Confusion regarding surface tension direction and force balance

[palaphys]

Homework Statement

Analyze the equilibrium of a liquid bubble of density sigma, such that it floats exactly half immersed in a fluid of density rho. The surface tension of the liquid is S. (You may assume that the contact angle between the fluid bubble interface is roughly 0)

Relevant Equations

F= Sl

so this is the scenario given in the question.
What I know: there are forces due to surface tension in the following interfaces, at the contact areas: bubble-liquid, bubble-air and liquid-air.

as the forces in action are in the vertical plane, I am ignoring the forces at play at the surface, which pull the liquid surface outward in the horizontal plane. ( please correct me if I am wrong. ).

I am very confused about the direction of surface tension in the fluid-bubble interface, and the air- bubble interface.
Please correct me if my interpretation of surface tension itself is incorrect, as what I have been taught is that surface tension acts to "minimize surface energy" and thus surface area. However I am unable to apply this concept here.

 

[haruspex]

Your diagram does not represent "the contact angle between the fluid bubble interface is roughly 0". You show it as 90°.

But the question puzzles me… I would think what we need to know is the surface tension of the medium in which the bubble is floating, not that of the bubble.

 

[palaphys]

Your diagram does not represent "the contact angle between the fluid bubble interface is roughly 0". You show it as 90°.

But the question puzzles me… I would think what we need to know is the surface tension of the medium in which the bubble is floating, not that of the bubble.

sorry about the diagram. It may be a bit inaccurate. also, we know the surface tension of the medium in which the bubble is floating. That is what I have mentioned in the question as S. Also, after a bit of thinking my intuition says that the surface tension force form the liquid must be upward, so as to minimize area contact with the surface, and likewise the surface tension force from the air(if any) must probably act downward for the same reason.

Is this right? Please let me know

 

[haruspex]

sorry about the diagram. It may be a bit inaccurate. also, we know the surface tension of the medium in which the bubble is floating. That is what I have mentioned in the question as S. Also, after a bit of thinking my intuition says that the surface tension force form the liquid must be upward, so as to minimize area contact with the surface, and likewise the surface tension force from the air(if any) must probably act downward for the same reason.

Is this right? Please let me know

I was confused by references to "liquid" and "fluid" as though these were different. I read it as a sphere of a liquid (but described as a bubble) floating on the surface of a different one (the fluid).
So, as I now understand it, we have only one liquid, with a spherical air bubble embedded in its surface. Is that right?

You still don’t seem to get the bit about zero contact angle. What do you think the join between the bubble and the surface of the surrounding liquid looks like? Try to draw a diagram in which the bubble has some thickness.

 

[Chestermiller]

I too find the problem statement very confusing. Is this the exact word-for-word statement of the problem that you were given? If not, please provide the exact statement.

 

[palaphys]

I was confused by references to "liquid" and "fluid" as though these were different. I read it as a sphere of a liquid (but described as a bubble) floating on the surface of a different one (the fluid).
So, as I now understand it, we have only one liquid, with a spherical air bubble embedded in its surface. Is that right?

You still don’t seem to get the bit about zero contact angle. What do you think the join between the bubble and the surface of the surrounding liquid looks like? Try to draw a diagram in which the bubble has some thickness.

I too find the problem statement very confusing. Is this the exact word-for-word statement of the problem that you were given? If not, please provide the exact statement.

I do not have the exact question right now. But I have modified the question a little bit here, the question mentions a BUBBLE which is exactly half immersed in another fluid, such that the bottom part is wet (that is the reason I said contact angle=0) . the question asks to determine the RADIUS of the bubble in terms of the surface tension of the FLUID, density of the bubble liquid and the same of the fluid. But to solve this an analysis of forces acting on the bubble is necessary.

 

[Steve4Physics]

I do not have the exact question right now.

That's a pity. I agree with @haruspex and @Chestermiller. The question is (still) unclear.

You might consider re-writing the question: imagine you are someone reading the question for the first time.

For example, you refer to ‘liquid’, ‘fluid’ and ‘medium’. It’s important to use terms unambiguously and consistently. So, if two different liquids are involved, you could call them A and B, (making sure it is clear which is which).

Also, I don’t think a bubble would be spherical in these circumstances. I wonder if something is mis-translated and the ‘bubble’ is actually meant to be a solid sphere - that might make more sense.

 

[haruspex]

Fwiw, I came to think of it as looking like a humpback bridge reflected in the river below.
The semicircular underside of the bridge and its reflection represent the inside surface of the bubble. The roadway arcs smoothly up (but soon becoming vertical) then forms a semicircle at a slightly greater radius. The nearly vertical bit, where surrounding surface transitions to bubble surface, constitutes the zero contact angle.
In this model, you can calculate the force of surface tension exerted by the surrounding liquid surface on the outer surface of the bubble and balance that against the buoyant force from the submerged half of the bubble.

