On the κ - θ model of cellular flames: Existence in the large and asymptotics

Abstract

Abstract. We consider the κ − θ model of flame front dynamics introduced in [6]. We show that a space-periodic problem for the latter system of two equations is globally well-posed. We prove that near the instability threshold the front is arbitrarily close to the solution of the Kuramoto-Sivashinsky equation on a fixed time interval if the evolution starts from close configurations. The dynamics generated by the model is illustrated by direct numerical simulation.2000 Mathematics Subject Classification. Primary: 35K55, 35B25; Secondary: 80A25.

Kuramoto-Sivashinsky Equation and Free-Interface Models in Combustion Theory

“…hence, the convergence to K-S, see [2,Section 3]. From a more physical viewpoint, all three of these models, K-S, κ−θ and Q-S share the same basic quality revealed by linear stability analysis, namely long-wave destabilization, which is suppressed by the dominant dissipative principal term for small wave lengths (see the discussion in [3] and Vukadinovic [36]).…”

“…Problem (2.2) is a parabolic system with smooth nonlinearities. Existence in the large can be proved thanks to the particular structure of the system: differentiating the ϕ-equation with respect to y, the coefficients of ϕ yy and θ y are the same (equal to ϕ y ) in both equations, see [3,Section 2]. It turns out that the κ-θ system is uniformly closed to the K-S equation when α=1+ε, where ε is a small positive parameter.…”

“…It turns out that the κ-θ system is uniformly closed to the K-S equation when α=1+ε, where ε is a small positive parameter. The trick in [3] is to reduce the κ−θ system to a scalar equation by expressing θ = ϕ yy − ϕ t − 1 2 ϕ 2 y from the first equation in (2.2), and substituting it into the second equation, to obtain an equivalent ultra-parabolic equation:…”

“…namely the K-S equation in the rescaled coordinates with ν = 1. The main result (see [3,Theorem 3.2]) is the existence of a constant C independent of 0 < ε ≤ ε 0 , such that:…”

Efficient Computation of Coordinate-Free Models of Flame Fronts

“…A related work is [5], in which solutions of two different weakly nonlinear models related to coordinate-free models of flame fronts are shown to remain close over time; one of these weakly nonlinear models is the KS equation. It is explicitly stated in [5] that the weakly nonlinear models are to be preferred because of the ease of numerical simulation; we demonstrate here that by our method, the full coordinate-free model may be simulated at essentially the same cost.…”