# On water, deposits and holes

Hi from Spain, I hope somebody could answer the following doubt,

In the typical problem of the water deposit open on the top and with a hole that allow the water to escape through, the theoretical velocity of water when it leaves the deposit and enter in the air does not depend on the size of the hole but only on the height diference betwen the exit hole and the water surface. This is the so called Torricelli’s theorem and can be derived by applying the Bernoulli’s formula between the top of the water and the exit, supposing the pressure to be the same in these two points (the atmospheric pressure). The same could be applied if we put a hose in the hole so that the water escapes from the deposit through the hose. According to this, the water velocity just after the outlet of the hose should be independent on its diameter (as far as the viscosity is negligible).

However, it is an empirical fact that if we narrow or block partially the outlet of the hose with the finger or anything, the water velocity increases (it is easy to see looking at the distance reached by the water). It seems as if the theory fails here. Why can’t we apply Bernoulli’s equation here in the same way as in the well-known problem of the deposit?. How could we calculate the exit velocity in this situation, given the size of the hole and the height of water in the deposit?
Actually we have viscosity but as far as I know its effect is to reduce the velocity, not to increase it, so I think viscosity is not the reason. Maybe turbulence?. Perhaps, but the increase in velocity can be seen also if we narrow the end of the hose smoothly, squasching it, and it does not seem to be a situation prone to turbulence.

Maybe the pressure of the water at the end of the hose is lower than atmospheric and the continuous flux breaks into drops that we perceive as a continuous stream?. If so, many Physics books would be lying us.

Regards

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Look at energy. If water escapes, the fluid level in the deposit goes down. I.e. we gain potential energy, water that was previously at fluid level is now down at the level of the hole or hose exit. This energy is (partially) converted to kinetic energy. In particular, if there is no energy loss (friction, viscosity) the kinetic energy is equal to potential energy, and it turns out that the water is exactly as fast as if it had fallen down from the top level of the fluid. There is no way to make it any faster because no more energy is available.
Practically, I guess the water comes out much slower, in particular if it has to flow through a hose. If you now squeeze the hose outlet, it gets faster, yes, but never faster than the limit found above. No way around that.
So we are left with the question why we can come closer to said limit by squeezing the hose outlet. Well, the amount of water coming out will decrease, so the water speed inside the hose is reduced, and we have less friction inside the hose. I.e. more of the available energy is left to accelerate the water. Guess if you had NO hose and just make the hole at the bottom of the deposit larger or smaller you wouldn't see a big effect on water velocity.

russ_watters
Mentor
Jarl said:
However, it is an empirical fact that if we narrow or block partially the outlet of the hose with the finger or anything, the water velocity increases (it is easy to see looking at the distance reached by the water).
That's not an empircal fact for the situation you outlined above. The water doesn't flow at a constant velocity regardless of orifice size because it is flowing through a long tube/pipe and there is an associated pressure loss.

However, for very small orifices, you will note that the idea holds - if you hold the hose vertically and slowly close it off with your thumb, you will notice after a certain point, the water doesn't shoot any higher the more you close it off. That's the point where you have dropped the flow enough for the pressure inside the hose to reach its maximum total pressure.

Clausius2
Gold Member
Jarl said:
Hi from Spain, I hope somebody could answer the following doubt,

Maybe the pressure of the water at the end of the hose is lower than atmospheric and the continuous flux breaks into drops that we perceive as a continuous stream?. If so, many Physics books would be lying us.

Regards
Hi, what part of spain? Your english is good!. I am from Madrid, but currently working at San Diego.

About your problem. Ok, you may still apply Bernoulli equation if the obstruction is not too severe for lossing too much stagnation pressure. BUT you should realise that the pressure at the end of the hose is still the atmospheric pressure whatever you block it with. Instead, the pressure just upstream the blocking agent wouldn't be the atmospheric pressure. If there is not a severe blocking (with sharp edges and so on) one may consider that the stagnant pressure holds conserved along the block, try to apply Bernoulli equation and continuity and realise that it is sensible it must be a pressure drop along the blocking piece and an increase on velocity.

The Bernoulli eqn holds for any sufficiently smooth flow at infinitely large Reynolds # and geometries which don't enhance boundary layer separation.

Mk
Clausius2 said:
I am from Madrid, but currently working at San Diego.
And you have a sweet ride

Clausius2
Gold Member
Mk said:
And you have a sweet ride
Yeahhh, it's a coool ride:rofl: !

Clausius2 said:
Hi, what part of spain? Your english is good!.
Currently in Badajoz. I'm afraid not so good, only if you give me some time to think my words!

Clausius2 said:
The Bernoulli eqn holds for any sufficiently smooth flow at infinitely large Reynolds # and geometries which don't enhance boundary layer separation.
I am not sure if I understand, do you mean that Bernoulli don't hold with a hose?. In my hose the flow is smooth. Turbunence? I don't know.

If I understand to ezskater and russ_watters, you say that the reason for the increase in velocity when we block the outlet (at least in some range of blocking) is the friction. If we narrow the outlet, maybe the friction just in the outlet is higher, but the friction inside the hose is lower because the velocity upstream is lower. Mmmm...interesting. Maybe. I don't see it clear. I will try to do more experiments at home.

Regarding the possibility of a pressure in the outlet below atmospheric one, I cannot find any serious reason against it, although I think it may happen only with very narrow holes and enough velocity, like in manual sprays.

russ_watters
Mentor
Jarl said:
I am not sure if I understand, do you mean that Bernoulli don't hold with a hose?. In my hose the flow is smooth. Turbunence? I don't know.
It isn't as smooth as you think.
If I understand to ezskater and russ_watters, you say that the reason for the increase in velocity when we block the outlet (at least in some range of blocking) is the friction. If we narrow the outlet, maybe the friction just in the outlet is higher, but the friction inside the hose is lower because the velocity upstream is lower. Mmmm...interesting. Maybe. I don't see it clear. I will try to do more experiments at home.
No, the thing with friction is that there are losses, so when friction is involved, the hose can't deliver the same pressure from the pipe in your street to the hose in your backyard when the flow-rate is high. The other reason (simpler than I explained before...) is orifice size vs pipe size - the vavle has a smaller diameter than the hose and lets less water through than the hose is really capable of holding. The water slows down and expands to fill the hose.

russ_watters said:
No, the thing with friction is that there are losses, so when friction is involved, the hose can't deliver the same pressure from the pipe in your street to the hose in your backyard when the flow-rate is high.
Well, I understand this, but I don't see the conection with my doubt.

russ_watters said:
The other reason (simpler than I explained before...) is orifice size vs pipe size - the vavle has a smaller diameter than the hose and lets less water through than the hose is really capable of holding. The water slows down and expands to fill the hose.
I still think this does not answer the question clearly:
What is exactly the reason why Bernoulli does not work in this case?, and
Is it possible to predict the velocity increase in a theoretically accurate way?

Up! .