Finite Element/Level Set Modeling of Surface Tension with Marangoni Effects and Dynamic Contact Lines

Abstract

In this work we provide a complete variational framework for the modeling of capillary effects, starting from the virtual-work principle and arriving at a variational formulation suitable for numerical treatment by means of the finite element method. This formulation is then coupled with a level-set representation of the interface and suitable approximation spaces and stabilization terms. An interesting aspect is that the Marangoni force arising from surface tension gradients is automatically incorporated. This is illustrated by solving the thermocapillary migration of a droplet under a constant temperature gradient, which has an analytical solution. The treatment of contact lines is also addressed from within the variational framework, in particular the imposition of the static contact angle and of local dissipation laws. Some numerical examples of spreading drops are used to clarify controversial issues of this challenging problem.