On the existence of high Lewis number combustion fronts. (English) Zbl 1245.80008
Summary: We study a mathematical model for high Lewis number combustion processes with the reaction rate of the form of an Arrhenius law with or without an ignition cut-off. An efficient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with infinite Lewis number.
References:
| [1] | Balasuriya, S.; Gottwald, G.; Hornibrook, J.; Lafortune, S., High Lewis number combustion wavefronts: a perturbative Melnikov analysis, SIAM J. Appl. Math., 67, 464-486 (2007) · Zbl 1121.80007 |
| [2] | Bates, P. W.; Fife, P. C.; Gardner, R. A.; Jones, C. K.R. T., The existence of travelling wave solutions of a generalized phase-field model, SIAM J. Math. Anal., 28, 60-93 (1997) · Zbl 0867.35041 |
| [3] | Berestycki, H.; Larrouturou, B.; Roquejofre, J.-M., Mathematical investigation of the cold boundary difficulty in the flame propagation theory, (Dynamical Issues in Combustion. Dynamical Issues in Combustion, IMA, Springer-Verlag, Berlin (1991)) · Zbl 0751.76066 |
| [4] | Berestycki, H.; Nicolaenko, B.; Scheurer, B., Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16, 1207-1242 (1985) · Zbl 0596.76096 |
| [5] | Billingham, J., Phase plane analysis of one-dimensional reaction diffusion waves with degenerate reaction terms, Dyn. Stab. Syst., 15, 23-33 (2000) · Zbl 0954.34041 |
| [6] | Buckmaster, J. D.; Ludford, G. S.S., Theory of Laminar Flames (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0557.76001 |
| [7] | P. Clavin, A. Linan, Theory of gaseous combustion, in: Nonequilibrium Cooperative Phenomena in Physics and Related Fields, Velarde, ed. NATO ASI Series, Physics B, 116, Plenum, 1983.; P. Clavin, A. Linan, Theory of gaseous combustion, in: Nonequilibrium Cooperative Phenomena in Physics and Related Fields, Velarde, ed. NATO ASI Series, Physics B, 116, Plenum, 1983. |
| [8] | Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equat., 31, 55-98 (1979) · Zbl 0476.34034 |
| [9] | Ghazaryan, A., Nonlinear stability of fronts in high Lewis number combustion, Indiana Univ. Math. J., 58, 181-212 (2009) · Zbl 1201.35039 |
| [10] | Ghazaryan, A.; Gordon, P.; Jones, C. K.R. T., Traveling waves in porous media combustion: uniqueness of waves for small thermal diffusivity, J. Dynam. Differ. Equat., 19, 951-966 (2007) · Zbl 1139.34040 |
| [11] | Ghazaryan, A.; Jones, C. K.R. T., On the stability of high Lewis number combustion fronts, J. Discrete Contin. Dynam. Syst., 24, 809-826 (2009) · Zbl 1175.35014 |
| [12] | A. Ghazaryan, Yu. Latushkin, S. Schecter, A.J. De Souza, Convective stability of combustion waves in one-dimensional solids, preprint.; A. Ghazaryan, Yu. Latushkin, S. Schecter, A.J. De Souza, Convective stability of combustion waves in one-dimensional solids, preprint. · Zbl 1248.35209 |
| [13] | Jones, C. K.R. T., Geometric singular perturbation, Dynam. Syst. Springer Lecture Notes Math., 1609, 44-120 (1995) · Zbl 0840.58040 |
| [14] | Logak, E., Mathematical analysis of a condensed phase combustion model without ignition temperature, Nonlin. Anal. Theor. Methods Applic., 28, 1-38 (1997) · Zbl 0866.34018 |
| [15] | Logak, E.; Loubeau, V., Travelling wave solutions to a condensed phase combustion model, Asympt. Anal., 12, 259-294 (1996) · Zbl 0865.34016 |
| [16] | Marion, M., Qualitative properties of a nonlinear system for laminar flames without ignition temperature, Nonlinear Anal., 9, 1262-1292 (1985) · Zbl 0648.76051 |
| [17] | Simon, P. L.; Merkin, J. H.; Scott, S. K., Bifurcations in non-adiabatic flame propagation models, (Jiang, S. Z., Focus on Combustion Research (2006), Nova Science Publishers: Nova Science Publishers New York), 315-357 |
| [18] | Varas, F.; Vega, J. M., Linear stability of a plane front in solid combustion at large heat of reactions, SIAM J. Appl. Math., 62, 1810-1822 (2002) · Zbl 1006.80003 |
| [19] | Weber, R. O.; Mercer, G. N.; Sidhu, H. S.; Gray, B. F., Combustion waves for gases (Le=1) and solids (Le→∞), Proc. R. Soc. Lond. A., 453, 1105-1118 (1997) · Zbl 0885.76105 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
