3.1. Terms of $\textit{O}(1)$: the set of possible rings

The $\textit{O}(1)$ expansion of (3.6) yields two equations

(3.8)\begin{equation} 0 = \sum_{n=1}^{N-1}\zeta^{n}f'\left(2r_{0}\sin\left({\rm \pi} n/N\right)\right)\sin\left({\rm \pi} n/N\right), \end{equation}

(3.9)\begin{equation}0 = \sum_{n=1}^{N-1}\zeta^{n}f'\left(2r_{0}\sin\left({\rm \pi} n/N\right)\right)\cos\left({\rm \pi} n/N\right). \end{equation}

Recall that $\zeta =\{ 1,-1\}$ for rings with nearest neighbours bouncing in-phase and out-of-phase, respectively, and note that $f'(0)=0$ which eliminates the $n=0$ term in each sum. Equations (3.8) and (3.9) express the requirement that the net radial and azimuthal waveforces on each drop in the ring be zero, respectively. For a given $\zeta$ and $N$, the azimuthal equation (3.9) is satisfied for all values of $r_0$ due to the azimuthal symmetry of the ring. However, the radial equation (3.8) will only be satisfied for a discrete set of $r_0$, whose values will depend on both $\zeta$ and $N$.

The possible radii, $r_0$, predicted by (3.8) are plotted as a function of $N$ in figures 8(a) and 8(b) for rings bouncing in-phase ($\zeta =1$) and out-of-phase ($\zeta = -1$), respectively. We note that the radii $r_0$ are independent of the impact-phase parameters $\mathcal {S}$ and $\mathcal {C}$. However, the values of $r_0$ do weakly depend on the bath's vibrational acceleration $\gamma$, due to the dependence on $\gamma$ of the spatial decay parameter $\xi$ that appears in the wave kernel $f(r)$ (see (3.3)). In figure 8, we plot the possible values of $r_0$ at $\gamma /\gamma _F = 0.7$, slightly below the theoretically predicted walking threshold of a single drop, $\gamma _W/\gamma _F = 0.74$, which is in the vicinity of the vibrational accelerations where the rings are expected to destabilize. A comparison between the possible values of $r_0$ at $\gamma /\gamma _F = 0.7$ and $\gamma /\gamma _F = 0.9$ is presented in appendix A.

Figure 8.

The possible ring radii, $r_0$, predicted by (3.8) as a function of drop number $N$ for (a) in-phase $(\zeta = 1)$ and (b) out-of-phase $(\zeta = -1)$ rings. Blue and red markers correspond to rings in which each drop bounces on a minimum or maximum of the wavefield produced by the neighbouring drops (see (3.11)), respectively. Rings with side length $s\ll \lambda _F$ are expected to have radii corresponding to the extrema $a_n$ of the function $[J_{0}(r)]^{2}$ (see appendix B), plotted in (c). Rings with side length $s\gg \lambda _F$ are expected to have side lengths $s_0 = 2r_{0}\sin ({\rm \pi} /N)$ corresponding to the extrema $b_n$ of the wave kernel $f(s)$ (3.3), plotted in (d). The wavefields generated by the three rings boxed in panel (a) are plotted in figure 9.

To gain insight into the solutions, $r_0$, of (3.8), consider the function

(3.10)\begin{equation} \bar{H}_{m}\left(\bar{\boldsymbol{x}}_{m},\bar{t}\right)= \frac{AM_{e}}{R}\sum_{n\ne m}\sigma_{n}\int_{-\infty}^{\bar{t}}\mathcal{S}_{n}\,f \left(\left|\bar{\boldsymbol{x}}_{m}\left(t\right)- \bar{\boldsymbol{x}}_{n}\left(s\right)\right|\right)\textrm{e}^{-\left(t-s\right)} \textrm{d}s, \end{equation}

which is the local wavefield beneath drop $m$ that includes the wavefield contributions from all neighbouring drops $n\ne m$ but excludes the wavefield produced by drop $m$ itself. It is important to note the distinction between the closely related quantities $h(\boldsymbol {x}_{\boldsymbol {m}},t)$, the net wavefield beneath drop $m$ produced by all drops, as defined in (3.2), and $H(\boldsymbol {x}_{\boldsymbol {m}},t)$. Consideration of the quantity $H_m$ is useful because it expresses the perturbation to a given drop's local wavefield induced by its neighbours.

