# Bubble in water

## The Attempt at a Solution

Pressure inside bubble = 2T/R
Buoyant force on bubble = $\frac{4}{3} \pi R^3 ρ_w g$

But I do not understand how surface tension is exerting force on the bubble .Also I do not understand the direction of surface tension force .

Thanks

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haruspex
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At the contact with the base, surface tension acts tangentially to the bubble surface and perpendicularly to the line of contact (i.e. to the tangent to the circle of contact). Consider vertical force components.

At the contact with the base, surface tension acts tangentially to the bubble surface and perpendicularly to the line of contact (i.e. to the tangent to the circle of contact). Consider vertical force components.
Sorry . I did not understand .

haruspex
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Sorry . I did not understand .
The surface tension should result in a force on the base of the bubble, pulling it down onto the base of the vessel. On any short section of the circle of contact, the tension should act tangentially to the bubble, not straight down. We want the integral of the force around the circle. By symmetry, only the vertical component of the force on any short section will be left uncancelled, so we can take that vertical component and simply multiply it by the circumference of the circle.

However, I am not able to arrive at any of the offered answers. I have started a thread in the Advisors' forum to see if anyone else has ideas.

haruspex
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Here's what I posted on the Advisors' forum:
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The buoyant force of the water on the bubble will not be $\frac 43\pi R^3\rho g$ because there is no water under the (almost) sphere. It will fall short by $P_w\pi r^2$, where $P_w$ is the pressure in the water at the base of the vessel.
Correspondingly, the gas pressure should exert a net upward force on the bubble of $P_g\pi r^2$, where $P_g$ is the pressure of the gas.
The two pressures should be related by $P_g=P_w+\frac {2T}R$, but that's taking the pressure in the water to be uniform around the bubble, which cannot be true for Archimedes principle to work. So I correct this to take the water pressure around the bubble as averaging $P_w-R\rho g$. I.e. $P_g=P_w-R\rho g+\frac {2T}R$
Finally, I take the surface tension along the contact with the base of the vessel as the force holding the bubble down. Allowing for the presumed angle of contact (taking the bubble to be a sphere with a cap removed), the net force from this is $2\pi rT\frac rR$.
Pulling all this together, I find T disappears leaving $r=R\sqrt{\frac 23}$.

Throwing all that away and crudely writing $\frac 43\pi R^3\rho g=2\pi rT\frac rR$ I do not get any of the offered answers. I would get one of them if I used $4\pi rT\frac rR$, as would be appropriate for a bubble out of water (so double shelled).
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I got a response from TSny, a contributor for whom I have the utmost respect. He agrees with my analysis.

Chestermiller
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I agree with your analysis also.

Please see the attached image . Do you agree that green vector correctly shows the direction of surface tension acting on the bubble along the circumference of circular part of the bubble in contact with the vessel ?

The surface tension should result in a force on the base of the bubble, pulling it down onto the base of the vessel.
The surface tension force is due to the flat circular part of the bubble on the large spherical part . Isn't this an internal force as far as complete sphere is concerned ? How can we consider this force while doing the force balance on the sphere ? In other words , how can an internal force pull down the bubble ?

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haruspex
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Please see the attached image . Do you agree that green vector correctly shows the direction of surface tension acting on the bubble along the circumference of circular part of the bubble in contact with the vessel ?

The surface tension force is due to the flat circular part of the bubble on the large spherical part . Isn't this an internal force as far as complete sphere is concerned ? How can we consider this force while doing the force balance on the sphere ? In other words , how can an internal force pull down the bubble ?
Yes, you have the vector correctly.
There is no flat circular part to the bubble wall. On that area, the gas is in direct contact with the dry base of the vessel.
I agree you have to be consistent as to what is considered the "free body". You can take it to be the wall of the bubble (i.e. the layer of liquid in contact with gas) or that plus the gas. You should get the same result either way. I chose wall+gas.
The surface tension does two things. Over the general spherical curve of the bubble wall, it is an internal force of the wall+gas but leads to the pressure difference between gas and surrounding liquid. This pressure difference does act on the wall+gas over the circular dry area.
It also pulls directly on the vessel base around the perimeter of the circle.

haruspex
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I agree with your analysis also.
Thanks Chet.
I just noticed something, though. The question specifies the point at which the bubble is about to rise. Nothing in my analysis requires that, it is just a general relationship that must obtain for the bubble to be stable. I would think that the borderline instability condition should lead to an expression that fixes R, not just relates it to r.

There is no flat circular part to the bubble wall. On that area, the gas is in direct contact with the dry base of the vessel.
Ok . I was getting it wrong .

I agree you have to be consistent as to what is considered the "free body". You can take it to be the wall of the bubble (i.e. the layer of liquid in contact with gas) or that plus the gas. You should get the same result either way. I chose wall+gas.
I think it should be only "gas" , wall of the bubble doesn't look meaningful . Forgive me if I do not make sense .

