4.1. Transition from harmonic to subharmonic pumping for $f \approx 90\ {\rm Hz}$

Figure 2(c) shows $\langle U \rangle$ as a function of $\varGamma$ for $f=90$ and 95 Hz. Above a threshold normalized acceleration $\varGamma _{sub} \simeq 5$ and $6$, respectively, we observe that the droplet transitions from a regime of harmonic pumping to a regime of subharmonic pumping, where the droplet responds at half the forcing frequency $f$. We also observed the same behaviour for $f=85$ Hz, not shown for clarity since the corresponding data are very close to those with $f=90$ Hz. This transition from harmonic to subharmonic response corresponds to a sharp increase of the sliding speed $\langle U \rangle$ for $f=85$ and 90 Hz. There is also an increase, albeit more moderate, at $f=95$ Hz.

Figure 4 shows the difference in the shapes and dynamics of two representative experiments performed at $f=90$ Hz and near $\varGamma _{sub}$, with a harmonic response for $\varGamma =4.7<\varGamma _{sub}$ and a subharmonic response for $\varGamma =5.5>\varGamma _{sub}$. Despite a relatively small change in the forcing amplitude, we observe a doubling of the speed of the droplet's centre of mass $\langle U \rangle$. Interestingly, one of the most obvious differences between the two droplets is regarding their basal diameter $d$. It shows little variation in the harmonic regime, evolving from 1.4 to 1.8 mm, compared with the subharmonic region, when it goes down to 1 and up to 2 mm. Its averaged value $\langle d \rangle$ is also smaller in the subharmonic case, and in fact figure 3(a) shows that the correlation between $\langle d \rangle$ and $\langle U \rangle$ previously discussed still holds. In Appendix A we show that this subharmonic response appears and also causes a jump in speed for thicker fibres or smaller tilt angles.

Figure 4.

Illustration of the harmonic and subharmonic droplet behaviours observed at $f=90$ Hz with $V=4\ \mathrm {\mu }$l, $b=200\ \mathrm {\mu }$m, $\alpha =27.5^\circ$ and (a,b) $A=0.14$ mm, $\varGamma =4.7$ (c,d) $A=0.17$ mm, $\varGamma =5.5$. (a,c) Snapshots showing that the shape of the droplet is periodic with period $T$ and $2T$, respectively. The dotted line is fixed in the laboratory frame and represents the maximum position of the fibre, which highlights its oscillations. (b,d) Corresponding evolution of the fibre's position (solid line) and basal diameter (dashed line). The mean value of the droplet's speed is (a,b) $\langle U \rangle =31\ {\rm mm}\ {\rm s}^{-1}$ and (c,d) $\langle U \rangle =58\ {\rm mm}\ {\rm s}^{-1}$. See supplementary movies 1 and 2 available at https://doi.org/10.1017/jfm.2023.462.

It is interesting to put these observations into perspective with prior work on droplets moving on a flat substrate. First, the existence of a subharmonic behaviour above a forcing threshold is reminiscent of the parametric Faraday instability (Miles & Henderson Reference Miles and Henderson1990) and is the result of a competition between inertia and capillarity. Subharmonic droplet deformations are commonly observed on forced sessile droplets on flat substrates (Chang et al. Reference Chang, Bostwick, Daniel and Steen2015; Chang, Daniel & Steen Reference Chang, Daniel and Steen2017). Maksymov & Pototsky (Reference Maksymov and Pototsky2019) showed, using an inertial thin film model, that, for small forcing accelerations, droplets respond harmonically, and that subharmonic Faraday waves only occur above a threshold acceleration. Costalonga & Brunet (Reference Costalonga and Brunet2020) observed experimentally the possibility of subharmonic response in their set-up of droplets on a horizontal substrate submitted to slanted vibrations. However, they report a transition from harmonic response for sliding droplets ($\langle U \rangle >0$) to subharmonic response for climbing droplets ($\langle U \rangle <0$), while we observe an acceleration of the descending speed in the subharmonic regime. We also note that they obtained a subharmonic regime for $f\simeq 1.5f_{pump}$; this 1.5 factor also matches our experiments ($1.5 f_{pump} \simeq 85$ Hz). Second, using droplets on tilted liquid infused substrates submitted to vertical vibrations, Sartori et al. (Reference Sartori, Guglielmin, Ferraro, Filippi, Zaltron, Pierno and Mistura2019) observed experimentally above a threshold acceleration a regime that they refer to as fast descending, where droplets slide much faster. This regime is associated with a basal diameter showing much more important variations than in the regular descending regime, similarly to our experiments. Finally, through numerical simulations, Ding et al. (Reference Ding, Zhu, Gao and Lu2018) reproduced the experiments of Brunet et al. (Reference Brunet, Eggers and Deegan2007) of droplets on tilted substrates with vertical vibrations. Their results suggest the strong importance of a non-sinusoidal evolution of the wetted area $S\approx {\rm \pi}b d$. We also see in figure 4(b,d) that $d$ switches from near-sinusoidal in the harmonic case to completely non-sinusoidal when the response is subharmonic.

