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You are standing in the middle of a circular drain: The drain has a 500 feet circumference.

A perfect cylinder of water crashes onto you and drains into the drain--this cylinder also has a 500 feet circumference, so it is a perfect fit for the drain.

You have an umbrella that is 20 feet in circumference. You open it to protect you from the falling water, allowing you a pocket of empty water.

My question is--since the water is acting as a giant barrier between you and the outside, would you end up suffocating, because no air can pass through? Or would dissolved oxygen in the water find its way to you?
 

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INTP 874 sx/sp VLEF melancholic
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Why don't you go test it out for us?
 
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If the umbrella could withstand the forces I believe oxygen (and any other water absorbed gasses) would find its way to you because of its release due to turbulence.

Here's why I believe so: many years ago I attempted to siphon the water in a pond over the dam to an elevation about twenty feet lower.

I used a 4" solid corrugated pipe that had perfect wall integrity. I filled the pipe looped on the hillside below the dam by siphoning water from the pond with a garden hose. When the 4" corrugated pipe was sufficiently full I had a helper put the upper end in the pond and I dragged the lower end down the side of the hill below the dam.

A gusher of water was produced, but soon air bubbles showed up sporadically in the mix, and finally, in about an hour, the water in the pipe had either slumped low enough with the air bubbles occupying the highest levels in the pipe or some similar phenomenon was taking place so that the water gusher just finally petered out.

I came to the conclusion that the turbulence produced by the water passing the ridges in the pipe (which occupied about 50% by profile of the pipe wall) so that the water passing across the ridges produced a great deal of turbulence, enough so that in the process the gasses absorbed in the water were being released as air bubbles

I repeated this process three times in total each time with identical results. There was, at least temporarily a huge rush of water, then lots of bubbles showed up, then the water gusher surged intermittently alternating with low ebbs and finally the water reached a low level in the pipe, and then quit.

My theory is, depending on the turbulence being produced as the water would by-pass the umbrella, a certain unknown quantity of absorbed atmospheric gasses would be released in the contained space under the umbrella. I hope that helps.

FYI- I used a one hundred foot roll of four inch plastic unperforated pipe like pictured in the attachment
 

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First, in the problem as stated , if mechanical forces did not destroy the setup, the umbrella would bouy up to the surface , hang close to one position, or move downward with the current - depends on the velocity of water. This is similal to the terminal velocity problem in free falling objects.
The experiment: the corrugated pipe has about the highest pressure drop you could get. And it is almost impossible to have a syphon operate reliably with a 4 inch pipe without the lower end being under water. The slow velocity of water over the internal ripples allowed air to back up the pipe. Air bubbles if fiormed would be swept out with water in a stable syphon. so it quit not from air lock via collecting bubbles but from air backflow.
General: scaling up or down for an incompressable fluid - e.g.water - gives different results varying wih dimensins - some parameters vary as the square of dimensions and some as the cube. Surface tension effects are much greater as the diameter is decreased.

In summary, the problem needs to be defined in real terms and a proper fluid dynamics solution obtained. And oxygen does not come out of solution without a change of pressure and/or temperature. The amount is sufficient for a cold blooded fish but not for a warm-blooded land-dwelling mammal. Whales have a vast store of oxygenated blood - and still can drown.

When my sons were in Dongeons and Dragons with friends, they got frustrated that I could not solve their imaginary problems. I told them it was make believe (like your stated problem) and just go with the magic.
 

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On turbulance, that is how gasses are disolved in a liquid - turbulance as you described imncreased the surface and speeds up disolving - no reason for turbulance to drive gasses our of solution.
 
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