I'm studying fluid dynamics and I came up with an example that I find profoundly counterintuitive. I'd like someone more used to this kind of problems to confirm my guess.

We have two identical tanks with water $$A$$ and $$B$$. The only difference is $$A$$ has a hole, while $$B$$ has a long tube of the same width as the hole in $$A$$. I attach a diagram.

The problem I find is that if one applies Bernoulli, the velocity is different. Therefore, deposit $$B$$ empties much faster than $$A$$. This is very counterintuitive to me. How having a pipe facilitates the discharge? I would expect the opposite (although I acknowledge that this guess is motivated by the presence of friction the tube that I'm explicitly neglecting in this problem).

Is deposit $$B$$ discharging faster in reality? Why is so? How does water "know" that it must flow faster because there is a pipe somewhere below?

• It is not intuitive but it does discharge faster. Try draining a tank with a syphon hose, the lower you put the end the larger the flow
– user65081
Sep 2, 2019 at 17:36
• Does it help your intuition if you move tank A down so the holes are at the same altitude? If you wanted to plug those holes. How much force would each require to hold the plug in? Sep 2, 2019 at 19:35

It's not the pipe that's affecting the discharge speed. The only thing that matters in these equations* is the height of the water column above the hole.

The water at the opening in B has more water pushing down on it from directly above, so it makes sense that it should exit faster. But the fact that there's a pipe surrounding the column doesn't actually matter. To show this, suppose we construct a tank C, which looks the same as tank A (i.e. no pipe) but has a height $$h_1+h_2$$. If you use those same equations, you'll find that the water exiting the hole in the bottom of tank C (with no pipe) has the same velocity as the water exiting the hole in the bottom of tank B (with a pipe).

*Note that these equations are themselves a simplified model of the way actual fluids work. In particular, they're only reliable for completely inviscid liquids (i.e. those that flow without any internal resistance).

• This is of course only true for completely inviscid fluids. For viscous flow, viscous losses reduces output of $B$.
– Gert
Sep 2, 2019 at 14:24

I suspect that the velocity of fluid flowing from a container through a hole is $$\sqrt{2 g h}$$ only if the area of the hole is much smaller than the cross-section area of the container, otherwise the pressure cannot be maintained at the hole. Imagine that in case A the entire bottom falls out from the container, then the liquid will not move with the velocity of $$\sqrt{2 g h}$$. Case B is similar to that: the tube has no bottom.

EDIT (9/2/2019): See http://www.physics.princeton.edu/~mcdonald/examples/leaky_tank.pdf : " the velocity V of the eﬄuent stream from the tank is a function of the area ratio $$a/A$$", where $$a$$ is the area of the hole and $$A$$ is the cross-sectional area of the tank.