Drainage Time and Shape: Inequalities from Torricelli's Law
Eugen J. Ionascu
TL;DR
This work analyzes drainage time in solids under Torricelli’s law by formulating an integral framework: the drainage time is $T=\frac{F(H)}{K}$ with $F(x)=\int_0^x \frac{A(h)}{\sqrt{h}}\,dh$ and equivalently $T=\frac{2}{K}\int_0^{\sqrt{H}} A(s^2)\,ds$, where $A(h)$ is the cross-sectional area. It introduces the Torricelli number $\rho_{torr}=\frac{T_{max}}{T_{min}}$, derives universal bounds $\frac{V}{K\sqrt{h_C}} \le T \le \frac{\sqrt{2}\,V}{K\sqrt{h_C}}$ for centrally symmetric solids, and computes explicit $\rho_{torr}$ values for cubes, octahedra, tetrahedra, and icosahedra, illustrating how geometry controls drainage. The paper also defines a turn-up number $\rho_{\ell}$ for a fixed orientation and provides a concrete cone example with $\rho_{\ell}=\tfrac{8}{3}$, highlighting orientation-dependent drainage. Finally, it shows how to realize $\rho=1$ with non-symmetric solids by constructing cross-sectional area functions $g(y)$ satisfying $\int_0^1 g(s^2)\,ds=\int_0^1 g(1-s^2)\,ds$, and offers a crisp angular characterization for $\rho=1$ via a Fourier-like integral condition.
Abstract
We derive integral inequalities governing drainage time in convex solids, inspired by Torricelli's Law, and introduce the Torricelli number as a shape invariant. We use these considerations to construct a class of solids that can be used in building asymmetrical clepsydrae.