Abstract

We derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumetric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time.

Keywords

  1. nonlinear wave equation
  2. nonadiabatic
  3. flame propagation
  4. extinction

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References

1.
B. J. Matkowsky, G. I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J. Appl. Math., 37 (1979), 686–699
2.
G. I. Sivashinsky, Diffusional-thermal theory of cellular flames, Combustion Sci. Tech., 15 (1977), 137–145
3.
G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations, Acta Astronaut., 4 (1977), 1177–1206
4.
B. J. Matkowsky, L. J. Putnick, G. I. Sivashinsky, A nonlinear theory of cellular flames, SIAM J. Appl. Math., 38 (1980), 489–504
5.
Bernard J. Matkowsky, David O. Olagunju, Propagation of a pulsating flame front in a gaseous combustible mixture, SIAM J. Appl. Math., 39 (1980), 290–300
6.
B. J. Matkowsky, D. O. Olagunju, Travelling waves along the front of a pulsating flame, SIAM J. Appl. Math., 42 (1982), 1138–1156
7.
Stephen B. Margolis, Gregory I. Sivashinsky, Flame propagation in vertical channels: bifurcation to bimodal cellular flames, SIAM J. Appl. Math., 44 (1984), 344–368
8.
G. I. Sivashinsky, On self-turbulization of laminar flames, Acta Astronautica, 6 (1979), 569–591
9.
Moshe Matalon, Thomas Erneux, Expanding flames may delay the transition to cellular structures, SIAM J. Appl. Math., 44 (1984), 734–744
10.
G. Joulin, G. I. Sivashinsky, On the dynamics of nearly-extinguished non-adiabatic cellular flames, Combustion Sci. Tech., 31 (1983), 75–90
11.
S. B. Margolis, B. J. Matkowsky, Nonlinear stability and bifurcation in the transition from laminar to turbulent flame propagation, Combustion Sci. Tech., 34 (1983), 45–77
12.
G. Joulin, P. Clavin, Linear stability analysis of nonadiabatic flames: a thermal-diffusional model, Comb. and Flame, 35 (1979), 139–145
13.
J. I. Blackshear, J. W. Mapp, M. Gorman, An experimental study of pulsating low pressure flames, Combustion Sci. Tech., 35 (1984), 311–315
14.
J. I. Blackshear, J. W. Mapp, M. Gorman, Nonsteady, nonplanar modes of propagation in premixed burner-stabilized flames, Combustion Sci. Tech., 43 (1985), 217–225
15.
D. B. Spalding, A theory of inflammability limits and flame-quenching, Proc. Roy. Soc. London. Ser. A., 240 (1957), 83–100
16.
J. Buckmaster, The quenching of deflagration waves, Comb. and Flame, 26 (1976), 151–162
17.
G. Joulin, P. Clavin, Analyse asymptotique des conditions d'extinction des flammes laminaries, Acts Astronautica, 3 (1976), 223–240
18.
B. J. Matkowsky, G. I. Sivashinsky, On the stability of nonadiabatic flames, SIAM J. Appl. Math., 40 (1981), 255–260
19.
Michael R. Booty, Stephen B. Margolis, Bernard J. Matkowsky, Interaction of pulsating and spinning waves in nonadiabatic flame propagation, SIAM J. Appl. Math., 47 (1987), 1241–1286
20.
Hans G. Kaper, G. K. Leaf, B. J. Matkowsky, W. E. Olmstead, Dynamics of nonadiabatic premixed flames in a gravitational field, SIAM J. Appl. Math., 47 (1987), 544–555
21.
L. M. Hocking, K. Stewartson, On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance, Proc. Roy. Soc. London Ser. A, 326 (1972), 289–313
22.
Hans G. Kaper, G. K. Leaf, B. J. Matkowsky, W. E. Olmstead, Dynamics of nearly extinguished nonadiabatic premixed flames in a gravitational field, SIAM J. Appl. Math., 48 (1988), 1054–1063

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 519 - 535
ISSN (online): 1095-712X

History

Submitted: 13 March 1987
Accepted: 12 August 1987
Published online: 10 July 2006

Keywords

  1. nonlinear wave equation
  2. nonadiabatic
  3. flame propagation
  4. extinction

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