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

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 approximate the singular uncompensated Young stress by a smooth function with a characteristic width epsilon. We show that the resulting model is consistent with the fundamental kinematics of the contact angle transport described by Fricke, Köhne and Bothe. We implement the model in the geometrical Volume-of-Fluid solver Basilisk using a ``free angle'' methodology. 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 is grid-independent and leads to a full regularization of the singularity at the moving contact line. 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 and the dynamic contact angle is determined by a balance between the uncompensated Young stress and an effective contact line friction. Away from the moving contact line, we confirm that the viscous bending of the interface is well-described by the asymptotic theory of Cox. A non-linear generalization of the original GNBC is proposed, which is closely related to the Molecular Kinetic Theory of wetting.