Sivashinsky equation for corrugated flames in the large-wrinkle limit

Joulin, G.; Denet, B., 2008: Sivashinsky equation for corrugated flames in the large-wrinkle limit. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics 78(1 Pt 2): 016315-016315

Sivashinsky's [Acta Astron. 4, 1177 (1977)] nonlinear integrodifferential equation for the shape of corrugated one-dimensional flames is ultimately reducible to a 2N -body problem, involving the 2N complex poles of the flame slope. Thual, Frisch, and Hénon [J. Phys. (France) 46, 1485 (1985)] derived singular linear integral equations for the pole density in the limit of large steady wrinkles (N>>1) , which they solved exactly for monocoalesced periodic fronts of highest amplitude of wrinkling and approximately otherwise. Here we solve those analytically for isolated crests, next for monocoalesced, then bicoalesced periodic flame patterns, whatever the (large) amplitudes involved. We compare the analytically predicted pole densities and flame shapes to numerical results deduced from the pole-decomposition approach. Good agreement is obtained, even for moderately large Ns . The results are extended to give hints as to the dynamics of supplementary poles. Open problems are evoked.