Geodesic
Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation
Claude-Michel Brauner1 ; Michael L. Frankel2 ; Josephus Hulshof3 ; Gregory I. Sivashinsky4
1Université de Bordeaux I, Talence, France
2Indiana University Purdue University Indianapolis, USA
3Vrije Universiteit, Amsterdam, Netherlands
4Tel-Aviv University, Israel
Interfaces and free boundaries, Tome 7 (2005) no. 2, pp. 131-146

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We consider a quasi-steady version of the κ-θ model of flame front dynamics introduced in [FGS03] . In this case the mathematical model reduces to a single integro-differential equation. We show that a periodic problem for the latter equation is globally well-posed in Sobolev spaces of periodic functions. We prove that near the instability threshold the solutions of the equation are arbitrarily close to these of the Kuramoto–Sivashinsky equation on a fixed time interval if the evolution starts from close configurations. We present numerical simulations that illustrate the theoretical results, and also demonstrate the ability of the quasi-steady equation to generate chaotic cellular dynamics.
DOI : 10.4171/ifb/117
Classification : 35-XX, 65-XX, 76-XX, 92-XX
Mots-clés : Combustion, premixed flame, Kuramoto-Sivashinsky equation, front dynamics, Sobolev spaces of periodic functions, stretched coordinates

Claude-Michel Brauner  1   ; Michael L. Frankel  2   ; Josephus Hulshof  3   ; Gregory I. Sivashinsky  4

1 Université de Bordeaux I, Talence, France
2 Indiana University Purdue University Indianapolis, USA
3 Vrije Universiteit, Amsterdam, Netherlands
4 Tel-Aviv University, Israel 
Claude-Michel Brauner; Michael L. Frankel; Josephus Hulshof; Gregory I. Sivashinsky. Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation. Interfaces and free boundaries, Tome 7 (2005) no. 2, pp. 131-146. doi: 10.4171/ifb/117
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     title = {Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the {Q-S} equation},
     journal = {Interfaces and free boundaries},
     pages = {131--146},
     year = {2005},
     volume = {7},
     number = {2},
     doi = {10.4171/ifb/117},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/117/}
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