Generalized Navier Boundary Condition for a Volume Of Fluid approach using a Finite-Volume method

TL;DR

This paper introduces an analytical Volume Of Fluid implementation of the Generalized Navier Boundary Condition within a finite-volume framework, validated through droplet simulations that match theoretical predictions and reveal spreading dynamics.

Contribution

It presents a novel VOF-based method for implementing the Generalized Navier Boundary Condition, validated with droplet simulations showing accurate shape and spreading behavior.

Findings

Good match between simulated and theoretical droplet shapes.

Observed Voinov-Tanner-Cox law in droplet spreading.

Spreading dynamics follow a $t^{1/2}$ scaling, indicating inertia-limited spreading.

Abstract

In this work, an analytical Volume Of Fluid (VOF) implementation of the Generalized Navier Boundary Condition is presented based on the Brackbill surface tension model. The model is validated by simulations of droplets on a smooth surface in a planar geometry. Looking at the static behavior of the droplets, it is found that there is a good match between the droplet shape resolved in the simulations and the theoretically predicted shape for various values of the Young's angle. Evaluating the spreading of a droplet on a completely wetting surface, the Voinov-Tanner-Cox law ($\theta \propto \text{Ca}^{1/3}$) can be observed. At later times scaling follows $r \propto t^{1/2}$, suggesting spreading is limited by inertia. These observations are made without any fitting parameters except the slip length.