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I have been taught that water levels will always equal out. However, now I find that sumps and some other setups allow for water not to become equal. What other arrangements allow for the water levels of two containers not to be equal? Also, and more importantly, why does this occur? I hope someone is able to explain this for an aspiring physicist.

Edit: Another example I have found is a phenomena called Heron's fountain, how does the water level rise instead of all of it pooling up in the lowest container?

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What do you mean equal out? –  Pranav Hosangadi Mar 4 at 6:41
    
To be at the same horizontal level –  Ryan Sinclair Mar 4 at 7:16
    
Different outside pressure levels at different parts of the water surface could cause them not to level out. –  Brandon Enright Mar 4 at 8:05

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Whenever it seems like two water levels should be equal but aren't, either there is a physical restriction preventing flow (like a dam keeping upstream waters higher than downstream, or surface tension causing meniscuses or capillary action), or there is energy being expended to put water back upstream as fast as gravity is pulling water downstream.

In a sump, the lower water level can be maintained despite water continously flowing into it is due to there being a pump at the bottom, consuming electrical energy to move water from that lower section back to the higher section. So there appears to be continuous flow of water yet no change in the different water level. The maximum height difference then depends on the power of the pump (how much work it can do per unit time, or technically how much water it can move back up per unit time). Note that since energy is flowing into the system at one point, the new equilibrium IS unequal water levels (it stays unequal so long as you keep the pump on). If you turned the pump off, the water level equalises quickly.

Same thing for a waterfall. Why is there always water at the top of the waterfall to keep it flowing even though all the water should logically be at the bottom? The sun is providing the energy to evaporate water from the bottom creating clouds and rain to put water back on top--rinse and repeat.

Always look for an energy source--it is the reason for a different equilibrium condition.

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So the source of energy is either electrical or gravity? Are there other ways that you can think of that cause differences? Like for example, (if you know) how does heron's fountain work? –  Ryan Sinclair Mar 4 at 7:22
    
heron's fountain is also powered by gravity. eventually the lower tank has to be emptied and the upper tank has to be refilled (with water). this entails work done against gravity--this potential energy is stored in the upper tank. when the fountain stops, you will notice that the effective movement of all the water in the system was downwards. –  gregsan Mar 4 at 11:10
    
Oh so eventually you have to manually put the water back in the top? –  Ryan Sinclair Mar 5 at 2:07
    
yes it definitely does not run forever –  gregsan Mar 5 at 5:09

If the water is frozen - it won't equal out. If the containers are not connected - it won't equal out. If water can't reach the connection equally easily from both containers - e.g. if the connection is high or low and the water is vapor - it won't equal out.

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Any system in which hydrostatic pressure is not the only pressure to consider could sustain a water level difference between connected containers.

To give an example: a system that isn't at a static, but rather a dynamic steady state can have unequal water levels. Think of two containers with one (tank 1) elevated above the other and a two hoses connecting the two. In hose 1 the water can flow freely, which will allow it to flow from tank 1 to tank 2 by gravity (i.e. hydrostatic head), but if there is a pump connected to hose 2 that pumps the same amount of water from tank 2 to tank 1 than you can sustain unequal levels. This is, on a much larger scale, also what happens with rivers: water flows down under the action of gravity, but it also evaporates and gets added back (among others) at the start of the river.

Another system that can have unequal water levels would be a container with two capillaries with small but unequal inner diameter (e.g. 0.1 and 1 mm). Capillary forces will cause the water to rise to a higher level in the small inner diameter tube than in the larger inner diameter tube.

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So, in your first example, it is the force of gravity that is causing the inequality? –  Ryan Sinclair Mar 4 at 7:15
    
No, gravity is responsible for the hydrostatic pressure. It is the pump that compensates this and sustains the inequality –  Michiel Mar 4 at 10:20

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