Sub-Shocks formation in the Shock Structure of Hyperbolic Systems with a Convex Extension with an application to a Binary Mixture of Gases

Abstract

Since the system of field equations is hyperbolic, the shock-structure solution is not always regular, and discontinuous parts (sub-shocks) can be formed when the shock velocity meets a characteristic velocity [1]. In particular, in the case of a hyperbolic system with a convex entropy (symmetric systems), a theorem by Boillat and Ruggeri [2] proved that a sub-shock surely arises when the shock’s velocity becomes greater than the maximum characteristic velocity in the unperturbed state. The question if there are sub-shocks also for shock velocities less than the maximum characteristic is still an open problem. An interesting case of the existence of sub-shocks for shock velocity smaller than the maximum velocity is offered by a binary mixture of polyatomic Eulerian gases with different degrees of freedom of a molecule based on the multi-temperature model of Rational Extended Thermodynamics [3]. For given values of the mass ratio and the specific heats of the constituents, we identify the possible sub-shocks as the Mach number of the shock wave and the concentration of the constituents change [4]. Namely, the regions with no sub-shocks, a sub-shock for only one component, or sub-shocks for both constituents are comprehensively classified. The most interesting case is that the lighter molecule has more degrees of freedom than the heavier one. In this situation, the topology of the various regions becomes different. We also numerically solve the system of the field equations using the parameters in the various regions and confirm whether the sub-shocks emerge. Finally, the relationship between an acceleration wave in one constituent and the sub-shock in the other is explicitly derived.

References:

[1] T. Ruggeri, “Breakdown of shock-wave-structure solutions,” Phys. Rev. E 47, 4135 (1993).

[2] G. Boillat and T. Ruggeri, “On the shock structure problem for hyperbolic system of balance laws and convex entropy,” Cont. Mech. Thermodyn. 10 285–292 (1998).

[3] T. Ruggeri and M. Sugiyama, Classical and Relativistic Rational Extended Thermodynamics of Gases (Springer, Cham, 2021).

[4] T. Ruggeri and S. Taniguchi, “A Complete Classification of Sub-Shocks in the Shock Structure of a Binary Mixture of Eulerian Gases with Different Degrees of Freedom,” Phys. Fluids 34, 066116 (2022).

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