Contact line dynamics in binary lattice Boltzmann simulations

We show that, when a single relaxation time lattice Boltzmann algorithm is used to solve the hydrodynamic equations of a binary fluid for which the two components have different viscosities, strong spurious velocities in the steady state lead to incorrect results for the equilibrium contact angle. We identify the origins of these spurious currents and demonstrate how the results can be greatly improved by using a lattice Boltzmann method based on a multiple-relaxation-time algorithm. By considering capillary filling we describe the dependence of the advancing contact angle on the interface velocity.

Figure 1

(Color online) The simulation geometry. The contact angle is defined by ${\theta }^{\mathrm{eq}}$. The crosses denote the points which are fit to the arc of a circle to obtain a numerical estimate of the equilibrium contact angle ${\theta }^{\mathrm{eq}}$. The grid is not to scale.

Figure 2

The equilibrium contact angle as a function of the gradient in $\varphi$ at the boundary (measured in lattice units) ${\phantom{\mid }{\partial }_{\perp }\varphi \mid }_b$, for (a) ${\tau }_{\alpha }={\tau }_{\beta }=1.0$ and (b) ${\tau }_{\alpha }=3$, ${\tau }_{\beta }=0.7$. Squares and crosses are simulation results obtained using single and multiple-relaxation-time lattice Boltzmann algorithms, respectively. The solid curve is the theoretical expression (22).

Figure 3

(a) The spurious flow field in steady state in a system consisting of a stripe of one fluid (light region) surrounded by another fluid (dark regions) held between two walls. Periodic boundary conditions are applied in the $x$ direction. The parameters used were ${\tau }_{\alpha }={\tau }_{\beta }=10$. (b) (Color online) An enlargement of the region near the interface indicating how the particle distribution function at A depends on contributions from lattice nodes in the lattice velocity vector directions.

Figure 4

(Color online) (a) The midlink bounce-back method, before and after the streaming step, along one lattice direction. (b) The lattice node bounce-back method. (c)–(f) Schematic diagrams showing the particle distribution functions for (c) a single-component fluid, (d) a single-component fluid under shear, (e) particle distribution function in the interface of a binary fluid before, and (f) after bounce back.

Figure 5

Lattice Boltzmann simulation results for capillary filling in smooth channels. Left: the length squared, in lattice units, of the column of the filling component (fluid $\alpha$) plotted against time for (a) single and (c) multiple-relaxation-time lattice Boltzmann simulations. The circles are the simulation results and the solid lines are fits to Washburn’s law. Right: the advancing contact angle of the liquid-liquid interface for (b) single and (d) multiple-relaxation-time simulations. The crosses are the simulation results and the solid lines are linear fits of $\cos {\theta }^a$ to the capillary number [25].