4.2. Transition from pumping to rocking for $f \geq 120\ {\rm Hz}$

Most droplets we have observed only exhibit a pumping mode, shown already in figures 1(b,c) and 4. For $f=120$ and 135 Hz and for high enough amplitude of vibrations, we observe a transition where the droplet can exhibit a combination of pumping and rocking modes. This is illustrated in figure 5. The existence of this rocking mode is particularly evident when considering the instantaneous speed ${\rm d}\kern0.06em x_{drop}/{\rm d}t$, figure 5(e). In the pumping-only mode, the droplet exhibits a near-constant velocity, showing variations of $\approx$20 % around the mean value $\langle U \rangle$: this is because pumping vibrations are mostly normal to the fibre. When the rocking mode appears, lateral vibrations become significant and the speed of the centre of mass of the droplet shows significant variations around the mean.

Figure 5.

Droplets with $f=120$ Hz, $V=4\ \mathrm {\mu }$l, $b=200\ \mathrm {\mu }$m, $\alpha =27.5^\circ$ and (a,b) $A=0.23$ mm, $\varGamma =13.4$ (c,d) $A=0.31$ mm, $\varGamma =18.0$. The difference in the shape of the droplet over one period is illustrated in (a,b). The dotted line represents the maximum value of $y_{fibre}$ and highlights the fibre's vibrations. Panels (c) and (d) show the time evolution of the basal diameter for the droplets in (a) and (b), respectively. (e) Instantaneous droplet speed. Averaged over one period, the mean value is (a) $\langle U \rangle =38\ {\rm mm}\ {\rm s}^{-1}$ and (b) $\langle U \rangle =83\ {\rm mm}\ {\rm s}^{-1}$. See supplementary movies 3 and 4 available in the supplementary material.

As shown in figure 2(d), this rocking mode significantly increases the sliding speed $\langle U \rangle$. This happens despite the fact that the averaged basal diameter $\langle d \rangle$ is larger in the presence of the rocking mode (figure 5c,d), and the correlation of figure 3 does not hold anymore. In this regime the combined effects of rocking and pumping cannot be captured solely by the change in wetted area, similarly to what is observed on droplets on horizontal flat surfaces submitted to slanted vibrations (see Costalonga & Brunet (Reference Costalonga and Brunet2020) and the discussion in § 1).

4.3. Transition between pumping and swinging for $30\ {\rm Hz} \lesssim f \lesssim 45\ {\rm Hz}$

Figure 2(e) shows the speed of droplets as a function of the amplitude of oscillations for $f=30,40$ and 45 Hz. For these three frequencies the droplet can respond by swinging across the fibre similarly to a pendulum, as illustrated in figure 6. This swinging motion is subharmonic at half the forcing frequency.

Figure 6.

Droplets at $f=30$ Hz $V=4\ \mathrm {\mu }$l, $b=200\ \mathrm {\mu }$m, $\alpha =27.5^\circ$ and $A=0.26$ mm, $\varGamma =0.96$. (a,b) Vibrating-only regime where the droplet slides at $\langle U \rangle =18\ {\rm mm}\ {\rm s}^{-1}$, (c,d) Subharmonic swinging regime where the droplet slides at $\langle U \rangle =31\ {\rm mm}\ {\rm s}^{-1}$. (a,c) View from the side and (b,d) from the top, looking down at the droplet. (e) Time evolution of the basal diameter $d$ extracted from the movies corresponding to (a,b) and (c,d). In the latter case, when the droplet swings, we did not extract $d$ with automatic image processing but did manual measurements combining both the side and top views as needed. We could not extract accurate data for $0< t<0.5T$ and $T< t<1.5T$. See supplementary movies 5–8 available in the supplementary material.

