Modeling capillary flows by conservation of acceleration and surface energy

Abstract

Modeling of capillary flows on a surface or at the interface of two fluids is approached from a formulation with two contributions, one described by the divergence of the normal to the surface and the second represented by its curl. These are the two curl-free and divergence-free components of a Helmholtz–Hodge decomposition of the capillary acceleration; the first is the gradient of the scalar potential of the acceleration and the second the dual curl of the vector potential. Proposed formulation is characterized by (i) definition of a directional curvature based on the dihedral angle instead of an average curvature, (ii) an intrinsic anisotropic surface tension per unit mass excluding the presence of density in the capillary terms of the equation of motion and (iii) introduction of a capillary potential, an energy per unit mass. This energetic approach leads to an equation of capillary motion of a surface without thickness; the same formalism can be integrated as a source in an equation of motion of immiscible fluids.