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Stability of pole solutions for planar propagating flames: II. Properties of eigenvalues/eigenfunctions and implications to stability

Vaynblat, Dimitri and Matalon, Moshe (2000) Stability of pole solutions for planar propagating flames: II. Properties of eigenvalues/eigenfunctions and implications to stability. SIAM Journal on Applied Mathematics, 60 (2). pp. 703-728. ISSN 0036-1399. doi:10.1137/S0036139998346440.

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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.

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Additional Information:© 2000 Society for Industrial and Applied Mathematics. Received by the editors October 29, 1998; accepted for publication March 23, 1999; published electronically February 2, 2000. This research was partially supported by the National Science Foundation under grants DMS9703716 and CTS9521022.
Subject Keywords:flame front, pole decomposition, coalescent pole solution, stability, eigenvalues and eigenfunctions, hydrodynamic instability, dynamics
Issue or Number:2
Record Number:CaltechAUTHORS:VAYsiamjam00b
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:859
Deposited By: Tony Diaz
Deposited On:31 Oct 2005
Last Modified:08 Nov 2021 19:05

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