Free Boundary Problems and Dynamical Geometry Associated with Flames

  • Michael L. Frankel
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 35)


We introduce several simplified free boundary problems capable of generating basic dynamical patterns that are peculiar to flame propagation. The evolution of free boundaries can in turn be modeled by appropriate equations of dynamical geometry that relate the normal velocity (or higher “normal” time derivatives) of the surface to its instantaneous geometrical characteristics. The discussion is aimed to initiating numerical simulation and rigorous study of these models.


Free Boundary Hopf Bifurcation Flame Front Flame Propagation Free Boundary Problem 
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  1. [1]
    G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. Part 1. Derivation of basic equations, Acta Astronautica, 34 (1977), pp. 1177–1206.CrossRefGoogle Scholar
  2. [2]
    M. L. Frankel and G. I. Slvashinsky, On the equation of curved flame front, Physica D 30 (1988), pp. 28–42.CrossRefGoogle Scholar
  3. [3]
    B. J. Matkowsky and G. I. Sivashinsky, An asymptotic derivation of two models inflame theory associated with constant density approximation, SIAM J. Appl. Math., 37 (1979), pp. 686–699.CrossRefGoogle Scholar
  4. [4]
    M. L. Frankel, On a free boundary problem associated with combustion and solidification, Math. Mod. and Num. Analysis, 23 (1989), pp. 283–291.Google Scholar
  5. [5]
    M. L. Frankel, On the nonlinear dynamics of flame fronts in condensed combustible matter, Phys. Let. A, 140 (1989), No. 7,8, pp. 405–410CrossRefGoogle Scholar
  6. [6]
    G. I. Sivashinsky and P. Clavin, On the nonlinear theory of hydrodynamic instability in flames, J. de Phys., 48, (1987), pp. 193–198.CrossRefGoogle Scholar
  7. [7]
    M. L. Frankel, An equation of surface dynamics modeling the flame fronts as density discontinuities in potential flows, (to appear).Google Scholar
  8. [8]
    M. L. Frankel, On the nonlinear evolution of solid-liquid interface, Phys. Let. A, 128 (1988), pp. 57–60.CrossRefGoogle Scholar
  9. [9]
    M. L. Frankel, Qualitative approximation of oscillatory flames in premixed gas combustion by a local equation of front dynamics, (to appear).Google Scholar
  10. [10]
    B. J. Matkowsky and G. I. Sivashinsky, Propagation of pulsating reaction front in solid fuel combustion, SIAM J. Appl. Math., 35 (1978), 465–478.CrossRefGoogle Scholar
  11. [11]
    V. Roytburd, A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics, J.Diff. Eqs., 56 No.1 (1985), pp. 40–62.CrossRefGoogle Scholar
  12. [12]
    M. Frankel and V. Roytburd, A study of dynamics of free boundary in a problem associated with pulsating flames, (to be published).Google Scholar
  13. [13]
    M. Frankel, On the equation of pulsating flame fronts in solid fuel combustion, Phys. D, (to appear).Google Scholar
  14. [14]
    L. D. Landau, On the theory of slow combustion, Acta Physicochim. USSR, 19 (1944), pp. 77–85.Google Scholar
  15. [15]
    D. Michelson and G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. Part 2. Numerical experiments, Acta Astronautica, 34 (1977), pp. 1207–1221.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Michael L. Frankel
    • 1
  1. 1.Department of Mathematical SciencesIndiana University — Purdue University at IndianapolisIndianaUSA

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