Phase-field-theory-based lattice Boltzmann equation method for $N$ immiscible incompressible fluids

From the phase field theory, we develop a lattice Boltzmann equation (LBE) method for $N$ ($N\ge 2$) immiscible incompressible fluids, and the Cahn-Hilliard equation, which could capture the interfaces between different phases, is also solved by LBE for an $N$-phase system. In this model, the interface force of $N$ immiscible incompressible fluids is incorporated by chemical potential form, and the fluid-fluid surface tensions could be directly calculated and independently tuned. Numerical simulations including two stationary droplets, spreading of a liquid lens with and without gravity and two immiscible liquid lenses, and phase separation are conducted to validate the present LBE, and numerical results show that the predictions by LBE agree well with the analytical solutions and other numerical results.

Figure 1

The distribution of $c_3$ along the centerline of the droplets at $y$ direction with three different mesh sizes $80\times 128,100\times 160$, and $200\times 320$.

Figure 2

Physical configuration.

Figure 3

Equilibrium profiles of liquid lens with (a) $\left|\mathit{g}\right|={10}^{-6}$, (b) $\left|\mathit{g}\right|={10}^{-5}$, and (c) $\left|\mathit{g}\right|=6\times {10}^{-5}$.

Figure 4

Comparison of puddle thickness versus gravity between LBE simulations and the de Gennes theory [48].

Figure 5

Equilibrium profiles of liquid lens with (a) ${\sigma }_{12}=5\times {10}^{-3}$, (b) ${\sigma }_{12}={10}^{-2}$, and (c) ${\sigma }_{12}=2\times {10}^{-2}$.

Figure 6

Comparison of puddle thickness versus ${\sigma }_{12}$ between LBE simulations and the de Gennes theory [48].

Figure 7

Equilibrium profiles of liquid lens with (a) ${\rho }_r=0.1$, (b) ${\rho }_r=0.5$, and (c) ${\rho }_r=0.8$.

Figure 8

Comparison of puddle thickness versus ${\rho }_r$ between LBE simulations and the de Gennes theory [48].

Figure 9

Physical configuration of four phase system.

Figure 10

Equilibrium profiles of two immiscible liquid lenses with (a) ${\sigma }_{23}=0.005$, (b) ${\sigma }_{23}=0.01$, and (c) ${\sigma }_{23}=0.015$. From top to bottom rows correspond to $\left|\mathit{g}\right|=0,{10}^{-5}$ and $5\times {10}^{-5}$.

Figure 11

Time evolution of phase separation with (a) $N=3$, (b) $N=4$, and (c) $N=5$. From the top to bottom rows correspond to $t=0,t={10}^5\delta t$, and $t=3\times {10}^5\delta t$, respectively.

Figure 12

Time evolution of typical domain length scale $L\left(t\right)$ in spinodal decomposition. (a) Diffusive regime, (b) hydrodynamic regime. Dashed lines are numerical predictions and solid lines are the theory.