Consider a container that is partially submerged in water as shown in illustration at the bottom of the question. A, B, C and D are valves, that are currently closed. The container is full of water, because before the experiment we pumped air out of it through valve D (let's pretend that the air pressure outside the container can push the water up in the container above the valve A under these circumstances - also the container is secured in its position as pictured).

The experiment proceeds as follows: we open all four valves at the same time and observe what happens. The question is - will the water during experiment flow through valves A B or C out of the container, or will the air rush into the container through these valves?

Illustration of described experiment

In my opinion the latter is correct - the air will rush in through all valves - I imagine the situation is the same as having low pressure zone below the container. However, the water surrounding the container makes me a bit unsure - does it affect the results of the experiment?

  • $\begingroup$ As you have drawn it, it looks like water will rush in all holes. Try drawing the picture again with a tiny opening on the bottom and large holes on the top and sides. Water will rush out the sides. $\endgroup$
    – mmesser314
    Nov 1 '19 at 15:57
  • $\begingroup$ @mmesser314 do you mean air will rush in all holes? I am not sure if I understand the "water will rush in" part of your comment $\endgroup$
    – usr1939
    Nov 1 '19 at 16:50
  • $\begingroup$ More detail is needed. As indicated below, the answer will depend on the relative sizes of the tank, valves, heights above each valve, etc. The answer will also depend on how quickly the valves can be opened. $\endgroup$ Nov 1 '19 at 18:21
  • $\begingroup$ @DavidWhite - I am afraid I can't provide any concrete info as the problem in this question popped up as a brainteaser, not something observed in the real world. Could you please elaborate more on the relations between sizes of holes and direction of flow (possibly in some answer)? $\endgroup$
    – usr1939
    Nov 1 '19 at 18:25

Because the water in the system is assumed to be static and connected, we know the pressure at any particular level is the same. So the pressure near the bottom of the vessel (below valve D) is equal to that at the surface outside the vessel, which is atmospheric.

As you go up the vessel, the pressure inside decreases with the pressure lapse rate of water, $\rho g h$ or $9807 \text{Pa/m}$, while the pressure outside is nearly constant.

So the internal pressure at any point above the surface is less than atmospheric. The action of opening any valve will allow air to rush in, equalizing the pressure.

After opening a valve, the water will start moving and you get a dynamic system with changing pressures inside the vessel. Analyzing the motion of the water at that point becomes more complex.

An alternate way to visualize this is to imagine how the situation was created. The pressure is initially atmospheric, but lowered as air is pumped out. The pressure at any level inside reduces until water reaches it, and is constant from then onward. So every point inside will be below atmospheric.

  • $\begingroup$ Just to sum this answer up - you are saying that even in case of opening all the valves air will go in through all of them? $\endgroup$
    – usr1939
    Nov 1 '19 at 18:50
  • $\begingroup$ Yes, the number of valves opened does not matter. $\endgroup$
    – BowlOfRed
    Nov 1 '19 at 18:53

My initial answer, as pointed out by @SV, was incorrect. My apologies! I wrote:

"You are probably imagining that there is a "suction" operating due to the weight of the water, but in fact there is only positive pressure both inside and outside the container, and water flow is simply due to the pressure difference.

Consider the pressure difference between the inlet and outlet of, e.g., valve A. At the moment when all valves are opened, there is a higher pressure at the inlet (the container side) of valve A than at the outlet (the air side). So, water will flow out of valve A until the pressure is the same at both its inlet and outlet. The same is true of valves B and C."

I tried the experiment, and indeed water did not flow out of A, B, and C. Here is my explanation:

Water pressure inside the container decreases linearly from ambient air pressure at the external surface of the water (0 feet), up to zero at a height of about 33 feet. Air pressure, on the other hand, decreases very slowly with height above the water surface. So, the pressure inside the container is less than the air pressure at every height. Since the flow direction is from high pressure to low pressure, when the valves are opened air will flow into the container through all valves D, A, B, and C.

  • $\begingroup$ Ok, so if I understand correctly - air will rush in only through valve D? How about the difference between flows from valves A,B and C - will the flow from C be stronger than from A because of the pressure difference between bottom and top of the container? $\endgroup$
    – usr1939
    Nov 1 '19 at 15:12
  • $\begingroup$ This answer can not be correct. Correct answer should depends on valve sizes, water level, etc. For example, what will happen if valve D is tiny (like a needle used by nurses), and valve A is huge? $\endgroup$
    – verdelite
    Nov 1 '19 at 16:09
  • $\begingroup$ @verdelite What do you think would be the difference between the case of "large D + small A" and case "large A and small D"?"When would air actually go inside through A? $\endgroup$
    – usr1939
    Nov 1 '19 at 16:53
  • $\begingroup$ @S.McGrew your answer is incorrect. The pressure at all of the valves is higher on the air side. The pressure in a body of (static) water is the same at any given height, that is a basic concept of hydrostatics. That means that at the same height inside the container the pressure is the same as the air pressure. All valves are above the outer water level and that means that the water pressure inside is lower than the ambient pressure (for all practical purposes we can assume that the air pressure is the same everywhere because of its low density). $\endgroup$
    – user137661
    Nov 1 '19 at 17:12
  • $\begingroup$ Are you saying that the pressure at the bottom of the container is identical to the pressure of air at the surface of the water? And that at a height of, say, 10 inches inside the container the water pressure is identical to that of air pressure 10 inches above the water? If that is true, then my answer is indeed incorrect. But can you provide a convincing argument to that effect? $\endgroup$
    – S. McGrew
    Nov 1 '19 at 17:36

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