- #1

Saitama

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## Homework Statement

A horizontally oriented tube AB of length ##l## rotate with a constant angular velocity ##\omega## about a stationary axis OO' passing through the end A. 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.

## Homework Equations

$$P+\rho gh+\frac{1}{2}\rho v^2=\text{constant}$$

## The Attempt at a Solution

The book I refer to states that the Bernoulli equation (stated in relevant equations) is applicable in irrotational flow. But in the given problem, the fluid rotates too. I am thinking of switching to rotating frame where the liquid moves radially so there is no problem of "irrotational flow". Is this is a valid step?

Applying Bernoulli at the open surface and near the orifice,

$$P_o=P_o+\int_{l-h}^l \rho \omega^2x\,dx+\frac{1}{2}\rho v^2$$

Solving for ##v## gives

$$v=\omega\sqrt{h^2-2hl}$$

According to the answer key, the above is incorrect. The correct answer is ##v=\omega\sqrt{2hl-h^2}##. What is wrong with my method?

Also, is it possible to find the position of CoM of remaining liquid as a function of time?

Any help is appreciated. Thanks!