= Drainage time of a conical tank
{title2=$T\sim\frac{2h_0^{5/2}}{5\epsilon^2\sqrt{2g}}$}
For a cone of half angle $\pi/4$ draining through a small circular aperture of radius $\epsilon$, the horizontal area at height $h$ is $\pi h^2$. <Torricelli's law> gives speed $\sqrt{2gh}$, so <conservation of mass> gives $h^2\dot h=-\epsilon^2\sqrt{2gh}$. Integration yields the displayed leading drainage time when $h_0\gg\epsilon$. The final layer $h=O(\epsilon)$ is outside the quasi-steady approximation and contributes only order $\sqrt{\epsilon/g}$, compared with the leading time of order $h_0^{5/2}/(\epsilon^2\sqrt g)$. This is an ideal inviscid estimate without a discharge coefficient.
Back to article page