For drops in a symmetric ring of radius $r$, the wavefield beneath each drop is identical. Thus, without loss of generality, we consider the perturbation to the wavefield beneath drop $m=0$, as produced by the neighbouring drops $n=1,2,\ldots ,N-1$:

(3.11)\begin{equation} \bar{H}_{0}\left(\bar{r}\right)=\frac{AM_{e}\mathcal{S}}{R} \sum_{n=1}^{N-1}\zeta^{n}f\left(2\bar{r}\sin\left( \frac{{\rm \pi} n}{N}\right)\right). \end{equation}

Note that the locations of the extrema of the function $\bar {H}_{0}$ in (3.11) correspond to the possible radii $r_0$ predicted by (3.8). Thus, the solutions $r_0$ in figure 8 correspond to each drop bouncing on either a maximum (red markers) or minimum (blue markers) of the wavefield produced by their neighbours at the time of impact. Figure 9 shows the wavefields generated by the three rings highlighted by grey squares in figure 8(a). We emphasize that in figure 9 we are plotting the net wavefield $h$, which is not to be confused with the perturbation wavefield $H$. Although the drops in figure 9 are seen to be bouncing on maxima of the net wavefield $h$, each drop is still in fact bouncing in a minimum of the wavefield $H$ produced by its neighbours. This is due to the fact that the large central peak in the wavefield produced by drop $m$ at position $\boldsymbol {x}_m$ overpowers the minimum in $H$, produced by drops $n\ne m$, at $\boldsymbol {x}_m$.

Figure 9.

The wavefields produced at $\gamma /\gamma _F = 0.7$ by in-phase rings with (a) $N=18$, $r_0/\lambda _F = 0.884$$(s_0/\lambda _F = 0.307)$, (b) $N=13$, $r_0/\lambda _F = 2.94$$(s_0/\lambda _F = 1.41)$, and (c) $N=6$, $r_0/\lambda _F = 3.61$$(s_0/\lambda _F = 3.61)$, boxed for reference in figure 8(a). Droplet positions are marked by black circles.

As was the case for droplet pairs (Couchman et al. Reference Couchman, Turton and Bush2019), we expect values of $r_0$ corresponding to maxima of (3.11) to be unstable at all $\gamma$. Conversely, values of $r_0$ corresponding to minima of (3.11) are expected to remain stable up until some critical vibrational acceleration $\gamma ^*$. In § 3.2, we compute the threshold and type of instability for each possible solution $r_0$ of (3.8).

To gain further insight into the pattern of permissible radii $r_0$ shown in figure 8, we consider two limiting cases corresponding to rings having either a large or small side length, $s_0 = 2r_{0}\sin ({\rm \pi} /N)$, relative to the Faraday wavelength, $\lambda _F$. In the limit of $s_0\gg \lambda _F$, each droplet in the ring is predominately influenced by its two nearest neighbours, as the wave-kernel $f(d)$ decays to ${\lesssim } 5\,\%$ of its original amplitude for inter-drop distances $d/\lambda _{F} \gtrsim 4$. In this limit, the ring alters its radius so that its side length corresponds to one of the stationary inter-drop spacings of a droplet pair, which occur at extrema of $f(s)$, as plotted in figure 8(d). The possible ring radii are then $r_{0}=s_{0}/(2\sin ({\rm \pi} /N))$ where $s_0$ satisfies $f'(s_0)=0$. An example of the wavefield produced by a ring in this limit is shown in figure 9(c). Note that the wave profile $f(s)$ in figure 8(d) is plotted at the time of impact of a drop at $s=0$. Therefore, for in-phase rings, side lengths corresponding to minima ($s_0=b_1, b_3, b_5,\ldots$) and maxima ($s_0=b_2, b_4, b_6,\ldots$) of $f(s)$ are expected to be stable and unstable, respectively. However, for out-of-phase rings, the neighbouring droplets now experience a flipped wave profile of shape $-f(s)$, as they impact the standing wave generated at $s=0$ half a cycle later. Thus, for out-of-phase rings, side lengths corresponding to maxima and minima of $f(s)$ are expected to be stable and unstable, respectively.

