# What is the relationship between the angular speed and the diameter of the eye of a water vortex?

1. Sep 23, 2014

### gianamar

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
L = r x p = I \omega

2. Sep 23, 2014

### haruspex

Please describe your experimental set up in detail.

3. Sep 23, 2014

### 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.

4. Sep 23, 2014

### 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.

5. Sep 23, 2014

### 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.

6. Sep 24, 2014

### haruspex

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?

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