Flame Propagation in Vertical Channels: Bifurcation to Bimodal Cellular Flames

We derive a nonlinear evolution equation and associated boundary conditions which, under certain circumstances, describe downward adiabatic flame propagation in long, vertical channels. We analyze this problem for the case of axial symmetry and show that there exist two critical parameters, the channel radius R and a gravitational acceleration parameter G, which determine the stability and bifurcation characteristics of the flame. In particular, we show that there exist discrete critical points in the $(G,R)$ plane at which either one or three solutions corresponding to bimodal cellular flames bifurcate from the basic planar solution. We perform a nonlinear stability and bifurcation analysis in arbitrary neighborhoods of these critical points and show that either zero, one or two of the branches are stable, and the remaining branches are unstable. We thus describe new stable modes of cellular flame propagation.