The Asymptotic Derivation of Isola and S Responses for Chambered Diffusion Flames

Abstract

We consider the asymptotic solution of the nonlinear ordinary differential equation with two-point boundary conditions which describes a steady diffusion flame in a chamber. The solution is characterized by the unburnt fuel fraction R as a function of the fuel injection rate M. The shape of the corresponding response curve depends on the Damköhler number D (representing either the constant chamber pressure or the chamber’s length) for which two critical values, $D_0 $ and $D_a $, are uncovered. For $D < D_a $ the only possibility is $R \equiv 1$ (extinguished states). When $D_a < D < D_0 $, a closed curve called an isola is also a possible response, its base lying on $R = 0$ (complete burning). As D approaches $D_a $ the isola shrinks to a point and disappears. But when D is increased towards $D_0 $ it expands until its left side swells into the end of the strip $M > 0,0 < R < 1$. Then it breaks and joins the extinguished response (which is now limited on the left) to form an S, which is the nature of the response when $D > D_0 $. The transitions between the various responses require special asymptotics which are also described herein.