Hmm, this problem is trickier than the one with rotating spheres. The physics is essentially the same as for the rotating sphere case, but the complication is that fluids don't behave like hard objects. In particular, when we think of a fluid exerting a force, we think of the fluid's pressure as the thing responsible for what's exerting the force (as opposed to the normal force, in the case of the spheres). Unlike the normal force for the sphere, which is a force acting only at a point, the fluid pressure acts everywhere the fluid is touching the walls of the cone, and in particular can vary from place to place. In order to understand how this pressure varies, you need two things: an equation of state for the fluid, which tells you how the density of the fluid is related to its temperature and pressure, and the equation of hydrostatic equilibrium, which tells you how the pressure must change as a function of position and applied forces (e.g. gravity, the centrifugal force). For a simple case, you can consider an incompressible fluid, so that the density is constant everywhere. Then you just need to worry about the equation of hydrostatic equilibrium to understand how the pressure varies throughout the fluid as a function of the speed at which you're rotating the cylinder.

This is a tricky (but interesting!) problem, and I'm not sure offhand how easy it is to do. There may be shortcuts that make it simpler, but I can't think of any offhand. If I do think of any, I'll let you know.