Caterpillar like motion of droplet in a shear flow

This paper is devoted to a specific motion observed for glycerin droplets sliding on a horizontal hydrophobic substrate under the influence of a shear flow. In this regime, the droplet elongates in the flow direction, adopting a rivulet shape. Waves develop on the droplet sheared surface, resulting in a wavy contracting and stretching motion mechanism, similar to the movement of a caterpillar. If long enough, the droplet can break up into several droplets that can be submitted to a pearling instability. Furthermore, these droplets can also coalesce.

Figure 1

(a) Side and (b) bottom views of a glycerol droplet sliding along a hydrophobic glass surface under the influence of a sheared airflow, $V_d=60\mu \text{L}$ and $U_{\infty }=22\mathrm{m}/\mathrm{s}$.

Figure 2

Time evolution of (a) advancing (downstream) $X_a$ and receding (upstream) $X_r$ contact line positions of the droplet and (b) dimensionless width $\overline{W}=W/\left(2R_0\right)$, length $\overline{L}=L/\left(2R_0\right)$, and maximal height $\overline{H}=H/R_0$ of a glycerol droplet sliding on a hydrophobic glass surface under the influence of a sheared airflow, $V_d=60\mu \text{L}$ and $U_{\infty }=20\mathrm{m}/\mathrm{s}$.

Figure 3

Evolution of the ratio between caterpillar length at the beginning of the equilibrium stage (iii) ($L_{\text{app}}$) and capillary length (${\ell }_c$) for different volumes of the droplet. Open symbols indicate $U_{\infty }=20$ m/s while solid symbols represent $U_{\infty }=22$ m/s.

Figure 4

(a) Time evolution of the centroid position $X_c$ for a droplet of $V_d=60\mu \text{L}$ and wind velocity of $U_{\infty }=20\mathrm{m}/\mathrm{s}$, and (b) one cycle of wave movement during the caterpillar droplet locomotion.

Figure 5

(a) Time evolution of advancing and receding angles of a glycerol droplet sliding on a hydrophobic glass surface under the influence of a sheared airflow, $V_d=60\mu \text{L}$ and $U_{\infty }=20\mathrm{m}/\mathrm{s}$ and (b) dimensionless side view contours at the instants $t_1$, $t_2$, and $t_3$ of Fig. 4, where the symbol $\times$ marks the location of the droplet's centroid $(X_c,Y_c)$.

Figure 6

The schematic of a two-dimensional (2D) caterpillar droplet on a glass surface, subject to the uniform air shear flow parallel to the substrate.

Figure 7

Time evolution of (a) the airflow variation speed of the wind tunnel for $U_{\infty }=20$ and 22 m/s and (b) the frequencies of the wave of glycerol droplet $V_d=60\mu \text{L}$. The dashed line is a fit curve realized on all the data.

Figure 8

The experimental frequencies and velocities of the interface wave on the droplet. (a) $U_{\infty }=20\mathrm{m}/\mathrm{s}$, (b) $U_{\infty }=22\mathrm{m}/\mathrm{s}$, and (c) the mean values of the frequencies and velocities measured in each droplet: The continuous line reports the gravity-capillary wavelength ${\lambda }_c=2\pi {\ell }_c$. Open symbols: $U_{\infty }=20\mathrm{m}/\mathrm{s}$; solid symbol: $U_{\infty }=22\mathrm{m}/\mathrm{s}$.

Figure 9

Evolution of the ratio between mean wave velocity $\overline{U_w}$ and droplet mean velocity $\overline{U_d}$ times the dimensionless height $h$ for different droplet volumes and airflow rates.

Figure 10

(a) Time evolution of the dimensionalized length over the capillary length of a glycerol droplet sliding on a hydrophobic glass surface under the influence of a sheared airflow, $V_d=60\mu \text{L}$. Different images of the glycerol droplet at three different instants, (b) $U_{\infty }=20$ m/s and (c) $U_{\infty }=22$ m/s, where in this case (1) and (3) show a droplet breakup, whereas (2) depicts a droplet coalescence.

Figure 11

Phase diagram $L_{\text{max}}/{\ell }_c-N_d$ reporting the different observed behaviors. $N_d=1$ indicates that the droplet is not experiencing any breakup.

Figure 12

Side views of a line of daughter glycerol droplets sliding from left to right along a hydrophobic glass surface under the influence of a laminar shear airflow. The red arrow indicates the small droplet that is absorbed by the last droplet on the left after coalescence. $V_d=100\mu \text{L}$ and $U_{\infty }=20\mathrm{m}/\mathrm{s}$.