Collective vibrations of confined levitating droplets

We report the results of our investigation of a fluid-based driven-dissipative oscillator system consisting of a lattice of millimetric fluid droplets bouncing on a vertically vibrating liquid bath and bound within an annular ring. We characterize the system behavior as it is energized through a progressive increase in the bath's vibrational acceleration. Depending on the number of drops, the onset of motion of the lattice may take the form of either out-of-phase oscillations or a striking solitary wavelike instability. Theoretical modeling demonstrates that these behaviors may be attributed to different bifurcations at the onset of instability. The results presented here demonstrate the potential and utility of the walking droplet system as a platform for investigating wave-mediated, inertial, nonequilibrium particle dynamics at the macroscale.

Figure 1

(a) Oblique perspective of a lattice of 40 equispaced droplets of silicone oil confined to an annular channel. (b) Schematic of the channel geometry with inner radius $R_1=24$ mm and outer radius $R_2=31$ mm. A shallow layer of depth $h=0.56\text{mm}$ acts as a damper so that wave motion is largely confined to the channel where the fluid depth $H=5.12\text{mm}$. The entire assembly is vibrated vertically with maximum acceleration $\gamma$ and frequency $f$.

Figure 2

$N=20$ droplets. (a) Evolution of the polar angle $\theta$ of six adjacent droplets undergoing out-of-phase oscillations with frequency $F=2$ Hz at vibrational acceleration $\widehat{\gamma }=0.962$. The trajectories are colored according to the droplets' instantaneous angular velocity $\dot{\theta }$. Note that neighboring droplets have equal and opposite $\dot{\theta }$ for all time $t$. (b) A snapshot of the polar positions of each droplet in the lattice at a time when each droplet attains its maximum angular speed of $|\dot{\theta }|=0.16$ rad ${\mathrm{s}}^{-1}$. The value of $\widehat{\gamma }$ is the same as in (a). The solid lines emphasize the net out-of-phase oscillation of two decahedral sublattices and the dashed lines delineate the channel geometry. (c) Plot of the mean oscillation amplitude $\langle A\rangle$ as a function of $\widehat{\gamma }$ shows a monotonic increase of $\langle A\rangle$ beyond a critical value of ${\widehat{\gamma }}_{c_1}=0.956$. The error bars represent the standard deviation of the mean. The red curve is a square-root fit of the data. Inset: a schematic of the reversible supercritical bifurcation path underpinning the observed instability. The solid red line indicates the stable branch consisting of static then oscillating droplets; the dashed red line is unstable.

Figure 3

$N=40$ droplets. (a) Evolution of the angular position $\theta$ of individual droplets in the lattice in the solitary-wave regime at vibrational acceleration $\widehat{\gamma }=0.890$. The trajectories are colored according to the droplets' instantaneous angular velocity $\dot{\theta }$. Prior to the arrival of the wave at event A, a droplet is stationary at ${\theta }_{-}\approx 0.08$ rad. In the interval AB, it is pulled in the direction of decreasing $\theta$ over a time of order 0.5 s, reaching a maximum angular speed of $|\dot{\theta }|\approx 0.38$ rad ${\mathrm{s}}^{-1}$. At event B, its angular speed returns to zero and thereafter the droplet settles down to ${\theta }_{+}\approx 0$ rad via underdamped oscillations. (b) Polar positions of the droplets in the lattice at $t=1.0$ and $35.0$ s. Each droplet is colored according to its instantaneous angular velocity. The value of $\widehat{\gamma }$ is the same as that in (a). The black marker highlights the overall clockwise rotation of the lattice and black arrows show the direction of wave propagation. See [47] for a color-coded animation of the polar droplet positions. (c) Droplet instantaneous angular velocity $\dot{\theta }$ as a function of droplet position at $t=1$ s. We fit a cubic spline interpolant through the maximum of each spike, revealing a well-defined wave packet propagating in an anticlockwise direction through the lattice. (d) Plot of $\omega \left(\widehat{\gamma }\right)$ (black dots). We find $\omega$ varies monotonically with $\widehat{\gamma }$ but drops to zero at ${\widehat{\gamma }}_{c_3}=0.839$. The best fit red curve emphasizes the nonlinearity of $\omega \left(\widehat{\gamma }\right)$. The maximum error in $\omega$ was found to be $\pm 0.01$ rad ${\mathrm{s}}^{-1}$. Inset: a schematic sketch of the subcritical bifurcation diagram underpinning the solitary wave instability. The upper solid red line (symbolic of the best fit line in the main plot) denotes the stable solitary wave branch, the dashed black curve a hypothetical unstable branch.

Figure 4

Stability threshold $M_c$ versus droplet number $N$ predicted by the model (1), disallowing for radial motion. Triangles mark supercritical bifurcations $[\text{Re}\left({\beta }_2\right)>0]$ and squares mark subcritical bifurcations $[\text{Re}\left({\beta }_2\right)<0]$ as predicted by Eq. (2). Red zones and circles denote droplet configurations predicted to be unstable for all $M$, and so inaccessible in the laboratory. Green zones denote configurations predicted to be stable up to a critical value of $M$, and thus potentially accessible in the laboratory. Figure adapted, with permission, from [40].