Contact angle adjustment in equation-of-state-based pseudopotential model

The single component pseudopotential lattice Boltzmann model has been widely applied in multiphase simulation due to its simplicity and stability. In many studies, it has been claimed that this model can be stable for density ratios larger than 1000. However, the application of the model is still limited to small density ratios when the contact angle is considered. The reason is that the original contact angle adjustment method influences the stability of the model. Moreover, simulation results in the present work show that, by applying the original contact angle adjustment method, the density distribution near the wall is artificially changed, and the contact angle is dependent on the surface tension. Hence, it is very inconvenient to apply this method with a fixed contact angle, and the accuracy of the model cannot be guaranteed. To solve these problems, a contact angle adjustment method based on the geometry analysis is proposed and numerically compared with the original method. Simulation results show that, with our contact angle adjustment method, the stability of the model is highly improved when the density ratio is relatively large, and it is independent of the surface tension.

Figure 1

The simulated contact angle with different $G_w$ for different density ratios.

Figure 2

The density distribution near the wall for different $G_w$ when density ratio is 10: (a) $G_w=--0.766$; (b) $G_w=--2.43$.

Figure 3

The influence of Martys and Chen's contact angle adjustment method on the density distribution: (a) liquid phase; (b) vapor phase.

Figure 4

Stationary droplet on the surface with different $\kappa$ when density ratio is 10: (a) $\kappa =0.5$; (b) $\kappa =0.9$; (c) $\kappa =0.98$.

Figure 5

Contact angle obtained from different iterations: (a) with 0 times iteration; (b) with 4 times iteration; (c) with 10 times iteration.

Figure 6

Contact angles obtained with the geometry method when the density ratio is equal to 100: (a) $\theta ={15}^{\circ }, {\theta }^{\prime }=15.2^{\circ }$; (b) $\theta ={30}^{\circ }, {\theta }^{\prime }={31}^{\circ }$; (c) $\theta ={45}^{\circ }, {\theta }^{\prime }={45}^{\circ }$; (d) $\theta ={90}^{\circ }, {\theta }^{\prime }={90}^{\circ }$; (e) $\theta ={135}^{\circ }, {\theta }^{\prime }=134.5^{\circ }$; (f) $\theta ={150}^{\circ }, {\theta }^{\prime }={149}^{\circ }$.

Figure 7

Contact angles obtained with the geometry method when the density ratio is equal to 1000: (a) $\theta ={15}^{\circ }, {\theta }^{\prime }=15.7^{\circ }$; (b) $\theta ={30}^{\circ }, {\theta }^{\prime }={32}^{\circ }$; (c) $\theta ={45}^{\circ }, {\theta }^{\prime }={46}^{\circ }$; (d) $\theta ={90}^{\circ }, {\theta }^{\prime }={90}^{\circ }$; (e) $\theta ={135}^{\circ }, {\theta }^{\prime }={135}^{\circ }$; (f) $\theta ={150}^{\circ }, {\theta }^{\prime }={154}^{\circ }$.

Figure 8

The influence of the interface width on the contact angles across the interface.

Figure 9

The influence of geometry contact angle adjustment method on the density distribution: (a) liquid phase; (b) vapor phase.

Figure 10

Contact angles obtained with the geometry method for different surface tensions: (a) $\theta ={45}^{\circ }, \kappa =0.5$; (b) $\theta ={45}^{\circ }, \kappa =0.9$; (c) $\theta ={45}^{\circ }, \kappa =0.98$; (d) $\theta ={90}^{\circ }, \kappa =0.5$; (e) $\theta ={90}^{\circ }, \kappa =0.9$; (f) $\theta ={90}^{\circ }, \kappa =0.98$; (g) $\theta ={135}^{\circ }, \kappa =0.5$; (h) $\theta ={135}^{\circ }, \kappa =0.9$; (i) $\theta ={135}^{\circ }, \kappa =0.98$.

Figure 11

Two-dimensional capillary rise.

Figure 12

Capillary rise simulation results with different $1-\kappa$ values at 70 000 time steps: (a) $1-\kappa =1$; (b) $1-\kappa =0.5$; (c) $1-\kappa =0.25$.

Figure 13

Geometrical setup of the numerical LBE.

Figure 14

Evolution of the interface position for $\theta ={60}^{\circ }$ and $\theta ={30}^{\circ }$.

Figure 15

The contact angle deformation of the moving liquid (the set angle is ${30}^{\circ }$, and the measured one is about ${45}^{\circ }$).