Lattice Boltzmann model for ternary fluids with solid particles

On the basis of phase-field theory, we develop a lattice Boltzmann model for ternary fluids containing solid. We develop a modified bounce-back method to describe the interactions between the solid and N-phase $(N\ge 2)$ fluids. We derive a wetting boundary condition for three-phase flows from the point of mass conservation and propose a scheme for implementing the wetting condition on curved boundaries. We develop a diffuse interface method to compute the capillary force acting on the moving solid objects at the ternary fluids-sold contact lines. In addition, this model can deal with problems involving high density and viscosity contrasts. The proposed method is examined through several test cases. We test the modified bounce-back scheme, wetting boundary condition, and capillary force model in three different cases, and the numerical results agree well with the analytical solutions. Finally, we apply the model to two three-dimensional problems to assess its numerical accuracy and stability.

Figure 1

Schematic of the modified bounce-back method at a boundary link.

Figure 2

Geometric relation among the vectors at the contact line: vector tangent to the interface, $t$; normal vector of the particle surface, $n$; local normal to the fluid-fluid interface, $m$.

Figure 3

Schematic of the contact angle and unit normal vectors.

Figure 4

Schematic of interpolation to obtain the unknown phase-field value. (a) First difference method; (b) second difference method.

Figure 5

Initial setup for two droplets on a circular cylinder.

Figure 6

Grid independence of the results for two droplets on a circular cylinder. Comparison between two resolutions: 240 × 240 (solid red line) and 360 × 360 (solid blue line).

Figure 7

Initial setup for a compound droplet on a circular cylinder.

Figure 8

Equilibrium configurations of a compound liquid at a cylinder surface with different contact angles and spurious currents distribution. (a) $\left({\theta }_{13},{\theta }_{23}\right)=\left({60}^{\circ },{60}^{\circ }\right)$; (b) $\left({\theta }_{13},{\theta }_{23}\right)=\left({60}^{\circ },{120}^{\circ }\right)$; (c)$\left({\theta }_{13},{\theta }_{23}\right)=\left({90}^{\circ },{60}^{\circ }\right)$.

Figure 9

Capillary flow of ternary fluids along a pipe.

Figure 10

Velocity streamline and pressure profile for the case of $\left({\theta }_{13},{\theta }_{23}\right)=\left({60}^{\circ },{45}^{\circ }\right)$ at $t=180000$, the cure illustrates the pressure variation at the centerline of the capillary tube.

Figure 11

The shapes of different phases in Case 2 $\left({\theta }_{13},{\theta }_{23}\right)=\left({60}^{\circ },{45}^{\circ }\right)$ at different time steps, (a) $t=0$, (b) $t=160000$, (c) $t=280000$.

Figure 12

Comparison of the present numerical results for interface position with the theoretical solutions.

Figure 13

Geometry of a particle at the interface between ternary fluids.

Figure 14

Particle center as a function of time.

Figure 15

Variation of the total force exerted on the particle.

Figure 16

Equilibrium profile of two rectangular cylinders at the interfaces.

Figure 17

Pressure distribution, (a) Case 1; (b) Case 2.

Figure 18

Time evolution of the vertical component of the capillary force and the hydrodynamic force exerted on the circular cylinders. (a) Capillary force; (b) hydrodynamic force.

Figure 19

Spurious currents distribution, (a) Case 1; (b) Case 2.

Figure 20

Setup of simulation for a particle impact on a pool of liquid with multiple fluid layers.

Figure 21

Snapshots of a spherical particle plunging into a liquid pool with multiple fluid layers.

Figure 22

Time evolution of the vertical position of the particle.

Figure 23

Time evolution of the vertical component of the hydrodynamic and capillary forces exerted on the particle.

Figure 24

Initial setup for interaction of an oil droplet with solid particles.

Figure 25

Snapshots of an oil droplet rising and interacting with particles and a water-air interface.

Figure 26

Three typical particle trajectories indicated by order number.

Figure 27

Total kinetic energy of fluids as a function of time.