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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 Peclet number, Pe≫1 ) and arbitrary reaction rate (arbitrary Damkohler number Da ).
We identify three regimes corresponding to the distinguished limits Da=O(Pe −1 ) , Da=O((logPe) −1 ) and Da=O(Pe) and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of Pe and Da . Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on Da and 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 Da for Pe≫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.
We identify three regimes corresponding to the distinguished limits Da=O(Pe −1 ) , Da=O((logPe) −1 ) and Da=O(Pe) and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of Pe and Da . Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on Da and 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 Da for Pe≫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 |
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Pages (from-to) | 1789-1816 |
Journal | Siam Journal on Applied Mathematics |
Volume | 75 |
Issue number | 4 |
Early online date | 20 Aug 2015 |
DOIs | |
Publication status | Published - Sep 2015 |
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Dive into the research topics of 'FKPP fronts in cellular flows: the large-Péclet regime'. Together they form a unique fingerprint.Projects
- 1 Finished
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Passive scalars in complex fluid flows: variability and extreme events
1/10/11 → 30/11/14
Project: Research
Profiles
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Jacques Vanneste
- School of Mathematics - Personal Chair in Fluid Dynamics
Person: Academic: Research Active