Concerning isothermal self-similar blast waves

Abstract

We investigate the two-dimensional self-similar flow behind a blast wave from a line explosion in a medium whose density varies with distance asr ^−ω with the assumption that the flow is isothermal. If ω<0, a continuous solution passing through the origin and the shock does not exist. If 16/9>ω>0, no critical points exist and a continuous solution passing through both the origin and the shock is shown to exist. If 16/9<ω, two new critical points appear. To be physically acceptable the flow must by-pass both critical points. We show that a continuous solution passing through both the origin and the shock and by-passing both critical points exists in this case. If ω≥2 no physically acceptable solution exists since the mass behind the shock is infinite. The dependence of the solutions which have zero flow velocity at the origin on the parameter ω is analytic for ω>0 so that interpolation between neighboring values of ω is permitted.

We investigate the stability of these isothermal blast waves to two-dimensional but non-self-similar perturbations. If 0<ω<4/3, the solutions are shown to be linearly unstable against short wavelength perturbations near the origin. If 2>ω>1, the self-similar solutions are shown to be globally unstable in a fully non-linear sense; while if 1>ω>0 the solutions are globally unstable unless the solution curves cross the shock with a normalized velocity of precisely (2/ω)_1/2.

Thus, for 2>ω>0 the solutions are always unstable somewhere (globally for certain in 2>ω>1; locally in a WKB sense near the origin in 0<ω<4/3). Since there is no characteristic time scale in the system, the local WKB instability near the origin grows as a power law in time rather than exponentially.

The existence of these instabilities implies that initial perturbations do not decay and the system does not tend to a self-similar form.