Promenading pairs of walking droplets: Dynamics and stability

We present the results of an integrated experimental and theoretical investigation of the promenade mode, a bound state formed by a pair of droplets walking side by side on the surface of a vibrating fluid bath. Particular attention is given to characterizing the dependence of the promenading behavior on the vibrational forcing for drops of a given size. We also enumerate the different instabilities that may arise, including transitions to smaller promenade modes or orbiting pairs. Our theoretical developments highlight the importance of the vertical bouncing dynamics on the stability characteristics. Specifically, quantitative comparison between experiment and theory prompts further refinement of the stroboscopic model [A. U. Oza et al., J. Fluid Mech. 737, 552 (2013)] through inclusion of phase adaptation and reveals the critical role that impact phase variations play in the stability of the promenading pairs.

Figure 1

Plan view of promenading pairs of order (a) $N=1$, (b) $N=2$, (c) $N=1.5$, and (d) $N=2.5$. The black arrows indicate the direction of motion and white lines the droplet trajectories. Note that the drops bounce in phase for integer $N$ and out of phase for half-integer $N$. (e) Regime diagram [19] of different bouncing and walking states for single drops, for 20 cSt silicone oil, and vibrational frequency $f=80$ Hz, as a function of the vibration number $V_i=2\pi f\sqrt{\rho R^3/\sigma }$ and dimensionless forcing acceleration $\gamma /g$, where $g$ is the gravitational acceleration. The walking regime is indicated in red, the bouncing regime in green and blue, and the Faraday threshold ${\gamma }_F=4.2g$. The region shaded in white corresponds to a drop of radius $R=0.38\pm 0.01$ mm, for which $V_i=0.793\pm 0.032$. The dashed arrows indicate the range of forcing acceleration over which the transition from the ${(2,1)}^2$ to the $(4,2)$ walking mode is expected to arise. The relevant variables are defined in Table 1.

Figure 2

(a) Strobed trajectories of an $N=2$ stable promenading pair. (b)–(d) Oblique view of the strobed trajectory of an $N=1$ promenade at different instants. The colors in the bath are due to the introduction of a colored striped transparent film between the lighting source and bath, which facilitates visualization of the surface deformations [20]. White lines indicate the trajectories of the droplets. See video 1 in [21].

Figure 3

(a) Top view of the lid, paddles, and launcher used to generate the $N=2$ promenade mode. The red curves indicate the trajectory of each of the droplets. (b) Schematic of one of the rotating paddles. (c) Oblique view of the apparatus. (d)–(f) Protocol followed to generate the promenading pairs: (d) drop trapping, (e) release of the droplets from the holding pens, and (f) emergence of the desired promenading pair.

Figure 4

Different stable promenading trajectories found in experiments at $\gamma /{\gamma }_F=0.85\pm 0.01$: (a) $N=1$, (b) $N=1.5$, (c) $N=2$, (d) $N=2.5$, and (e) $N=3$. Trajectories are colored according to instantaneous speed $|{\dot{\mathit{x}}}_i|$, nondimensionalized by the free walking speed $u_0$. All the promenaders were generated with a drop size of $R=0.38$ mm, except the $N=3$ mode, for which $R=0.35$ mm.

Figure 5

(a) Observed stability of the promenading pairs as a function of the forcing acceleration $\gamma /{\gamma }_F$ for drops of radius $R=0.38$ mm. Green indicates stable promenaders. Red denotes drop splitting, with the data points indicating values of $\gamma /{\gamma }_F$ above which the trajectories diverge for $N=1$ and 1.5 and below which trajectories diverge for $N=3$ and 3.5. Gray and blue denote consecutive and nonconsecutive cascades, respectively. The horizontal line at $\gamma /{\gamma }_F=0.85$ indicates the parameter regime explored in Fig. 4. (b)–(g) Classification of the different observed behaviors. Arrows indicate the sense of motion. (b) Stable $N=2.5$ promenade. (c) $N=2.5$ promenade splitting. (d) Cascade from $N=2.5$ to $N=1.5$. (e) Cascade from an $N=2$ promenade to an $N=1$ orbit. (f) Cascade from an $N=3.5$ promenade to an $N=1.5$ orbit. (g) Cascade from the $N=3$ to the $N=1$ promenade mode, accompanied by a shift in direction. Trajectories are colored according to speed.