But since you say you added the bit about zero contact angle, a key reason for that interpretation goes away.

 

[palaphys]

That's a pity. I agree with @haruspex and @Chestermiller. The question is (still) unclear.

You might consider re-writing the question: imagine you are someone reading the question for the first time.

For example, you refer to ‘liquid’, ‘fluid’ and ‘medium’. It’s important to use terms unambiguously and consistently. So, if two different liquids are involved, you could call them A and B, (making sure it is clear which is which).

Also, I don’t think a bubble would be spherical in these circumstances. I wonder if something is mis-translated and the ‘bubble’ is actually meant to be a solid sphere - that might make more sense.

I have found a question which very closely resembles the question which I encountered:

 

[haruspex]

I have found a question which very closely resembles the question which I encountered:
View attachment 359516

That matches my original interpretation in post #2, and resolves the issue I flagged there: the surface tension given is that of the surrounding liquid, not of the liquid in the droplet.
So, can you now draw the diagram with the zero contact angle between the two liquids?

Btw, the question is still flawed in that it makes an unstated assumption about the units of the densities.

 

[palaphys]

Btw, the question is still flawed in that it makes an unstated assumption about the units of the densities.

what unstated assumption do you mean?

also for zero contact angle I think the diagram above is more accurate (the one on the top left is just a reference for defining contact angle, please ignore it)

 

[haruspex]

what unstated assumption do you mean? View attachment 359519
also for zero contact angle I think the diagram above is more accurate (the one on the top left is just a reference for defining contact angle, please ignore it)

No, that's not it.
You are told the drop is spherical and half immersed, so the surrounding liquid surface must contact it somewhere near its "equator". But the contact angle is zero, so what must that surface do near the droplet?

Oh, and there is a second flaw in the question: it has two solutions.
(No, the other "solution" I had in mind is a 180° contact angle.)

 

[Steve4Physics]

I have found a question which very closely resembles the question which I encountered:
View attachment 359516

Can I add to what @haruspex has said.

Your Post #9 question might be better-written as:

"A spherical object of density $\rho$ is half immersed in a liquid of density $\sigma$ (where values of $\rho$ and $\sigma$ are to be expressed in units of kgm$^{-3}$)...".

[Edit: The Post #9 question should also say that the contact angle is approximately zero.]

You need to identify all forces (including their directions) on the sphere. Look at this diagram (taken from https://www.geeksforgeeks.org/surface-tension/):

A key step is to draw a diagram for your problem and to use it to identify the direction of the force (adhesion) of the liquid on the sphere.

 

[palaphys]

Can I add to what @haruspex has said.

Your Post #9 question might be better-written as:

"A spherical object of density $\rho$ is half immersed in a liquid of density $\sigma$ (where values of $\rho$ and $\sigma$ are to be expressed in units of kgm$^{-3}$)...".

You need to identify all forces (including their directions) on the sphere. Look at this diagram (taken from https://www.geeksforgeeks.org/surface-tension/):View attachment 359524
A key step is to draw a diagram for your problem and to use it to identify the direction of the force (adhesion) of the liquid on the sphere.

I think that the surface tension force from the liquid of density $\sigma$ acts in an upward direction, as surface tension minimizes surface area in contact. In addition to that, there would be a buoyant force and a weight force. Balancing these would give me the answer. correct?

 

[palaphys]

No, that's not it.
You are told the drop is spherical and half immersed, so the surrounding liquid surface must contact it somewhere near its "equator". But the contact angle is zero, so what must that surface do near the droplet?

Oh, and there is a second flaw in the question: it has two solutions.

the surface near the droplet must be flat like this right? the contact angle being zero means that the bubble is ''wetted'' right?

 

[Steve4Physics]

I think that the surface tension force from the liquid of density $\rho$ acts in an upward direction,

That's confusing. Do you mean 'from the liquid of density $\sigma$'?

Note that the force at the boundary between the sphere and liquid is called adhesion; it is incorrect to call it surface tension.

At the boundary:
a) the liquid exerts a force of adhesion on the sphere;
b) the sphere exerts a force of adhesion on the liquid.
These two forces have equal magnitudes and opposite directions (Newton's 3rd law). The liquid is being pulled in one direction and the sphere is being pulled the opposite direction.

A suitable diagram helps you find the directions of the two forces - which is why you are being encouraged to present a diagram!

as surface tension minimizes surface area in contact.

If you have a single liquid, surface tension minimises its surface area. That's not relevant here.

In addition to that, there would be a buoyant force and a weight force. Balancing these would give me the answer. correct?

Yes! You will find it best to work with symbols: use 'g' for acceleration due to gravity and 'S' for surface tension. Don't use numerical values till the end.

