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Efflux Velocity of Fluid through a Small Orifice in a Tube

Let's discuss a problem based on efflux velocity of fluid through a small orifice in a tube. Try it yourself first, then read the solution.

The Problem:

A horizontal tube of length (L), open at (A) and closed at (B), is filled with an ideal fluid. The end (B) has a small orifice. The tube is set in rotation in the horizontal plane with angular velocity (\omega) about a vertical axis passing through (A). Show that the efflux velocity of the fluid is given by $$ v=\omega l\sqrt{\frac{2L}{l}-1}$$ where (l) is the length of the fluid.

Solution:

Consider a mass element (dm) of the fluid at a distance (x) from the vertical axis. The centrifugal force on (dm) is
$$ Df=dm\omega^2x$$ $$=dm\frac{dv}{dt}$$ $$=dm \frac{dv}{dx}v$$
$$ vdv=\omega^2 xdx$$
$$ \int vdv=\omega^2 \int xdx$$ $$
\frac{v^2}{2}=\frac{\omega^2}{2} \int_{L}^{L-l}
$$
So,
$$ v=\omega l\sqrt{\frac{2L}{l}-1}$$

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