Abstract
We study a mathematical model of combustion processes in an inert porous media filled with a combustible gaseous mixture. We focus on the phenomenon of a combustion wave driven by a local pressure elevation. In this article, we are concerned with subsonic pressure-driven flames and with the case of a quadratic dependence of the friction force on the velocity of the gaseous mixture. After a suitable non-dimensionalization, the resulting mathematical model includes three nonlinear ordinary differential equations (ODEs). The system contains an unknown parameter V that represents the traveling wave speed. The existence of the traveling wave is proven in this study. It means that the parameter V can be chosen so that the corresponding phase trajectory satisfies the boundary conditions. Moreover, under reasonable assumptions about the monotonicity of the flame front, we prove the uniqueness of the pressure-driven wave.
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References
Babkin VS (1993) Filtration combustion of gases. Present state of affairs and prospects. Pure Appl Chem 64: 335–344
Gol’dshtein V, Shreiber I, Sivashinsky G (1994) On creeping detonation in filtration combustion. Shock Waves 4: 109–112
Brailovsky I, Gol’dshtein V, Shreiber I, Sivashisky G (1997) On combustion wave driven by diffusion of pressure. Combust Sci Technol 124: 145–165
Brailovsky I, Frankel M, Sivashinsky G (2000) Galloping and spinning modes of subsonic detonation. Combust Theory Model 4: 47–60
Brailovsky I, Sivashinsky G (2000) Hydraulic resistance as a mechanism for deflagration-to-detonation transition. Combust Flame 122: 492–499
Goldfarb I, Gol’dshtein V, Kuzmenko G (1999) Pressure driven flame in porous media. Phys Lett A 251: 394–403
Nigmatullin RI (1990) Dynamics of multiphase media. Hemisphere Publishing Corporation, New York
Dullien FA (1992) Porous media, 2nd edn. Academic Press, San Diego
Nield DA, Bejan A (1992) Convection in porous media. Springer, New York
Krapivnik N, Gol’dshtein V (2010) Existence of pressure driven wave. Russ J Phys Chem B 4(4): 574–579
Bykov V, Goldfarb I, Gol’dshtein V (2005) Multi-scale analysis of pressure driven flames. Singular perturbations and hysteresis. SIAM, Philadelphia, pp 257–298
Semenov NN (1928) Zur Theorie des Verbrennungsprozesses. Zeitschr Phys 48: 571–581
Frank-Kamenetskii DA (1969) Diffusion and heat exchange in chemical kinetics, 2nd edn. Plenum Press, New York
Bykov V, Goldfarb I, Gol’dshtein V (2004) Inertia effect on a structure of pressure driven flames in porous media. J Eng Math 49: 77–97
Bykov V, Goldfarb I, Gol’dshtein V, Kagan L, Sivashinsky G (2004) Effects of hydraulic resistance and heat losses on the detonability and flammability limits. Combust Theory Model 8: 413–424
Gordon P, Kagan L, Sivashinsky G (2003) Fast subsonic combustion as a free-interface problem. Interfaces Free Bound 5(1): 47–62
Penner SS, Williams FA (1961) The theory of steady, one-dimensional, laminar flame propagation for one-step chemical reactions. Astron Acta 7: 171
Zel’dovich YB (1940) Zurn Exp Teor Fiz 10:542–568 (English translation: In: Ostriker JP, Barenblatt GI, Sanaev RA (eds) Selected studies by Yakov Borisovich Zel’dovich, vol 1. Princeton University Press, Princeton, 1992)
Hartman P (1964) Ordinary differential equation. Wiley, New York
Bautin NN, Leontovich EA (1976) Methody i priemy kachestvennogo issledovaniya dinamicheskih sistem na ploskosti. Moskva, “Nauka” (Russian)
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Gol’dshtein, V., Krapivnik, N. Existence and uniqueness of the traveling front in premixed combustion of porous media. J Eng Math 72, 177–186 (2012). https://doi.org/10.1007/s10665-011-9474-4
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DOI: https://doi.org/10.1007/s10665-011-9474-4