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Annali di Matematica Pura ed Applicata

, Volume 183, Issue 2, pp 173–239 | Cite as

Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

  • Claudia LedermanEmail author
  • Jean-Michel Roquejoffre
  • Noemi Wolanski
Article

Abstract

This paper is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [17]. The paper [17] derives, by means of a three-scale matched asymptotics, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis Number – i.e. the ratio between thermal and molecular diffusion – to be strictly less than unity. If ε is the inverse of the – reduced – activation energy, the idea underlying the construction of [17] is that (i) the time scale of the radius motion is ε-2, and that (ii) at each time step, the solution is ε-close to a steady solution.

In this paper, we give a rigorous proof of the validity of this model under the restriction that the Lewis number is close to 1 – independently of the order of magnitude of the activation energy. The method used comprises three steps: (i) a linear stability analysis near a steady – or quasi-steady – solution, which justifies the fact that the relevant time scale is ε-2; (ii) the rigorous construction of an approximate solution; (iii) a nonlinear stability argument.

Keywords

half derivatives high activation energies linear and nonlinear stability combustion 

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  • Claudia Lederman
    • 1
    Email author
  • Jean-Michel Roquejoffre
    • 2
  • Noemi Wolanski
    • 1
  1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Université Paul SabatierToulouseFrance

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