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Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

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Abstract

The authors consider a free interface problem which stems from a gas-solid model in combustion with pattern formation. A third-order, fully nonlinear, self-consistent equation for the flame front is derived. Asymptotic methods reveal that the interface approaches a solution to the Kuramoto-Sivashinsky equation. Numerical results which illustrate the dynamics are presented.

Keywords

Asymptotics Free interface Kuramoto-Sivashinsky equation Pseudo-differential operator Spectral method 

Mathematics Subject Classification

35B40 35R35 35B35 35K55 80A25 

References

  1. 1.
    Berestycki, H., Brauner, C.-M., Clavin, P., et al.: Modélisation de la Combustion, Images des Mathématiques. CNRS, Paris (1996). Special Issue Google Scholar
  2. 2.
    Brauner, C.-M., Frankel, M.L., Hulshof, J., et al.: On the κ-θ model of cellular flames: existence in the large and asymptotics. Discrete Contin. Dyn. Syst., Ser. S 1, 27–39 (2008) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brauner, C.-M., Frankel, M.L., Hulshof, J., Sivashinsky, G.I.: Weakly nonlinear asymptotics of the κ-θ model of cellular flames: the Q-S equation. Interfaces Free Bound. 7, 131–146 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brauner, C.-M., Hulshof, J., Lorenzi, L.: Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinet. Relat. Models 2, 109–134 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brauner, C.-M., Hulshof, J., Lorenzi, L.: Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem. Interfaces Free Bound. 13, 73–103 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brauner, C.-M., Hulshof, J., Lorenzi, L., Sivashinsky, G.I.: A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete Contin. Dyn. Syst., Ser. A 27, 1415–1446 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brauner, C.-M., Lorenzi, L., Sivashinsky, G.I., Xu, C.-J.: On a strongly damped wave equation for the flame front. Chin. Ann. Math. 31B(6), 819–840 (2010) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brauner, C.-M., Lunardi, A.: Instabilities in a two-dimensional combustion model with free boundary. Arch. Ration. Mech. Anal. 154, 157–182 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buckmaster, J.D., Ludford, G.S.S.: Theory of Laminar Flames. Cambridge University Press, Cambridge (1982) CrossRefzbMATHGoogle Scholar
  10. 10.
    Eckhaus, W.: Asymptotic Analysis of Singular Perturbations. Studies in Mathematics and Its Applications, vol. 9. North-Holland, Amsterdam (1979) zbMATHGoogle Scholar
  11. 11.
    Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser, Basel (2006) CrossRefzbMATHGoogle Scholar
  12. 12.
    Hyman, J.M., Nicolaenko, B.: The Kuramoto-Sivashinsky equation: a bridge between PDEs and dynamical systems. Physica D 18, 113–126 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kagan, L., Sivashinsky, G.I.: Pattern formation in flame spread over thin solid fuels. Combust. Theory Model. 12, 269–281 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lions, J.-L.: Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. Lect. Notes in Math., vol. 323. Springer, Berlin (1970) Google Scholar
  15. 15.
    Lorenzi, L.: Regularity and analyticity in a two-dimensional combustion model. Adv. Differ. Equ. 7, 1343–1376 (2002) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lorenzi, L.: A free boundary problem stemmed from combustion theory. I. Existence, uniqueness and regularity results. J. Math. Anal. Appl. 274, 505–535 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lorenzi, L.: A free boundary problem stemmed from combustion theory. II. Stability, instability and bifurcation results. J. Math. Anal. Appl. 275, 131–160 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lorenzi, L.: Bifurcation of codimension two in a combustion model. Adv. Math. Sci. Appl. 14, 483–512 (2004) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lorenzi, L., Lunardi, A.: Stability in a two-dimensional free boundary combustion model. Nonlinear Anal. 53, 227–276 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lorenzi, L., Lunardi, A.: Erratum: “Stability in a two-dimensional free boundary combustion model”. Nonlinear Anal. 53(6), 859–860 (2003). Nonlinear Anal. 53(2), 227–276 (2003). MR1959814 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel (1995) CrossRefzbMATHGoogle Scholar
  22. 22.
    Matkowsky, B.J., Sivashinsky, G.I.: An asymptotic derivation of two models in flame theory associated with the constant density approximation. SIAM J. Appl. Math. 37, 686–699 (1979) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sivashinsky, G.I.: On flame propagation under conditions of stoichiometry. SIAM J. Appl. Math. 39, 67–82 (1980) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sivashinsky, G.I.: Instabilities, pattern formation and turbulence in flames. Annu. Rev. Fluid Mech. 15, 179–199 (1983) CrossRefGoogle Scholar
  25. 25.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Applied Mathematical Sciences, vol. 68. Springer, New York (1997) CrossRefzbMATHGoogle Scholar
  26. 26.
    Zik, O., Moses, E.: Fingering instability in combustion: an extended view. Phys. Rev. E 60, 518–531 (1999) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.Institut de Mathématiques de BordeauxUniversité de BordeauxTalence cedexFrance
  3. 3.Dipartimento di Matematica e InformaticaUniversità degli Studi di ParmaParmaItaly

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