Phase-field-based lattice Boltzmann model for axisymmetric multiphase flows

In this paper, a phase-field-based lattice Boltzmann (LB) model is proposed for axisymmetric multiphase flows. Modified equilibrium distribution functions and some source terms are properly added into the evolution equations such that multiphase flows in the axisymmetric coordinate system can be described. Different from previous axisymmetric LB multiphase models, the added source terms that arise from the axisymmetric effect contain no additional gradients, and therefore the present model is much simpler. Furthermore, through the Chapmann-Enskog analysis, the axisymmetric Chan-Hilliard equation and Navier-Stokes equations can be exactly derived from the present model. The model is also able to deal with flows with density contrast. A variety of numerical experiments, including planar and curve interfaces, an elongation field, a static droplet, a droplet oscillation, breakup of a liquid thread, and dripping of a liquid droplet under gravity, have been conducted to test the proposed model. It is found that the present model can capture accurate interface and the numerical results of multiphase flows also agree well with the analytical solutions and/or available experimental data.

Figure 1

Schematics of the computational geometry and boundary schemes. $r=0$ represents the symmetry axis.

Figure 2

Order parameter profiles with different values of $M$. (a) Planar interface test; (b) curve interface test.

Figure 3

Results of the elongation field test during one period by the present model.

Figure 4

Results of the elongation field test during one period by the axisymmetric model of Premnath et al. [22].

Figure 5

Static droplet tests with different values of $M$. (a) Density profiles across the interface; (b) pressure distributions across the interface.

Figure 6

Evolution of an oscillating droplet with ${\rho }_l=10.0$, ${\rho }_g=1.0$, $R_r=30$, $R_z=60$, $\sigma =0.3$.

Figure 7

Comparison of the evolutions of the amplitude at different surface tensions, ${\rho }_l=10.0,{\rho }_g=1.0,R_r=30,R_z=60$.

Figure 8

Comparison of the evolutions of the amplitude at different equilibrium radii, ${\rho }_l=10.0,{\rho }_g=1.0,\sigma =0.2$.

Figure 9

Evolutions of the instability of a liquid thread at different wave numbers, (a) $k=0.63$, (b) $k=0.31$, (c) $k=0.21$. Time is normalized by the capillary time $\sqrt{R^3{\rho }_l/\sigma }$.

Figure 10

The resulting droplet sizes at different wave numbers. Here $R^{*}$ represents the dimensionless droplet size scaled by $R$.

Figure 11

Snapshots of the dripping droplet with $\sigma =0.057$ at different times. Time is normalized by the capillary time $\sqrt{R^3{\rho }_l/\sigma }$.

Figure 12

Snapshots of the dripping droplet with $\sigma =0.05$ at different times. Time is normalized by the capillary time $\sqrt{R^3{\rho }_l/\sigma }$.