Juggling soliton: A new kind of wave-particle entity

We present an experimental study of a new kind of dual drop-wave entity existing on a localized structure in a water Faraday-wave system. A nonpropagating hydrodynamic soliton can juggle a single drop of $\sim$2–3.5 mm diameter for about ${10}^2$–${10}^4$ rebounds. By analyzing the drop trajectories, several regimes are observed: periodic bouncing, period doubling, period tripling, a sawtooth state, and chaotic/erratic trajectories. We present evidence that the most stable cases result from detuning of the drop self-oscillations and synchronization with the soliton's sloshing motion. This synchronization ensures stability and thus longer lifetimes. We analyzed the lifetime of the drop, concluding that the periodic behavior, which appears for the lowest-amplitude solitons, is the most stable state. Further analysis shows that lifetimes follow a Weibull distribution.

Figure 1

Typical images of the two kinds of parametric instabilities that can be observed in the experimental setup: (a) Faraday waves and (b) NPHS. Both images were acquired using back-light illumination. For visualization purposes, a few drops of black ink are added to the liquid mixture (distilled water plus surfactant). The second instability has transverse movement, so a thinning of the liquid at the top of the soliton can be observed. As a reference, the depth of the layer of fluid is $2\mathrm{cm}$. Images obtained from Gordillo [42].

Figure 2

(a) Illustration of the experimental setup, including the light-emitting diode arrangement (i), a white acrylic diffusive screen (ii), the acrylic basin (iii), the electromagnetic shaker (iv), and a high-speed camera (v). The driving acceleration is measured with a piezoelectric accelerometer. (b) NPHS characteristics as a function of reduced detuning $\nu =(f/2-f_o)/f_o$, where $f_o=5.43$ Hz is the resonant frequency of the first transverse mode. Its normalized amplitude $A_s/H$ (width $w_s/H$) decreases (increases) as $\nu \to 0$.

Figure 3

Drop-diameter histogram. Drops are generated using a micropipette with a volume fixed to $10\mu \mathrm{l}$. Most drops $(84\%)$ are concentrated in the range $D=2.54\pm 0.10\mathrm{m}\mathrm{m}$.

Figure 4

(a) Front-view image of a $\sim 3.5\mathrm{m}\mathrm{m}$ diameter drop that bounces over a NPHS, with forcing frequency $f=10.4$ Hz and acceleration $\mathrm{\Gamma }=0.085$ (the same for all panels). (b) Sequence of images of a 2.7-mm-diameter drop bouncing over a NPHS, showing an almost complete cycle of the drop motion (total time 0.2 s, soliton period $1/5.2\approx 0.192$ s; time lapse between frames is 25 ms). The drop strongly deforms during the rebound, as well as the soliton's surface. The drop oscillates in its fundamental mode all the time, although more strongly during the free-flight period. (c) Side view of the same drop shown in (b). Here, half a period is shown, with the same time lapse between frames. Surface waves generated at the collision are visible.

Figure 5

(a) Trajectory of a 2.7-mm-diameter drop in the $(y,z)$ plane obtained by a lateral view, with forcing frequency $f=10.4$ Hz and acceleration $\mathrm{\Gamma }=0.085$ [same drop shown in Fig. 4]. Arrows indicate the drop motion direction. The solid red line shows a parabola as a guide to the eye for the drop's free-flight period and the horizontal black dashed line shows the liquid-surface quiescent position $(z=0)$. The transverse and vertical positions vs time are presented in (b) and (c).

