Resonant and antiresonant bouncing droplets

When placed onto a vibrating liquid bath, a droplet may adopt a permanent bouncing behavior, depending on both the forcing frequency and the forcing amplitude. The relationship between the droplet deformations and the bouncing mechanism is studied experimentally and theoretically through an asymmetric and dissipative bouncing spring model. Antiresonance phenomena are evidenced. Experiments and theoretical predictions show that both resonance at specific frequencies and antiresonance at Rayleigh frequencies play crucial roles in the bouncing mechanism. In particular, we show that they could be exploited for bouncing droplet size selection.

Figure 1

BD for frequency $f=50$ Hz, radius $a=0.76$ mm, and viscosity $\nu =5$ cSt. One observes that the droplet adopts periodically oblate and prolate shapes. Note that the deformation may be, for instance, asymmetric when the droplet takes off as denoted with a white arrow.

Figure 2

Bouncing threshold (logarithmic scale) as a function of the dimensionless Rayleigh frequency ${\Omega }_2$ for droplets of different viscosities: (a) 5cSt, (b) 20 cSt, and (c) 50cSt. Black dots represents the experimental data. The curves corresponds to analytical models. Dotted gray line: Couder model [1], dashed orange line: Eichwald model [18], plain red line: our ABS model capturing two extrema while Eichwald captures only resonant behaviors.

Figure 3

(a) Droplet deformed with $\delta =-0.318$ resulting in an oblate-shaped droplet. The left side is the experimental picture, the right one being the Prosperetti model. (b) Prolate shaped droplet with $\delta =0.437$. The value of $\delta$ in each case has been obtained by fitting the shape of the droplet. (c) Evolution of the relative surface energy for a droplet deformed with the $Y_2^0$ spherical harmonic, $E_0$ being the energy of an undeformed droplet (black dots). The plain curve is a guide to the eyes showing the parabolic behavior. The horizontal axis measures the deformation $\delta$. Note also the asymmetric shape of the curve for large $\delta$.

Figure 4

Numerical and experimental spatiotemporal diagrams for BD with $\nu =5$ cSt (in arbitrary units). The white region in the experimental diagrams represents the motion of the liquid surface while the black region corresponds to the droplet elongation. The gray curve in numerical simulation is for the elongation of the spring. The white line drawn in the spring motion represents its center of mass. The numerical parameters are those indicated in Table 1. (a) Resonant ABS/BD. (b) Antiresonant ABS/BD. One observes in (a) the propulsion of the spring or droplet at the maximum height of the surface motion. (b) The phase opposition of the spring or droplet elongation with the surface oscillation at antiresonance is illustrated.