Figure 6

Promenading trajectories for different values of $\gamma /{\gamma }_F$, colored according to the instantaneous drop speed: (a)–(c) $N=1$ and (d)–(f) $N=1.5$. Arrows indicate the sense of motion. Trajectories are shown (a) and (d) in the ${(2,1)}^2$ walking mode, (b) and (e) just after the transition to the $(4,2)$ walking mode, and (c) and (f) that arise following the onset of chaos in the vertical dynamics.

Figure 7

Dependence of various promenade characteristics on the forcing acceleration $\gamma /{\gamma }_F$ and mode $N$: (a)–(c) dependence on $N$ for $\gamma /{\gamma }_F=0.85\pm 0.01$ and (d)–(f) dependence on $\gamma /{\gamma }_F$ for the indicated values of $N$. (a) Mean interdrop distance ${\overline{x}}_N/{\lambda }_F$. (b) Pair speed. Here $u_N$ denotes the mean total speed, $v_N$ the mean translational speed, and $u_0$ the free walking speed. (c) Period of oscillation $T_N/T_F$. (d) Dependence of the mean translational speed $v_N$ on $\gamma /{\gamma }_F$ for different promenade modes $N=1$ through $N=2.5$. Note that $v_N$ increases monotonically with increasing $N$. The shaded regions indicate droplets in the $(4,2)$ vertical mode and the unshaded regions indicate the ${(2,1)}^2$ mode. Data for each promenade mode are reported until the critical memory at which it becomes unstable. Average drop speed $u_N$ and mean translational speed $v_N$ as a function of $\gamma /{\gamma }_F$ for modes (e) $N=1$ and (f) $N=1.5$.

Figure 8

Impact phase adaptation. (a) Observed translational velocities used in the base state algebraic equations (5). (b) Impact phases $\mathcal{S}=sin\mathrm{\Phi }$ provided by the base state equation (5), as a function of the promenade mode $N$ and memory $\gamma /{\gamma }_F$. (c) Best-fit curve for the phase dependence indicated in (b), $sin\mathrm{\Phi }=p_3{\mathrm{\Gamma }}_{\mathrm{eff}}^2+p_2{\mathrm{\Gamma }}_{\mathrm{eff}}+p_1$, as a function of the effective forcing ${\mathrm{\Gamma }}_{\mathrm{eff}}=\gamma /{\gamma }_F-p_4\tilde{h}$. The parameter values are $p_1=0.466,p_2=-1.679,p_3=1.6705$, and $p_4=0.251$. (d) Simulation of an $N=2$ promenade mode at $\gamma /{\gamma }_F=0.84$, with the time evolution of the impact phase $\mathcal{S}\left(t\right)$ indicated in the color bar.

Figure 9

Comparison between theory and experiment for a drop of radius $R=0.38$ mm. (a) Data showing the dependence on $\gamma /{\gamma }_F$ of the dimensionless average distance ${\overline{x}}_N/{\lambda }_F$ and amplitude of oscillation (indicated by error bars) for different promenade modes $N$ colored according to the legend. Solid lines indicate results of the numerical simulation of Eq. (4) with phase adaptation consistent with Eq. (6). Shaded regions indicate maximum and minimum computed separation distance. (b) Stability characteristics predicted by the variable-phase stroboscopic model (vertical bars), color coded according to the color scheme used in Fig. 5. The background is shaded according to experimental observations. The predicted onset of transverse oscillations is indicated for the $N=1$ mode. Red lines indicate drop trajectories crossing, owing to shortcomings in modeling the wave height near the droplet.

Figure 10

(a) The $x$ component of the force ${\mathit{F}}_i\left(t\right)$, as defined by Eq. (9), exerted by the wave field on one of the $N=3.5$ promenading droplets as a function of its position $x_2\left(t\right)$ at different values of the memory $\gamma /{\gamma }_F$. Note that the area inside the curves represents the work done by the $x$ component of the wave force during one oscillation period of the promenader. (b) The $x$ component of the wave force as a function of the interdroplet distance $x_N$ for out-of-phase promenaders at $\gamma /{\gamma }_F=0.83$. (c) Cascade from $N=3.5$ to $N=1.5$. The red dot represents the base state used in simulations.