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

TL;DR

The paper addresses dynamic wetting under a sharp-interface framework by replacing the singular uncompensated Young stress with a finite-width regularization, yielding the Contact Region GNBC (CR-GNBC). It combines a rigorous kinematic transport of the dynamic contact angle with a free-angle reconstruction in a geometrical Volume-of-Fluid implementation (Basilisk), establishing grid-independent, regularized solutions for moving contact lines. Key contributions include the CR-GNBC formulation, the derivation of a consistent contact-angle evolution law, a finite-curvature regularization at the contact line, and a non-linear extension motivated by Molecular Kinetic Theory. The approach provides a robust, thermodynamically consistent framework for simulating dynamic wetting with improved physical fidelity and numerical stability, with clear pathways to non-flat surfaces and broader transient regimes.

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, Köhne 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 $\dotθ_d=0$) 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.

Figures (14)

• Figure 1: Mathematical notation for the withdrawing tape setup.
• Figure 3: Extrapolation of the contact angle $\mathop{\mathrm{\theta_d}}\nolimits$ using the angle above$\theta_a$ located $3/2 \: \Delta$ away from the wall.$h_0$ to $h_2$ denote the horizontal heights.
• Figure 4: Validation of the free angle extrapolation method for varying grid sizes. (a) Temporal evolution of the contact angle. (b) Temporal evolution of the curvature. In both panels, blue corresponds to $D/\Delta = 32$, orange to $D/\Delta = 64$, green to $D/\Delta = 128$, red to $D/\Delta = 256$, and the black dashed line is the analytical solution. (c) Convergence of both the angle and curvature errors with $L_2$ and $L_\infty$ norms, showing quasi-second-order convergence for the angle and quasi-first-order convergence for the curvature.
• Figure 5: Steady-state meniscus example for $\mathop{\mathrm{Ca}}\nolimits=0.1$ using the present CR-GNBC with $\varepsilon=0.05$. The image is in the contact line's reference frame, where the left plate is pulled up with $\bar{U_w} = \sqrt{\mathop{\mathrm{Ca}}\nolimits}$. The inset shows a zoom around the contact line with streamlines highlighting a stagnation point in the upper phase
• Figure 6: Vertical height of the contact line as a function of time for different capillary numbers $\mathop{\mathrm{Ca}}\nolimits$, presented separately for (a) simple Navier boundary condition and (b) CR-GNBC. Steady-state heights are achieved, and a transition $\mathop{\mathrm{Ca_{tr}}}\nolimits$ is observed, beyond which the liquid film rises continuously. In both (a) and (b), $\mathop{\mathrm{Ca_{tr}}}\nolimits = 0.13$. Simulations are conducted with $\varepsilon = 0.05$, $\mathop{\mathrm{\theta_e}}\nolimits = 90^{\degree}$, and a resolution of $\varepsilon / \Delta = 5.12$.