On roll, square and hexagonal cellular flames

We consider the behaviour of a premixed flame anchored on a flat burner. For Lewis numbers L < L* < 1, stationary spatially periodic solutions corresponding to stationary cellular flames bifurcate from the basic solution which corresponds to a steady planar flame. We study the existence and stability of two-dimensional patterns which correspond to certain imposed symmetries by considering the evolution of N pairs of wave vectors, each of which is separated from the next by angle π/N. In the neighbourhood of the critical Lewis number L*, we derive evolution equations for the amplitudes corresponding to N = 2, which corresponds to square patterns, and N = 3, which corresponds to triangular or hexagonal patterns. We determine existence and stability results in terms of m(0, 1), the flow rate of the fuel, and K > 2/e, the scaled heat loss to the burner. Square patterns exist for L < L* and are stable for values of m and K above a stability boundary in the m−K plane, which has a maximum at K = K* 4.77, so that for K > K* square patterns are stable for all m. The stability of the square patterns does not vary with L. Hexagonal patterns exist for L < L_H, where 1 > L_H > L*. The size of the stability region increases with decreasing L < L*. For a range of values of L there is bistability, that is, for given parameter values rolls and hexagons are simultaneously stable, each with its own domain of attraction.