# Torricelli's law generalized

Since we never faced in a deep way the dynamics of fluids, and related subjects, I was wondering about this: as a very particular case of Bernoulli's theorem, we get Torricelli's law for some fluid coming out from a hole in a recipient.

$$v = \sqrt{2gh}$$

But I was actually thinking about this:

1) Shouldn't the velocity depends upon the largeness of the exit hole?

2) What about the density or the viscosity of the fluid? It's quite obvious that if we take water and sulfuric acid, or water and liquid mud we don't get the same velocity.

So.. does anybody know what is the "correct" Torricelli's law? References are good enough, unless you want to give me some answer to cut it shortly.

## 1 Answer

Toricelli's law has several simplifying assumptions in it. For the derivation to work, the cross-sectional area of the hole that the water is escaping from must be very small relative to the cross-sectional area of the tank that contains the water. This means that a large exit hole requires a very large cross-sectional area for the tank that the water is escaping from.

In addition, the law includes implicit assumptions of zero effects from viscosity and zero effects from fluid drag forces. However, the density is included in the law, because the Bernoulli equation (which is where the Toricelli derivation starts) is an equation that describes pressure of a fluid, and the static pressure of a fluid is given by the formula P = (rho)gh, where rho is the density of the fluid.

In principle, additional variables can be included in the derivation for Toricelli's law, but the derivation would need somewhat specific qualifications, such as the length of the pipe that water is flowing from, the diameter of that pipe, the relative roughness of the pipe, the fluid viscosity, etc. This would make for a somewhat less general law that would be substantially more complicated than what exists now.