 

[palaphys]

That's confusing. Do you mean 'from the liquid of density $\sigma$'?

Note that the force at the boundary between the sphere and liquid is called adhesion; it is incorrect to call it surface tension.

At the boundary:
a) the liquid exerts a force of adhesion on the sphere;
b) the sphere exerts a force of adhesion on the liquid.
These two forces have equal magnitudes and opposite directions (Newton's 3rd law). The liquid is being pulled in one direction and the sphere is being pulled the opposite direction.

A suitable diagram helps you find the directions of the two forces - which is why you are being encouraged to present a diagram!


If you have a single liquid, surface tension minimises its surface area. That's not relevant here.


Yes! You will find it best to work with symbols: use 'g' for acceleration due to gravity and 'S' for surface tension. Don't use numerical values till the end.

1. yes, I meant sigma. (edited now)
2. You say there are forces of adhesion in action. But what about surface tension forces then? (at the interface)
3. How do I determine the direction of the adhesion forces here intuitively?

 

[Steve4Physics]

2. You say there are forces of adhesion in action. But what about surface tension forces then? (at the interface)

I was overcomplicating things. Sorry. Surface tension and adhesion, (and also cohesion and capillary force) are useful terms - you can read about them for yourself.

So forget about adhesion. Go bck to thinking in terms of weight, upthrust and surface tension force.

How do I determine the direction of the adhesion [surface tension] forces here intuitively?

1) Draw a diagram, making sure the meniscus shows the correct contact angle.

2) Look at the shape of the meniscus.

3) Is the meniscus above or below the main liquid level? If the mensicus is above the main liquid level, it's because the sphere is pulling the liquid up. Corespondingly if the meniscus is below the main liquid level. Then apply Newton's 3rd law to get the direction of the force on the sphere.

Minor edit.

 

[palaphys]

I was overcomplicating things. Sorry. Surface tension and adhesion, (and also cohesion and capillary force) are useful terms - you can read about them for yourself.

So forget about adhesion. Go bck to thinking in terms of weight, upthrust and surface tension force.


1) Draw a diagram, making sure the meniscus shows the correct contact angle.

2) Look at the shape of the meniscus.

3) Is the meniscus above or below the main liquid level? If the mensicus is above the main liquid level, it's because the sphere is pulling the liquid up. Corespondingly if the meniscus is below the main liquid level. Then apply Newton's 3rd law to get the direction of the force on the sphere.

Minor edit.

2. where is the meniscus here? are you talking about the one which is above the main liquid level? as the spherical object is fully wetted I think there is no meniscus below the liquid level.
This seems to imply that the surface tension is downward. But the solution to the problem suggests otherwise.

 

[Steve4Physics]

2. where is the meniscus here?

The meniscus in this problem is the curved liquid surface where the liquid meets the sphere's equator (at the sphere-liquid-air boundary). A cross-sectional view (with contact angle about zero) looks like this:

This seems to imply that the surface tension is downward. But the solution to the problem suggests otherwise.

The force on the sphere (from the surface tension effect) is downwards.

The circumference of the sphere's equator has length $2\pi R$. If $S$ is the surface tension and $\theta$ is the contact angle, then the downwards force on the sphere from the surface tension effect is $2\pi R S \cos(\theta)$. If you are not clear where this formula comes from, read-up/research capillary action for yourself, as it is closely related.

 

[palaphys]

The meniscus in this problem is the curved liquid surface where the liquid meets the sphere's equator (at the sphere-liquid-air boundary). A cross-sectional view (with contact angle about zero) looks like this:
View attachment 359565


The force on the sphere (from the surface tension effect) is downwards.

The circumference of the sphere's equator has length $2\pi R$. If $S$ is the surface tension and $\theta$ is the contact angle, then the downwards force on the sphere from the surface tension effect is $2\pi R S \cos(\theta)$. If you are not clear where this formula comes from, read-up/research capillary action for yourself, as it is closely related.

this is the given solution. Is it incorrect? Very confused now.

 

[haruspex]

View attachment 359585
this is the given solution. Is it incorrect? Very confused now.

At last, that clarifies the intent. It seems the author omitted to mention that the contact angle is 180°. The diagram in post #20 is correct except that the meniscus curves the wrong way: it should dip down to meet the droplet, not up.

 

[palaphys]

At last, that clarifies the intent. It seems the author omitted to mention that the contact angle is 180°. The diagram in post #20 is correct except that the meniscus curves the wrong way: it should dip down to meet the droplet, not up.

Okay, now I am a bit more clear, but still got some doubts . How do I accurately predict the direction of surface tension in ANY case, and what is the formal definition of contact angle? how exactly to measure that?

Also, what is the logic behind the fact that the direction of the meniscus helps in finding the direction of surface tension?
Why is the fact that ''surface tension aids in minimizing surface energy and hence surface area'' not useful here?

