Vibration-Induced Climbing of Drops

We report an experimental study of liquid drops moving against gravity, when placed on a vertically vibrating inclined plate, which is partially wetted by the drop. The frequency of vibrations ranges from 30 to 200 Hz, and, above a threshold in vibration acceleration, drops experience an upward motion. We attribute this surprising motion to the deformations of the drop, as a consequence of an up or down symmetry breaking induced by the presence of the substrate. We relate the direction of motion to contact angle measurements. This phenomenon can be used to move a drop along an arbitrary path in a plane, without special surface treatments or localized forcing.

Figure 1

Side view of a climbing drop (and its reflection) on a vibrating plate inclined at $\alpha =45°$ at different phases $\varphi$ of the cycle. (b) and (d) show the maximum lower and upper contact angle, respectively. Parameters are $V=5\mu \mathrm{l}$, $f=60\mathrm{Hz}$ ($f/f_0=1.18$), $a/a_0=1.03$, and $\nu =31{\mathrm{mm}}^2/\mathrm{s}$. The lower plot shows acceleration versus phase and the corresponding acceleration for each of the images.

Figure 2

Phase diagram of drop motion for $V=5\mu \mathrm{l}$, $\alpha =45°$, and $\nu =31{\mathrm{mm}}^2/\mathrm{s}$. The normalization factors are $f_0=50.77\mathrm{Hz}$ and $a_0=174\mathrm{m}/{\mathrm{s}}^2$.

Figure 3

(a) The capillary number Ca vs $a/a_0$ ($f=60\mathrm{Hz}$, $V=5\mu \mathrm{l}$, $\nu =31{\mathrm{mm}}^2/\mathrm{s}$), measured along the dotted-dashed line in Fig. 2. Inset: Magnified view of data below the climbing threshold. $\mathrm{Ca}={10}^{-3}$ corresponds to a speed of $1.79\mathrm{mm}/\mathrm{s}$. (b) Estimate of the force averaged over a cycle due to the difference between the upper and the lower contact angles.

Figure 4

Views from above of (1) sliding, (2) static, and (3)–(5) climbing drops. As the speed of the drop increases from (3) to (5), the trailing end transitions to a corner (4) and then to pearling. $a/a_0=0.46$ (1), 0.67 (2), 0.96 (3), 1.13 (4), and 1.39 (5). The scale bar equals 1 mm.

Figure 5

Drops resting on a horizontal plate (or hanging below it) but shaken at an angle. The drop moves in the direction of the shake pointing normal toward the drop.

Figure 6

Left panel—Displacement of the upper (▵) and lower (▿) contact lines for a climbing drop during 12 periods ($a/a_0=1.03$). Capillary number (top center and right) and contact angle cubed (bottom center and right) versus time for one period of oscillation for a sliding drop (center) and a climbing drop (right). The average capillary number is denoted by a dashed line in the top plots. ${\theta }_a^3$ and ${\theta }_r^3$ are denoted by dotted and dotted-dashed lines, respectively, in the bottom plots. $f=60\mathrm{Hz}$, $V=5\mu \mathrm{l}$, and $a/a_0=0.45$ (center) and 1.03 (right).

Figure 7

A mechanical model for moving drops. The support is vibrated at an angle $\alpha$ relative to the vertical. In equilibrium, surface tension holds the drop in a symmetric position $\varphi =0$. In the horizontal direction, a nonlinear frictional force $-F_0{\dot{x}}^{\beta }$ acts between the base of the drop and the support.