A theoretical derivation of cellular structures of flames

Using an equation for the formation of flame fronts derived by Sivashinsky and augmented by an additional term describing buoyancy effects we present an analytical treatment of the formation of cellular structures of flames formed by plane burners. In particular we find rectangular and square patterns. We first study the stability of the plane flame front by linear stability analysis and then transform the basic equation into a set of equations for the amplitudes of the stable and unstable modes. The amplitudes of the stable modes can be eliminated by the slaving principle so that generalized Ginzburg-Landau equations result which in a general frame were previously derived by one of us (H.H.). These equations are then solved explicitly and the stability of the resulting pattern is proven.