Phase-field lattice Boltzmann model with singular mobility for quasi-incompressible two-phase flows

In this paper, a lattice Boltzmann for quasi-incompressible two-phase flows is proposed based on the Cahn-Hilliard phase-field theory, which can be viewed as an improved model of a previous one [Yang and Guo, Phys. Rev. E 93, 043303 (2016)]. The model is composed of two LBE's, one for the Cahn-Hilliard equation (CHE) with a singular mobility, and the other for the quasi-incompressible Navier-Stokes equations (qINSE). Particularly, the LBE for the CHE uses an equilibrium distribution function containing a free parameter associated with the gradient of chemical potential, such that the variable (and even zero) mobility can be handled. In addition, the LBE for the qINSE uses an equilibrium distribution function containing another free parameter associated with the local shear rate, such that the large viscosity ratio problems can be handled. Several tests are first carried out to test the capability of the proposed LBE for the CHE in capturing phase interface, and the results demonstrate that the proposed model outperforms the original LBE model in terms of accuracy and stability. Furthermore, by coupling the hydrodynamic equations, the tests of double-stationary droplets and droplets falling problems indicate that the proposed model can reduce numerical dissipation and produce physically acceptable results at large time scales. The results of droplets falling and phase separation of binary fluid problems show that the present model can handle two-phase flows with large viscosity ratio up to the magnitude of ${10}^4$.

Figure 1

Comparison of the interface shape after one period and ten periods with the initial shape at $Pe=2000$ for the droplet diagonal translation test. From left to right, the results predicted by the YG-LBE, SCH-YG-LBE, and the present models. The black solid line denotes $t=0$, the red dashed line denotes $t=nT$. (a) $n=1$, (b) $n=10$.

Figure 2

The absolute error between the theoretical and numerical values of the center of mass.

Figure 3

Interface shapes after ten periods at different $Ma$ number predicted by (a) the YG-LBE model, (b) the present model.

Figure 4

Relative error of $L_2$-norm for the diagonal translation problem at different $Ma$ number after ten periods.

Figure 5

Comparison of the interface shape at $t=\left(2L_0\right)/U_0$ and $t=\left(10L_0\right)/U_0$ with initial shape for Zalesak's disk problem at (a) $Pe=20$ and (b) $Pe=2000$. From left to right, the results predicted by the YG-LBE model and the present model.

Figure 6

Interface shapes for different $Ma$ number predicted by (a) the YG-LBE model, (b) the present model at $t=\left(10L_0\right)/U_0$.

Figure 7

Results of single-vortex problem at $Pe=500$ and $n=6$ for (a) the YG-LBE model (b) the present model. From left to right $T/4,T/2,3T/4,T$.

Figure 8

Interface shapes predicted by the present model at $Pe=5000$. (a) $n=2$, (b) $n=4$, (c) $n=6$.

Figure 9

The snapshots of two static droplets at different time obtained by the YG-LBE model. (a) $T_r=0$, (b) $T_r=2000$, (c) $T_r=2800$, (d) $T_r=3200$.

Figure 10

The snapshots of two static droplets at different time obtained by the present model. (a) $T_r=2000$, (b) $T_r=3200$.

Figure 11

The phase variable (a) and transport flux (b) profiles across the droplet center at $T_r=2800$.

Figure 12

Time history of the mass ratio of the small droplet to the large droplet.

Figure 13

Interfacial evolution of falling droplets at $R_2/R_1=5$. The solid and dashed lines are the results obtained by the present and YG-LBE models, respectively. (a) $t=0$, (b) $t=4.0\times {10}^4$, (c) $t=1.0\times {10}^5$, (d) $t=1.5\times {10}^5$, (e) $t=2.0\times {10}^5$.

Figure 14

Interfacial evolution of falling droplets at $R_2/R_1=5/3$. The solid and dashed lines are the results obtained by the present and YG-LBE models, respectively. (a) $t=0$, (b) $t=4.0\times {10}^4$, (c) $t=1.0\times {10}^5$, (d) $t=1.5\times {10}^5$, (e) $t=2.0\times {10}^5$.

Figure 15

The disrtibution of velocity (a) and pressure (b) for different $B$ values at ${\mu }_r=10$ and $t=100$.

Figure 16

Interfacial evolution of falling droplets obtained from the present model for different viscosity ratio. (a) ${\mu }_r=10$, (b) ${\mu }_r={10}^2$, (c) ${\mu }_r={10}^3$, (d) ${\mu }_r={10}^4$.

Figure 17

The distribution of velocity along $y/L_y=0.5$ for the phase separation problem at $t/t_{\nu }=750$.

Figure 18

Time evolution of the phase-variable distribution during the phase separating process. (a) $t_{\nu }=0$, (b) $t_{\nu }=125$, (c) $t_{\nu }=250$, (d) $t_{\nu }=500$, (e) $t_{\nu }=750$, (f)$t_{\nu }=1000$.