Flows from Bins: New Results - Selected Topics of Computational and Experimental Fluid Mechanics - page 156

Fig. 1 Frontispiece to
“ Lezioni accademiche
d'Evangelista Torricelli ... ”,
published in 1715 ( http://
www.photolib.noaa.gov/
htmls/libr0366.htm )

This last expression comes from equating the kinetic energy gained by an element
of mass m mv 2 2, with the potential energy lost, mgh , and solving for v . The lawwas

discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in
1641. It was later shown to be a particular case of Bernoulli's principle.
Torricelli was Galileo's assistant and companion during the last two years of the
elder scientist's life, and he succeeded Galileo in the post of grand ducal mathemati-
cian. In his Opera Geometrica , published at the expense of Grand-Duke Ferdinand II,
Torricelli elucidated and diffused the difficult geometry of Cavalieri, thereby gaining
himself widespread recognition throughout Europe. The first part, compiled around
1641, studies figures arising through rotation of a regular polygon inscribed in or cir-
cumscribed about a circle around one of its axes of symmetry. In the second section,
De moto gravium , Torricelli continued Galileo's study of the parabolic motion of
projectiles (Torricelli 1641 ). The treatise includes several significant contributions
to mechanics, the calculus and ballistics. It also refers to the movement of water
in a paragraph so important that Ernst Mach proclaimed Torricelli the founder of
hydrodynamics.
Torricelli's law can be demonstrated in the spouting can experiment, which is
designed to show that in a liquid with an open surface, pressure increases with depth.
It consists of a tube with three separate holes and an open surface. The three holes are
blocked, then the tube is filled with water. When it is full, the holes are unblocked.
The jets become more powerful, the fluid exit's velocity is greater the further down
the tube they are (Fig. 2 a). Instead, in a bin filled with dry sand and having staggered
holes of diameter D , the sand exit's velocity is the same and thus the shapes of the
sand jets are similar between them (Fig. 2 b).
In a container of vertical walls the pressure due to dry sand changes very slowly
with height when the level of filling, respect to the bottom, overcomes a critical value,
ʻ

, /

(Janssen 1895 ). This is the first hint that the sand jets must be very different respect