Self-similar evolution of wind-blown bubbles with mass loading by hydrodynamic ablation

Abstract

We present similarity solutions for adiabatic bubbles that are blown by winds having time independent mechanical luminosities and that are each mass-loaded by the hydrodynamic ablation of distributed clumps. The mass loading is `switched-on' at a specified radius (with free-expansion of the wind interior to this point) and injects mass at a rate per unit volume proportional to M^delta r^lambda where delta = 4/3 (1) if the flow is subsonic (supersonic) with respect to the clumps. In the limit of negligible mass loading a similarity solution found by Dyson (1973) for expansion into a smooth ambient medium is recovered. The presence of mass loading heats the flow, which leads to a reduction in the Mach number of the supersonic freely-expanding flow, and weaker jump conditions across the inner shock. In solutions with large mass loading, it is possible for the wind to connect directly to the contact discontinuity without first passing through an inner shock. For a solution that gives the mass of swept-up ambient gas to be less than the sum of the masses of the wind and ablated material, lambda < -2. Maximum possible values for the ratio of ablated mass to wind mass occur when mass loading starts very close to the bubble center and when the flow is supersonic with respect to the clumps over the entire bubble radius. The maximum temperature in the bubble often occurs near the onset of mass loading, and in some cases can be many times greater than the post-inner-shock temperature. Our solutions are relevant to eg stellar wind-blown bubbles, galactic winds, etc. This work complements Pittard et al (2001) where it was assumed that clumps were evaporated through conductive energy transport.