A contact theory for surface tension driven systems

Abstract

This paper presents a general model description for the contact of surface tension driven systems. The example system of a liquid droplet in contact with a deformable solid substrate is considered. This can be easily modified to consider two liquids or two solids in contact. The surface kinematics, essential to the modeling of surface tension, are described here in curvilinear coordinates. In particular modeling focus are the contact conditions at the contact boundary, where a wetting ridge may develop. It is shown that in the case of quasi-statics and hyperelasticity the governing equations can be derived from a global potential that accounts for contact as well as the energy storage within the bulk and surface domains. Altogether, 21 Euler–Lagrange equations are derived in this manner. Apart from these strong form equations, the governing weak form as well as its complete linearization, which are required for computational methods, are also discussed. It is shown that the governing equations can be further simplified into a reduced set of equations that are then suitable for an efficient computational implementation of the system. Computational solution methods are not discussed here, as the present focus is on the theory and its implications. A few remarks on analytical solutions, as well as a simple computational example, are given nonetheless. An auxiliary benefit of this work is a summary of the variation and linearization of the kinematical and constitutive equations of the system.