In fluid mechanics, the Young-Dupré relation relates the contact angle of a droplet at rest on a surface to the surface tensions of the interfaces. Let , and respectively be the energy costs of the liquid-gas, solid-liquid and solid-gas interfaces:
The derivation is simple: this is the only expression that maintains the droplet’s boundaries when you account for the surface tension force pulling along each interface.
A more general derivation is possible by using the calculus of variations. In 2D, the upper surface of the droplet is denoted by . Consider the following Lagrangian , with the two first terms respectively being the energy costs of the top and bottom surfaces:
And the last term comes from the constraint that the volume of the droplet must be constant:
The total energy to be minimized is thus given by the following functional, where the endpoints of the droplet are and :
In this optimization problem, the endpoint is a free parameter, i.e. the -value of the optimum is unknown and must be found. In such cases, the optimum needs to satisfy the so-called transversality condition at the variable endpoint, in this case :
Due to the droplet’s shape, we have the boundary condition , so the last term vanishes. We are thus left with the following equation:
At the edge of the droplet, imagine a small right-angled triangle with one side on the -axis, the hypotenuse on having length , and the corner between them being the contact point with angle . Then, from the definition of the cosine:
When inserted into the above transversality condition, this yields the Young-Dupré relation.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.