Material Science – Confusion in Derivation of Excess Pressure in a Cylindrical Drop

continuum-mechanicsliquid-statematerial-sciencepressuresurface-tension

I have recently learnt about surface tension and have developed a list of key points to solve problems:-

  1. Surface tension acts on the surface where a surface is defined as the interface(flat or curved) between a liquid and another substance which could be a solid, a different liquid or gas.
  2. It acts tangential(or along) the surface.
  3. It is defined as force per unit length and for any given line/curve lying on the surface, surface tension acts in both directions normal (and in same plane to surface)to the line/curve. However, because the line/curve may be surrounded by different substances on both sides,so the net resultant force normal to the line may or may not be zero. For example, in:-
    enter image description here
    Net force is 0 on AB because it is surrounded by the same liquid on both sides but net force is not equal to 0 on CD because it is surrounded by liquid on 1 side and the moving rod on the other.Overall, the liquid provides greater force so resultant force is towards liquid.

Using the above concepts, I have been able to calculate excess pressure inside a spherical drop. But, I am unable to derive it inside a cylindrical drop. My attempt is as follows:-

The cylinder has 3 surfaces, 2 flat and 1 curved. surface tension acts along all 3 surfaces. We draw a rectangle passing the midpoint of the cylinder and show the forces due to surface tension.

Now, we split the cylinder into 2 halves and we draw the forces on one half due to the other half. So, we have:-

enter image description here

Now,
$$ Tl+Tl+T(2R)+T(2R)=P(2Rl)$$

where P is the excess pressure.

On solving, we get
$$P={T\over R}+{2T\over l}$$
However, various sources indicate $P={T\over R}$ and looking at their proof, they omit the force due to the flat faces i.e.
$$Tl+Tl=P(2Rl)$$

Please can you explain 2 things:-

  1. Whether my list of points is correct and sufficient (to solve elementary problems)
  2. Mistakes in my proof

Thank you.

Best Answer

Simple Answer: Assume that $l>>R$. You will get the answer in single step.

Complicated Reality: Liquid would not form cylendrical shape on its own. The ends are not going to be flat plates that you have assumed.

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