Time evolution of the bubble radius (green curve) and velocity (blue curve) for an exponentially growing luminosity source powered by an exponentially growing BH (black curve) for the energy-driven case. The source parameters and vertical dotted lines are as in Fig. 2. The velocity of the bubble reaches a minimum around a few Salpeter times, after which the shell starts accelerating. Rayleigh-Taylor instabilities arise at this stage (see Sect. 5).
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