Thought experiment: Rising bubble in rigid container

[K^2]

256bits, it doesn't have to be perfectly incompressible. Compared to a gas, just about any real fluid will be near enough incompressible to get the same behavior as stated in the problem. So yes, real fluid will give a little, and the pressure in the bubble will drop slightly, but the change will be negligible, and you can easily have a setup where that pressure drop is much smaller than pressure differential between bottom and top of the container, resulting in an overall pressure increase in fluid as the bubble rises.

 

[sophiecentaur]

@k^2
OK, allowing for some compressibility in the liquid takes us away from a total nonsense scenario.
But the requirement is that the bubble volume doesn't change either so the pressure on it must remain the same. So the increase in pressure at the bottom of the container must be a factor of nearly two, as I described earlier.
Perhaps the trap door into this problem lies in asking how the bubble got to the bottom in the first place. It would need to have been introduced at that level as some liquid was drawn off through a hole in the top. Would that have involved Work Done? and a consequent change in Potential? I think so. It would correspond to an amount of liquid, the volume of the bubble being lifted up to the top(??) of the container. (Or possibly to just above the bubble). This volume of liquid would then fall to the bottom of the container, displacing the bubble and reducing the potential again.

 

[K^2]

sophiecentaur, you are making it way too complicated. In this entire problem, the only work being done is the work done by rising bubble against viscosity of the fluid. The exact nature of that process is complex, but we know that fluid will reach a steady state, and absorb all that work as heat. (And yes, real fluid will expand slightly due to the added heat, slightly reducing volume of the bubble, but these are tiny, tiny corrections.)

The only problems the initial statement suffers are problems of idealization, which are typical for this type of problem. If you consider constraint forces during collision of two rigid bodies, you run into the same kind of problem. What do you do? You consider energy and momentum. Same thing here. Except what we get to work with are energy and ideal gas law.

How the bubble actually gets to the bottom is irrelevant. We have an initial state. Question is how that state evolves.

 

[sophiecentaur]

Yes, you are right but I am thinking of the non ideal case. The relative nature of a gas and a liquid (and steel) allows this, I think. So, although it may be an irrelevant conclusion, it seems that the pressure must nearly double in the non ideal case.

 

[K^2]

Only if at the beginning of experiment the top of the fluid was not under pressure, and all of the pressure was accounted for by the gradient. That's not necessarily the case. Say, I start with a 1m tall cylinder filled with air at 1atm and sealed, with exception of a small inlet valve. Through that valve, I start forcing in liquid until the cylinder is 90% filled with liquid. I then allow the temperature in the cylinder to equalize with room temperature. The pressure of the gas is now 10atm. The pressure gradient in fluid is just under 0.1atm. If I now quickly invert the cylinder, perhaps by the top end so that centrifugal effect helps me to end up with bubble almost perfectly at the bottom, I have the setup for our problem. Bottom of the cylinder is at 10atm, while top is at 9.9atm. Once the bubble rises, the top is at 10atm, and the bottom is at 10.1atm. Increase of only about 1%.

 

[chingel]

I understand that this would be the scenario since there doesn't seem to be another explanation. But if the fluid column is of the same height, the gas exerts the same pressure, where does the extra force come from to the bottom?

 

[K^2]

Reaction force to the increased force on the top.

 

[sophiecentaur]

Only if at the beginning of experiment the top of the fluid was not under pressure, and all of the pressure was accounted for by the gradient. That's not necessarily the case. Say, I start with a 1m tall cylinder filled with air at 1atm and sealed, with exception of a small inlet valve. Through that valve, I start forcing in liquid until the cylinder is 90% filled with liquid. I then allow the temperature in the cylinder to equalize with room temperature. The pressure of the gas is now 10atm. The pressure gradient in fluid is just under 0.1atm. If I now quickly invert the cylinder, perhaps by the top end so that centrifugal effect helps me to end up with bubble almost perfectly at the bottom, I have the setup for our problem. Bottom of the cylinder is at 10atm, while top is at 9.9atm. Once the bubble rises, the top is at 10atm, and the bottom is at 10.1atm. Increase of only about 1%.

Yes, I agree. The fractional pressure increase is less and less as the pressure goes up. Actually, what you wrote gets it all in proportion. Good comment. I wish you'd made it ten pages back.

 

[chingel]

Oh, I see. When the bubble is at the bottom, the force felt at the top is the bubble pressure minus the pressure of the fluid column and at the bottom is the bubble pressure, when the bubble is at the top, the pressure at the bottom is the bubble pressure plus the fluid column pressure and at the top is the bubble pressure.