Bifurcations in non-adiabatic flame propagation models

The study of premixed laminar flames is a central question in combustion theory. Here the differential equations describing steady flame propagation are investigated mathematically. The emphasis is on analytic results, hence a simple model, a onestep exothermic reaction with linear or radiative heat loss is studied. The equations governing our model can be considered as travelling wave equations, where the unknowns are the fuel and temperature profiles and the flame velocity. The main parameters of the system are the Lewis number (ratio of thermal to mass diffusivity), the activation energy and the heat loss parameter. How the number, velocity and stability of travelling wave solutions change as the parameters are varied is studied. Much is already known about the change of planar flame velocity. The heat loss parameter – flame velocity curve has a turning point, obtained previously theoretically, by asymptotic expansions for large activation energy as well as numerically. For Lewis numbers close to unity the upper branch of the curve gives stable flames, the lower branch unstable flames, and the turning point is a saddle-node bifurcation point. For larger values of the Lewis number there is a Hopf bifurcation point on the upper branch of the curve, dividing it into stable and unstable sections. The number and stability of propagating flame solutions can also change as the parameters are varied. For a fixed value of the activation energy parameter, the Lewis number – heat loss parameter plane is divided into several regions by different bifurcation curves according to the number and stability of travelling wave solutions. These regions are the following: Lewis number/heat loss parameter pairs to which there is no flame solutions, parameter pairs to which there is a stable and an unstable planar flame, parameter pairs to which there is only an unstable planar flame and an oscillatory flame or a cellular flame. The bifurcation curves