LINEAR-ANALYSIS OF DIFFUSIONAL-THERMAL INSTABILITY IN-DIFFUSION FLAMES WITH LEWIS NUMBERS CLOSE TO UNITY

A general theory of diffusional-thermal instability for diffusion flam es is developed by considering the diffusion-flame regime of activatio n-energy asymptotics. Attention is focused on near-extinction flames i n a distinguished limit in which Lewis numbers deviate from unity by a small amount. This instability analysis differs from that of premixed flames in that two orders of the inner reaction-zone analyses are req uired to obtain the dispersion relation. The results, illustrated for a one-dimensional convective diffusion dame as a model, exhibit two ty pes of unstable solution branches, depending on whether Lewis number i s less than or greater than unity. For flames with Lewis numbers suffi ciently less than unity, a cellular instability is predicted, which ca n give rise to stripe patterns of the flame-quenching zones with maxim um growth rate occuring at a finite wavelength comparable with the thi ckness of the mixing layer. The result for the critical Lewis number s hows that the tendency toward cellular instability diminishes as the P eclet number of the flame decreases. On the other hand, for dames with Lewis numbers sufficiently greater than unity, a pulsating instabilit y is predicted, which occurs most strongly when the Peclet number is s mall. For this type of instability, the planar disturbance is found to be most unstable with a real grow rate, and a conjugate pair of compl ex solutions bifurcates from the turning point of the real-solution br anch and extends to higher wave numbers. An increase of the reaction i ntensity is found to stabilize the dame at all wavelengths. Employing the Peclet number as a small parameter, an approximate dispersion rela tion is derived as a quadratic equation, which correctly predicts all of the qualitative characteristics of the instability.