Reason for Calculating Surface tension

[Vivek98phyboy]

When calculating force due to surface tension across a hemispherical drop, we look at only the circumference and multiply it by the value of surface tension. When we know that it is the surface tension which is responsible for the curved surface of the liquid drop, why don't we calculate the force due to surface tension along the curved surface as well?

 

[trurle]

When calculating force due to surface tension across a hemispherical drop, we look at only the circumference and multiply it by the value of surface tension. When we know that it is the surface tension which is responsible for the curved surface of the liquid drop, why don't we calculate the force due to surface tension along the curved surface as well?

You can calculate survace tension across any curve drawn on droplet; the difference would be is what most of curves would require to take integrals including as inputs curvature of surface, slope and gravity - much more complicated than simple multiplication at the perimeter of droplet.

 

[Vivek98phyboy]

You can calculate survace tension across any curve drawn on droplet; the difference would be is what most of curves would require to take integrals including as inputs curvature of surface, slope and gravity - much more complicated than simple multiplication at the perimeter of droplet.

But as far as I've checked all the books and internet, there is no such example of using integral over the hemispherical surface. All they did was calculating T×2πr by only considering forces due to the other hemispherical part along the periphery.

 

[trurle]

But as far as I've checked all the books and internet, there is no such example of using integral over the hemispherical surface. All they did was calculating T×2πr by only considering forces due to the other hemispherical part along the periphery.

If you did not found such integral in literature, write it yourself. There is no rule against.

 

[Vivek98phyboy]

If you did not found such integral in literature, write it yourself. There is no rule against.

If i were to integrate it across the curved surface then i would get T×2πr² but it won't give you the answer

 

[Vivek98phyboy]

I haven't got any good response till now. Can someone help me understand this

 

[Lord Jestocost]

You can get the sample chapter "Fluid Interfaces and Capillarity" of John C. Berg's book "An Introduction to Interfaces and Colloids - The Bridge to Nanoscience" on https://www.worldscientific.com/worldscibooks/10.1142/7579.
To my mind, the relationships become clearer when one considers the force balance on a tiny patch of a curved surface (see Fig. 2-25 in that chapter).

 

[Vivek98phyboy]

You can get the sample chapter "Fluid Interfaces and Capillarity" of John C. Berg's book "An Introduction to Interfaces and Colloids - The Bridge to Nanoscience" on https://www.worldscientific.com/worldscibooks/10.1142/7579.
To my mind, the relationships become clearer when one considers the force balance on a tiny patch of a curved surface (see Fig. 2-25 in that chapter).

They directly stated that pressure on the patch= surface tension along perimeter

But I am expecting an explanation for not considering the surface tension in curved part

 

[gleem]

why don't we calculate the force due to surface tension along the curved surface as well?

There are three forces responsible for forming and maintaining the spherical shape and size of a bubble. The pressure inside forcing the bubble to expand and the surface tension and external pressure constraining the surface from expanding beyond a certain point. The final size occurs when the force of the internal gas pressure is in equilibrium with the resisting force due to the surface tension and the external pressure. The surface tension at any point on the bubble is in all directions within the bubble surface resulting in sphere of fixed radius. The force on the surface due to each pressure is everywhere the same (the internal pressures is greater than the external pressure ). One only needs to find a place that readily relates these three forces. The equatorial path on the surface of the bubble dividing the bubble in two hemispheres is the natural place to evaluate this duel of forces. The internal pressure at this location is trying to pull the two hemispheres joined here apart and the force it creates is equal to the component of the pressure perpendicular to the equatorial plane times the area of the equatorial plane. The surface tension along this path is everywhere on this path attempting to keep the hemispheres together and is also perpendicular to this plane but only on it perimeter and is equal to the surface tension force time the circumference of the the bubble. The external pressure directly acts against the internal pressure with its net force equal to the pressure times the area of the equatorial plane.