Shuffling gait motion of an aerodynamically driven wall-bound drop

Drop motion on a dry solid substrate is determined by the lateral adhesion force associated with the contact angle hysteresis and by any applied external forces. In this experimental and computational study the lateral adhesion force is measured directly, through dragging of a drop along a surface by a cantilever. These force measurements are compared with the estimated value of the aerodynamic force needed to displace a wall-bound drop in a channel flow. The scaled lateral adhesion and aerodynamic forces are functions of the capillary number. Their values agree for capillary numbers higher than ${10}^{-4}$. For smaller capillary numbers, $\mathrm{Ca}<{10}^{-4}$, the value of the required aerodynamic force can be smaller than the lateral adhesion force. This unexpected result is explained by the complex three-dimensional periodic motion of the drop, similar to a shuffling gait. During the shuffling gait propagation, at any instant only a part of the drop contact line propagates, which requires a smaller force to displace the drop. Alternating motion of different parts of the contact line forms a continuous shuffling gait propagation. As soon as the aerodynamic force exceeds a threshold value $F^{*}$ the drop slides nearly steadily.

Figure 1

Schematic of the channel flow used for the aerodynamically driven drop motion.

Figure 2

Wall-bound drop in a channel flow. Definition of the main geometrical parameters. The length and width of a spherical cap of the same volume are used.

Figure 3

Computations of the velocity field around a truncated sphere: the magnitude of the velocity and the streamlines of the velocity field (a) and the pressure field (b) in the symmetry plane. The unperturbed air velocity far from the sphere on the channel centerline is 20 m/s.

Figure 4

Computations of the drag coefficient on a wall-bound obstacle in the form of the truncated sphere. Solid lines correspond to the contact angle $\theta ={30}^{\circ }$, dashed lines to $\theta ={60}^{\circ }$, and dotted lines to $\theta ={90}^{\circ }$.

Figure 5

(a) Sketch of the drop adhesion force instrument (DAFI). A glass cantilever is pressed against the tip of a force sensor. It transmits the lateral force from the drop to the sensor. The contact angles and contact length of the drop are determined from side and front views by two cameras. While the stage moves to the right side, the glass cantilever deflects. (b) The side and front views of a sliding 10 $\mu \mathrm{l}$ water drop at a velocity of 10 mm/s ($\mathrm{Ca}=0.12\times {10}^{-3}$) on a silanized silicon wafer.

Figure 6

Normalized lateral adhesion force over the drop capillary number from the DAFI measurements. The solid curve represents a power law fit $F/\sigma d_0\sim 0.67+0.33{\mathrm{Ca}}^{1/3}$.

Figure 7

Normalized aerodynamic drag force $F/\sigma d_0$ as a function of the drop capillary number estimated from the channel flow experiment for water drops of various volumes: (a) comparison with the fit $F\sim \mathrm{Ca}$, (b) a zoomed portion of the plot (a) for smaller capillary numbers in comparison with the fit for the DAFI measurements, shown in Fig. 6.

Figure 8

Six consecutive shadowgraph images of a sessile 20 $\mu \mathrm{l}$ water drop on an aluminum substrate $({\mathrm{\Theta }}_{\mathrm{adv},cr}={90}^{\circ };{\mathrm{\Theta }}_{\mathrm{rec},cr}={33}^{\circ })$. The air flow is from left to right with ${\mathrm{Re}}_{\mathcal{H}}=14.5\times {10}^3$. The green lines represent the surface of the substrate. The white line shown in all $t>0$ images represents the initial contour of the drop at $t=0$. The contact line of the drop is pinned. See the Supplemental Material [52] for high-speed observations of the oscillatory drop propagation.

Figure 9

Evolution of the position of the advancing edge $x_{\mathrm{adv}}$ of a 10 $\mu \mathrm{l}$ drop, propagating in the wind tunnel on a polished aluminium substrate, in the wind tunnel and the corresponding apparent advancing contact angle ${\mathrm{\Theta }}_{\mathrm{adv}}$ at this position: the time averaged drop velocity corresponds to the relatively high capillary number $\mathrm{Ca}=2.2\times {10}^{-3}$ (a) and low capillary number $\mathrm{Ca}=1.6\times {10}^{-4}$ (b).

Figure 10

Evolution of the velocity of the advancing edge $v_{\mathrm{adv}}$ of a 10 $\mu \mathrm{l}$ drop, propagating in the wind tunnel on a PMMA substrate, and the corresponding apparent advancing contact angle ${\mathrm{\Theta }}_{\mathrm{adv}}$ at this position. The capillary number is $\mathrm{Ca}=1.06\times {10}^{-4}$.

Figure 11

Comparison of the computations and experimental results for the position $x_{\mathrm{drop}}$ of a 25 $\mu \mathrm{l}$ drop on PMMA substrate as a function of time. Also the values of the continuously growing Reynolds number of the air flow in the wind tunnel are shown.

Figure 12

CFD computations of the water drop propagation by shuffling gait. The corresponding video can be seen in the Supplemental Material [52].