A consistent treatment of dynamic contact angles in the sharp-interface framework with the generalized Navier boundary condition

TL;DR

This paper introduces a regularized, physically consistent model for dynamic contact angles in sharp-interface simulations, implemented in Basilisk, which regularizes singularities and aligns with fundamental kinematic principles.

Contribution

The authors develop the Contact Region GNBC (CR-GNBC), a smooth, physically consistent model for dynamic contact angles that regularizes singularities and is implemented in a Volume-of-Fluid solver.

Findings

The model achieves grid-independent solutions and regularizes the moving contact line singularity.

The dynamic contact angle is determined by a balance between Young stress and contact line friction.

The model's curvature at the contact line is finite and mesh converging.

Abstract

In this work, we revisit the Generalized Navier Boundary condition (GNBC) introduced by Qian et al.\ in the sharp interface Volume-of-Fluid context. We replace the singular uncompensated Young stress by a smooth function with a characteristic width $\varepsilon > 0$ that is understood as a physical parameter of the model. Therefore, we call the model the ``Contact Region GNBC'' (CR-GNBC). We show that the model is consistent with the fundamental kinematics of the contact angle transport described by Fricke, K\"ohne and Bothe. We implement the model in the geometrical Volume-of-Fluid solver Basilisk using a ``free angle'' approach. This means that the dynamic contact angle is not prescribed but reconstructed from the interface geometry and subsequently applied as an input parameter to compute the uncompensated Young stress. We couple this approach to the two-phase Navier Stokes solver and study the withdrawing tape problem with a receding contact line. It is shown that the model allows for grid-independent solutions and leads to a full regularization of the singularity at the moving contact line, which is in accordance with the thin-film equation subject to this boundary condition. In particular, it is shown that the curvature at the moving contact line is finite and mesh converging. As predicted by the fundamental kinematics, the parallel shear stress component vanishes at the moving contact line for quasi-stationary states (i.e. for $\dot{\theta}_d=0$) and the dynamic contact angle is determined by a balance between the uncompensated Young stress and an effective contact line friction. Furthermore, a non-linear generalization of the model is proposed, which aims at reproducing the Molecular Kinetic Theory of Blake and Haynes for quasi-stationary states.