No...A black hole is not really like a drain pipe! In fact, if it consumes matter at too ferocious a rate, the radiation pressure generated by the infalling matter provides tremendous resistance to the flow of matter.

The rate at which matter can fall into a black hole is regulated to what is called the Eddington Accretion Rate, which depends on the mass of the black hole. Also, it all depends on whether the star has been captured into orbit or is just passing by. Orbital capture means that the black hole, over the course of millions of years, can leisurely nibble away at the star 'without choking' at the Eddington Accretion Rate.

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