Question

A horizontally oriented tube AB of length l rotates with a constant angular velocity ω about a stationary vertical axis OO' passing through the end A (Fig. 1.86). The tube is filled with an ideal fluid. The end A of the tube is  open, the closed end B has a very small orifice. Find the velocity of the fluid relative to the tube as a function of the column "height" h.

Answer 1

In a rotating frame (with constant angular velocity) the Eulerian equation is 

In the frame of rotating tube the liquid in the "column* is practically static because the orifice is sufficiently small. Thus the Eulerian Eq. in projection form along vector r (which is the position vector of an arbitrary liquid element of length dr relative to the rotation axis reduces to

Then applying Bemouirs theorem at the orifice for the points just inside and outside of the end B