In the opposite limit, $s_0\ll \lambda _F$, the net wavefield is well approximated by that produced by a continuous ring of wave sources with radius $r_0$. In appendix B, we show that for in-phase ($\zeta = 1$) rings with radius $r_0$, the resulting wavefield is $h(r) \propto J_{0}(r)J_{0}(r_{0})$, a radially symmetric wavefield centred on the ring's centre of mass. In this limit, the possible values of $r_0$ correspond to the extrema of $[J_{0}(r)]^{2}$, as plotted in figure 8(c). An example of the wavefield produced by a ring in this limit is shown in figure 9(a). Radii corresponding to minima ($r_0=a_1, a_3, a_5,\ldots$) and maxima ($r_0=a_2, a_4, a_6,\ldots$) of $[J_{0}(r)]^{2}$ are expected to be stable and unstable, respectively. A similar analysis is not possible for out-of-phase rings, as a continuous ring of wave sources of alternating phases would result in a net wavefield of zero.

We note that in the intermediate regime where $s_0 \approx \lambda _F$, the wavefields are similar in form to those reported by Sungar et al. (Reference Sungar, Sharpe, Pilgram, Bernard and Tambasco2018) in their study of the hydrodynamic Talbot effect generated by a circular array of pillars. An example of the wavefield produced in this intermediate regime is plotted in figure 9(b).

3.2. Terms of $\textit{O}(\epsilon )$: the type and threshold of instability

At $\textit{O}(\epsilon )$ in the expansion of (3.6), we obtain the following $6N \times 6N$ linear system, $\dot {\boldsymbol {q}}=\boldsymbol {Aq}$, as derived in appendix C

(3.12)\begin{equation} \frac{\textrm{d}}{\textrm{d}t}\underset{\boldsymbol{q}_{6N\times1}}{\underbrace{ \begin{pmatrix}\boldsymbol{\delta R}\\ \boldsymbol{\delta \varTheta}\\ \boldsymbol{\delta r}\\ \boldsymbol{\delta \theta}\\ \boldsymbol{\delta u}\\ \boldsymbol{\delta v} \end{pmatrix}}}= \underset{\boldsymbol{A}_{6N\times6N}}{\underbrace{\begin{pmatrix}-\boldsymbol{I} & \textbf{{0}} & \boldsymbol{I} & \textbf{{0}} & \textbf{{0}} & \textbf{{0}}\\ \textbf{{0}} & -\boldsymbol{I} & \textbf{{0}} & \boldsymbol{I} & \textbf{{0}} & \textbf{{0}}\\ \textbf{{0}} & \textbf{{0}} & \textbf{{0}} & \textbf{{0}} & \boldsymbol{I} & \textbf{{0}}\\ \textbf{{0}} & \textbf{{0}} & \textbf{{0}} & \textbf{{0}} & \textbf{{0}} & \boldsymbol{I}\\ \varphi\boldsymbol{U} & -\varphi\boldsymbol{V} & \varphi c_{1}\boldsymbol{I} & \textbf{{0}} & -\frac{1}{\kappa}\boldsymbol{I} & \textbf{{0}}\\ \frac{\varphi}{r_{0}^{2}}\boldsymbol{V} & \frac{\varphi}{r_{0}^{2}}\boldsymbol{W} & \textbf{{0}} & \frac{\varphi}{r_{0}^{2}}c_{2}\boldsymbol{I} & \textbf{{0}} & -\frac{1}{\kappa}\boldsymbol{I} \end{pmatrix}}}\begin{pmatrix}\boldsymbol{\delta R}\\ \boldsymbol{\delta \varTheta}\\ \boldsymbol{\delta r}\\ \boldsymbol{\delta \theta}\\ \boldsymbol{\delta u}\\ \boldsymbol{\delta v} \end{pmatrix}. \end{equation}

The $6N\times 1$ vector $\boldsymbol {q}$ is composed of six $N\times 1$ vectors: $\boldsymbol {\delta R}(t)=(\delta R_{1}(t),\delta R_{2}(t),\ldots,\delta R_{N}(t))$, ${\boldsymbol {\delta \varTheta }}(t)=(\delta \varTheta _{1}(t),\delta \varTheta _{2}(t),\ldots ,\delta \varTheta _{N}(t))$,…, $\boldsymbol {\delta v}(t)=(\delta v_{1}(t),\delta v_{2}(t),\ldots ,\delta v_{N}(t))$, where

