Interfaces endowed with non-constant surface energies revisited with the d'Alembert-Lagrange principle

The equation of motions and the conditions on surfaces and edges between fluids and solids in presence of non-constant surface energies, as in the case of surfactants attached to the fluid particles at the interfaces, are revisited under the principle of virtual work. We point out that adequate behaviors of surface concentrations may drastically modify the surface tension which naturally appears in the Laplace and the Young-Dupr\'e equations. Thus, the principle of virtual work points out a strong difference between the two revisited concepts of surface energy and surface tension.