At $f=30$ Hz, the swinging motion is only observed for high enough amplitude of oscillations and when droplets are significantly perturbed or after being deposited on the fibre (e.g. due to the detachment from the micropipette, or by flicking the oscillating structure). When they are gently deposited on a still fibre with a slowly increasing amplitude of oscillations, only the harmonic pumping response is observed. However, once they enter the swinging mode they do not return to harmonic pumping. Henceforth, for $f=30\ {\rm Hz}$, the harmonic pumping response is unstable to finite perturbations. Figure 2(e) shows that the transition to the swinging mode significantly increases the sliding speed.

Figure 6(e) shows that, in the swinging mode, the basal diameter $d$ and hence the wetting area $S$ are significantly smaller than in the pumping case. This can be understood at least partly by considering the centrifugal acceleration induced by the swinging motion. The droplet has an equivalent spherical radius $r=(3V/4{\rm \pi} )^{1/3} \simeq 1$ mm and swings at $f/2$ so that its angular velocity can be approximated, on average, as $\omega /2$. The resulting centrifugal acceleration is $r\omega /2 \approx 9\ {\rm m}\ {\rm s}^{-2}$, which is comparable to both the gravitational acceleration and to the acceleration induced by the fibre motion (here $\varGamma \simeq 1$ and hence $A\omega ^2 \simeq g=9.8\ {\rm m}\ {\rm s}^{-2}$). This supports the idea that the centrifugal acceleration due to swinging pushes the droplet away from the fibre, diminishing its wetting area. However, the data would not collapse on figure 3(a) showing the correlation between $\langle d \rangle$ and $\langle U \rangle$; this is not surprising and shows that energy is dissipated differently as the flow inside the droplet is very different when swinging as compared with when it is pumping.

When increasing the frequency to $f=40$ Hz, harmonic pumping becomes more and more unstable and eventually cannot be reached anymore. Here, we only observe subharmonic swinging even at low amplitude of oscillations. Increasing again the frequency to $f=45$ Hz, we also only observed subharmonic swinging at low amplitude. However, upon increasing the amplitude the droplet switches to a subharmonic pumping mode. At this frequency ($\,f=45$ Hz) it is now the swinging mode that becomes unstable at high amplitude; this transition also increases drastically the sliding speed (figure 2e). Data with a different fibre diameter and different tilt angles shown in Appendix A also exhibit this behaviour.

While the two previously discussed regimes of rocking and subharmonic pumping are also observed on flat vibrating substrates, swinging droplets can only occur on fibres. Next, we aim to rationalize the existence of this swinging mode.

5.1. Experiments

In order to obtain a clearer picture of the pendulum-like swinging droplet motion illustrated in figure 6, we focus on droplets on horizontal fibres ($\alpha =0^\circ$) to decouple the droplet's sliding motion from its response to oscillations. We construct a regime map from experiments, where we observe three different behaviours: harmonic vibrations, subharmonic vibrations and swinging. This is shown in figure 7, confirming the observations in § 3 on tilted fibres, namely, the existence of a subharmonic swinging regime for $f$ ranging from approximately 30 and up to 45 Hz, and a subharmonic vibration starting near $80$ Hz. This regime map is obtained as follow: at a given frequency, the amplitude of oscillations is increased from 0 and up to a change of regime of vibrations or detachment of the droplet from the fibre. Care was taken to ensure that the rate of the amplitude sweep, ${\rm d}A/{\rm d}t$, was small enough to not influence the measured threshold represented as symbols in figure 7. For each threshold, this procedure was repeated five times to obtain standard deviations shown by the error bars. For thresholds corresponding to transitions between two regimes of vibrations, the same procedure was then repeated starting from a high amplitude of oscillations and slowly lowering it. The two procedures, increasing or lowering the amplitude of oscillations, give slightly different thresholds; this explains the overlap between regions of harmonic and subharmonic pumping for $f\geq 80$ Hz. Figure 7 also shows a region where both a harmonic vibrating droplets and subharmonic swinging droplets can coexist. In the latter regime, subharmonic swinging only takes place if a finite perturbation is introduced in the system (e.g. blowing gently on the droplet, depositing the droplet on an already vibrating fibre or flicking the vibrating structure), or when decreasing the amplitude of vibrations from the region where only swinging motion occurs. Once a droplet is in the subharmonic swinging mode it enters a stable state, and we never observed a droplet switching from a swinging to a non-swinging motion. For $f=25$ and $30$ Hz, the structure was flicked manually to see if swinging could take place. At these two frequencies the region of swinging motion is bounded from above below the ejection threshold: this is not because the droplet switches to harmonic vibrations upon increasing the amplitude, but because it detaches from the fibre when the swinging mode is excited at higher amplitudes.

