The whirlpools that are drawn around black holes (as in the above illustration) are accretion disks. Accretion disks form because the infalling matter will, in all likelihood, have a nonzero net angular momentum. The angular momenta of the infalling objects all cancel each other in every direction but one (the axis of the disk). The matter left in the disk goes into orbit around the black hole.
The accretion disk of a black hole actually wouldn't look like a normal accretion disk as shown above. This is because light bends around the black hole, with the result that if you are looking at the near side of the disk almost edge-on, you'll be looking at the far side of the disk face-on. It looks sort of like a misshapen LP record with a 90-degree fold in it. (See an example here.)
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