(3.13a,b)\begin{gather} \delta R_{m}\left(t\right) = \int_{-\infty}^{t}\delta r_m \left(s\right)\textrm{e}^{-\left(t-s\right)}\textrm{d}s, \quad \delta \varTheta_{m}\left(t\right) = \int_{-\infty}^{t}\delta \theta_m \left(s\right) \textrm{e}^{-\left(t-s\right)}\textrm{d}s, \end{gather}

(3.14a,b)\begin{gather} \delta u_{m} = \delta \dot{r}_{m}, \quad \ \delta v_{m} = \delta \dot{\theta}_{m}. \end{gather}

Each block of the matrix $\boldsymbol {A}$ is an $N\times N$ matrix where $\boldsymbol {I}$ and $\textbf {{0}}$ denote the identity and zero matrices, respectively, and $\boldsymbol {U}$, $\boldsymbol {V}$ and $\boldsymbol {W}$ are defined in (C 25), (C 26) and (C 27) of appendix C, respectively. The scalars $c_1$ and $c_2$ are defined in (C 22) and (C 23) of appendix C, respectively, and

(3.15)\begin{equation} \varphi =-\frac{AM_{e}\beta}{4R\kappa}\mathcal{S}_{0}\mathcal{C}_{0}. \end{equation}

In (3.15), $\mathcal {S}_{0}$ and $\mathcal {C}_{0}$ are the values of the phase parameters for a ring in its initial base state with radius $r_0$. In order to compute $\mathcal {S}_{0}$ and $\mathcal {C}_{0}$ for a given ring, we follow the procedure outlined by Couchman et al. (Reference Couchman, Turton and Bush2019). Based on (3.2), the non-dimensional local wave amplitude $\bar {h}_0$ beneath each drop in the ring is

(3.16)\begin{equation} \bar{h}_{0}=\frac{AM_{e}\mathcal{S}\left(\gamma,h_{0}\right)}{R}\sum_{n=0}^{N-1}\zeta^{n}f\left(2r_{0}\sin\left(\frac{{\rm \pi} n}{N}\right)\right). \end{equation}

Equations (3.5a) and (3.16) can be solved simultaneously to obtain $\mathcal {S}_0=\mathcal {S}(\gamma ,h_{0})$ and $h_0$. Having obtained $h_0$, $\mathcal {C}_{0}=\mathcal {C}(\gamma ,h_{0})$ can then be computed directly using (3.5b). It will be seen that the values of $\mathcal {S}_{0}$ and $\mathcal {C}_{0}$ are significantly different for different ring geometries, as are characterized by $N$, $r_0$ and $\zeta$. In turn, $\mathcal {S}_0$ and $\mathcal {C}_0$ influence the ring's stability due to the dependence of the matrix $\boldsymbol {A}$ on $\varphi$ in (3.12).

We proceed by using (3.12) to determine the threshold and type of instability for each ring identified in figure 8. For a given ring, we start at $\gamma /\gamma _F = 0.7$ and increase $\gamma$ progressively while tracking the evolution of the eigenvalues of $\boldsymbol {A}$ (3.12), as computed numerically. Below $\gamma /\gamma _F = 0.7$, the assumption that the drops are in a period-doubled $(2,1)$ bouncing mode is not expected to be valid (Wind-Willassen et al. Reference Wind-Willassen, Moláček, Harris and Bush2013). We note that, as discussed in $\S$3.1, $r_0$ depends weakly on $\gamma$, which obliges us to recompute the corresponding value of $r_0$ at each value of $\gamma$ using (3.8). The matrix $\boldsymbol {A}$ always has three eigenvalues equal to zero, reflecting the translational ($x, y$) and rotational ($\theta$) invariance of the ring. The instability threshold $\gamma ^*$ corresponds to the value of $\gamma$ at which the real part of any eigenvalue transitions from negative to positive. By analysing the eigenvector corresponding to the most unstable eigenvalue, we deduce the type of instability.

