Evans function stability of non-adiabatic combustion waves

Abstract

In this paper we investigate the linear stability and properties of the planar travelling non-adiabatic combustion front for the cases of zero and non-zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as 'fast' and 'slow'. The Evans function approach extended by the compound-matrix method is employed to numerically solve the linear-stability problem for the travelling-wave solution. We demonstrate that the 'slow' branch of the solutions is unstable, whereas the 'fast' branch can be stable or exhibits Hopf or Bogdanov-Takens instability, depending on the parameter values.

Analysing instability of combustion waves using the Evans function

“…For convenience we assume u a = 0 . This assumption does not affect the qualitative properties of the travelling waves [7] and circumvents the cold boundary problem [16].…”

“…It can also be shown that V − (ξ), defined by (18) is the solution of (24) with the largest rate of exponential growth, µ (22) using standard tools for numerical integration, while avoiding the numerical difficulties usually encountered when dealing with stiff systems [6,12].…”

“…Equations (5)- (8) are solved numerically using shooting and relaxation methods [6]. Figure 1 illustrates how the combustion wave speed c varies with the parameter β, for the two cases Le = 5.0 and l = 1 × 10 −4 (Figure 1(a)) and l = 2 × 10 −4 ( Figure 1(b)).…”

“…We take ξ = 0 for definiteness. The foregoing discussion motivates the definition of the Evans function [2,6]:…”

Front propagation in a phase field model with phase-dependent heat absorption

“…This is a dynamical systems formulation of the stability problem which makes use of a complex valued function, called the Evans function, whose zeros correspond to the isolated eigenvalues of the linearization operator. The Evans function method has been successfully used to study linear longitudinal and transverse stability of solitary waves in various hydrodynamical contexts [6,7] and in systems describing the dynamics of chemical reactions [4] or combustion waves [12].…”