Spectral Analysis of a Fourth-Order Singular Differential Operator Arising in Combustion

摘要

Sivashinsky, Law, and Joulin (SLJ) present a mathematical model of stagnation-point flow combustion, where a premixed flame is stabilized in a combustible gas mixture flowing towards a flat plate. The configuration is two-dimensional Cartesian, the relevant coordinates being x and y. The plate is located in the plane y = 0. The flow, which is symmetric around x = 0, is coming in from the positive y-direction. At any point (x,y) element of (/-/infinity, infinity) /-/ (0, infinity), the flow velocity is given by the vector v = alpha (x, /-/y), where alpha > 0; alpha may be interpreted as a measure of the Reynolds number. Thus, as the flow approaches the plate, the streamlines bend outward away from the y-axis to become tangential at the plate. Under appropriate circumstances, a flame is stabilized parallel to the plate at some finite distance away from it. The flame is not necessarily planar; it may develop a spatial structure, which may evolve with time. Assuming that the position of the flame above the plate is described by a function y = /phi/(x,t), SLJ derived the following nonlinear fourth-order differential equation for /phi/: /phi//sub t/ + 4/phi//sub xxxx/ + /phi//sub xx/ /-/ 1/2 (/phi//sub x/) sup 2 + alpha (x/phi/)/sub x/ = 0, /-/infinity < x < infinity, t > 0. Our intention is to study the bifurcation of solutions for this equation from the trivial solution, using alpha as the bifurcation parameter. In this article we take the first step towards this goal and investigate the linearized equation /phi//sub t/ + 4/phi//sub xxxx/ + /phi//sub xx/ + alpha (x/phi/)/sub x/ = 0, /-/infinity < x < infinity, t > 0. 11 refs. (ERA citation 14:015602)