Acoustic stability of nonadiabatic high-energy-density shocks

As predicted by D'yakov [Shock wave stability, Zh. Eksp. Teor. Fiz. 27, 288 (1954)] and Kontorovich [On the shock waves stability, Zh. Eksp. Teor. Fiz. 33, 1525 (1957)], an initially disturbed shock front may exhibit different asymptotic behaviors, depending on the slope of the Rankine-Hugoniot curve. Adiabatic and isolated planar shocks traveling steadily through ideal gases are stable in the sense that any perturbation on the shock front decays in time with the power t(-3/2) (or t(-1/2) in the strong-shock limit). While some gases whose equations of state cannot be modeled as a perfect gas, as those governed by van der Waals forces, may induce constant-amplitude oscillations to the shock front in the longtime regime, fully unstable behaviors are seldom to occur due to the unlikely conditions that the equation of state must meet. In this paper, it has been found that unstable conditions might be found when the gas undergoes an endothermic or exothermic transformation behind the shock. In particular, it is reported that constant-amplitude oscillations can occur when the amount of energy release is positively correlated to the shock strength and, if this correlation is sufficiently strong, the shock turns to be fully unstable. The opposite highly damped oscillating regime may occur in negatively correlated configurations. The mathematical description then adds two independent parameters to the regular adiabatic index y and shock Mach number M-1, namely, the total energy added/removed and its sensitivity with the shock strength. The formulation in terms of endothermic or exothermic effects is extended but not restricted to include effects associated to ionization, dissociation, thermal radiation, and thermonuclear transformations, so long as the time associated to these effects is a much shorter time than the acoustic time.