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Fluid Rotating in a Cylinder
The diagram shows a cross section and a top view of a closed cylinder. Half of it is filled with a fluid. Modify the angular velocity and gravity to observe how the parabola of the surface changes. At high velocity and low gravity most of the bottom is dry and the rotating surface approximates a cylinder.



The cross section at high speeds has a smaller area than the cross section at rest. Their volumes, however, are equal if they are distinguished by a critical radius of or 0.707.
A cross section of the surface is a parabola or two segments of it. This parabola is shown in the left image by a red line plotting: . The volume of the shifted paraboloid can be obtained by: .


"Fluid Rotating in a Cylinder" from The Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/FluidRotatingInACylinder/
Contributed by: Michael Schreiber
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