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Consider a perfectly wetting fluid of surface tension S which is the be filled inside two vessels: A) A Cuboidal box B)A Cylindrical vessel of radius R.

The fluid, due to surface tension rises near the points of contact between the fluid and the surface. I now wish to calculate the pressure at the tip of the fluid in contact with the respective containers in both cases.

How will I apply Young-Laplace equation for the pressure at this point? Will the kind of container in both cases cause any change? ( In cuboidal container will one radius of curvature tend infinity and in circular container one of the radius be R? ) enter image description here

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Why do you want to use the Young-Laplace equation to find pressure? You would have to know curvature of the air-liquid interface shown, but this is hard to measure.

It is easier to measure height $h$ above the level where the liquid is at atmospheric pressure $p_0$, and express pressure there as function of $h$; it obeys the law of hydrostatic pressure:

$$ p(h) = p_0 - \rho g h. $$

Now we can use the Young-Laplace equation to find radius of curvature of the air-liquid interface at that height:

$$ p-p_0 = \sigma \bigg( \frac{1}{R_1} + \frac{1}{R_2}\bigg). $$

For water that has climbed up, pressure is lower than the atmospheric pressure $p_0$, so the bracket is negative, which means mean curvature of the surface is negative.

In case of the cuboid container, one term in the brackets vanishes and the other $R_i$ is due to radius of curvature of the liquid-air interface.

In case of a round container, no term vanishes, but one of them is $1/R$, $R$ being radius of the container circumference.

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