Finite-element lattice Boltzmann simulations of contact line dynamics

The lattice Boltzmann method has become one of the standard techniques for simulating a wide range of fluid flows. However, the intrinsic coupling of momentum and space discretization restricts the traditional lattice Boltzmann method to regular lattices. Alternative off-lattice Boltzmann schemes exist for both single- and multiphase flows that decouple the velocity discretization from the underlying spatial grid. The current study extends the applicability of these off-lattice methods by introducing a finite element formulation that enables simulating contact line dynamics for partially wetting fluids. This work exemplifies the implementation of the scheme and furthermore presents benchmark experiments that show the scheme reduces spurious currents at the liquid-vapor interface by at least two orders of magnitude compared to a nodal implementation and allows for predicting the equilibrium states accurately in the range of moderate contact angles.

Figure 1

Triangular finite elements centered at vertex ${\mathbf{x}}_k$ that form the set $\mathrm{\Omega }={\mathrm{\Omega }}_{\alpha }\cup {\mathrm{\Omega }}_{\beta }\cup {\mathrm{\Omega }}_{\gamma }$ at a solid boundary.

Figure 2

Triangular finite elements at a solid boundary. The value at the line segment $e_j$ is taken as ${\nabla \mu |}_{e_j}={0.5\left(\mathbf{\nabla }\mu \right|}_{{\mathrm{\Omega }}_{\alpha }}+\mathbf{\nabla }\mu {|}_{{\mathrm{\Omega }}_{\beta }})·\mathbf{n}$ and ${\nabla \mu |}_{e_k}=0$.

Figure 3

A thin slice of the unstructured grid of thickness $z=\pm 0.02R$ used for the contact angle simulations. The black dots mark the grid points and the solid line the initial contour $C=0.5$. There are $1.5\times {10}^6$ elements in the entire mesh.

Figure 4

Cross sections of the equilibrium contours $(C=0.1,C=0.5$, and $C=0.9)$ for a droplet with ${\theta }_{\text{eq}}={150}^{\circ }$ and contrasts (a) $M_{\rho }=20$; $M_{\nu }=1$ and (b) $M_{\rho }=100$; $M_{\nu }=40$.

Figure 5

(a) Contact angle measurements $\theta$ as a function of dimensionless wetting potential ${\mathrm{\Omega }}_c$ for $M_{\rho }=20,M_{\nu }=1$, and $M_{\rho }=100$; $M_{\nu }=40$ (b) Measurements given a thicker interface and finer resolution. The simulations are performed on a mesh containing $1.5\times {10}^6$ elements.

Figure 6

Time evolution of the average kinetic energy density in lattice units $\langle T\rangle =0.5{\sum }_i({\rho }_i{\mathbf{u}}_i·{\mathbf{u}}_i)/N$ for a droplet with $M_{\rho }=100,M_{\nu }=40$, and fixed contact angle ${\theta }_{\text{eq}}={120}^{\circ }$. The mobility $M$ is varied through $S=\sqrt{M{\rho }_l{\nu }_l}/R$. Time is scaled by the viscous time $t_0={\rho }_l{\nu }_{l}R/\sigma$.

Figure 7

(a) Simulation setup for capillary intrusion. (b) The length $y/y_0$ of the column of the intruding liquid as a function of time $t/t_0$. Units are scaled by the characteristic length $y_0=H$ and time $t_0=H{\eta }_l/\sigma$.