Solving a Bubble Puzzle: Understanding Surface Tension Force

In summary, the conversation discusses the surface tension force on a bubble at the point where it is about to rise. The direction of the surface tension force is shown to be tangential to the bubble's surface and perpendicular to the line of contact with the base of the vessel. The conversation also addresses the issue of whether this force is an internal force or not, and concludes that it is both an internal force on the wall of the bubble and a force on the vessel base. The analysis of the problem leads to a relationship between the pressure of the gas and the surrounding liquid, and the question of where the bubble is about to rise is raised.
  • #1
Vibhor
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Homework Statement



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Homework Equations

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 .

Please help me with the problem .

Thanks
 

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  • #2
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.
 
  • #3
haruspex said:
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 .
 
  • #4
Vibhor said:
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.
 
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  • #5
Here's what I posted on the Advisors' forum:
-----
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).
------
I got a response from TSny, a contributor for whom I have the utmost respect. He agrees with my analysis.
 
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  • #6
I agree with your analysis also.
 
  • #7
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 ?

haruspex said:
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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  • #8
Vibhor said:
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.
 
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  • #9
Chestermiller said:
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.
 
  • #10
haruspex said:
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 .

haruspex said:
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 :rolleyes: .
 
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  • #11
haruspex said:
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 :nb) . 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 ?
 
  • #12
Vibhor said:
I am so sorry but this surface tension has baffled me :nb) . 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.
 
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  • #13
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.
 
  • #14
Delta² said:
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
 
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  • #15
Delta² said:
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 .
 
  • #16
Vibhor said:
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).
 
  • #17
May be :smile: .
 
  • #18
Chestermiller said:
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
 
  • #19
haruspex said:
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.
 
  • #20
Vibhor said:
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.
 
  • #21
Delta² said:
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.
 
  • #22
Chestermiller said:
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.
There's also the issue of "angle of contact". Water likes to wet surfaces, i.e. tends to have a very low angle of contact.
 

Related to Solving a Bubble Puzzle: Understanding Surface Tension Force

1. What is surface tension?

Surface tension is the force that causes the molecules at the surface of a liquid to cling together, creating a sort of "skin" that allows the liquid to maintain its shape and resist external forces.

2. How does surface tension affect bubble puzzles?

In bubble puzzles, surface tension is the force that holds the bubbles together and allows them to form stable structures. Without surface tension, the bubbles would pop and the puzzle would not hold its shape.

3. What factors influence surface tension force?

The surface tension force is influenced by the type of liquid, temperature, and the presence of other substances in the liquid. For example, adding soap to water can decrease surface tension, making it easier for bubbles to form and pop.

4. How can we solve a bubble puzzle by understanding surface tension force?

By understanding the role of surface tension in bubble puzzles, we can manipulate the bubbles to create different structures and patterns. We can also use our knowledge of surface tension to predict how the bubbles will behave and adjust our strategies accordingly.

5. Are there any real-world applications of understanding surface tension force?

Yes, surface tension plays a crucial role in many natural phenomena such as the formation of raindrops, the floating of insects on water, and the shape of soap bubbles. It also has practical applications in industries such as pharmaceuticals, where surface tension is used to measure the purity of liquids.

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