Stable Solution of Nonlinear Flame Shape Equation

Abstract A modified form of Sivashinsky's nonlinear differential equation describing the evolution of cellular flames given by was investigated. First, a boundary-value problem with an unknown initial value was formulated to find steady, ultimate flame shapes numerically. The results of numerical integrations showed that a periodic solution with respect to x existed in a region of O≤*lambda;*le;O·895 and that there was no periodic solution if A exceeded a critical value, λ, where 0·8950*lt;λ**lt;0·8975. For a value of A slightly smaller than the critical value, two different flame shapes were found (the upper and lower branch solutions). Secondly, in order to examine the stability of these solutions, an analytical form of the upper branch solution was obtained for small values of A. Finally, the stability of the upper branch solution with a small but non-vanishing value of A was investigated. The result showed the upper branch solution to be stable, and the solution exhibited aspects of cellular flames ob...