The shape of a liquid drop is governed by what is known as the Young-LaPlace equation. It was derived more or less simultaneously by Thomas Young (1804) and Simon Pierre de Laplace (1805). Read here how they did it.
The shape of liquid drop is governed by what is known as the Young-LaPlace equation. It was derived more or less simultaneously by Thomas Young (1804) and Simon Pierre de Laplace (1805). A short derivation of this equation is presented here.
Consider a small section of a curved surface with carthesian dimensions x and y. On the normal n two arcs can be constructed that support the lengths of x and y:
When the radii are enlarged by a small amount dR the work dW performed equals:
Where ? is the surface energy. The work is done against a pressure. According to thermodynamics this work can also be written as:
So we obtain for P:
From the first equation it can simply be shown that:
Insertion of the results from (5) into eq (4 yields the Young-LaPlace equation:
The radii of curvature
An expression for R1 for the cylindrical symmetrical can be deduced as follows:
Consider a contour f(z) with derivative f’(z) equaling n1. The length of the segment dL over distance dz equals:
This is equated to α.R1 . The angle α is derived from the difference in derivatives in z and z+dz:
Upon completion we obtain for R1:
The second radius R2 is derived as follows: take a small segment from the circumference of the profile. From the length of this segment dC we can see that:
From the figure we learn that:
From which it can readily be deduced that:
As a final result we obtain or the Young-Laplace equation:
C. van Veen
Read the pdf file here.