Instability of a convergent spherical shock wave

Self-Similar Solutions to the Compressible Euler Equations and their Instabilities

“…6. The greatest Ω j is Ω 0 = r, this is associated with a gauge mode that we provide in (29). In case of SGSs this is a SLM (but this is not an instability).…”

“…In this paper we address the construction of SGSs as well as the stability of both SGSs and NSGSs with respect to radial perturbations, preserving the symmetry of the problem. The stability of shocks was extensively addressed in the literature [27][28][29][30][31]. Our interest in the characterization of perturbations with different levels of regularity is motivated by the fact that, in physics, we cannot guarantee the smoothness of a phenomenon even when the initial data is smooth.…”

Dissipative instability of converging cylindrical shock wave

“…This conclusion is similar to the conclusion obtained in [6], although in the mechanism of cumulation restriction considered in [5], the cylindrical shape of the shell may not change if its rotation is excited. At the same time, as with the weak power-law growth of perturbations in [7], [13], in [5], the intensification of the initially small rotation is due only to the process of decreasing the radius of the cylindrical shell itself and is determined only from the law of conservation of the angular momentum. Therefore, for relatively small time intervals, at which the radius of the shell can be considered practically unchanged, the value of the initial small rotational perturbation also remains practically constant.…”

“…Conclusions about the instability of a converging spherical shock wave in an ideal gas and in a Van der Waals gas were also obtained in [8], [13] and [11] respectively. As in [7], the increase in time of the amplitude of the radial disturbance of the shock wave front in [13] (where 3 / 5   and 5 / 7…”

“…It is assumed that converging shock waves are unstable with respect to extremely small perturbations, but due to the slow power-law (not exponential) growth of perturbations over time, such instability cannot prevent unlimited accumulation of pressure and temperature during implosion [10], [11]. Indeed, the linear instability of a converging spherical shock wave in an ideal gas with an adiabatic exponent 3 / 5   and 3   is established in [7]. In this case, the increase in the amplitude of small perturbations is inversely proportional to the radius of the converging shock wave ) (t R   , where the small index 1 0…”