In figure 10, we compare the theoretically predicted ring radii, threshold and type of instability with our experimental data from figure 2. For comparison, we also show the theoretical predictions obtained using a constant-impact-phase model, where the phase parameters were fixed at $\mathcal {S}_0 \mathcal {C}_0 = 0.35$, a value chosen to match the walking threshold, $\gamma _W$, predicted by our variable-impact-phase model. Only rings corresponding to the blue markers in figure 8 are plotted in the theoretical predictions shown in figures 10(b) and 10(c). Rings corresponding to the red markers in figure 8, indicating drops bouncing on the maxima of the wavefield produced by their neighbours, are found theoretically to be unstable already at $\gamma /\gamma _F = 0.7$, as anticipated. In figures 10(b) and 10(c), rings that are predicted to be unstable at the starting value of $\gamma /\gamma _F = 0.7$ are denoted by crosses. Rings that remain stable up to a critical vibrational acceleration, $\gamma ^*$, are denoted by coloured markers.

Figure 10.

A comparison of the type and threshold of ring instability as (a) observed experimentally (reproduced from figure 2) and predicted theoretically by (b) our variable-impact-phase model and (c) a constant-impact-phase model in which the phase parameters were fixed at $\mathcal {S}_0 \mathcal {C}_0 = 0.35$. As in figure 2, the rings are plotted as a function of the drop number $N$ and non-dimensional ring radius $r_0/\lambda _F$. The left and right columns correspond, respectively, to drops bouncing in-phase ($\zeta =1$) and out-of-phase ($\zeta =-1$) relative to their nearest neighbours. The marker colour denotes the vibrational acceleration, $\gamma ^*$, normalized by the walking threshold of a single drop, $\gamma _W$, at which the static ring is predicted to destabilize into a dynamic state. The marker shape denotes the type of instability observed: in-phase radial oscillations (), out-of-phase radial oscillations (), out-of-phase azimuthal oscillations (), irregular rearrangements () and more complex, superposed radial and azimuthal motions (). The black dots in rows (b and c) mark the rings observed experimentally in row (a) for reference, while the crosses denote rings that are already predicted to be unstable at $\gamma / \gamma _F= 0.7$. In the experimental data reported in (a), drops that are not bouncing in the $(2,1)^2$ mode, assumed in our theoretical analysis, are marked for reference. Comparison of (a) and (c) reveals that the principal shortcoming of the constant-impact-phase model is that it always predicts the onset of ring instability for $\gamma < \gamma _W$.

Five distinct types of instability were predicted by our linear stability analysis. In-phase radial oscillations (, see figure 3a), out-of-phase radial oscillations (, see figure 3d) and out-of-phase azimuthal oscillations (, see figure 3c) were all characterized by a pair of complex-conjugate eigenvalues crossing the real axis at the instability threshold $\gamma ^*$. An example of the eigenvalue spectrum for the $N=5$, $r_0 / \lambda _F = 0.440$, $\zeta = 1$ ring, predicted to go unstable to in-phase radial oscillations, is shown in figure 11. Inspecting the eigenvectors corresponding to the pair of unstable eigenvalues revealed the type of oscillation. For example, in-phase radial oscillations were characterized by an eigenvector with all azimuthal components, $\boldsymbol {\delta \theta }$, $\boldsymbol {\delta v}$ and $\boldsymbol {\delta \varTheta }$, equal to zero and non-zero radial components $\delta r_1 = \delta r_2 = \delta r_3\ldots , \delta u_1 = \delta u_2 = \delta u_3\ldots$ and $\delta R_1 = \delta R_2 = \delta R_3\ldots$ . Out-of-phase radial oscillations were characterized by alternating signs in the radial components: $\delta r_1 = -\delta r_2 = \delta r_3 \ldots\,$, $\delta u_1 = -\delta u_2 = \delta u_3 \ldots$ and $\delta R_1 = -\delta R_2 = \delta R_3 \ldots$ . More complex superpositions of radial and azimuthal motion were also observed (), corresponding to the crossing of two identical pairs of complex-conjugate eigenvalues, with corresponding eigenvectors that had both non-zero radial and azimuthal components. Finally, there were two in-phase rings, $N=10$, $r_0 / \lambda _F = 1.30$ and $N=17$, $r_0 / \lambda _F = 1.88$, for which the instability threshold corresponded to the crossing of a single, real eigenvalue. The corresponding eigenvector indicates that the drops initially move in an irregular set of directions causing the symmetric ring to break-up, which we classify as an irregular rearrangement (). The nonlinear analysis required to predict the shape of the resulting aggregate is beyond the scope of our study.

