Stability of Pole Solutions for Planar Propagating Flames: II. Properties of Eigenvalues/Eigenfunctions and Implications to Stability
Stability of Pole Solutions for Planar Propagating Flames: II. Properties of Eigenvalues/Eigenfunctions and Implications to Stability
复制标题
平面传播火焰极点解的稳定性：II。
DOI：
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发表时间：
1999
影响因子：
1.9
通讯作者：
M. Matalon
中科院分区：
文献类型：
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作者：
D. Vaynblat;M. Matalon
在上一篇论文（第一部分）中，我们将注意力集中在火焰传播过程中出现的极点解上。平面火焰锋面变得不稳定后的非线性发展可以用一个允许极点解作为平衡态的非线性微分方程来描述。具体来说，我们关注的是聚结稳态，它对应于在无限域中周期性扩展的稳定传播的单峰结构。这种模式是在实验中经常观察到的。为了检验这些平衡解的线性稳定性，我们在第一部分中提出了相应的特征值问题，并推导了谱和相应的特征函数的精确解析表达式。在本文中，我们研究了它们的性质，因为它们与稳定性问题有关。基于解析表达式，我们的结果解决了早期稳定性问题的数值研究所引起的争议。我们证明了对于任意周期2L，总是存在且只有一个稳定的共轭极解。我们还研究了特征值和特征函数对L的依赖性，它提供了对非线性PDE行为的洞察，因此，对火焰锋面的非线性动力学。
In a previous paper (Part I) we focused our attention on pole solutions that arise in the context of flame propagation. The nonlinear development that follows after a planar flame front becomes unstable is described by a single nonlinear PDE which admits pole solutions as equilibrium states. Specifically, we were concerned with coalescent steady states, which correspond to steadily propagating single-peak structures extended periodically over the infinite domain. This pattern is one that is commonly observed in experiments. In order to examine the linear stability of these equilibrium solutions, we formulated in Part I the corresponding eigenvalue problem and derived exact analytical expressions for the spectrum and the corresponding eigenfunctions. In this paper, we examine their properties as they relate to the stability issue. Being based on analytical expressions, our results resolve earlier controversies that resulted from numerical investigations of the stability problem. We show that, for any period 2L, there always exists one and only one stable steady coalescent pole solution. We also examine the dependence of the eigenvalues and eigenfunctions on L which provides insight into the behavior of the nonlinear PDE and, consequently, on the nonlinear dynamics of the flame front.