 Drainage time of a conical tank

For a cone of half angle $\pi /4$ draining through a small circular aperture of radius $ϵ$, the horizontal area at height $h$ is $\pi h^2$. Torricelli's law gives speed $\sqrt{2gh}$, so conservation of mass gives $h^2\dot{h}=-{ϵ}^2\sqrt{2gh}$. Integration yields the displayed leading drainage time when $h_0\gg ϵ$. The final layer $h=O(ϵ)$ is outside the quasi-steady approximation and contributes only order $\sqrt{ϵ/g}$, compared with the leading time of order $h_0^{5/2}/({ϵ}^2\sqrt{g})$. This is an ideal inviscid estimate without a discharge coefficient.