Mathematical strategies to quantify exactly the forces acting on the nonaxisymmetric capillary bridges
Résumé
The didactic object of these developments on differential geometry of curves and surfaces is to present fine and convenient mathematical strategies, adapted to the study of capillary bridges from experimental data and relatively simple to use in practice. The common thread is to be able to calculate accurately in any situation the bending stress over the free surface Σ, represented mathematically by the integral of the Gaussian curvature over the surface (called the total curvature) and also to obtain an information concerning the capillary tension forces by term by term integrating the generalized Young-Laplace equation. We mainly develop three convenient mathematical tools for assessing the physical properties in the field of the axisymmetric or not capillary bridges with convex or nonconvex plane boundaries, according to the local wettability and roughness effects: the unit speed reparameterization (or by arc length) of a regular curve and in particular for surfaces of revolution, the Fenchel's theorem and the Gauss-Bonnet-Binet theorem that expresses a relation between the integral of the Gaussian curvature over the surface, the topology of the surface and the integrals of the geodesic curvatures which are directly linked to the wetting angle at the contact lines. We express also the resultant of the bending energy only with respect to the wetting angles at the contact line.
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