 

[haruspex]

Okay, now I am a bit more clear, but still got some doubts . How do I accurately predict the direction of surface tension in ANY case, and what is the formal definition of contact angle? how exactly to measure that?

https://en.wikipedia.org/wiki/Contact_angle

Also, what is the logic behind the fact that the direction of the meniscus helps in finding the direction of surface tension?

The force from surface tension acts parallel to the surface and normally to the boundary of the surface.

Why is the fact that ''surface tension aids in minimizing surface energy and hence surface area'' not useful here?

It does not help here, except to say that the surface is trying to shrink, so is pulling on its boundary. It does determine the three dimensional shape of the meniscus.

 

[palaphys]

https://en.wikipedia.org/wiki/Contact_angle

The force from surface tension acts parallel to the surface and normally to the boundary of the surface.

It does not help here, except to say that the surface is trying to shrink, so is pulling on its boundary. It does determine the three dimensional shape of the meniscus.

In this diagram, I have constructed a tangent. Where is the contact angle? Unable to find it even after briefly going through the wiki page.

 

[Steve4Physics]

To follow-up what @haruspex said:
- the Post #1 problem says the contact angle is "roughly 0";
- the problem in Post #9 does not give any contact angle;
- the solution in Post #21 applies when the contact angle is 180$^o$.
Argh!

View attachment 359588
In this diagram, I have constructed a tangent. Where is the contact angle? Unable to find it even after briefly going through the wiki page.

Contact angles are usually shown for a meniscus meeting a flat surface. But in your problem, the meniscus meets a curved surface (the sphere). So you need to consider the angle between 2 tangents:

The red line is the tangent to the sphere where it meets the meniscus. (With the sphere half submerged, this line is vertical.)

The blue line is the tangent to the meniscus where it meets the sphere.

The contact angle ($\theta$) is the angle between them.

It's a useful exercise to visualise what happens as $\theta$ varies from 0 to 180$^o$.

In the above diagram the meniscus is being pulled upwards by the sphere. Therefore (Newton's 3rd law) the sphere is being pulled downwards by the liquid. This is true for $\theta \lt 90^o$.

 

[RicoGerogi]

Homework Statement: Analyze the equilibrium of a liquid bubble of density sigma, such that it floats exactly half immersed in a fluid of density rho. The surface tension of the liquid is S. (You may assume that the contact angle between the fluid bubble interface is roughly 0)
Relevant Equations: F= Sl

View attachment 359445
so this is the scenario given in the question.
What I know: there are forces due to surface tension in the following interfaces, at the contact areas: bubble-liquid, bubble-air and liquid-air.

as the forces in action are in the vertical plane, I am ignoring the forces at play at the surface, which pull the liquid surface outward in the horizontal plane. ( please correct me if I am wrong. ).

I am very confused about the direction of surface tension in the fluid-bubble interface, and the air- bubble interface.
Please correct me if my interpretation of surface tension itself is incorrect, as what I have been taught is that surface tension acts to "minimize surface energy" and thus surface area. However I am unable to apply this concept here.

Surface tension is like pulling to make less contact. It pulls up at the bubble liquid interface and down at the air bubble one, and it balances everything out vertically. Basically it’s just trying to shrink the exposed surface area.

 

[palaphys]

Surface tension is like pulling to make less contact. It pulls up at the bubble liquid interface and down at the air bubble one, and it balances everything out vertically. Basically it’s just trying to shrink the exposed surface area.

this is what I thought. But I was told that this is a (possibly) incorrect interpretation.

 

[palaphys]

To follow-up what @haruspex said:
- the Post #1 problem says the contact angle is "roughly 0";
- the problem in Post #9 does not give any contact angle;
- the solution in Post #21 applies when the contact angle is 180$^o$.
Argh!


Contact angles are usually shown for a meniscus meeting a flat surface. But in your problem, the meniscus meets a curved surface (the sphere). So you need to consider the angle between 2 tangents:

View attachment 359591

The red line is the tangent to the sphere where it meets the meniscus. (With the sphere half submerged, this line is vertical.)

The blue line is the tangent to the meniscus where it meets the sphere.

The contact angle ($\theta$) is the angle between them.

It's a useful exercise to visualise what happens as $\theta$ varies from 0 to 180$^o$.

In the above diagram the meniscus is being pulled upwards by the sphere. Therefore (Newton's 3rd law) the sphere is being pulled downwards by the liquid. This is true for $\theta \lt 90^o$.

But, it seems impossible to visualize a 0 degree or 180 degree contact angle. What would be the difference between the two, and what do they physically imply?

 

[palaphys]

@Steve4Physics

is this an accurate diagram? As the meniscus becomes flatter and flatter, eventually the contact angle seems to tend to 180. I do not understand how this is different from a zero contact angle