Confusion in Derivation of Excess Pressure in a Cylindrical Drop I have recently learnt about surface tension and have developed a list of key points to solve problems:-

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*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.

*It acts tangential(or along) the surface.

*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:-

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:-

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:-

*

*Whether my list of points is correct and sufficient (to solve elementary problems)

*Mistakes in my proof

Thank you.
 A: 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.
A: That's a good start, however there are some subtleties behind treating surface tension as a force. Indeed, there are some classic problematic cases that are often overlooked at first. The best way to approach it is by using the principle of virtual work, but if you insist on a Newtonian method and want to know more on pathological examples, check out the short paper: Why is surface tension a force parallel to the interface? by A. Marchand.
Your reasoning is not quite correct. What I think you are trying to calculate is the difference of pressure necessary to maintain the curvature of the cylinder at equilibrium. This is a local property, and you can use a similar trick to yours by using translational symmetry to calculate it, this is why you mustn't include the caps. In general, you have the famous Young-Laplace formula which will directly spit out the result. Btw, you typically need to multiply the result by $2$ if you're dealing with soap films for example since you have $2$ intefaces.
Another problematic point in your reasoning is that you are assuming a constant pressure difference to maintain your cylinder in equilibrium. This is not the case as the flat extremities require a null pressure difference while the lateral curved face needs a different pressure difference that is given in the sources. There is even a third pressure difference needed to maintain the "seams" that link the two kinds of surfaces as well, but I guess you could argue it away by invoking a wireframe to maintain the shape. Using this experimental setup, you can see that you cannot have a simultaneous flat top and cylindrical side. If the extremities are flat, the sides would cave in (forming a catenoid, as long as your cylinder is not too long), and if you pump some air in, you can straighten the sides, but now the extremities will bulge out.
Hope this helps and tell me if you need more details.
