Perturbed Riemann problem for a scalar Chapman-Jouguet combustion model

Abstract

The author considers the perturbed Riemann problem for a scalar Chapman-Jouguet combustion model which comes from Majda’s model with a modified, bump-type ignition function proposed in the results of Lyng and Zumbrun in 2004. Under the entropy conditions, the unique solution in a neighborhood of the origin on the (x, t) plane (t > 0) is obtained. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a strong detonation into a weak deflagration in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution does not contain combustion wave, which exhibits the instability for the unburnt state.

Keywords

Scalar Chapman-Jouguet combustion model / Perturbed Riemann problem, Detonation / Deflagration

References

[1] Bao W, Jin S. The random projection method for hyperbolic conservation laws with stiff reaction terms. J. Comput. Phys., 2000, 163: 216-248

[2] Bdzil J B, Stewart D S. The dynamics of detonation in explosive systems. Annual Review of Fluid Mechanics, 2007, 39: 263-292

[3] Chang T, Hsiao L. The Riemann Problem and Interaction of Waves in Gas Dynamics, 1989, Essex: Longman

[4] Chorin A J. Random choice methods with application to reacting gas flow. J. Comput. Phys., 1977, 25: 253-272

[5] Courant R, Friedrichs K O. Supersonic Flow and Shock Waves, 1948, New York: Interscience

[6] Fickett W. Detonation in miniature. Amer. J. Phys., 1979, 47: 1050-1059

[7] Li J, Zhang P. The transition from Zeldovich-von Neumann-Döring to Chapman-Jouget theories for a nonconvex scalar combustion model. SIAM J. Math. Anal., 2003, 34: 675-699

[8] Li T T. Global Classical Solutions for Quasilinear Hyperbolic Systems, 1994, New York: John Wiley and Sons

[9] Liu T P, Zhang T. A scalar combustion model. Arch. Rational Mech. Anal., 1991, 114: 297-312

[10] Lyng G, Zumbrun K. A stability index for detonation waves in Majda’s model for reacting flow. Physica D, 2004, 194: 1-29

[11] Lyng G, Zumbrun K. One-dimensional stability of viscous strong detonation waves. Arch. Rational Mech. Anal., 2004, 173: 213-277

[12] Lyng G, Raoofi M, Texier B, Zumbrun K. Pointwise Green function bounds and stability of combustion waves. J. Differential Equations, 2007, 233: 654-698

[13] Majda A. A qualitative model for dynamic combustion. SIAM J. Appl. Math., 1981, 41: 70-93

[14] Rosales R R, Majda A. Weakly nonlinear detonation waves. SIAM J. Appl. Math., 1983, 43: 1086-1118

[15] Shen C, Sun M N. The perturbation on initial binding energy for a Majda-CJ combustion model. Applicable Analysis, 2013, 92: 2115-2126

[16] Sheng W C, Sun M N, Zhang T. The generalized Riemann problem for a scalar nonconvex Chapman-Jouguet combustion model. SIAM J. Appl. Math., 2007, 68: 544-561

[17] Sheng W C, Zhang T. Structural stability of solutions to the Riemann problem for a scalar nonconvex combustion model. Discrete Contin. Dyn. Syst., 2009, 25: 651-667

[18] Sun M N. Entropy solutions of a Chapman-Jouguet combustion model. Mathematical Models and Methods in Applied Sciences, 2012, 22(9): 1250018

[19] Sun M N, Sheng W C. The ignition problem for a scalar nonconvex combustion model. J. Differential Equations, 2006, 231: 673-692

[20] Sun M N, Sheng W C. The generalized Riemann problem for a scalar Chapman-Jouguet combustion model. Z. Angew. Math. Phys., 2009, 60: 271-283

[21] Teng Z H, Chorin A J, Liu T P. Riemann problems for reacting gas with applications to transition. SIAM J. Appl. Math., 1982, 42: 964-981

[22] Williams C D. The detonation of explosives. Sci. Amer., 1987, 256: 98-104

[23] Ying L A, Teng Z H. Riemann problem for a reaction and convection hyperbolic system. Approx. Theory Appl., 1984, 1: 95-122

[24] Zhang P, Zhang T. The Riemann problem for scalar CJ-combustion model without convexity. Discrete Contin. Dynam. Systems, 1995, 1: 195-206

[25] Zhang T, Zheng Y X. Riemann problem for gas dynamic combustion. J. Differential Equations, 1989, 77: 203-230

[26] Zhang X T, Ying L A. Dependence of qualitative behavior of the numerical solutions on the ignition temperature for a combustion model. J. Comput. Math., 2005, 23: 337-350