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As a picture is worth a thousand words, here is my problem:

Picture here

It is a closed water column with a bit of air in the top section. If I run the pump and make the water flow from IN to OUT,

  1. Will the level stay the same and the water circulate indefinitely? or
  2. Will slowly start to go down in the column and and the left side rise until it overflow?
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up vote 2 down vote accepted

This will run indefinitely, and it is the basis of just about every water fountain produced commercially. In a given time, the pump will move a certain volume of water from IN to OUT. That lowers the height of the column at IN, and increases the headspace over that column. Both of those reduce the pressure that column of water provides at the interface between the two columns. The displacement of the water also increases the height of the column of water at OUT. That increases the pressure that that column provides at the interface between the two columns. Both of those effects lead to the column at OUT pushing water into the column at IN. Equilibrium is reestablished when exactly the volume of water the pump moved has been moved back into the IN column. That process happens for as long as the pump is running.

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When you will start the pump it will suck the water from the $IN$ side of the pump that will create a little bit a region of low pressure at the top as their is no entry for the air at the top so the low pressure will try to pull the water column upward in order to maintain pressure equilibrium due to which the water at the other side will go down and when the sucked water will come out from the $OUT$ side it will again equalise the water level on the left side. So the level stay's the same and the water circulate indefinitely. But the condition for overflow of the water from the other side also depend's on the power of the pump you are using. Hope it cleared your doubt.

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