Flame propagation in a porous medium. (English) Zbl 1512.35050
Summary: We consider a specific parameter regime in a model for combustion in hydraulically resistant porous media. We prove the existence of the traveling fronts in this regime and investigate their stability. The existence is proved using Geometric Singular Perturbation theory. The stability analysis involves locating the spectrum of the linearization of the underlying system about the front by numerical computation of the Evans function combined with energy-like estimates. Furthermore, a conclusion about nonlinear stability is also obtained.
MSC:
| 35B35 | Stability in context of PDEs |
| 35C07 | Traveling wave solutions |
| 35K57 | Reaction-diffusion equations |
| 80A25 | Combustion |
Keywords:
combustion modeling; traveling front; singularly perturbed system; reaction-diffusion system; geometric singular perturbation theoryReferences:
| [1] | Gordon, P., Quenching and propagation of combustion fronts in porous media, Commun. Math. Sci., 4, 471-479 (2007) · Zbl 1119.80012 |
| [2] | Gordon, P.; Ryzhik, L., Traveling fronts in porous media: existence and a singular limit, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462, 1965-1985 (2006) · Zbl 1149.76683 |
| [3] | Gordon, P., Recent mathematical results on combustion in hydraulically resistant porous media, Math. Model. Nat. Phenom., 2, 56-76 (2007) · Zbl 1337.35070 |
| [4] | Brailovsky, I.; Goldshtein, V.; Shreiber, I.; Sivashinsky, G., On combustion waves driven by diffusion of pressure, Combust. Sci. Tech., 124, 145-165 (1997) |
| [5] | Ghazaryan, A.; Gordon, P., KPP type flame fronts in porous media, Nonlinearity, 21, 973-992 (2008) · Zbl 1145.80010 |
| [6] | Ghazaryan, A.; Gordon, P.; Virodov, A., Stability of fronts and transient behavior in KPP systems, Proc. R. Soc. A, 466, 1769-1788 (2010) · Zbl 1201.35040 |
| [7] | Fisher, R. A., The wave of advance of advantageous genes, Ann. Eugenics, 7, 353-369 (1937) · JFM 63.1111.04 |
| [8] | Kolmogorov, A. N.; Petrovsky, I. G.; Piskunov, N. S., A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Mosc. Univ. Math. Mech., 1, 1-26 (1937) |
| [9] | Billingham, J.; Needham, D. J., The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates: I. Permanent form travelling waves, Phil. Trans.: Phys. Sci. Eng., 334, 1-24 (1991) · Zbl 0739.35033 |
| [10] | Ghazaryan, A.; Lafortune, S., Combustion waves in hydraulically resistant porous media in a special parameter regime, Physica D, 332, 23-33 (2016) · Zbl 1415.80005 |
| [11] | Ghazaryan, A.; Gordon, P.; Jones, C., Traveling waves in porous media combustion: Uniqueness of waves for small thermal diffusivity, J. Dyn. Differ. Equ., 19, 951-966 (2007) · Zbl 1139.34040 |
| [12] | Ghazaryan, A.; Lafortune, S.; McLarnan, P., Stability analysis for combustion fronts traveling in hydraulically resistant porous media, SIAM J. Appl. Math. (2015) · Zbl 1320.35137 |
| [13] | Sandstede, B., Stability of travelling waves, (Fiedler, B., Handbook of Dynamical Systems II: Towards Applications (2002), Elsevier), 983-1055 · Zbl 1056.35022 |
| [14] | Jones, C., Geometric singular perturbation theory, (Dynamical Systems (Montecatini Terme, 1994). Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., vol. 1609 (1994), Springer: Springer Berlin), 44-118 · Zbl 0840.58040 |
| [15] | Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 55, 763-783 (1979) |
| [16] | Williams, F., Combustion Theory. The Fundamental Theory of Chemically Reacting Flow Systems (1985), Perseus Books: Perseus Books Reading, MA |
| [17] | Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0456.35001 |
| [18] | Simon, P.; Merkin, J.; Scott, S., Bifurcations in non-adiabatic flame propagation mODEls, (Jiang, S. Z., Focus on Combustion Research (2006), Nova Science: Nova Science New York), 315-357 |
| [19] | Pego, R.; Weinstein, M., Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. A, 340, 47-94 (1992) · Zbl 0776.35065 |
| [20] | Barker, B.; Humpherys, J.; Lyng, G.; Lytle, J., Evans function computation for the stability of travelling waves, Phil. Trans. R. Soc. A, 2117, Article 20170184 pp. (2018) · Zbl 1402.65052 |
| [21] | Afendikov, A. L.; Bridges, T. J., Instability of the Hocking-Stewartson pulse and its implications for the three-dimensional Poiseuille flow, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 457, 257-272 (2001) · Zbl 0984.76022 |
| [22] | Allen, L.; Bridges, T. J., Numerical exterior algebra and the compound matrix method, Numer. Math., 92, 197-232 (2002) · Zbl 1012.65079 |
| [23] | Bridges, T. J., The Orr-Sommerfeld equation on a manifold, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 455, 3019-3040 (1999) · Zbl 0983.37112 |
| [24] | Bridges, T. J.; Derks, G.; Gottwald, G., Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework, Physica D, 172, 190-216 (2002) · Zbl 1047.37053 |
| [25] | Brin, L. Q., Numerical testing of the stability of viscous shock waves, Math. Comp., 70, 1071-1088 (2001) · Zbl 0980.65092 |
| [26] | Derks, G.; Gottwald, G. A., A robust numerical method to study oscillatory instability of gap solitary waves, SIAM J. Appl. Dyn. Syst., 4, 140-158 (2005) · Zbl 1062.65138 |
| [27] | Ng, B.; Reid, W., An initial-value method for eigenvalue problems using compound matrices, J. Comput. Phys., 30, 125-136 (1979) · Zbl 0408.65061 |
| [28] | Simon, P.; Kalliadasis, S.; Merkin, J.; Scott, S., Evans function analysis of the stability of non-adiabatic flames, Combust. Theor. Model, 7, 545-561 (2003) · Zbl 1068.80531 |
| [29] | Gubernov, V.; Mercer, G. N.; Sidhu, H. S.; Weber, R. O., Evans function stability of combustion waves, SIAM J. Appl. Math., 63, 1259-1275 (2003) · Zbl 1024.35044 |
| [30] | Bridges, T. J.; Derks, G., Hodge duality and the Evans function, Phys. Lett. A, 251, 363-372 (1999) · Zbl 0942.35154 |
| [31] | Ghazaryan, A.; Latushkin, Y.; Schecter, S., Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction mODEls, SIAM J. Math. Anal., 42, 2434-2472 (2010) · Zbl 1227.35057 |
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