Premixed flames in closed cylindrical tubes

We consider the propagation of a premixed flame, as a two-dimensionalsheet separating unburned gas from burned products, in a closedcylindrical tube. A nonlinear evolutionequation, that describes the motion of theflame front as a function of its meanposition, is derived. The equation containsa destabilizing term that results from thegas motion induced by thermal expansionand has a memory term associated withvorticity generation. Numerical solutionsof this equation indicate that, whendiffusion is stabilizing, the flameevolves into a non-planar form whose shape, and its associated symmetry properties, aredetermined by the Markstein parameter, andby the initial data. In particular, weobserve the development of convexaxisymmetric or non-axisymmetric flames, tulip flames and cellular flames.