Figure 6

Examples of possible drop's height-time trajectories, obtained from front-view images. (a) Regular periodic rebounds, $f=10.2\mathrm{Hz}, \mathrm{\Gamma }=0.1122$, and $D=2.43\mathrm{m}\mathrm{m}$. (b) Double period rebounds, $f=10.3\mathrm{Hz}, \mathrm{\Gamma }=0.097$, and $D=2.52\mathrm{m}\mathrm{m}$. (c) Chaotic/erratic rebounds, $f=10.3\mathrm{Hz}, \mathrm{\Gamma }=0.1122$ and $D=2.49\mathrm{m}\mathrm{m}$. The solid (red) lines show the parabolic fits. The averages for all the adjusted effective accelerations are (a) $g_{\mathrm{eff}}=9.1\pm 0.1\mathrm{m}{\mathrm{s}}^{-2}$, (b) $g_{\mathrm{eff}}=9.1\pm 0.2\mathrm{m}{\mathrm{s}}^{-2}$, and (c) $g_{\mathrm{eff}}=9.4\pm 0.2\mathrm{m}{\mathrm{s}}^{-2}$. The horizontal dashed lines show (a) the regular height average and (b) two distinctive heights for the doubling period case. (d) Height fluctuations $\mathrm{\Delta }{H}_{\max }=H_{\max }^i-\langle H_{\max }\rangle$ vs the normalized time difference between successive maxima, with $\mathrm{\Delta }\tau =t_{\max }^i-t_{\max }^{i-1}$ the time difference between successive maxima, and $T_s=f_s^{-1}$ the soliton period, for regular periodic rebounds ($\circ$), double period rebounds ($\square$), and erratic bouncing ($\diamond$).

Figure 7

(a) Histogram of adjusted effective gravitational acceleration $g_{\mathrm{eff}}$ for all measured drop parabolic trajectories. Most cases are indeed close to the gravitational acceleration $g=-9.81\mathrm{m}{\mathrm{s}}^{-2}$. (b) $g_{\mathrm{eff}}$ vs maximum drop height $H_{\max }$. The horizontal dashed line corresponds to the gravitational acceleration $g$. Lower trajectories have a smaller effective acceleration, probably due to a stronger viscous drag by the air layer between the drop and the NPHS surface.

Figure 8

Experimental phase diagram $(\nu ,\mathrm{\Gamma })$. The single drop juggling soliton is stable in a subregion of the stability region of the NPHS alone. Most of the drops are in the range $D=2.44$–2.64 mm. The data points show the $(\nu ,\mathrm{\Gamma })$ values that have been studied where single drop stable juggling solitons are found, with the conditions explained in the text (stable realizations recorded for 2 s, equivalent to 8–10 rebounds).

Figure 9

Examples of possible drop dynamics, visualized through return maps $(H_n,H_{n+1})$, for all $\mathrm{\Gamma }$ and (a) $f=10.2\mathrm{Hz}$, (b) $f=10.3\mathrm{Hz}$, (c) $f=10.4\mathrm{Hz}$, (d) $f=10.5\mathrm{Hz}$, (e) $f=10.6\mathrm{Hz}$, and (f) $f=10.7\mathrm{Hz}$. The periodic, doubling period, and erratic cases are shown in red, green, and blue, respectively.

Figure 10

Histograms of height deviations $H_n-\langle H_n\rangle$, for (a) $f=10.2\mathrm{Hz}$, (b) $f=10.3\mathrm{Hz}$, and (c) $f=10.4\mathrm{Hz}$. The periodic, doubling period, and erratic cases are shown in red, green, and blue, respectively.

Figure 11

Occupation fraction of each dynamical regime as function of $\mathrm{\Gamma }$, for (a) $f=10.2\mathrm{Hz}$, (b) $f=10.3\mathrm{Hz}$, and (c) $f=10.4\mathrm{Hz}$. The symbols correspond to regular periodic ($\circ$), periodic doubling ($\square$), and erratic ($\diamond$).

Figure 12

Drop oscillations during two rebounds at $f=10.4\mathrm{Hz}$ and $\mathrm{\Gamma }=0.112$. (a) Drop center of mass vs time; the parabolic flights are fitted with $g_{\mathrm{eff}}=9.82\pm 0.10\mathrm{m}{\mathrm{s}}^{-2}$ and $g_{\mathrm{eff}}=9.81\pm 0.08\mathrm{m}{\mathrm{s}}^{-2}$, respectively. (b) Drop diameter $D$ vs time. The complete data set is fitted with a single sinusoidal function, $D\left(t\right)=D+a_{d}sin(2\pi f_{d}t+\varphi )$, with $D=2.670\pm 0.004\mathrm{m}\mathrm{m}, a_d=0.143\pm 0.007\mathrm{m}\mathrm{m}$, and $f_d=52.04\pm 0.07\mathrm{Hz}$. (c) Soliton amplitude $A_s$ as a function of time. (d) The cell's base oscillation.

