Geodesic
Equilibrium mathematical model of multicomponent heterogeneous media
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 4, pp. 49-57

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In this paper, a mathematical model of an equilibrium two-phase mixture is constructed based on the general equations of conservation of heterogeneous multicomponent mixtures. This mathematical model was studied for the presence of hyperbolicity and invariance regarding the Galilean transformation. Hyperbolicity of the mathematical model of the equilibrium two-phase mixture was demonstrated, which proved the possibility of calculating high-speed processes, for example, the processes of initiation of detonation in condensed explosives by strong shock waves. Hyperbolicity of the mathematical model of equilibrium two-phase mixture leads to the fact that velocity of propagation of disturbances (sound velocity) in the mixture is a finite value. This fact is very important in analyzing the processes of the output of initiating shock waves into the detonation regime. The assumption of the equilibrium of the mixture for calculating the initiation of detonation greatly simplifies the general mathematical model of heterogeneous multicomponent mixtures. It is shown that the system of conservation laws in an equilibrium mathematical model of two-phase mixture can be reduced to the system of conservation laws for a mixture, when the closing equations are the equations of state for specific internal energy and phase pressure, as well as the ratios that are usual for heterogeneous mixtures. Within the frameworks of the equilibrium mathematical model of two-phase mixture, justification for the coordination of energy of phase transitions was carried out. It was taken into account that phase transitions in a detonation wave occur at a constant volume. Analysis of the equilibrium mathematical model of two-phase mixture for the presence of invariance regarding the Galileo transformation showed its invariance, which confirms the correctness of the assumptions made in this paper.
Keywords: mathematical model, two-phase mixture, equation of state, high-speed process, shockwave. 
@article{VYURM_2018_10_4_a5,
     author = {Yu. M. Kovalev and F. G. Magazov and E. S. Shestakovskaya},
     title = {Equilibrium mathematical model of multicomponent heterogeneous media},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {49--57},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2018_10_4_a5/}
}
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Yu. M. Kovalev; F. G. Magazov; E. S. Shestakovskaya. Equilibrium mathematical model of multicomponent heterogeneous media. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 4, pp. 49-57. http://geodesic.mathdoc.fr/item/VYURM_2018_10_4_a5/