Ratcheting droplet pairsC. A. Galeano-Rios, M. M. P. Couchman, P. Caldairou, and J. W. M. Bush

Citation: Chaos 28, 096112 (2018); doi: 10.1063/1.5032116View online: https://doi.org/10.1063/1.5032116View Table of Contents: http://aip.scitation.org/toc/cha/28/9Published by the American Institute of Physics

CHAOS 28, 096112 (2018)

Ratcheting droplet pairsC. A. Galeano-Rios,1,a) M. M. P. Couchman,2,b) P. Caldairou,2 and J. W. M. Bush2,c)

1Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 2 April 2018; accepted 31 July 2018; published online 20 September 2018)

Millimetric droplets may be levitated on the surface of a vibrating fluid bath. Eddi et al. [Europhys.Lett. 82, 44001 (2008)] demonstrated that when a pair of levitating drops of unequal size are placednearby, they interact through their common wavefield in such a way as to self-propel through a ratch-eting mechanism. We present the results of an integrated experimental and theoretical investigationof such ratcheting pairs. Particular attention is given to characterizing the dependence of the ratchet-ing behavior on the droplet sizes and vibrational acceleration. Our experiments demonstrate that thequantized inter-drop distances of a ratcheting pair depend on the vibrational acceleration, and thatas this acceleration is increased progressively, the direction of the ratcheting motion may reverse upto four times. Our simulations highlight the critical role of both the vertical bouncing dynamics ofthe individual drops and the traveling wave fronts generated during impact on the ratcheting motion,allowing us to rationalize the majority of our experimental findings. Published by AIP Publishing.https://doi.org/10.1063/1.5032116

We consider droplet pairs of unequal size bouncing onthe surface of a vibrating bath. As originally reported byEddi et al.,1 when the pair is sufficiently close, the asym-metry in the wavefield along their line of centers causesthem to propagate through a ratcheting mechanism. Wereport the results of a combined experimental and theoret-ical investigation of ratcheting pairs of bouncing droplets.Particular attention is given to rationalizing the rever-sals in directions observed as the vibrational forcing isincreased progressively. Our study highlights the short-comings of the stroboscopic models of bouncing droplets insituations where the vertical bouncing dynamics are vari-able and droplets interact through the propagating wavefronts generated during impact.

I. INTRODUCTION

Coalescence of a millimetric droplet with an underly-ing bath of the same fluid can be avoided by vibrating thebath vertically2 with acceleration γ sin ωt, provided the vibra-tional frequency, ω, is comparable to the natural oscillationfrequency of the drop, ωd ∼

√σ/ρR3, where σ , ρ, and R

denote, respectively, the surface tension, density, and radiusof the drop. As the droplet bounces on the free surface, itinteracts with waves triggered by its previous impacts. Fora drop of a given size, there is a critical vibrational acceler-ation γB, the bouncing threshold, below which it coalescesand above which it bounces. The bouncing mode dependson γ , and is characterized by the ordered pair (m, n): in the(m, n) mode, the drop bounces n times in m periods of thebath oscillation. For γ just above γB, the drop bounces inplace with the same frequency as the driving in the (1, 1)

mode. As γ is further increased, the bouncing mode changes

a)C.A.Galeano.Rios@bath.ac.ukb)couchman@mit.educ)bush@math.mit.edu

according to a sequence that depends on drop size, which mayinclude (2, 2), (4, 4), or (4, 3) modes.3–6 Eventually, once thedrop’s bouncing amplitude is sufficiently large, it bounces attwice the period of the driving, in the (2,1) mode. The dropthen achieves resonance with the most unstable wave modeof the bath, namely, the subharmonic Faraday waves excitedby its impact. Consequently, this period-doubling transitionis accompanied by a dramatic increase in the amplitude of thedroplet’s wavefield.2 Further increasing γ beyond the walkingthreshold, γW , may serve to destabilize the resonant bouncerinto a dynamic state, the so-called walker, in which the dropletself-propels by virtue of a resonant interaction with its ownwavefield.7 Increasing γ beyond the Faraday threshold, γF ,prompts the emergence of a standing field of subharmonicFaraday waves throughout the bath. The Faraday wavelength,λF , is calculated by Kumar8 for a viscous fluid, and for aweakly viscous fluid is well approximated by the standardwater-wave dispersion relation.

