INSTABILITY OF POLE SOLUTIONS FOR PLANAR PROPAGATING FLAMES IN SUFFICIENTLY LARGE DOMAINS

It is well known that the partial differential equation (PDE) describi ng the dynamics of a hydrodynamically unstable planar flame front has exact pole solutions for which the PDE reduces to a set of ordinary di fferential equations (ODEs). The paradox, however, lies in the fact th at the set of ODEs does not permit the appearance of new poles in the complex plane, or the formation of cusps in the physical space, as obs erved in experiments. The validity of the PDE itself has thus been que stioned. We show here that the discrepancy between the PDE and the ODE s is due to the instability of exact pole solutions for the PDE. In pr evious work we have reported that most exact pole solutions are indeed unstable for the PDE but, for each interval of relatively small lengt h L, there remains one solution (up to translation symmetry) which is neutrally stable. The latter is a one-peck, coalescent solution for wh ich the poles (whose number is maximal) are steady. The front undergoe s bifurcations as the length of the domain considered increases: the o ne-pole, one-peak coalescent solution is first neutrally stable. As th e length of the interval increases, it becomes unstable and the two-po le one-peak coalescent solutionis, in turn, neutrally stable. This phe nomenon occurs once again: as the two-pole solution becomes unstable, the three-pole solution becomes stable. The contribution of the presen t work is to show that subsequent bifurcations are of a different natu re. As the interval length increases, the steady one-peak, coalescent solutions whose number of poles is maximal are no longer stable and bi furcations to unsteady states occur. In all cases, the appearance of n ew poles is observed in the unsteady dynamics. We also show analytical ly that such an instability is not permitted in the ODEs for which all steady one-peak, coalescent solutions are neutrally stable.