Stability of Pole Solutions for Planar Propagating Flames: I. Exact Eigenvalues and Eigenfunctions
Stability of Pole Solutions for Planar Propagating Flames: I. Exact Eigenvalues and Eigenfunctions
复制标题
平面传播火焰极点解的稳定性：I. 精确特征值和特征函数
DOI：
10.1137/s0036139998346439
复制
发表时间：
1999
期刊：
影响因子：
--
通讯作者：
M. Matalon
中科院分区：
文献类型：
--
作者：
D. Vaynblat;M. Matalon
众所周知，描述流体动力不稳定平面火焰锋面动力学的非线性偏微分方程允许精确极点解作为平衡态。这样的解对应于在实验中通常观察到的稳定传播的尖状结构。在这项工作中，我们研究了这些平衡状态的线性稳定性-稳定凝聚极点解。在以往的类似研究中，要么对截断线性系统的特征值进行数值求解，要么对线性化偏微分方程的初值问题进行数值积分，以考察初始小扰动在时间上的演化。相反，我们的结果是基于特征值和相应特征函数的精确解析表达式。本文导出了特征值和特征函数的表达式。它们的性质及其对极解稳定性的影响将在后面的一篇文章中讨论。
It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states---the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.