Question

A cylindrical vessel with water is rotated about its vertical axis with a constant angular velocity ω. Find:

(a) the shape of the free surface of the water;

(b) the water pressure distribution over the bottom of the vessel along its radius provided the pressure at the central point is equal to P_o.

Answer 1

The Euler’s equation is in the space fixed frame where  downward. We assume incompressible fluid so ρ is constant. 

Then   where z is the height vertically upwards from some fixed origin. We go to rotating frame where the equation becomes

the additional terms on the right are the well known coriolis and centrifugal forces. In the frame rotating with the liquid vector v=0 so