Combustion fronts in non-diffusing disordered premixtures. I: Single-channel curved flames

We consider curved flames in model solid-like premixtures, when the initial reactant content and/or reactivity are maxima along the axis of straight channels. Hydrodynamics and fuel diffusion are neglected. The one-step reaction has an explicit dependence on coordinates, and follows such a generalized Arrhenius law that the flat flame problem in homogeneous media is solved exactly for any activation exponent, n. In the large-n, Arrhenius-like, limit we produce a PDE for the flame shape and speed, U. This indicates that: (i) 2D Lorentzian transverse reactivity profiles yield uniform reaction temperatures for any effective channel widths (L 1 , L 2 ), and give elliptic-paraboloidal fronts; (ii) too small L i s induce extinction, at known n-dependent values; and (iii) reactivity and initial-composition gradients play similar roles if a certain combination thereof is maintained fixed. For (1) values of n, we revisit the problem via a δ-function model - tailored to reproduce as many exact results as possible. By means of generalized elliptic coordinates, we analytically confirm and sharpen items (i)-(iii). Finite-difference direct numerical simulations, validated through comparisons with linear stability analyses, are finally developed. They also confirm (i)-(iii) for moderate ns and show how accurate the δ-model is, even then. More sensitives rates (n > 5) yield a whole hierarchy of instabilities close to extinction: Hopf bifurcation, travelling waves then hot spots, period doubling, premature extinction; yet the time-averaged U stays close to what the δ-model gave. Non-Lorentzian (e.g. Gaussian) reactivity profiles lead to nearly identical conclusions. Open problems, and implications as to larger-scale propagations in disordered media, are evoked.