Figure 11.

The spectrum of the thirty eigenvalues of the matrix $\boldsymbol {A}$ in (3.12) at the vibrational acceleration $\gamma = 1.26 \gamma _W$, just beyond the instability threshold $\gamma ^*$, for the ring $N=5$, $r_0 / \lambda _F = 0.440$, $\zeta = 1$. Black, red and blue markers indicate a single eigenvalue, two identical eigenvalues and three identical eigenvalues, respectively. The arrows indicate that, in this case, a complex-conjugate pair of eigenvalues crosses the real axis at $\gamma ^*$. Inspection of the corresponding eigenvectors indicates that this ring goes unstable to in-phase radial oscillations.

Our theory is found to accurately predict the stable ring radii, $r_0$, observed experimentally. It also rationalizes why we were unable to construct rings experimentally in certain regions of the $(N,r_0)$ plane. For instance, in the case of $\zeta = -1$ rings, the theory predicts that the $N=4$, $r_0/\lambda _F = 1.45$ ring is unstable, explaining the gap apparent in the experimental data. Similarly, the theory predicts that for $\zeta = -1$, only one $N=10$ ring is stable and that no $N=12$ rings are stable, as found experimentally. In other cases, the theory predicts the existence of stable rings that were not found experimentally. For example, in the case of $\zeta = 1$ rings, we see that the theory predicts several stable rings with $N \ge 10$ that we did not find in the laboratory. This discrepancy is presumably due to the manner in which we created the rings: we either pushed droplets together one after another, or constructed a linear chain of droplets and then wrapped the chain around a partially submerged rod. Both techniques preferentially select certain radii based on the initial inter-drop separation distances. If the drops could be simultaneously released onto the bath in a ring formation with the desired radius, these theoretically stable states might be attainable in the laboratory.

Figure 10(b) reveals that our theoretical predictions slightly underpredict the stable ring radii, $r_0$, with the discrepancy between theory and experiment being larger for smaller rings, but no greater than 15 %. We expect that this discrepancy may arise from our stroboscopic wave model failing to capture the influence of the radially propagating wave fronts generated at droplet impact (Eddi et al. Reference Eddi, Sultan, Moukhtar, Fort, Rossi and Couder2011b; Damiano et al. Reference Damiano, Brun, Harris, Galeano-Rios and Bush2016), that are captured by more detailed wave models (Milewski et al. Reference Milewski, Galeano-Rios, Nachbin and Bush2015; Durey & Milewski Reference Durey and Milewski2017; Galeano-Rios et al. Reference Galeano-Rios, Milewski and Vanden-Broeck2017; Galeano-Rios, Milewski & Vanden-Broeck Reference Galeano-Rios, Milewski and Vanden-Broeck2019). As shown by Galeano-Rios et al. (Reference Galeano-Rios, Couchman, Caldairou and Bush2018), these fronts influence droplet–droplet interactions when the drops are both in contact with the bath for a significant fraction of the Faraday period, $T_F$, and are in close proximity. For the ratcheting pairs considered by Galeano-Rios et al. (Reference Galeano-Rios, Couchman, Caldairou and Bush2018), the drops were in low energy $(1,1)$ and $(2,2)$ bouncing modes, with contact times $\tau _C \approx T_F/2$, and the fronts had the most notable effect on pairs with the smallest inter-drop spacing, approximately $\lambda _F/2$. In our experiments, the drops are in a high energy $(2,1)^2$ bouncing mode, with contact time $\tau _C \lesssim T_F/4$, and so the effect of the travelling fronts are expected to be relatively minor. Nevertheless, the discrepancy between theory and experiment being maximal for rings with radii $r_0 < \lambda _F$ suggests that travelling fronts may be influencing the dynamics of the most closely packed rings.

