Coalescence problems in the theory of expanding wrinkled premixed flames

In the framework of a Michelson-Sivashinsky (MS) evolution equation for the front shape we study cylindrically-expanding premixed flames, focusing attention on the spontaneous dynamics of collections of finite-amplitude wrinkles. To begin with, we consider sharp crests described by pole-decomposed solutions to the MS equation, for which the flame dynamics is reduced to a finite set of Complex ODE's. The latter are simplified in the limit of large flame radii to result in a restricted N-body problem (with long-range interactions) for the real angular locations of the wrinkle crests. The corresponding statistical problem of coalescence/expansion competition is treated approximately, by a mean-held method, and yields analytical predictions for the cell-size distributions vs. time. Comparisons with spectral integrations of the MS equation and with a simulation of the restricted N-body problem reveal fair agreements. Next we consider initial conditions outside the previous class, yet again representing sharp, alike crests; provided a certain crest ''weight'' is suitably fitted, the above mean-field model still gives fair predictions. However, we also show that small differences in initial conditions can suddenly induce late implants, at least when the mean flame speed is considered constant. Finally, we evoke unsolved problems e.g. difficulties about attributing the recently suggested t(3/2) flame size growth to the unfolding of initial conditions and we give hints on how to account for crest implants within a mean-field method.