The Problem with Bathtubs

Bengt is in the bath. It's very hot, because Bengt thinks real men bathe in at least 50°C, but that has nothing to do with the problem. Suddenly he notices a little bit of lint in the water. Yuck! He pulls out the plug to let the water out and get rid of the lint. We can assume that the lint is in a random location and has the same density as the water; it behaves essentially like a randomly selected water molecule. But Bengt is annoyed at how slowly the water is flowing, so he considers whether it might possibly help to add more water. That way the pressure and therefore the rate of flow would increase, right?

1. Is there any way adding water could decrease the expected time for the lint to disappear? Assume that the added water appears uniformly; that is, we can think of it in an entirely statistical way, with water molecules being added and removed at random. The only physics involved is the increased rate of flow from the pressure.
2. What if we involve a little more physics? Presumably Bengt has the sense to add the new water at the opposite end of the bathtub from the drain. Could it reasonably be expected to be a good idea?