Thought experiment: Rising bubble in rigid container

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Discussion Overview

The discussion revolves around a thought experiment involving a bubble filled with an ideal gas rising in a perfectly rigid jar filled with an incompressible liquid. Participants explore the implications of buoyancy, pressure differentials, and the behavior of gases under these conditions, raising questions about the nature of the bubble's expansion and the effects of pressure on its ascent.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the bubble must rise due to buoyancy, regardless of whether it expands or not.
  • Others argue that if the fluid is incompressible, the bubble's volume remains constant, leading to a constant pressure within the bubble as it rises.
  • A participant questions whether the pressure differential between the top and bottom of the bubble would create a net downward force, suggesting that a larger force would be needed to prevent the bubble from expanding at the top.
  • Another viewpoint suggests that the bubble is effectively weightless and that static cases should be considered to understand the energy dynamics involved.
  • Some participants express confusion about the implications of temperature changes in the bubble, questioning whether the process is isothermal and how heat flow might affect the situation.
  • There is a discussion about the rigidity of the jar and the implications for bubble expansion, with some asserting that the bubble cannot expand due to the incompressible fluid, while others suggest it may still compress or deform.
  • One participant mentions that the total volume of gas must remain constant, leading to the conclusion that the pressure in the bubble cannot decrease.
  • Another perspective posits that the bubble can be treated as a void, allowing the liquid to fall and the bubble to rise, although concerns are raised about the speed of this ascent.
  • Some participants speculate on the behavior of the bubble in terms of elongation and fragmentation as it rises through the liquid.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the behavior of the bubble, particularly concerning its expansion, the effects of pressure, and the nature of buoyancy in this scenario. The discussion remains unresolved, with no consensus reached on the implications of the rigid container and incompressible fluid.

Contextual Notes

Participants highlight limitations in their understanding, particularly regarding the assumptions about incompressibility, the behavior of gases under pressure, and the implications of temperature changes. The discussion reflects a range of interpretations of the thought experiment without resolving these complexities.

  • #31
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.
 
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  • #32
@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.
 
  • #33
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.
 
  • #34
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.
 
  • #35
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%.
 
  • #36
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?
 
  • #37
Reaction force to the increased force on the top.
 
  • #38
K^2 said:
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. :wink:
 
  • #39
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.
 

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