Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio

Within the phase-field framework, we present an accurate and robust lattice Boltzmann (LB) method for simulating contact-line motion of immiscible binary fluids on the solid substrate. The most striking advantage of this method lies in that it enables us to handle two-phase flows with mass conservation and a high density contrast of 1000, which is often unavailable in the existing multiphase LB models. To simulate binary fluid flows, the present method utilizes two LB evolution equations, which are respectively used to solve the conservative Allen-Cahn equation for interface capturing, and the incompressible Navier-Stokes equations for hydrodynamic properties. Besides, to account for the substrate wettability, two popular contact angle models including the cubic surface-energy model and the geometrical one are incorporated into the present method, and their performances are numerically evaluated over a wide range of contact angles. The contact-angle hysteresis effect, which is inherent to a rough or chemically inhomogeneous substrate, is also introduced in the present LB approach through the strategy proposed by Ding and Spelt [J. Fluid Mech. 599, 341 (2008)]. The present method is first validated by simulating droplet spreading and capillary intrusion on the ideal or smooth pipes. It is found that the cubic surface-energy and geometrical wetting schemes both offer considerable accuracy for predicting a static contact angle within its middle region, while the former is more stable at extremely small contact angles. Besides, it is shown that the geometrical wetting scheme enables us to obtain better accuracy for predicting dynamic contact points in capillary pipe. Then we use the present LB method to simulate the droplet shearing processes on a nonideal substrate with contact angle hysteresis. The geometrical wetting model is found to be capable of reproducing four typical motion modes of contact line, while the surface-energy wetting scheme fails to predict the hysteresis behaviors in some cases. At last, a complex contact-line dynamic problem of three-dimensional microscale droplet impact on a wettable solid is simulated, and it is found that the numerical results for droplet shapes agree well with the experimental data.

Figure 1

Schematics of fluid lattice node and solid wall boundary.

Figure 2

The predicted droplet equilibrium shapes using the present LB method coupled with the surface-energy (A) and geometrical (B) wetting schemes under conditions of wide range of prescribed contact angles, (a) $\theta ={30}^{\circ }$, (b) $\theta ={60}^{\circ }$, (c) $\theta ={90}^{\circ }$, (d) $\theta ={120}^{\circ }$, (e) $\theta ={150}^{\circ }$, (f) $\theta ={170}^{\circ }$.

Figure 3

Schematic of the capillary flow along a pipe.

Figure 4

The predicted interface position in the capillary pipe by the present LB approach with various density ratios: (a) ${\rho }_l/{\rho }_g=1$; (b) ${\rho }_l/{\rho }_g=10$; (c) ${\rho }_l/{\rho }_g=100$; (d) ${\rho }_l/{\rho }_g=1000$.

Figure 5

The shearing of a droplet on the substrate using the LB method coupled with the surface-energy (A) and geometrical (B) wetting schemes under different hysteresis windows, ${\rho }_l/{\rho }_g=1000$: (a) $({\theta }_R,{\theta }_A)=(0^{\circ },{180}^{\circ })$, (b) $({\theta }_R,{\theta }_A)=(0^{\circ },{110}^{\circ })$, (c) $({\theta }_R,{\theta }_A)=({70}^{\circ },{180}^{\circ })$, (d) $({\theta }_R,{\theta }_A)=({70}^{\circ },{110}^{\circ })$.

Figure 6

The shearing of a droplet on the substrate using the LB method coupled with the surface-energy (A) and geometrical (B) wetting schemes under different hysteresis windows, ${\rho }_l/{\rho }_g=10$: (a) $({\theta }_R,{\theta }_A)=(0^{\circ },{180}^{\circ })$, (b) $({\theta }_R,{\theta }_A)=(0^{\circ },{110}^{\circ })$, (c) $({\theta }_R,{\theta }_A)=({70}^{\circ },{180}^{\circ })$, (d) $({\theta }_R,{\theta }_A)=({70}^{\circ },{110}^{\circ })$.

Figure 7

Snapshots of microscale droplet impact on solid surface obtained by the present LB method coupled with the surface-energy (A) and geometrical (B) wetting schemes, $\theta ={107}^{\circ }$: ${\rho }_l/{\rho }_g=844$ and ${\mu }_l/{\mu }_g=48.5$.

Figure 8

Snapshots of microscale droplet impact on solid surface obtained by the present LB method coupled with the surface-energy (A) and geometrical (B) wetting schemes, $\theta ={31}^{\circ }$: ${\rho }_l/{\rho }_g=844$ and ${\mu }_l/{\mu }_g=48.5$.

Figure 9

The time evolutions of the spreading ratio $D^{*}$ and dimensionless droplet height $H^{*}$ obtained by the present LB method coupled with surface-energy and geometrical wetting schemes: (a) $\theta ={107}^{\circ }$, (b) $\theta ={31}^{\circ }$.