What is the Young-Laplace equation and its relationship to surface tension?

[Shreya]

Homework Statement

Why does surface tension seems to have direction in these images? Isn't it a scalar?

Relevant Equations

S=Fl/Av

 

[ergospherical]

It is correct that surface tension is a scalar $\gamma(\mathbf{r})$. If you have a surface $\mathcal{S}$ containing a bounding contour $\mathcal{C}$ between two different phases, the surface tension force exerted by one phase on the other is $\displaystyle{\int}_{\mathcal{C}} \gamma(\mathbf{r}) \mathbf{n} ds$ where $\mathbf{n}$ is a unit vector tangent to $\mathcal{S}$ but orthogonal to $\mathcal{C}$.

In your diagram (Young's law), a force balance is being done on a tiny mass element at the lower left corner. Keep in mind that there are three different interfaces.

 

[haruspex]

The images are not illustrating surface tension; they are illustrating forces arising from surface tension.
In the same way, tension in a string is an intensive state of the string, not a force. It results in forces at the ends.

 

[Shreya]

surface tension force exerted by one phase on the other is ∫Cγ(r)nds where n is a unit vector tangent to S but orthogonal to C.

illustrating forces arising from surface tension.

Thank you erogospherical and haruspex! It makes sense now!

 

[Shreya]

containing a bounding contour

Can you explain what you mean by a contour?
And what is Young's law ? (I haven't studied it yet)

 

[Chestermiller]

Like pressure and rope tension, surface tension is a 2nd order tensor quantity, with bi-directional character. The scalar we call surface tension is just the magnitude of this tensor. Physically, surface tension acts within a free surface, and acts perpendicular to each arbitrary curve or line within the surface.

 

[Shreya]

to each arbitrary curve or line within the surface.

Thank you Chester, I get it now !

 

[ergospherical]

Can you explain what you mean by a contour?
And what is Young's law ? (I haven't studied it yet)

- the word contour, in that context, is just a fancy word for a curve :)
- Young's law gives the relationship between the three surface tensions; can you deduce it?

 

[Shreya]

Young's law gives the relationship between the three surface tensions; what is it?

S(la)cos (theta) + S(sa) = S(sl), right?
I actually have studied it, but the name of law was not specified

 

[Shreya]

Tangent to surface would mean 2 directions. So why do we consider a particular direction here. i e how do i know which one of the two to choose?

 

[haruspex]

Tangent to surface would mean 2 directions. So why do we consider a particular direction here. i e how do i know which one of the two to choose?

Posts #2 and #6 both mention that the force is in relation to a given line element within the surface, and acts orthogonally to it within the surface.

 

[Shreya]

Thank you all for helping me understand.

 

[hutchphd]

Posts #2 and #6 both mention that the force is in relation to a given line element within the surface, and acts orthogonally to it within the surface.

And it helps me to remember the units [force/length] in this context. The force "supplied" by the tension across that line element is proportional to the tension "times" the length of the small element.
And the other very useful relationship in this context is the Young -Laplace equation for curved surfaces:https://en.wikipedia.org/wiki/Laplace_pressure

 

[Shreya]

Young -Laplace equation

Thank you, hutchphd! I had actually learned this, but the textbook didn't mention the name.