A cylindrical open-topped can (exposed to standard atmospheric pressure p_atm) is partly filled with water (of density rho, assumed to be incompressible). The can is now spun about a vertical axis coinciding with the axis of symmetry until the whole system is rigidly rotating at constant angular speed w. Find the pressure p in the water as a function of radius r and height z, where the origin is at the surface of the water on the axis of symmetry.
Solution: We consider a small cube of water at position (r,z). Radially the pressure difference times the cross-sectional area of the cube has to equal the the centripetal force, and vertically we have to balance buoyancy. At the hole, p = p_atm. Thus p(r,z) = p_atm + 0.5*rho*(w*r)^2 - rho*g*z. Note that r varies from 0 to R (the radius of the can), whereas z varies from -z_min to +z_max. Here z_max is found by putting p=p_atm to get the usual parabolic surface profile z_max = (w*r)^2/2g, and z_min can be found from conservation of the volume V of water.
Comment: The expression for p looks just like what Bernoulli’s equation predicts. But it isn’t clear why Bernoulli should apply. The streamlines are circles. Bernoulli comes from work-energy and no work is being done in steady state. However, could one consider some simple model of how the water transiently distorts from its initially flat-surface nonrotating equilibrium to its final parabolic-surface rotating steady state to explain why Bernoulli’s equation gives the final pressure? Not obvious to me, but maybe someone else has an idea.
Follow-up: Next suppose we consider instead a cylindrical closed can with a hole punched at the center of its top face. Water is dribbled into the can until it is totally full. We now rotate it as before. Now what will be p(r,z)? I’m tempted to say it’s the same as before except now z varies from 0 to H where H is the height of the can. Although the surface can no longer adopt a parabolic shape, there is a radial distribution of the normal force from the top surface of the can down onto the “centrifugally” spun flat water surface. But I’m not sure I’m right about p(r,z). Can anyone confirm or correct me?