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 ignitiontemperature, first-order reaction model of thermo-diffusive combustion, in the case of high Lewis numbers (Le > 1). The system of two parabolic PDEs is characterized by a free interface at which ignition temperature Θi 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 = Θi/(1 − Θi) and a perturbation parameter ε = 1/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 → 0.2010 Mathematics Subject Classification. Primary: 35R35; Secondary: 35K55, 35B35, 80A25.

Double free boundary problem for defaultable corporate bond with credit rating migration risks and their asymptotic behaviors

“…where K 0 = µ 1 if y > 0, µ 2 if y < 0. By well-known estimates for linear parabolic PDEs with discontinuous coefficients whose principal part is in divergence form (see [24,Chapter III,5]), and the proof of [28, Theorem 1.1], the claim of this lemma follows. Now, we are in position to prove the smoothness of η.…”

“…Numerical results support our theoretical approach. The stability of traveling wave equation will be studied in our future work [3] using the techniques of [1,4,5].…”

Instability of free interfaces in premixed flame propagation

“…Models with stepwise ignition-temperature kinetics (see [4]) are substantially different from those arising in conventional thermo-diffusive combustion with the standard Arrhenius kinetics at large Zeldovich number. Here, we are going to focus on a zero-order stepwise kinetics model, see [11] for a model with stepwise ignitiontemperature kinetics and a first-order reaction.…”

“…Here, 0 < θ i < 1 is the ignition temperature and A > 0 is a normalizing factor. For the first-order stepwise kinetics, the reaction rate is more standard and reads W (T, Y ) = AY H(T − θ i ), where H stands for the Heaviside function (see [4], [11]). There are two principal differences with Arrhenius kinetics.…”

“…Moreover, at the free interface(s), the temperature and mass fraction gradients are this time continuous. In this survey, we point out that the general method of [8], which was developed initially for solving thin flame problems, works equally well on thick flame models with ignition-temperature kinetics, see [1,5,11]. In this respect, the method is quite general and suitable for a wide range of free interface problems.…”