Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

Abstract

We consider a free interface problem which stems from a solid-gas model in combustion with pattern formation. We derive a third-order, fully nonlinear, self-consistent equation for the flame front. Asymptotic methods reveal that the interface approaches a solution of the Kuramoto-Sivashinsky equation. Numerical results are presented which illustrate the dynamics. 2000 Mathematics Subject Classification. 35B40, 35R35, 35B35, 35K55, 80A25.

Modeling and simulation of fingering pattern formation in a combustion model

“…Hereafter in Part II, we will take asymptotically large as in Ref. 5. Thus, the parameters of the problem are α and .…”

“…3 where we perform a linear stability analysis (we refer to the forthcoming paper Ref. 6 for the nonlinear stability). Theorem 3.1 states that the threshold of linear stability occurs at α c , see (1.9).…”

“…7, we revisit the fourth order pseudo-differential equation derived in Ref. 5 within the framework of a simplified, quasi-steady version of (1.10)-(1.13): 14) with the notation γ = α − ln U , which eventually enables a simple computation of the char. …”

“…5. In this respect, we use a simplified, quasi-steady version of system (1.10)-(1.13), motivated from the fact that not far from the instability threshold, the time derivatives exhibit a relatively small effect: the dynamics appears to be essentially driven by the front.…”

“…5, in the equations satisfied by u, v and ϕ, we keep only linear and second-order terms for ϕ, and first-order terms for u and v. The final system for u and v reads:…”