Lattice Boltzmann simulation of water droplet impacting a hydrophobic plate with a cylindrical pore

This work uses a nonorthogonal multiple-relaxation-time lattice Boltzmann (LB) method to simulate a water droplet impacting on a hydrophobic plate with a cylindrical pore. The LB method is based on a pseudopotential multiphase model with tunable contact angles. It is first validated against experimental data and then employed to study the dynamics of a water droplet impacting a perforated hydrophobic plate. A comprehensive parametric study is carried out by varying the pore size, plate thickness, and Weber number. Based on these results, a phase diagram is constructed to describe the different regimes of droplet penetration behaviors after impingement on the perforated plate. Three distinctive regimes are found, that is, continuously dripping, totally crossing, and hanging, and the mechanisms behind them are scrutinized through the analysis of pressure and energy balance. The competition between the dynamic pressure and the capillary pressure determines the formation of the liquid jet below the plate. Viscous dissipation hinders the development of liquid fingering. Based on the balance of the droplet dynamic pressure, capillary pressure, and the viscous pressure losses, the boundary between different regimes is established.

Figure 1

Sketch of a liquid drop impacting a perforated solid surface.

Figure 2

Change of the maximum speed in the central line under different mesh resolutions, where the relative error between $D_x=300$ and $D_x=500$ is 2.52%.

Figure 3

Validation of the present model against experimental data [17]. Below the experimental capture Weber number, all of the droplet remains in the plate. The droplet escapes and breaks up when higher than the experimental capture Weber number is used. The solid curve $\mathrm{R}{\mathrm{e}}^{*}\left(\mathrm{W}{\mathrm{e}}^{*}-3.6\right)=5.1\mathrm{W}{\mathrm{e}}^{*}$ represents the theoretical function of the critical Weber number and the critical Reynolds number.

Figure 4

The phase diagram for the $h/R=0.4$ case. The triangle indicates partial crossing of the droplet, while the major droplet is hanging on the plate. The square represents the case of the droplet totally crossing the plate. The circle describes continuously dripping liquid from the droplet. The solid line represents Eq. (15), and the dashed line stands for Eq. (17).

Figure 5

The phase diagram for the $h/R=0.6$ case. The triangle indicates partial crossing of the droplet while the major droplet is hanging on the plate. The circle describes continuously dripping liquid from the droplet. The solid line represents Eq. (15).

Figure 6

The evolution of the hanging droplet under $h/R=0.6$, $r/R=0.4$, and $\mathrm{We}=46$.

Figure 7

The evolution of the totally crossing droplet under $h/R=0.4$, $r/R=0.4$, and $\mathrm{We}=46$.

Figure 8

The evolution of the continuously dripping droplet under $h/R=0.6$, $r/R=0.19$, and $\mathrm{We}=46$.

Figure 9

The ideal model of droplet equilibrium: the left indicates the spread, and the right represents the rebound.

Figure 10

The velocity vector field during the evolution. The left figure indicates the velocity vector field inside the droplet in the spread stage. The right figure represents the velocity vector field outside the droplet in the permeation stage.

Figure 11

The surface energy (left $Y$-axis, represented by line) and total useful energy (right $Y$-axis, represented by line and symbol) evolution of the droplet. The total useful energy: 1. the continuously dripping case, 2. the hanging case, 3. totally crossing case; The surface energy: 4. the continuously dripping case, 5. the hanging case, 6. the totally crossing case.

Figure 12

The kinetic energy (right $y$ axis, represented by line) and viscous dissipation rate (left $y$ axis, represented by line and symbol) for the rebound and permeation stages. The kinetic energy: 1. the continuously dripping case, 2. the hanging case, and 3. the totally crossing case. The viscous dissipation rate: 4. the continuously dripping case, 5. the hanging case, and 6. the totally crossing case.