Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface

2019/01/04 by Claude-Michel Brauner, Brauner, Claude-Michel, Luca Lorenzi +3
Computer Science · Engineering · #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1901.01123

Abstract

In this paper we analyze the stability of the traveling wave solution for an ignition-temperature, first-order reaction model of thermo-diffusive combustion, in the case of high Lewis numbers (\rm Le >1). The system of two parabolic PDEs is characterized by a free interface at which ignition temperature Θ_i is reached. We turn the model to a fully nonlinear problem in a fixed domain. When the Lewis number is large, we define a bifurcation parameter m=Θ_i/(1-Θ_i) and a perturbation parameter ε= 1/\rm Le. The main result is the existence of a critical value m^c(ε) close to m^c=6 at which Hopf bifurcation holds for ε small enough. Proofs combine spectral analysis and non-standard application of Hurwitz Theorem with asymptotics as ε→ 0.

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