The fast-time instability of diffusion flames

Abstract

A diffusion flame burns with varying intensity if the pressure is varied, and the nonlinear steady-state response is typically S-shaped; that is, multiple solutions exist for some range of pressure. The physically relevant branch can only be determined by unsteady analyses, and in this paper we discuss the stability of a large class of diffusion flames when the activation energy characterizing the reaction rate is large. Specifically, we examine the evolution from the stationary solution on a time scale so short that changes are confined to the thin flame sheet where all the reaction occurs. In this region time derivatives are added to the steady state equations, which otherwise describe a balance between diffusion and chemical reaction. The stability problem is then reduced to the determination of the spectrum of Schrödinger's equation, defined on the infinite interval, with a potential that is not of one sign on this interval. In this way certain conclusions about extinction are drawn and certain past misconceptions are clarified.