Figure 13

Example of drop oscillations for a larger drop, $f=10.4\mathrm{Hz}$ and $\mathrm{\Gamma }=0.115$; in this case, for nine rebounds in the period doubling regime. Each subset is fitted with a single sinusoidal function, $D\left(t\right)=D+a_{d}sin(2\pi f_{d}t+\varphi )$, with $D=3.56\pm 0.01\mathrm{m}\mathrm{m}, a_d=0.23\pm 0.01\mathrm{m}\mathrm{m}$ for the larger oscillations of the drop (dashed line) and $D=3.50\pm 0.01\mathrm{m}\mathrm{m}, a_d=0.17\pm 0.09\mathrm{m}\mathrm{m}$ for the smaller oscillations of the drop (solid line). In both cases, the adjusted drop frequency is $f_d=31.20\pm 0.02\mathrm{Hz}$. The mean diameters are different by $\approx 1.7\%$. We believe this is due to nonlinear deformations of the drop (very elongated states), which are probably different between two consecutive rebounds in the period doubling state.

Figure 14

Normalized oscillation frequency $f_d/f_s$ vs drop diameter $D$, for $f=10.2\mathrm{Hz} (\circ ), 10.3\mathrm{Hz} (\square ), 10.4\mathrm{Hz} (\diamond )$, and $10.5\mathrm{Hz} \left(▵\right)$. The solid lines correspond to the theoretical normalized fundamental frequency $f_o/f_s$, given by Eq. (3). The measured possible drop's oscillation frequencies are $f_d=9f_s, 10f_s, 11f_s, 12f_s, 13f_s$, and $14f_s$ for the smaller drops, and $f_d=6f_s$ for the larger ones. Synchronization and detuning are evident from the levels and frequency variations with respect to $f_o/f_s$. For each large drop in the period doubling regime, their diameters and oscillation frequencies were averaged.

Figure 15

(a) Measured lifetimes $\tau$ as a function $\mathrm{\Gamma }$. Symbols represent the individual $\tau$ measurements ($\circ$) and their averages $\langle \tau \rangle$ ($\diamond$) for different accelerations. (b) Cumulative distribution function for $\mathrm{\Gamma }=0.103$ ($\circ$) and $\mathrm{\Gamma }=0.164$ ($\diamond$). The solid lines are fits to the Weibull function $F\left(\tau \right)=1-e^{-{(\tau /{\tau }_o)}^{\alpha }}$, with ${\tau }_o=385.1\pm 16.7\mathrm{s}, \alpha =1.34\pm 0.10$, and $R^2=0.985$ for $\mathrm{\Gamma }=0.103$, and ${\tau }_o=14.7\pm 1.1\mathrm{s}, \alpha =0.78\pm 0.07$, and $R^2=0.923$ for $\mathrm{\Gamma }=0.164$, where $R^2$ is the regression coefficient. (c) Double plot of the scale and shape factors, ${\tau }_o$ ($▵$) and $\alpha$ ($\square$), as a function of $\mathrm{\Gamma }$. (d) Lifetime cumulative distribution function for all of the 117 realizations. The solid line is the fit to the Weibull function, with ${\tau }_o=120.7\pm 9.6\mathrm{s}, \alpha =0.47\pm 0.02$, and $R^2=0.980$.

Figure 16

Example of a sawtooth state, $f=10.3$ Hz and $\mathrm{\Gamma }=0.118$. Small symbols $(\circ )$ joined by thin lines correspond to the drop's center-of-mass vertical position vs time. Just the values around the local maxima are shown. Each parabola corresponds to a drop free flight above the NPHS. Larger symbols $(\diamond )$ joined by thick lines are the local maxima. About 50 rebounds are observed in a 10 s time lapse.

Figure 17

(a) Plot of height variations $\mathrm{\Delta }{H}_{max}$, with respect to the sawtooth maximum value, vs the phase difference $\mathrm{\Delta }\varphi$, for each video acquisition obtained every few minutes during the whole juggling soliton lifetime. Each symbol color represents a different video acquisition. The solid line shows the jumps from one branch to another for one of the video acquisitions. (b) Same as in (a), but additional to the sawtooth state ($\circ$), we present two period doubling realizations $(*,\diamond )$ and two tripling period states ($▵$). All these short living states collapse on either of the two branches of the sawtooth state or on extensions of them.