# Modelization of partial wetting by macroscopic approach in discrete mechanics

## Modélisation du mouillage partiel par une approche macroscopique en mécanique discrète

Jean-Paul Caltagirone
University of Bordeaux

Roger Prud’homme
Sorbonne-Université

Published on 30 October 2023   DOI : 10.21494/ISTE.OP.2023.1019

### Mots-clés

The partial wetting is generally defined by a contact angle between the liquid and the surface in the case of a static equilibrium excluding other types of actions such as gravity, inertia, viscosity, etc. When these last effects are no longer negligible, the modeling of two-phase flows governed by capillary forces cannot be reduced to simple geometrical laws on the surface tensions between the phases. The triple line is subject to accelerations that combine in a complex way to fix in time its motion on the surface. The macroscopic approach adopted is based on the representativity of the discrete equation of motion derived from the fundamental law of dynamics expressed in terms of accelerations. The formalism leads to a wave equation whose form corresponds to the two components of a Helmholtz Hodge decomposition, the first to the curl-free and the second to the divergence-free. Like all other contributions, the capillary effects are expressed in two terms of the capillary potential, an energy per unit mass. The longitudinal and transverse surface tensions allow for possible anisotropy effects in the tangent plane at the interface. The assignment of the surface tension values on the triple line related to the contact angle allows to take into account the partial wetting effects in a dynamic context. Two examples illustrate the validity of this approach.

The partial wetting is generally defined by a contact angle between the liquid and the surface in the case of a static equilibrium excluding other types of actions such as gravity, inertia, viscosity, etc. When these last effects are no longer negligible, the modeling of two-phase flows governed by capillary forces cannot be reduced to simple geometrical laws on the surface tensions between the phases. The triple line is subject to accelerations that combine in a complex way to fix in time its motion on the surface. The macroscopic approach adopted is based on the representativity of the discrete equation of motion derived from the fundamental law of dynamics expressed in terms of accelerations. The formalism leads to a wave equation whose form corresponds to the two components of a Helmholtz Hodge decomposition, the first to the curl-free and the second to the divergence-free. Like all other contributions, the capillary effects are expressed in two terms of the capillary potential, an energy per unit mass. The longitudinal and transverse surface tensions allow for possible anisotropy effects in the tangent plane at the interface. The assignment of the surface tension values on the triple line related to the contact angle allows to take into account the partial wetting effects in a dynamic context. Two examples illustrate the validity of this approach.