Numerical investigation of vibration-induced droplet shedding on smooth surfaces with large contact angles

In this work, numerical simulations are performed to study the droplet response to the vertical vibration of the substrate, under various frequencies and amplitudes using the multiphase lattice Boltzmann method. First, the numerical results are validated against published experimental data. The effects of droplet size, surface wettability, amplitude, and frequency of the vibrating substrate on droplet detachment are studied. For high contact angles, regardless of the droplet size, when the vibration frequency matches the droplet resonance frequency the droplet is easily removed from the surface. For lower contact angles, the resonance frequency is higher and the detachment amplitude increases significantly. It was also found that viscous forces do not affect the resonance frequency, but have a noticeable impact on the detachment amplitude. The findings of this study can be useful in applications where droplet shedding is crucial, e.g., condensation heat transfer.

Figure 1

Droplet height at different equilibrium contact angle values for a sessile droplet is shown. Solid black line shows the analytical droplet height and the circular symbols show the numerical results.

Figure 2

Capillary rise of liquid column in a vertical tube under zero gravity conditions at $\mathrm{Oh}=0.0138$, ${\theta }_{\mathrm{eq}}={47}^o$, and $R_t=0.1\mathrm{mm}$. The L-W law is shown by the solid black line and the LBM predicted height is depicted by triangular symbols.

Figure 3

Schematic of the simulation domain. Periodic BC is imposed on the left and right sides of the domain. The upper side is an open boundary, and is kept at constant pressure. The lower wall moves vertically with a velocity of $U_w$.

Figure 4

Snapshots of vibrated droplet ($V=1.5\mu \mathrm{L}$, ${\theta }_{\mathrm{eq}}={140}^o$, $f=80\mathrm{Hz}$ and $A^{*}=0.42$). (a) Experimental results from Ref. [17] and (b) Numerical results, (c) comparison of droplet height, $H/H_0$, and contact diameter, $D/D_0$ between numerical results and experimental data, with $H_0$ and $D_0$ being the initial droplet height and diameter, respectively.

Figure 5

Droplet shape evolution during vibration at (a) 120 Hz, (b) 80 Hz, and (c) 40 Hz ($V=2\mu \mathrm{L}$ and ${\theta }_{\mathrm{eq}}={140}^o$). The vibration amplitude is incrementally increased at each cycle until droplet detachment occurs. $t_j^{*}$ denotes the time at which the droplet jumps away from the surface and $\mathrm{\Delta }{t}^{*}=t^{*}-t_j^{*}$.

Figure 6

Dimensionless droplet velocity, height and contact diameter vs dimensionless time vibrated at (a) 40 Hz, (b) 80 Hz, and (c) 120 Hz. The dashed red line is the wall velocity, which is incrementally increased until droplet detaches from the surface. The solid blue line shows the vertical velocity of the droplet. The droplet height and contact diameter are depicted by dash-dotted lines with red triangles and grey circles, respectively.

Figure 7

Apparent contact angle vs dimensionless time at (a) 40 Hz, (b) 80 Hz, and (c) 120 Hz.

Figure 8

Left: Pressure distribution (darker areas have higher pressures), Right: Streamlines for a $2\mu L$ droplet vibrated at $80\mathrm{Hz}$. The droplet interface is shown at $\varphi =0.5$.

Figure 9

Minimum required amplitude for droplet detachment as a function of frequency for different droplet sizes.

Figure 10

Transmissibility ratio as a function of frequency for different droplet sizes.

Figure 11

Required amplitude for droplet detachment as a function of frequency for different equilibrium contact angles for a $0.1\mu \mathrm{L}$ droplet.

Figure 12

Transmissibility ratio as a function of frequency for different equilibrium contact angles for a $0.1\mu \mathrm{L}$ droplet.

Figure 13

The numerical results for resonance frequency $f_r$ are compared with the theory of Ref. [40]. The theoretical resonance frequency is depicted by the solid line and the numerically obtained resonance frequencies are shown by the square symbols.