FKPP fronts in cellular flows: The Large-Péclet Regime

Alexandra Tzella, Jacques Vanneste

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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.

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Original languageEnglish
Pages (from-to)1789–1816
Number of pages28
JournalSIAM Journal on Applied Mathematics
Issue number4
Early online date20 Aug 2015
Publication statusPublished - 2015


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


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