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Efflux Velocity of Fluid through a Small Orifice in a Tube

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 oriffice. 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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