Lifetime of a bouncing droplet

When a droplet is gently laid onto the surface of the same liquid, it stays at rest for a moment before coalescence. The coalescence can be delayed and sometimes inhibited by injecting fresh air under the droplet. This can happen when the surface of the bath oscillates vertically. In this case the droplet basically bounces on the interface. The lifetime of the droplet has been studied with respect to the amplitude and the frequency of the excitation. The lifetime decreases when the acceleration increases. The thickness of the air film between the droplet and the bath has been investigated using interference fringes obtained when the system is illuminated by low-pressure sodium lamps. Moreover, both the shape evolution and the motion of the droplet center of mass have been recorded in order to evidence the phase offset between the deformation and the trajectory. A short lifetime is correlated to a small air-film thickness and to a large phase offset between the maximum of deformation and the minimum of the vertical position of the center of mass.

Figure 1

(Color online) (Left) Experimental setup for measuring the droplet trajectories and deformations. The setup is composed of the electromagnetic shaker GW-V55, a plate on which the container filled with the oil bath is attached, an incandescence lamp, and a high-speed video camera (REDLAKE). (Right) In order to observe interference fringes, the container is lighted homogeneously by three sodium low-vapor-pressure lamps. The droplet is filmed from the top using a high-speed video camera.

Figure 2

Amplitude-frequency phase diagram for a droplet of $1.5\mathrm{mm}$ of diameter and made up of oil of $50\mathrm{cSt}$. Top and bottom curves correspond to the Faraday ${\Gamma }_F$ and Couder ${\Gamma }_C$ acceleration thresholds, respectively (see text). The data points represent the experimental conditions that have been investigated in the present work in order to cover at best the region where the droplet can bounce. Two large data points correspond to the cases I and II described in the text.

Figure 3

Snapshots of a droplet: (left) entirely visible, (right) at its maximum of deformation. The experimental contour of the droplet is represented by a white curve. The height $h$ and the width $L$ are indicated by the double arrows while the center of mass is shown by the white circle.

Figure 4

Cumulative distribution $F$ of the bouncing droplet lifetimes for three situations: (i) $f=50\mathrm{Hz}$ and $\Gamma =3.0$ (black circles and solid line), (ii) $f=75\mathrm{Hz}$ and $\Gamma =6.1$ (open circles and dashed line), and (iii) $f=100\mathrm{Hz}$ and $\Gamma =11.8$ (crossed squared and dot line). The lines correspond to the fit by two parameters Weibull distribution.

Figure 5

Medians of lifetimes $\tau$ as a function of the reduced acceleration $\Gamma$. The symbols correspond to different considered frequencies (see legend). The vertical arrows represent the Couder thresholds according to the different frequencies. The solid line is the fit from Eq. (2). The large points correspond to cases I and II described in the text.

Figure 6

Interference fringes obtained when the air film located between the droplet and the bath is the most squeezed; the number of fringes is maximum. The different pictures have been obtained for the four frequencies: (a) $26\mathrm{Hz}$, (b) $46\mathrm{Hz}$, (c) $74\mathrm{Hz}$, and (d) $98\mathrm{Hz}$.

Figure 7

Phases ${\varphi }_c$ when the fringes are the closest as a function of the reduced acceleration $\Gamma$. The different frequencies considered have been represented by different symbols (see legend). The dashed curve is a guide for the eyes.

Figure 8

Comparison of (a) the dimensionless position $z∕A$ of the center of mass and (b) the aspect ratio $A_r$ variation with respect to the reduced time $(\phi =\omega t)$ for cases I and II (see text). Cases I and II are represented by ● and ◆, respectively. The solid curve gives an indication concerning the phase of the plate.