Why does Torricelli's law seems to fail when water speeds up after I put my finger in a hose?

We are taught that the speed of the fluid through a hole in a water filled tank obeys Torricelli law: $v=\sqrt{2gh}$, thus the speed is independent of the size of the hole. Why is it then that making the hole of the end of a hose smaller (for instance, by using my finger) the speed increases? (note, I was told that this is because the situation is different, and what I have is constant pressure from the street, but this is not the case in my house where we have a tank on the roof)

What you have given as toricelli's law is a general simplication. The actual law is stated as this: $$v = \sqrt\frac{2gh}{1-\frac{a^2}{A^2}}$$ Here $$a$$ is the area of the smaller hole and $$A$$ is the area of the bottom of the tank. This complete formula gives the dependence on the area you make (or decrease) by using your finger. While generalising, we often consider $$A\gg a$$ and hence neglect the denominator to get what you wrote earlier, $$i.e$$, $$v = \sqrt{2gh}$$

Torricelli's law is just a restatement of the conservation of energy of a non-viscous, non-turbulent and incompressible liquid flow. Thus, the maximum speed that can be obtained by making water flow through a hose (purely by the force of gravity, ie. a tank on the roof) is the formula that is given.

The water flows slower when it moves through a hose with a larger diameter because of the turbulence of the flow, reducing it's speed from the ideal limit.

• Even with a non-viscuous non turbulent flow, Torricelli law is still a simplification that does not take into account the continuity equation and assuming a large water reservoir. May 18 at 17:35

Torricelli's formula can be derived for water jet running out from a tank, where water does not move much and obeys laws of hydrostatics.

Inside a hose, all the water moves roughly with the same velocity and the Toricelli's formula does not apply, because moving water experiences considerable friction as it moves along the hose and behaves in a more complicated way that make laws of hydrostatics inapplicable. The pressure in a water element decreases as it moves along the hose.

When you restrict the opening of the hose, more friction happens at the end, which decreases water throughput and therefore also speed of water inside the hose. The friction then is not as strong and the conditions are closer to hydrostatic case. The speed of water at the end gets closer to the maximum possible value, given by Torricelli's formula.

If you severely reduce the cross-section of the hose by putting your finger into it, you increase the pressure drop across the restriction and thus the flow rate decreases (you can block flow altogether too, of course, if your finger tightly fits or covers the hose's open end, or by 'kinking' the hose).

This causes the volumetric flow rate $\dot{Q}$ ($\mathrm{m^3/s}$) to be reduced but it also needs to flow through a smaller cross-section. Roughly we can calculate $\dot{Q}$ as follows:

$$\dot{Q}=vA,$$

or:

$$v=\frac{\dot{Q}}{A}.$$

with $v$ ($\mathrm{m/s}$) the flow speed and $A$ ($\mathrm{m^2}$) the cross-section.

So to maintain the same volumetric flow rate, at smaller $A$, $v$ needs to increase. Thus, seemingly paradoxically perhaps, you can reach high $v$ at low $\dot{Q}$.

When you're blocking the end of the tube gradually more and more, flow speed $v$ will first increase, until you've blocked the opening completely, so then $\dot{Q}=0$ and thus also $v=0$.

For 'reasonable' values of $A$ Torricelli's Law is respected.

The above also holds true for water drawn from the mains (constant pressure).

• "So to maintain the same volumetric flow rate [...]" Why SHOULD the same volumetric flow rate be maintained? Jun 12 at 14:25

Torricelli's law indicates the maximum dynamic pressure (proportional to velocity squared) that a certain hydrostatic pressure can provide. If you put a thin diameter (assume frictionless) nozzle on a non-frictionless hose, because the maximum speed in the thin nozzle is restricted to that of Torricelli's law for the given head (height) at that section in the pipe, $$v=\sqrt{2gh}$$, the flow speed in the much wider hose has to decrease due to volumetric flow rate, and because the hose is the only source of friction, the energy lost to friction is less because the energy loss to friction in a section of pipe is proportional to velocity squared.

Increasing the pressure at an open outlet so it approaches the source pressure only has the effect of ensuring that there's less dynamic pressure in the supply pipework meaning less flow velocity and less loss due to friction. If it's zero then all the energy is dynamic pressure and the rest friction. If it's close to the source pressure then only a fraction of the energy is taken up by dynamic pressure and friction, with all the remaining static pressure being able to be converted to dynamic pressure and not friction assuming a frictionless nozzle (due to the small length of a nozzle, the friction loss in a nozzle will still be low enough that the system won't come close the friction loss of the nozzleless system).

The maximum dynamic pressure (assuming 100% of the static pressure energy is converted to dynamic pressure) is the limit provided by Torricelli's law.

As the nozzle width approaches zero, the static pressure before the nozzle approaches the static pressure $$\rho gh$$ of the supply because the flow rate in the pipes approaches zero, and in a frictionless nozzle, the velocity in the nozzle will approach $$\sqrt{2gh}$$, except for the width of zero where v=0. In a nozzle with friction, as width approaches zero, the velocity in the nozzle approaches $$\sqrt{2gh}$$ but it reverses and approaches zero as the width of the nozzle gets really small as the effect of friction in the nozzle becomes significant.

Like decreasing the width of the nozzle and keeping the diameter of the supply pipe the same, if you keep the nozzle the same and increase the width of the pipe supplying it then the static pressure before the nozzle will approach the maximum hydrostatic pressure for the height (because flow speed will decrease because diameter increases and the volume needs to be the same as the volume coming out of the nozzle due to volumetric flow rate, which is limited by Torricelli's law because there can only be a certain speed in the nozzle, and it has a fixed diameter. So it has the same effect, except increasing the width of the supply pipe does mean that the speed of water coming out of the nozzle starts to converge on the limit of Torricelli's law at larger diameter nozzles, meaning a higher flow rate (speed*diameter) can be achieved with the wider nozzles (and that includes no nozzle at all) compared to when the wider nozzles are connected to a thinner supply pipe.

If you had a pipe of 2cm diameter and a nozzle of 1cm diameter, and a pipe of 4cm diameter and a nozzle of 2cm, the static pressure before the nozzle in the 2nd configuration would actually be slightly higher than before the nozzle in the first configuration because the supply pipe being physically wider means there is less friction loss, so increasing the pipe diameter doesn't only reduce the dynamic pressure in the pipe, which reduces friction loss, but a wider pipe actually causes less friction loss per unit of speed on the water that is flowing.