Two-dimensional cellular burner-stabilized flames

We consider the behavior of a premixed flame anchored on a flat burner. For Lewis numbers $L < {L^*} < 1$, one-dimensional stationary spatially periodic solutions corresponding to stationary one-dimensional cellular flames (rolls) bifurcate from the basic solution which corresponds to a steady planar flame. We derive and analyze an equation for the evolution of the amplitude of the roll solution just beyond the critical Lewis number ${L^*}$. That is, we consider the case of supercritical bifurcation $\left ( {L < {L^*}} \right )$ and determine the ranges of wave numbers of perturbations corresponding to both the Eckhaus instability (to longitudinal perturbations) and the zigzag instability (to transverse perturbations) of the bifurcating solution. We determine these ranges in terms of the flow rate $m \in \left ( 0, 1 \right )$ and the scaled heat loss to the burner $K > 2/e$. For wave numbers $k < 0.25$ we find that the zigzag instability occurs for all allowed values of $K$ and for $m$ bounded away from 1 and 0. As $k$ increases, the range of values of $m$ and $K$ for which this instability occurs decreases. For $k \ge 0.4$ the zigzag instability no longer occurs for any allowed value of $m$ and $K$. For each value of $L$ there is a minimum value $m = {m_*}\left ( L \right )$ above which the Eckhaus instability does not occur. As $L$ approaches ${L^*}, {m_*}\left ( L \right )$ increases.