FKPP Fronts in Cellular Flows: The Large-Péclet Regime

Alexandra Tzella,Jacques Vanneste

doi:10.1137/15m1006714

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 Peclet number, $\text{Pe} \gg 1$) and arbitrary reaction rate (arbitrary Damkohler 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 $\text{Pe}$ an...

Highlights

In a wide variety of environmental and engineering applications, chemical or biological reactions in fluids propagate in the form of localized, strongly inhomogeneous structures associated with reactive fronts [43, 29]
We study the classic problem of Fisher–Kolmogorov– Petrovskii–Piskunov (FKPP) front propagation in a cellular flow
We examine in detail the asymptotic form of the front speed c in the limit of large Peclet number Pe corresponding to a diffusion that is weak compared to advection, and for arbitrary values of the Damkohler number Da, i.e., arbitrary reaction rate

Summary

Introduction

In a wide variety of environmental and engineering applications, chemical or biological reactions in fluids propagate in the form of localized, strongly inhomogeneous structures associated with reactive fronts [43, 29]. This eigenvalue can be interpreted in the framework of large-deviation theory: it is the Legendre dual of the rate function g(c) associated with the probability density function for the position of fluid particles that have been displaced – by advection and diffusion – to a distance ct in a time t 1 These particles control the concentration near the leading edge of the front which, by linearisation, is approximately of the form exp(−t(g(x/t) − Da)), whence the front speed c = x/t = g−1(Da) is obtained. We rely on the form (3.10) of the front speed: this makes direct contact with recent large-deviation results obtained in [20, 21] for the problem of a non-reacting passive scalar (i.e., Da = 0) in an unbounded cellular flow which we use in our treatment of Regimes I and II. This requires to analyse three distinguished regimes defined by distinct distinguished scalings of q, c and Da

Regime I

Regime II

Regime III

Findings

Conclusion