Figure 7.

Regime map of the three different droplet behaviours ($V=4\ \mathrm {\mu }$l) on a horizontal fibre ($\alpha =0^\circ$) of thickness $b=200\ \mathrm {\mu }$m. The region between the horizontal line $\varGamma =0$ and the circles, in blue, represents the regime of harmonic vibrations. The region between triangles, in green, represents the regime of subharmonic vibrations at $f/2$. The region between squares, in red, represents the regime of subharmonic swinging at $f/2$. The grey hatched region between 45 and 47 Hz corresponds to a narrow range of frequencies where it is challenging to obtain reproducible results. The top axis represents the dimensionless frequency $\omega /\omega _p$, with $f_p=\omega _p/2{\rm \pi}$ the pendulum frequency discussed in the text. We have used $f_p=15$ Hz.

5.2. Analogy with a forced pendulum

The swinging motion of the droplet is reminiscent of a pendulum. We illustrate this analogy in figure 8(a,b) between a droplet hanging below an oscillating fibre and a pendulum with constant length $L_{eq}$, mass $m$, in the gravitational field $g$ and submitted to vertical oscillations of its support as $A\cos (\omega t)$. By making this analogy we assume the droplet to be a solid with a motion described by a single degree of freedom, the angle $\phi$ with respect to the vertical direction, and ignore any interface deformations. This system obeys the following damped Mathieu equation (Kovacic, Rand & Sah Reference Kovacic, Rand and Sah2018), which is the classical pendulum equation written in the frame of the oscillating support:

(5.1)\begin{equation} \phi''(t) + 2{c_p}{\omega_p}\phi'(t)+\omega_p^2(1-\varGamma\cos(\omega t))\phi(t)=0,\quad \omega_p=(g/L_{eq})^{1/2}, \end{equation}

where $({\cdot })'={\rm d}({\cdot } )/{\rm d}t$ denotes the derivative with respect to time, and $\omega _p$ is the natural frequency of pendulum oscillations. We have assumed small angles with $\vert \phi (t) \vert \ll 1$, and to account for dissipation we have included a linear damping term characterized by the dimensionless friction coefficient $c_p$. Equation (5.1) is an archetype for parametric instabilities with connections to a wide range of physical phenomena and has been studied extensively (Kovacic et al. Reference Kovacic, Rand and Sah2018). The equilibrium position $\phi =0$ is unstable for large enough values of the forcing $\varGamma$ at driving frequencies $\omega \approx 2\omega _p/n$, with $n=1,2,3,\ldots$ a positive integer. The most unstable mode corresponds to $n=1$, $\omega \simeq 2\omega _0$, where the pendulum then swings subharmonically at $\omega /2$. A stability diagram of (5.1) is shown in figure 8(d) where the greyed regions represent the so-called Arnold tongues of instability, where the position $\phi =0$ is linearly unstable and the oscillations of the substrate make the pendulum swings. The Arnold tongues would go down to $\varGamma =0$ in the undamped case ($c_p=0$), the main effect of damping is to shift the tongue of instabilities to higher values of $\varGamma$.

Figure 8.

