Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method

In this paper, we develop a conservative phase-field method for interface-capturing among N $(N\ge 2)$ immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N $(N\ge 2)$ immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.

Figure 1

The steady profiles of order parameter $c_i$ and spurious currents. (a) Equilibrium profile of $c_i$ along the vertical centerline; (b) $c_3$ along the vertical centerline with $m_0={10}^{-4},{10}^{-3}$, and ${10}^{-2}$; and (c) spurious currents versus $D$ with $m_0={10}^{-4},{10}^{-3}$, and ${10}^{-2}$.

Figure 2

Evolution of the density contours of binary fluids Rayleigh-Taylor instability.

Figure 3

Evolution of the bubble front and spike tip positions at $\mathrm{Re}=256$.

Figure 4

Evolution of the density contours of ternary fluids Rayleigh-Taylor instability.

Figure 5

Evolution of the average location of each fluid for ternary fluids RTI.

Figure 6

Physical configuration of spreading of liquid lens.

Figure 7

Equilibrium shapes of liquid lens with (a) ${\sigma }_{12}:{\sigma }_{13}:{\sigma }_{23}=1:1:1$, (b) ${\sigma }_{12}:{\sigma }_{13}:{\sigma }_{23}=1:\sqrt{2}:1$, (c) ${\sigma }_{12}:{\sigma }_{13}:{\sigma }_{23}=1:\sqrt{3}:1$, and (d) streamlines near three-phase interfaces.

Figure 8

Equilibrium shapes of two immiscible liquid lenses with (a) ${\sigma }_{14}={\sigma }_{24}={10}^{-2}$, (b) ${\sigma }_{14}=\sqrt{2}\times {10}^{-2}$, and (c) ${\sigma }_{14}={\sigma }_{24}=\sqrt{3}\times {10}^{-2}$.

Figure 9

Equilibrium profiles of two immiscible liquid lenses. The top to bottom rows correspond to ${\sigma }_{13}=0.008$, 0.01, and 0.012, respectively, and from left to right the columns correspond to ${\sigma }_{23}=0.008$, 0.01, and 0.012, respectively.

Figure 10

Time evolution of spinodal decomposition 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 $t={10}^6\delta t$, respectively.

Figure 11

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