Simulating shape of spinning water surface

[Swamp Thing]

I would like to simulate a simplified version of this phenomenon:



where I will assume that the viscosity is zero and the liquid can therefore swirl around "laminarly" forever according to some velocity profile that I specify.

How can I calculate the shape of the surface, at least in this simplified case, and hopefully estimate the buoyancy force on a ball that is held at some position along the central axis?

Another question: there were some comments referring to Bernoulli pressure differences due to differences in velocity. But is it not the case that Bernoulli has no role in this scenario, since the cross section does not vary along the direction of flow?

What is happening below the ball at 05:22 ?

 

[LURCH]

To address your final question first; Bernoulli’s principle is not directly caused by surfaces having different shapes, but by differences in the lateral motion of fluids. The change in shape only comes into play because it causes the fluid to flow at different rates.

In this case, the difference is between water that is stationary and water that is moving (in a circle).