Analogy between (a) a droplet swinging of a fibre – visualized in a plane perpendicular to the cross-section of the fibre – and (b) a simple pendulum or (c) an elastic pendulum with spring constant $k$, with a point mass $m$ and an equilibrium length $L_{eq}$. The system is placed in the gravitational field $g$, the support oscillates as $A\cos \omega t$ and $\phi$ denotes the angle with respect to the vertical. (d) Linear stability diagrams of (5.2) for a simple pendulum and an elastic pendulum for different values of $\varOmega =\omega _p/\omega _s$, the ratio of the natural frequency of the pendulum motion $\omega _p=(g/L_{eq})^{1/2}$ to that of the spring motion $\omega _s=(k/m)^{1/2}$. The damping coefficients are $c_s=c_p=0.15$, and $\varGamma =A\omega ^2/g$ is the dimensionless forcing acceleration.

We assume that a droplet sagging on a fibre can be mimicked as a simple pendulum and can then be modelled by (5.1), with $L_{eq}=I/a$, $\omega _p=(mga/I)^{1/2}$ for a physical pendulum with a homogeneous distribution of mass. Here, $a$ is the distance from the axis of rotation to the centre of mass of the droplet and $I$ the moment of inertia of the droplet about the axis of rotation. We estimate $a$ and $I$ from experimental images of a droplet on a static fibre ($\varGamma =0$) and find $f_p = \omega _p/2{\rm \pi} \approx 15$ Hz for a $4\ \mathrm {\mu }$l water droplet on a $200\ \mathrm {\mu }$m horizontal nylon fibre. From the discussion above we would then expect to observe a swinging motion centred around $f\approx 2f_p=30$ Hz. Experiments show that the range of unstable frequencies is $30\ {\rm Hz} \lesssim f\lesssim 45\ {\rm Hz}$.

One explanation to this discrepancy could be an inaccurate estimation of the natural frequency $f_p$. Another experimental procedure to measure it, distinct from the geometrical method described above, is to induce the swinging motion of the droplet on a still fibre and directly observe the period of oscillations. When perturbing the droplet using a mechanical impulse or blowing on it on a still fibre we, however, did not manage to induce significant oscillations without detaching the droplet. Instead, we oscillated the fibre for the droplet to naturally swing, and suddenly stopped the oscillations. The measured oscillation frequency is in this case $21$ Hz, which is different than the previous estimate $f_p=15$ Hz. This would predict a swinging motion with a tongue of instability centred around $21$ Hz, which might be considered to be in better agreement with our experimental results but still does not capture some of the experimental features, and in particular the fact that swinging cannot be observed above 45 Hz. With this method, however, the droplet is both swinging and vibrating in a pumping mode, and we cannot exclude an effect of the vibrations on the swinging itself. This is in fact always true and hints at the fact that the droplet pumping mode of response should be considered to explain the swinging region in the phase diagram shown in figure 7.

5.3. Analogy with a forced elastic pendulum

To account for the vibrations of the droplet itself, in addition to the possibility of swinging, we add in the analogy discussed above a linear spring (figure 8c). This spring allows for a modulation of the pendulum length, analogous to the pumping mode of vibration of the droplet that stretches and flattens it, and its spring constant $k$ plays the role of the surface tension coefficient. This is a first-order model for the effect of surface tension and cannot capture all the features of droplet vibrations, but we use this minimal toy model to gain insights into the coupling between the swinging and pumping modes of vibrations. In Appendix B we show that, for small angles $\phi$, the dynamics of an elastic pendulum is governed by the following:

(5.2a)\begin{gather} \ell(t)=\frac{-L_{eq} \varOmega^2}{(1-(\omega/\omega_s)^2)^2 +4c_s^2(\omega/\omega_s)^2}[\varGamma (1-(\omega/\omega_s)^2)\cos(\omega t)+ 2c_s (\omega/\omega_s)\sin(\omega t) ], \end{gather}

