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- Thread starter Jahnavi
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To achieve equilibrium, I think you need to have the work done by the gas if the bubble expands be equal to the energy needed to expand the cylindrical bubble because of surface tension. And in the case of a bubble, there is an inside surface and an outside surface. Are you familiar with the equation ## dW=S \, dA ## for surface tension? (I believe I have this correct, but I'm going to need to google it, because only on occasion have I worked a surface tension problem). ## \\ ## Edit: And yes, the cylindrical shape does get a different answer than the spherical shape, and I have also verified the calculation gets the correct answer for a sphere. You need to write ## dW=(P_2-P_1)A \,dr=S \, dA' ##, where ## A=(2 \pi r) \, L ## and ## A'=2 \, ( 2 \pi r) \, L ##. Does that seem logical? ## \\ ## Using the same type of calculation, you can also derive the result for a spherical bubble that you mentioned, where ## A= 4 \pi r^2 ##, and ## A'=2 (4 \pi r^2) ##. Bubbles are normally spherical, because the spherical shape has minimum surface area for a given volume. Thereby, the potential energy from the surface tension for a given volume is minimized.

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Chestermiller

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Pressure force on the half cylinder = ##\Delta P (2RL)##

Surface force on the half cylinder = ##4SL##

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I think it might have made sense if it had been specified as very short, so just a circular band between parallel plates.

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You need to write ## dW=(P_2-P_1)A \,dr=S \, dA' ##, where ## A=(2 \pi r) \, L ## and ## A'=2 \, ( 2 \pi r) \, L ##. Does that seem logical? ## \\ ##

What is A' and dA' ?

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Pressure force on the half cylinder = ##\Delta P (2RL)##

Surface force on the half cylinder = ##4SL##

Nice

Why does cylinder have to be "long" ?

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## A ## is the surface area of the sphere or cylinder. ## A' ## is the surface area that forms the surface tension. For a spherical droplet ## A =A' ##, but for a soap bubble with two surfaces,(inside and outside), and also for this cylindrical bubble that has two surfaces, ## A'=2 A ##.What is A' and dA' ?

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OK .

But how is A and A' appearing in the same expression i.e in dW = P2 - P1 = Adr = SdA' ?

But how is A and A' appearing in the same expression i.e in dW = P2 - P1 = Adr = SdA' ?

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## dU=S \, dA' ## is the potential energy obtained when the surface area ## A' ## is increased by ## dA' ##. Meanwhile ## dW= (P_2-P_1)A \, dr ## is the work available (to do work on increasing the surface area ## A' ##), as the bubble is expanded with ## r ## increasing by ## dr ##. When the conditions are such that these two quantities (differentials) are equal, you will have equilibrium. (This is similar to setting forces equal, where ## -\nabla U=\vec{F} ## etc.). (When ## dW >dU ## the bubble will continue to expand, etc., until ## dW=dU ##).## \\ ## You could also think of it as being a system that consists of two springs: ## dW=F_1 \, dr ##, (where ## F_1=(P_2-P_1)A ##), describes one spring system created by the excess pressure of the gas inside, and it is counterbalanced by what is essentially an elastic type material that has a potential energy ## U ## that obeys ## dU=S \, dA' ## The problem is to balance these two elastic forces. ## \\ ## @Chestermiller has an alternative and clever approach to set these forces equal, since the surface tension force is really at right angles to the pressure force on the walls of the bubble.

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Yes.Awould be the outer surface area of the bubble , the one that is exposed to atmospheric pressure . Right ?

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Why do you think the cylinder has to be "very long " ?

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They were just trying to try to avoid the effects of putting end faces on it. It really doesn't need to be long. It can be a cylindrical bubble contained between two metal discs as @haruspex points out above. I agree with his input that otherwise, a cylindrical bubble is very hypothetical and simply does not occur in the real world. In many ways, these equations could be describing a cylindrical (elongated) balloon though, so it can be worthwhile to perform the computation as it may have other real world applications. (The tension in a balloon will increase with increasing dimensions, so I am only making a loose comparison here).Why do you think the cylinder has to be "very long " ?

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They were just trying to try to avoid the effects of putting end faces on it. It really doesn't need to be long. It can be a cylindrical bubble contained between two metal discs as @haruspex points out above. I agree with his input that otherwise, a cylindrical bubble is very hypothetical and simply does not occur in the real world. In many ways, these equations could be describing a cylindrical (elongated) balloon though, so it can be worthwhile to perform the computation as it may have other real world applications. (The tension in a balloon will increase with increasing dimensions, so I am only making a loose comparison here).

OK .

Thanks @Charles Link and @Chestermiller .

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