On the sub-shock formation in extended thermodynamics

In hyperbolic dissipative systems, the solution of the shock structure is not always continuous and a discontinuous part (sub-shock) appears when the velocity of the shock wave is greater than a critical value. In principle, the sub-shock may occur when the shock velocity s reaches one of the characteristic eigenvalues of the hyperbolic system. Nevertheless, Rational Extended Thermodynamics (ET) for a rarefied monatomic gas predicts the sub-shock formation only when s exceeds the maximum characteristic velocity of the system evaluated in the unperturbed state λ0max. This fact agrees with a general theorem asserting that continuous shock structure cannot exist for s>λ0max. In the present paper, first, the shock structure is numerically analyzed on the basis of ET for a rarefied polyatomic gas with 14 independent fields. It is shown that, also in this case, the shock structure is still continuous when s meets characteristic velocities except for the maximum one and therefore the sub-shock appears only when s>λ0max. This example reinforces the conjecture that, the differential systems of ET theories have the special characteristics such that the sub-shock appears only for s greater than the unperturbed maximum characteristic velocity. However, in the second part of the paper, we construct a counterexample of this conjecture by using a simple 2 × 2 hyperbolic dissipative system which satisfies all requirements of ET. In contrast to previous results, we show the clear sub-shock formation with a slower shock velocity than the maximum unperturbed characteristic velocity.