(5.2b)\begin{gather}{\phi}''(t) + 2 \left(c_p\omega_p +\frac{\ell'(t)}{L_{eq}+\ell(t)}\right) {\phi}'(t) + \omega_p^2 \frac{1-\varGamma\cos(\omega t)}{1+\ell(t)/L_{eq}}\phi(t) = 0, \end{gather}

with a one-way coupling between the vibrations characterized by the extension $\ell (t)$ and the swinging characterized by the angle $\phi (t)$ (see figure 8c). Here, $\omega _s=(k/m)^{1/2}$ is the natural frequency of the spring and $c_s$ is a damping coefficient introduced to model viscous dissipation. We define $\varOmega =\omega _p/\omega _s$, the ratio between the natural pendulum frequency of the droplet to its natural frequency of vibrations. For a droplet, we expect $\omega _s \sim (\sigma /\rho r^3)^{1/2}$ and $\omega _p\sim (g/r)^{1/2}$ with $r\sim V^{1/3}$ a characteristic size of the droplet, so that $\varOmega \sim Bo ^{1/2}$ with $Bo =\rho g r^2/\sigma$ the Bond number that compares gravitational and capillary effects.

When $\ell (t)/L_{eq}$ remains small, (5.2) simplifies to the damped Mathieu equation (5.1) and the analysis discussed previously in § 5.2 applies. However, when $\ell (t)/L_{eq}$ can vary significantly, the coefficients in (5.2) remain periodic but depend on the evolution of $\ell (t)$. This regime of significant variation of the pendulum length is expected to be relevant close to the resonance frequency of the spring. We can then anticipate a possible effect on the instability diagram when the main unstable frequency of the pendulum $2\omega _p$ is close to the resonance frequency of the spring $\omega _s$, i.e. for $\varOmega \approx 1/2$.

We have discussed experimental measurements of $\omega _p$ in § 5.2, yielding $\omega _p \approx 15\unicode{x2013}22$ Hz. In § 2 we discussed the natural frequency of the first pumping mode of the droplet: $f_{pump} \simeq 57$ Hz. Taking $\omega _s \approx \omega _{pump}$, this gives a ratio $\varOmega \approx 0.26\unicode{x2013}0.37$. We fix the values of $c_p=c_s=0.15$ arbitrarily and compute numerically the linear stability diagrams of the equilibrium position $\phi =0$ in the $(\omega /\omega _p-\varGamma )$ plane for chosen values of $\varOmega$. We explain in Appendix B how we have obtained these stability diagrams, and the results are shown in figure 8 for $\varOmega$ ranging from 0.2 to 0.5. As expected, for $\varOmega =0.2$ there is little influence of the spring on the stability of the swinging motion. As $\varOmega$ gets closer to 0.5, however, the modulation of the pendulum length due to the spring makes the position $\phi =0$ more easily unstable and lowers the main tongue of instability down to $\varGamma =0$, despite the presence of damping. Interestingly, the high-frequency limit of the instability band near $\omega =2\omega _p$ becomes very steep: this feature is also observed in our experimental phase diagram shown in figure 7 where the swinging motion is observed for all amplitudes at $f=45$ Hz, but is never reached at $47$ Hz and above. This band is also not necessarily centred near $\omega = 2\omega _p$ but is shifted to higher values of $\omega$ for $\varOmega <0.5$; this is consistent with the apparent underestimation of $\omega _p$ from geometric considerations.

The damping coefficients we have chosen are arbitrary. We let $c_s=c_p=c$ and are limited to large values such that $c_s\gtrsim \varOmega ^2/2 \approx 0.15$, otherwise (5.2) would yield a non-physical negative pendulum length. We show (see Appendix B, figure 13) that there are no qualitative differences in the instability diagrams when varying $c$ in the range $0.15 \leq c \leq 0.3$, and in particular all the features discussed above still hold. A simple way to measure experimentally the coefficient $c_s$ is to submit a droplet to a step acceleration and to fit the obtained data to the exponentially decaying motion expected for a harmonic oscillator. We estimate $c_s \approx 0.01$, which is too small to model with (5.2). A more realistic droplet model would involve a nonlinear spring showing stiffening as it elongates and allowing for smaller damping coefficients; a Duffing oscillator indeed models well a forced droplet (Deepu, Chowdhuri & Basu Reference Deepu, Chowdhuri and Basu2014). However, doing so requires additional parameters and the added nonlinearity would greatly complicate the analysis of the system.