Uniqueness of traveling fronts in premixed flames with stepwise ignition-temperature kinetics and fractional reaction order

Abstract

In this paper, we consider a reaction-diffusion system describing the propagation of flames under the assumption of ignition-temperature kinetics and fractional reaction order. It was shown in [3] that this system admits a traveling front solution. In the present work, we show that this traveling front is unique up to translations. We also study some qualitative properties of this solution using the combination of formal asymptotics and numerics. Our findings allow conjecture that the velocity of the propagation of the flame front is a decreasing function of all of the parameters of the problem: ignition temperature, reaction order and an inverse of the Lewis number.

Figures (14)

• Figure 1: Sketch of a traveling front solution for system \ref{['eq:i1']} with $\alpha\ge 1$. Flame (red) and deficient reactant (blue) fronts propagating from burned ($(u,v)=(1,0)$) to unburned ($(u,v)=(0,1)$) states at $\xi\to \infty$ and $\xi\to -\infty$ respectively with the constant speed $c>0$. The arrow indicates the direction of propagation.The position of an ignition interface where the temperature is equal to an ignition one ($u=\theta$) is indicated by $\xi_{ign}$.
• Figure 2: Sketch of a traveling front solution for system \ref{['eq:i1']} with $0<\alpha< 1$. Flame (red) and deficient reactant (blue) fronts propagating from burned ($(u,v)=(1,0)$) to an unburned ($(u,v)=(0,1)$) states at $\xi\to \infty$ and $\xi\to -\infty$ respectively with the constant speed $c>0$. The arrow indicates the direction of propagation.The position of an ignition interface where the temperature is equal to an ignition one ($u=\theta$) is indicated by $\xi_{ign}.$ The position of the trailing interface, the point to the right of which temperature and concentration of deficient are identically one and zero ($u=1,v=0$), respectively is indicated by $\xi_{tr}.$
• Figure 3: Numerical solutions of problem \ref{['eq:16']} for $\alpha=1/4$ (blue), $\alpha=1/2$ (orange) and $\alpha=3/4$ (green).
• Figure 4: Functions $\phi$ and $\zeta$ for $\alpha=0.75$ (blue), $\alpha=0.5$ (orange) and $\alpha=0.25$ (green) and $\Lambda=1$. The arrow indicates direction of increase of $\alpha$.
• Figure 5: Functions $\phi$ and $\zeta$ for $\Lambda=5$ (blue), $\Lambda=1$ (orange) and $\Lambda=0.2$ (green) and $\alpha=1/2$. The arrow indicates direction of increase of $\Lambda$.

Theorems & Definitions (14)

• Theorem 2.1
• Lemma 2.1
• Proposition 2.1
• proof : Proof of Theorem \ref{['t:main']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.1
• Lemma 3.3