The Sivashinsky equation for corrugated flames in the large-wrinkle limit

TL;DR

This paper analytically solves the pole density problem for the Sivashinsky equation describing corrugated flames, extending previous work to various flame patterns and comparing results with numerical simulations.

Contribution

It provides analytical solutions for pole densities of isolated crests and periodic flame patterns in the large-wrinkle limit, enhancing understanding of flame shape dynamics.

Findings

Analytical pole densities match numerical results well.

Solutions extend to different flame pattern configurations.

Insights into pole dynamics and open problems are discussed.

Abstract

Sivashinsky's (1977) nonlinear integro-differential equation for the shape of corrugated 1-dimensional flames is ultimately reducible to a 2N-body problem, involving the 2N complex poles of the flame slope. Thual, Frisch & Henon (1985) derived singular linear integral equations for the pole density in the limit of large steady wrinkles $(N \gg 1)$, 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.