Interfaces and Free Boundaries


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Volume 7, Issue 2, 2005, pp. 131–146
DOI: 10.4171/IFB/117

Published online: 2005-06-30

Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation

Claude-Michel Brauner[1], Michael L. Frankel[2], Josephus Hulshof[3] and 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

We consider a quasi-steady version of the $\kappa-\theta$ model of flame front dynamics introduced in \cite{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.

Keywords: Combustion, premixed flame, Kuramoto-Sivashinsky equation, front dynamics, Sobolev spaces of periodic functions, stretched coordinates

Brauner Claude-Michel, Frankel Michael, Hulshof Josephus, Sivashinsky Gregory: Weakly nonlinear asymptotics of the kappa-theta model of cellular flames: the Q-S equation. Interfaces Free Bound. 7 (2005), 131-146. doi: 10.4171/IFB/117