Spinning Paraboloid

The liquid makes a dish shape known as a circular paraboloid. The horizontal cross-sections are circles. The vertical cross sections are parabolas, the same curve a ball makes when thrown through the air.

When the cylinder spins, the liquid furthest away from the centre is moving faster, which requires a larger horizontal force to keep it moving in a circle. You may have felt this force in a fast car making a turn.

The horizontal force is provided by the liquid pushing upwards towards the surface. When the surface is steeper there is more horizontal force, so the liquid ends up creating a curve. Specifically, the liquid is a parabolic curve. This is because the liquid that is twice as far away from the centre, needs twice the horizontal force, which makes the curve is twice as steep.