How Does Hole Diameter Affect Angular Speed in Water Vortices?

[gianamar]

Homework Statement

In a water vortex formed in a plastic bottle, what is the relationship between the angular speed of the water and the diameter of the hole in the cap? I would have expected that according to the conservation of angular momentum, the smaller the hole the faster the water speed (L = r x p). However, I've been trying to test this relationship out but seem to get that the larger the hole the faster the water goes.

Homework Equations

L = r x p = I \omega

 

[haruspex]

Please describe your experimental set up in detail.

 

[gianamar]

First I drilled holes of different diameters into bottle caps. Then, I held a 2L bottle with the bottom cut off, upside-down, in place with a clamp stand and filled it around 2/3 of the way up with water, covering the hole in the cap. I swirled the bottle to create a vortex and once the vortex was established, I dropped a small plastic ball into the water and used a stop-motion camera to film the experiment. However, the number of revolutions of the plastic ball per second increased as I used caps with larger diameters, whilst I expected that the angular frequency would decrease to conserve the angular momentum of the vortex.

 

[haruspex]

Where is the plastic ball when you measure its rotation rate? It's not going through the hole yet, right?
Outside of the central vortex, you should have irrotational flow (yes?). That means the rotation rate is in inverse ratio of the radius, so it matters at what radius the ball sits. I don't see how the size of exit hole in the cap is going to affect the vortex higher up in the bottle (much). Maybe I still don't have the right picture.

 

[gianamar]

No, the plastic ball doesn't go through the hole - it is dropped in from the top of the apparatus when the vortex is formed.

 

[haruspex]

No, the plastic ball doesn't go through the hole - it is dropped in from the top of the apparatus when the vortex is formed.

Then why should the rate of rotation of the ball around the vortex be related in any way to the size of the exit hole?
You didn't completely answer my question about the position of the ball. Where does it sit radially? Is it always near the wall of the bottle?