Diffusional-thermal instability of diffusion flames

Abstract

The diffusional–thermal instability, which gives rise to striped quenching patterns that have been observed for diffusion flames, is analysed by studying the model of a one-dimensional convective diffusion flame in the diffusion-flame regime of activation-energy asymptotics. Attention is focused principally on near-extinction conditions with Lewis numbers less than unity, in which the reactants with high diffusivity diffuse into the strong segments of the reaction sheet, so that the regions between the strong segments become deficient in reactant and subject to the local quenching that leads to the striped patterns. This analysis differs from other flame stability analyses in that the complete description of the dispersion relation is obtained from a composite expansion of the results of an analysis with the conventional convective-diffusive scaling and one with reaction-zone scaling. The results predict that striped patterns will occur, for flames sufficiently close to quasi-steady extinction, with a finite wavenumber that in convective–diffusive scaling is proportional to the cube root of the Zel'dovich number. The convective–diffusive response contributes to the stabilization of long-wavelength disturbances by through positive excess enthalpies by which the flame becomes more resistant to instability, while the reaction-zone response provides stabilization of short-wavelength disturbances by transverse diffusion, within the reactive inner layer, which relaxes the perturbed scalar fields towards their unperturbed states. As quasi-steady extinction is approached, marginal stability arises first at an intermediate range between these two scalings. Parametric results for this bifurcation point are obtained through numerical solutions of the associated generalized eigenvalue problems. Comparisons with measured pattern dimensions for different sets of reactants and diluents reveal excellent qualitative agreement.