Communications in Mathematical Physics

, Volume 259, Issue 2, pp 451–474 | Cite as

Traveling Fronts in a Reactive Boussinesq System: Bounds and Stability

Article

Abstract

This paper considers a simplified model of active combustion in a fluid flow, with the reaction influencing the flow. The model consists of a reaction-diffusion-advection equation coupled with an incompressible Navier-Stokes system under the Boussinesq approximation in an infinite vertical strip. We prove that for certain ignition nonlinearities, including all that are C2, and for any domain width, planar traveling front solutions are nonlinearly and exponentially stable within certain weighted H2 spaces, provided that the Rayleigh number ρ is small enough. The same result holds for bistable nonlinearities in unweighted H2 spaces. We also obtain uniform bounds on the Nusselt number, the bulk burning rate, and the average maximum vertical velocity for chemistries that include bistable and ignition nonlinearities.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berestycki, H.: The influence of advection on the propagation of fronts in reaction-diffusion equations. In: Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, Berestycki, H., Pomeau, Y. (eds.), Doordrecht, NL: Kluwer Acad. Publ., 2003, pp. 11–48Google Scholar
  2. 2.
    Constantin, P., Kiselev, A., Ryzhik, L.: Fronts in reactive convection: bounds, stability and instability. Commun. Pure and Appl. Math. 56, 1781–1803 (2003)CrossRefGoogle Scholar
  3. 3.
    Fisher, R. A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 355–369 (1937)Google Scholar
  4. 4.
    Frank-Kamenetskii, D. A.: Diffusion and Heat Transfer in Chemical Kinetics, New York: Plenum Press, 1969Google Scholar
  5. 5.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. New York: Springer-Verlag, 1981Google Scholar
  6. 6.
    Hodgkin, A. L., Huxley, A. F.: A quantitative description of membrane current and its application to conduction and excitation in nerves. J. Physiol. 117, 500–544 (1952)PubMedGoogle Scholar
  7. 7.
    Kolmogorov, A. N., Petrovskii, I. G., Piskunov, N. S.: Étude de l’équation de la chaleur avec croissance de la quantité de matière et son application à un problème biologique. Bull. Moskov. Gos. Univ Mat. Mekh. 1(6), 1–25 (1937)Google Scholar
  8. 8.
    Malham, S., Xin, J.: Global solutions to a reactive Boussinesq system with front data on an infinite domain. Comm. Math. Phys. 193, 287–316 (1998)CrossRefGoogle Scholar
  9. 9.
    Peters, N.: Turbulent Combustion. Cambridge, UK: Cambridge University Press, 2000Google Scholar
  10. 10.
    Texier-Picard, R., Volpert, V.: Problèms de réaction-diffusion-convection dans des cylindres non bornés. C. R. Acad. Sci. Paris Sr. I Math. 333, 1077–1082 (2001)Google Scholar
  11. 11.
    Texier-Picard, R., Volpert, V.: Reaction-diffusion-convection problems in unbounded cylinders. Revista Matematica Complutense 16(1), 233–276 (2003)Google Scholar
  12. 12.
    Sattinger, D. H.: On the Stability of Waves of Nonlinear Parabolic Systems. Adv. in Math. 22, 312–355 (1976)CrossRefGoogle Scholar
  13. 13.
    Vladimirova, N., Rosner, R.: Model flames in the Boussinesq limit: the effects of feedback. Phys. Rev. E. 67, 066305 (2003)CrossRefGoogle Scholar
  14. 14.
    Volpert, A., Volpert, V., Volpert, V.: Traveling Wave Solutions of Parabolic Systems. Translations of Mathematical Monographs 140, Providence, RI: Amer. Math. Soc., 1994Google Scholar
  15. 15.
    Winn, B.: Doctoral Thesis. Chicago: The University of Chicago Press, 2005, to appearGoogle Scholar
  16. 16.
    Xin, J.: Front propagation on heterogeneous media. SIAM Rev. 42, 161–230 (2000)CrossRefGoogle Scholar
  17. 17.
    Zel’dovich, Ya. B., Barenblatt, G. I., Librovich, V. B., Makhviladze, G. M.: The Mathematical Theory of Combustion and Explosions. New York: Consultants Bureau, 1985Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations