read More: http://epubs.siam.org/doi/abs/10.1137/15M1006714

Original language | English |
---|---|

Pages (from-to) | 1789–1816 |

Number of pages | 28 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 75 |

Issue number | 4 |

Early online date | 20 Aug 2015 |

DOIs | |

Publication status | Published - 2015 |

## Keywords

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