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

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 Mr where δ = 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, in agreement with previous hydrodynamic simulations. In such circumstances, the flow may or may not remain continuously supersonic with respect to the clumps. 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, λ ∼ < −2, meaning that the exponent of the density profile of the interclump medium must be at most slightly positive, with negative values preferred. 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. Whilst mass loading always reduces the temperature of the shocked wind, it also tends to reduce the emissivity in the interior of the bubble relative to its limb, whilst simultaneously increasing the central temperature relative to the limb temperature. 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 potentially relevant to a wide range of astrophysical objects, including stellar wind-blown bubbles, galactic winds, starburst galaxy superwinds, and the impact of an AGN wind on its surrounding environment. This work complements the earlier work of Pittard et al. (2001) in which it was assumed that clumps were evaporated through conductive energy transport.