Linear analysis of diffusional-thermal instability in diffusion flames with Lewis numbers close to unity

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A general theory of diffusional-thermal instability for diffusion flames is developed by considering the diffusion-flame regime of activation-energy asymptotics. Attention is focused on near-extinction flames in 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 required to obtain the dispersion relation. The results, illustrated for a one-dimensional convective diffusion flame as a model, exhibit two types of unstable solution branches, depending on whether Lewis number is less than or greater than unity. For flames with Lewis numbers sufficiently less than unity, a cellular instability is predicted, which can give rise to stripe patterns of the flame-quenching zones with maximum growth rate occuring at a finite wavelength comparable with the thickness of the mixing layer. The result for the critical Lewis number shows that the tendency toward cellular instability diminishes as the Peclét number of the flame decreases. On the other hand, for flames with Lewis numbers sufficiently greater than unity, a pulsating instability is predicted, which occurs most strongly when the Peclét number is small. For this type of instability, the planar disturbance is found to be most unstable with a real grow rate, and a conjugate pair of complex solutions bifurcates from the turning point of the real-solution branch and extends to higher wave numbers. An increase of the reaction intensity is found to stabilize the flame at all wavelengths. Employing the Peclét number as a small parameter, an approximate dispersion relation is derived as a quadratic equation, which correctly predicts all of the qualitative characteristics of the instability.