I´m a little confused with a fluid-dynamics exercise.
We have a cylinder of radius $R$ rotating with respect to the vertical axis with angular velocity $\omega $.
The cylinder has a small hole in the wall almost in the bottom. The hole is small, thus we can neglect the velocity of the surface of the fluid.
We have as data that the lower part of the surface is at $z=H$.
Now we have to calculate the radial velocity of the fluid when it leaves the cylinder.
My attempt to solve the exercise was: as a first step, I calculated the shape of the surface using a non-intertial frame of reference with a centripetal force.
Doing this I obtained that the surface is paraboloid shaped. More precisely it is a revolution paraboloid defined by this equation:
Now, I'm confused about the next step. How can I calculate the exit velocity using Bernoulli and the maximum altitude of the fluid in the parabola? I know that Bernoulli is applicable only to streamlines, but I can't even imagine the course of these.