Our comparison of the theoretical predictions of a variable- and constant-impact-phase model in figure 10 highlights that variations in the vertical dynamics significantly influence the stability of droplet rings. In particular, our variable-impact-phase model predicts that rings remain stable above the instability threshold of a single drop, $\gamma _W$, in agreement with our experimental observations. Conversely, the constant-impact-phase model, in which the impact phase is assumed to be the same for all rings, predicts that the rings destabilize below $\gamma _W$. The impact phase influences our stability analysis through the quantities $\mathcal {S}_0$ and $\mathcal {C}_0$ that appear in (3.15), the values of the phase parameters in the base state of each ring. As shown in figure 12, $\mathcal {S}_0$ and $\mathcal {C}_0$ change significantly with the drop number $N$ and ring radius $r_0$. By comparing figures 10(b) and 12, it is evident that the stabilization of the rings above $\gamma _W$ is generally due to a decrease in the product $\mathcal {S}_0 \mathcal {C}_0$ of the base state phase parameters. Note that in the trajectory equation (3.1), $\mathcal {C}$ alters the horizontal waveforce exerted on each drop; in the wavefield equation (3.2), $\mathcal {S}$ alters the amplitude of the wave generated at each impact. Therefore, a decrease in the product $\mathcal {S}_0 \mathcal {C}_0$ results in a decrease in the local wave gradient and so the lateral force experienced by each droplet at impact, thereby enhancing the ring's horizontal stability.

Figure 12.

The base state values of the phase parameters $\mathcal {S}_0 \mathcal {C}_0$ for each theoretically predicted stable ring in figure 10(b). Plots (a and b) correspond, respectively, to drops bouncing in-phase ($\zeta =1$) and out-of-phase ($\zeta =-1$) relative to their nearest neighbours. The product $\mathcal {S}_0 \mathcal {C}_0$ influences the stability of each ring through the quantity $\varphi$ in the linear system (3.12). In particular, rings with relatively small $\mathcal {S}_0 \mathcal {C}_0$ are generally found to have a higher instability threshold $\gamma ^*$ in figure 10(b).

In figure 13(a), we replot the data in figure 10 as a function of the instability threshold and the depth of the wavefield well beneath each drop that is produced by its neighbours, $H$ (3.11). As reflected by both the experimental data (green circles) and the predictions of our variable-impact-phase model (red crosses), we see that rings are more stable when droplets bounce in a deeper minimum of the wavefield produced by their neighbours. Figure 13(b) highlights that this stabilization is due to the product $\mathcal {S}_0 \mathcal {C}_0$ decreasing with decreasing $H$. The constant-impact-phase model (blue crosses), in which the phase parameters are fixed at $\mathcal {S}_0 \mathcal {C}_0 = 0.35$, is unable to capture this behaviour. We note further that in figure 13(a), the data points characterized by $H$ tending to zero and $\gamma ^* / \gamma _W$ tending to one correspond to loosely bound rings with a large $r_0$, as each drop is not being strongly influenced by its neighbours’ wavefields.

Figure 13.

(a) The instability threshold of each ring plotted as a function of the depth of the wavefield well beneath each drop, $\bar {H}$ (3.11), as produced by its neighbours. Green circles correspond to the experimentally observed rings in figure 10(a), red crosses to the predictions of the variable-impact-phase model in figure 10(b), and blue crosses to the predictions of the constant-impact-phase model in figure 10(c), in which $\mathcal {S}_0 \mathcal {C}_0 = 0.35$ is fixed. (b) The values of $\mathcal {S}_0 \mathcal {C}_0$ corresponding to each of the red crosses in (a), that influence the stability of each ring through the variable $\varphi$ (3.15) appearing in the linear system (3.12). The decrease in $\mathcal {S}_0 \mathcal {C}_0$ with decreasing $H$ is responsible for the increased stability of rings in geometries with smaller $H$. We note that in (a) there are more theoretical than experimental data points, as there were several theoretically predicted stable rings in figure 10 that were not considered in the laboratory.

