On the Consistency of Dynamic Wetting Boundary Conditions for the Navier-Stokes-Cahn-Hilliard Equations

TL;DR

This work analyzes the sharp-interface limit of the Navier--Stokes--Cahn--Hilliard (NSCH) equations for binary fluids with moving contact lines. It introduces dynamic wetting boundary conditions together with a generalized Navier boundary condition (GNBC) to achieve consistent limiting behavior as the diffuse-interface thickness $\varepsilon$ and mobility $m$ vanish, and examines how the mobility scaling $m\propto\varepsilon^{\alpha}$ with $0<\alpha<3$ affects convergence. A formal sharp-interface limit is presented, showing the Ginzburg–Landau energy concentrates on the fluid–ambient interface with surface tension $\sigma_{la}$ and that the diffuse GNBC converges to the sharp-interface counterpart; numerically, 2D Couette-flow simulations validate convergence between diffuse- and sharp-interface models under GNBC and appropriate mobility scaling, including in triple-wedge regimes. The results yield a robust benchmark framework for dynamic-wetting simulations and have practical implications for multi-phase flows in thin channels, such as inkjet printing.

Abstract

We investigate the limiting behavior of the Navier-Stokes-Cahn-Hilliard model for binary-fluid flows as the diffuse-interface thickness passes to zero, in the presence of fluid-fluid-solid contact lines. Allowing for motion of such contact lines relative to the solid substrate is required to adequately model multi-phase and multi-species fluid transport past and through solid media. Even though diffuse-interface models provide an inherent slip mechanism through the mobility-induced diffusion, this slip vanishes as the interface thickness and mobility parameter tend to zero in the so-called sharp-interface limit. The objective of this work is to present dynamic wetting and generalized Navier boundary conditions for diffuse-interface models that are consistent in the sharp-interface limit. We concentrate our analysis on the prototypical binary-fluid Couette-flow problems. To verify the consistency of the diffuse-interface model in the limit of vanishing interface thickness, we provide reference limit solutions of a corresponding sharp-interface model. For parameter values both at and away from the critical viscosity ratio, we present and compare the results of both the diffuse- and sharp-interface models. The close match between both model results indicates that the considered test case lends itself well as a benchmark for further research.

Figures (8)

• Figure 1: A schematic of the considered physical setting of two fluid regions in a thin channel is displayed, located around the fluid-fluid interface. The zoom highlights the diffuse-interface representation of the setting, with the indicated phase-field variable and interface thickness.
• Figure 2: Schematic of the considered contact-line configuration pertaining to the sharp-interface GNBC (\ref{['eq:SI-GNBC']}), including nomenclature. The $(\xi_1,\xi_2)$ plane is orthogonal to the contact line. Adapted from Gerbeau:2009km.
• Figure 3: Schematic overview of the considered Couette test case. The dashed red line indicates the $\{\varphi(t=0) = 0\}$ contour line of the initial phase-field. The solid red line illustrates the equilibrated $\{\varphi(t=\infty)=0\}$ contour, representing the interface between the two fluids, corresponding to wall velocity $\overline{u}_\textsc{w}$ at the wetting boundaries $\Gamma_{\textsc{w}}$. Quantities of interest are the contact-point displacement, $\Delta x$, and macroscopic contact angle, $\tilde{\vartheta}$.
• Figure 4: No-slip NSCH (top) model with different mobility-induced slip lengths ($s_m$) and interface widths ($\varepsilon$), and sharp-interface model with different slip lengths ($s_\nu$). Showing velocity fields with streamlines.
• Figure 5: No-slip NSCH (top) and sharp-interface (bottom) models with different slip lengths ($s_\nu,s_m$) and wall velocities ($\overline{u}_\textsc{w}$). Showing velocity fields with streamlines.