Acceleration waves, shock formation and stability in a gravitating atmosphere

Acceleration waves and oscillating gas bubbles modelled by rational extended thermodynamics

“…The analysis of the phenomena related to the acceleration waves reveals that together with hyperbolicity, the presence of dissipative terms is also mandatory. In fact, it has been shown that conservation laws provide for the transformation of an AW into a shock wave for any value of the initial amplitude [3,7,12], in contrast to the experimental evidence [13]. In a better model of balance laws, on the contrary, it is usually possible to identify a critical value of the initial amplitude below which the acceleration wave does not turn into a shock [2–4,7–9].…”

“…In fact, it has been shown that conservation laws provide for the transformation of an AW into a shock wave for any value of the initial amplitude [3,7,12], in contrast to the experimental evidence [13]. In a better model of balance laws, on the contrary, it is usually possible to identify a critical value of the initial amplitude below which the acceleration wave does not turn into a shock [2–4,7–9]. This modelling is consistent with experimental evidence if the threshold values for jump formation are sufficiently high.…”

Acceleration Waves in Rational Extended Thermodynamics of Rarefied Monatomic Gases

“…They can be generated by a small compressive disturbance in a gas that occurs, for example, in a gas flow induced by the motion of a piston advancing with finite acceleration. If the gas behavior is described by a set of hyperbolic conservation laws as those of the Euler model, neglecting the effect of viscosity and conductivity, it is possible to show analytically that the acceleration wave transforms into a shock wave in a finite time, independently from the amplitude of the initial disturbance [3][4][5]. This mathematical prediction does not find evidence in experimental data.…”

“…However, this is not a sufficient condition: many authors stressed that a suitable dissipation is also required [3,4,7]. Under such conditions, the theory usually predicts the existence of a critical amplitude A cr and a critical time t cr [2,3,5,[7][8][9]. If the initial wave amplitude is greater than A cr , the evolution of the weak discontinuity, after the time t cr , brings to the shock formation.…”

“…with aN (120) i = − 1488375k 13 B T 13 + 4068225k 12 B mT 12λ2 i + 30230550k 11 B m 2 T 11λ4 i − 177007950k 10 B m 3 T 10λ6 i + + 395140095k 9 B m 4 T 9λ8 i − 482337405k 8 B m 5 T 8λ10 i + 349324020k 7 B m 6 T 7λ12 i − 154092468k 6 B m 7 T 6λ14 i + + 42307383k 5 B m 8 T 5λ16 i − 7316169k 4 B m 9 T 4λ18 i + 790870k 3 B m 10 T 3λ20 i − 51422k 2 B m 11…”