Two vertical plates submerged partially in a wetting liquid form a wedge with a very small angle `delta varphi`. The edge of this wedge is vertical. The density of the liquid is `rho`, its surface tension is `alpha`, the contact angle is `theta`. Find the height `h`, to which the liquid rises, as a function of the distance `x` from the edge.

Explanation

Let `h` be the height of the water level at a distance `x` from the edge. The the total energy of water in the wedge above the level outside is.
`E = int x delta varphi. dx. h . rho g((h)/(2)) -2 int dx .h. alpha cos 0`
=`int dx (1)/(2) x rho g delta varphi (h^2 - 2 (2 alpha cos theta)/(x rho g delta varphi) h)`
=`int dx (1)/(2) x rho g delta varphi[(h - (2 alpha cos theta)/(x rho g delta varphi)) - (4 alpha^2 cos^2 theta)/(x^2 rho^2 g^2 delta varphi^2)]`
This is minimum when `h = (2 alpha cos theta)/(x rho g delta varphi)`.
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