Clepsydrae, from Galilei to Torricelli

The whole article deals with free fall: the free fall of a solid particle carefully studied by Galileo Galilei and the free fall of a fluid particle along a stream line introduced by Evangelista Torricelli. Both limits are brought together in the problem of the vertical emptying of a cylindrical tube of diameter D₀ through a hole of diameter d, and one can move continuously from one limit to the other varying d, from d = D₀, to d << D₀. The limit, d = D₀, corresponds to Galilei's problem of the solid free fall, in which the velocity of the upper interface, ζV(i)ζ, increases as it comes closer to the hole following the law ζV(i)ζ = √2gζZ₀ - Z(i)ζ, where g is the acceleration due to gravity, Z(i) is the liquid height above the hole (defined by Z = 0), and Z₀ is the initial location of the interface. The opposite limit, d << D₀, corresponds to Torricelli's problem, in which the velocity of the interface slows down as it comes closer to the hole, following the law ζV(i)ζ = (d/D₀)²√2gζZ(i)ζ. Theoretically, the problem reduces to the integration of the differential equation: dv(i)²/dz(i) - [(D₀/d)⁴ - 1] (v(i)²/z(i)) + 1 = 0, where length and velocity have respectively been reduced by Z₀ and √2gZ₀, and the z-axis oriented from the hole z = 0, towards the initial interface location z = 1. With α ≡ (D₀/d)⁴ - 2, the above equation leads to the solution v(i) = -√z(i)/α(1 - z(i)(α)), when α ≠ 0, and v(i) = -√z(i) 1n 1/z(i) when α = 0. Galilei's and Torricelli's regimes correspond respectively to the limit α = -1 and α >> 1. These solutions are compared to experimental measurements conducted over a large range of geometrical and physical parameters. In a second stage, the model, developed for nonconstant cross sectional area, is compared to the experimental results obtained in conical Clepsydrae. (C) 2000 American Institute of Physics.