Finally, we note that as observed in our experiments, the most common type of instability predicted was radial in-phase oscillations. The theory also predicted the majority of other types of instabilities observed experimentally, including out-of-phase radial oscillations, out-of-phase azimuthal oscillations and complex motions consisting of coupled radial and azimuthal motion. However, the instabilities predicted by the theory often occurred in slightly different regions of the $(N,r_0)$ plane than observed in the laboratory. There are several factors that could account for this discrepancy. First, in the experimental data shown in figure 10(a), we note that for closely bound rings, the drops were in a $(4,2)$ bouncing mode, instead of the $(2,1)$ mode assumed in our theoretical model. Second, as previously discussed, our stroboscopic model neglects the influence of transient fronts, that may become important when modelling the dynamics of interacting droplets in close proximity (Galeano-Rios et al. Reference Galeano-Rios, Couchman, Caldairou and Bush2018). Third, the type of instability predicted by our stability analysis was found to be extremely sensitive to the drop size. For example, a small change in drop size from $R=0.36$ mm to $R=0.34$ mm changes the type of instability predicted for a variety of rings. For instance, orbital motion, which was not predicted theoretically with $R=0.36$ mm drops, emerges for in-phase $(N, r_0/\lambda _F)= (2, 0.30)$ and $(3, 0.35)$ rings composed of $R=0.34$ mm drops.

3.3. The nonlinear evolution of unstable rings

In addition to characterizing the linear stability of each ring, we performed numerical simulations using the fully nonlinear equations of motion (3.1) to investigate the behaviour of the unstable rings denoted by crosses in figure 10. At a fixed vibrational acceleration $\gamma /\gamma _F = 0.7$, we initialized the drops in each of the unstable ring configurations shown in figure 10, imposed small velocity perturbations to each drop, and simulated the subsequent system evolution. An example of the evolution of the unstable ring $N=16$, $r_0/\lambda _F = 1.39$, $\zeta = 1$ is shown in figure 14. We observe that the drops transition through two intermediate structures (see figure 14b,c) before finally settling into the stationary configuration shown in figure 14(d). In figure 14(e), at each instant we plot both the local wave amplitude beneath each drop, $h_m$, along with the local wave amplitude averaged over all drops, $\langle h_{m}\rangle =(1/N)\sum _{m=0}^{N-1}h_{m}$. Figure 14(e) makes clear that each re-arrangement of the drops leads to a decrease in this averaged local wave amplitude. This behaviour was generic: all of the unstable rings rearranged themselves so as to decrease $\langle h_{m}\rangle$ relative to the initial ring, suggesting that the underlying wavefield acts as a potential landscape that the system tries to minimize through rearrangement. A complete energetic description of the bouncing droplet system, and the utility of such a description in predicting the stability of various single- and multiple-droplet static and dynamic states, is left for future consideration.

Figure 14.

Time evolution of the unstable ring $N=16$, $r_0/\lambda _F = 1.39$, $\zeta = 1$ at the constant vibrational acceleration $\gamma /\gamma _F = 0.7$, as predicted by numerically simulating the trajectory equation (3.1). The drops are initialized in the ring shown in (a), evolve through the two transitional states shown in (b and c), before settling into the stationary configuration shown in (d). In (e), the evolution of the local wave amplitudes beneath each drop, $h_m$ (computed using (3.2)), is plotted in red, their mean $\langle h_{m}\rangle$ in black. The zoom box in (e) highlights that $\langle h_{m}\rangle$ decreases as the drops transition from state (c) to (d).

Guided by our theoretical findings, we returned to the laboratory in order to perform additional experiments to characterize the nonlinear behaviour of three of the theoretically unstable $\zeta = 1$ rings indicated in figure 10(b): $(N, r_0/\lambda _F)= (6, 1.55)$, $(8, 1.95)$ and $(9, 2.18)$. As described in § 2, we partially submerged a rod in the centre of each ring to stabilize the drops as the ring was constructed. Upon removal of the rod, the drops spontaneously rearrange into a stable structure. In figure 15, we find a good agreement between the experimentally observed and theoretically predicted final stable drop configurations. In each case, once again, the drops evolve so as to reduce $\langle h_{m}\rangle$.

Figure 15.

A comparison between the theoretically predicted and experimentally observed stable droplet configurations that emerge after initializing drops in the $\zeta = 1$ unstable rings $(N, r_0/\lambda _F)=$ (a) $(6, 1.55)$, (b) $(8, 1.95)$ and (c) $(9, 2.18)$. The first row shows the initially unstable ring in which the drops are initialized and the second row shows the final configuration, as predicted by numerically simulating the trajectory equation (3.1). For the theoretical data, the underlying wavefield is shown along with the non-dimensional local wave amplitude averaged over all of the drops, $\langle \bar {h}_{m}\rangle$. The third row shows a top view of the experimentally observed stable configurations into which each unstable ring evolved.