Evans function analysis of the stability of non-adiabatic flames

Abstract

The steady propagation of a planar laminar premixed flame, with a one-step exothermic reaction and linear heat loss, is studied. The corresponding travelling wave equations are solved numerically, and the temporal stability to longitudinal perturbations of any resulting flames is investigated using the Evans function. The dependence of the flame velocity on the heat loss parameter is determined for different values of the Lewis number. These curves have a turning point, as obtained previously by asymptotic expansions for large activation energy. For Lewis numbers close to unity the upper branch of the curve gives stable flames, the lower branch unstable flames, and the turning point is a saddle-node bifurcation point. For larger values of the Lewis number there is a Hopf-bifurcation point on the upper branch of the curve, dividing it into stable and unstable sections. The saddle-node and Hopf-bifurcation curves are also determined. The two curves have a common, Takens–Bogdanov, bifurcation point.