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

Abstract

In this paper we analyze the stability of the traveling wave solution for an ignition-temperature first-order reaction model of diffusional-thermal combustion in the case of high Lewis numbers ($\mathrm{Le}>1$). In contrast to conventional Arrhenius kinetics where the reaction zone is infinitely thin, the reaction zone for stepwise temperature kinetics is of order unity. The system of two parabolic PDEs is characterized by a free interface at which ignition temperature ${\mathrm{\Theta }}_i$ 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 $m={\mathrm{\Theta }}_i/(1-{\mathrm{\Theta }}_i)$ and a perturbation parameter $\epsilon =1/\mathrm{Le}$. The main result is the existence of a critical value $m^c(\epsilon )$ close to $m^c=6$ at which Hopf bifurcation holds for $\epsilon$ small enough. Proofs combine spectral analysis and non-standard application of Hurwitz's Theorem with asymptotics as $\epsilon \to 0$.