I still do not understand properly how this surface tension force is acting as an " external force on wall + gas " ?? It makes some sense if I consider only the gas as a system .

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This pressure difference does act on the wall+gas over the circular dry area.
It also pulls directly on the vessel base around the perimeter of the circle.
I am so sorry but this surface tension has baffled me . I am also feeling quite confused why this force should be pulling the bubble down instead of pulling it up ( i.e at 180 ° opposite to the green vector shown in image in post #7) ??

Why should it be pulling up the vessel ?

haruspex
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I am so sorry but this surface tension has baffled me . I am also feeling quite confused why this force should be pulling the bubble down instead of pulling it up ( i.e at 180 ° opposite to the green vector shown in image in post #7) ??

Why should it be pulling up the vessel ?
You do need to think of the bubble wall as a thin skin under tension.
Let's start with something simpler: a light rope over a pulley with a weight W on each end. The external forces on the rope are the weights and the normal force from the pulley. The tension in the rope is an internal force, but it leads to the force the rope exerts on the pulley and to the force it exerts on the weights. These are analogous to the tension in the bubble wall leading to the force it exerts on the gas inside the bubble, and to the force exerted on the vessel base around the circle perimeter, respectively.
Surface tension is much like a two dimensional version of tension in an elastic string, except that it is constant instead of being proportional to extension, so the potential energy is proportional to the area instead of to the square of the area.

Delta2
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Am I the only one that finds it extremely counterintuitive that the surface tension doesn't play a role in the final result ? (according to harupex's analysis).

Judging by the level of learning this exercise requires I think what the book expects one to do is what harupex says in the last two lines of post #5 ("Throwing all that away and crudely writing..."). It isn't 100% correct as probably harupex's detailed analysis is, but many books expect you to do some simplifying assumptions in order to deal with the problem more easily.

Nidum
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Am I the only one that finds it extremely counterintuitive that the surface tension doesn't play a role in the final result ? (according to harupex's analysis).
+1

Judging by the level of learning this exercise requires I think what the book expects one to do is what harupex says in the last two lines of post #5 ("Throwing all that away and crudely writing...")
Yes . This is what is to be done . Strangely , in this problem , none of the options are correct .

Delta2
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Yes . This is what is to be done . Strangely , in this problem , none of the options are correct .
Maybe it is just a typo , it is just a 2 missing from the numerator on choice 3).

May be .

I agree with your analysis also.
Chet ,

Acknowledging Haruspex's nice analysis and efforts to make me understand things , I am still unsure about a couple of things . The problem is solved but could you help me understand these two conceptual issues I am having .

1) How is force due to surface tension acting as an external force on the bubble as far as force balance is concerned (post#7) ?

2) How is force due to surface tension pulling down the bubble (post#11) ?

Thanks

Chestermiller
Mentor
Thanks Chet.
I just noticed something, though. The question specifies the point at which the bubble is about to rise. Nothing in my analysis requires that, it is just a general relationship that must obtain for the bubble to be stable. I would think that the borderline instability condition should lead to an expression that fixes R, not just relates it to r.
I think it must be the combination of the two. Anyway, in my judgment, this is not a very well-defined problem. If the water pressure is changing with depth and the pressure within the bubble is virtually constant, the bubble shape can't be perfectly spherical. I also don't think that the shape at the base is going to be close to spherical, with a slope the same as if you just cut off a sphere at that cross section.

Chestermiller
Mentor
Chet ,

Acknowledging Haruspex's nice analysis and efforts to make me understand things , I am still unsure about a couple of things . The problem is solved but could you help me understand these two conceptual issues I am having .

1) How is force due to surface tension acting as an external force on the bubble as far as force balance is concerned (post#7) ?

2) How is force due to surface tension pulling down the bubble (post#11) ?

Thanks
Surface tension acts as if there is an ideal thin stretched membrane between the two materials at the interface. That model should help to answer both these questions.

haruspex
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It isn't 100% correct
If my analysis is correct, then it's a lot worse than merely not 100% correct. The approximations made are so inappropriate as to lead to a completely invalid result.

My own analysis still makes approximations that are too crude. It shows that the assumption that r<<R leads to a contradiction. So it is wrong to ignore the fact that the displaced volume of water is not a complete sphere.
As Chet mentions, since the pressure is uniform inside the bubble but not outside, it will not even be spherical.

Finally, as I mentioned, all of this analysis is towards finding the relationship between r and R which applies for any stable bubble. The question specifies that the bubble is on the verge of detaching. That extra condition will almost certainly lead to a way to fix both r and R, though one might need to know extra physical details, such as the kinetic properties of the gas.

haruspex