Eddi et al.1 demonstrated that bouncing droplet pairs ofunequal size may self-propel through a ratcheting motion.Notably, this ratcheting motion arises below the walkingthreshold of the individual droplets, and so is due exclusivelyto the wave-mediated droplet interaction. They demonstratedthat as γ is increased progressively, the direction of motionof the ratcheting pairs along their line of centers may reversetwice, and argued that these reversals in direction were dueto changes in the vertical bouncing modes of the individualdrops. Specifically, at the lowest memory at which ratchetingoccurs, they found that both drops are in the (1,1) mode, andthe large drop follows the smaller drop. As γ was increasedbeyond the threshold at which the small drop period-doublesinto a (2,2) mode, they found that the smaller drop then fol-lows the larger one. Finally, upon further increasing γ , so thatthe larger drop also enters the (2,2) mode, they found that thelarge drop once again follows the small, but at a separationdistance larger than that arising for (1,1) bouncers. By explor-ing a broader parameter regime, we demonstrate here that up

1054-1500/2018/28(9)/096112/11/$30.00 28, 096112-1 Published by AIP Publishing.

096112-2 Galeano-Rios et al. Chaos 28, 096112 (2018)

to four reversals of direction may arise for a given dropletpair as γ is increased. Furthermore, we show that the major-ity of these reversals in direction are not caused by changesin the bouncing mode, but rather by changes in the impactphase within a given mode, which can alter both the magni-tude and direction of the net wave force exerted on the pair bythe waves during impact.

The stroboscopic model of Oza et al.11 has proven to besufficient for predicting the stability of a variety of single-walker states, including the bouncing and walking states, aswell as circular orbits in a rotating frame4,11 and in the pres-ence of a simple harmonic potential.12,13 However, the stro-boscopic model cannot be expected to capture the behaviorof ratcheting pairs for several reasons. First, the stroboscopicmodel does not account for variations in a drop’s verticaldynamics and so would be unable to capture the range ofbouncing modes, including (1, 1), (2, 2), and (2, 1) modes,seen in our experiments. Second, the stroboscopic model isunable to capture modulations in the bouncing phase, aneffect shown to be important when modeling droplet-dropletinteractions in orbiting14 and promenading15 pairs. Third, thestroboscopic model assumes implicitly that a standing field ofFaraday waves is generated at each impact and that the impactbetween the drop and bath is instantaneous. Experimentalmeasurements by Damiano et al.16 have shown that a droplet’swavefield actually has radially propagating wavefronts. In theparameter regime examined in our experiments, the contacttime between the drop and the bath is sufficiently long that thehorizontal dynamics of the drops is significantly influenced bythe traveling wave fronts generated during impact.

To characterize the dynamics of ratcheting pairs theo-retically, we adopt the model of Milewski et al.,5 whichuses weakly viscous quasi-potential theory to more accuratelycapture the wavefield generated by a droplet. Notably, thewavefield predicted by this model was shown to be in goodagreement with the experimental wavefield measurements ofDamiano et al.16 Specifically, it captures the radially prop-agating wavefronts generated at each drop impact. Further,the model of Milewski et al.5 is able to capture the varietyof bouncing modes observed in the experiment, by explic-itly modeling the drop’s vertical dynamics and the drop’sinteraction with the bath surface during contact.

In Sec. II, we detail our experimental protocol and presentthe results of our experimental characterization of ratchet-ing pairs. In Sec. III, we briefly recap the model being usedin our accompanying theoretical study, highlight the mech-anism responsible for the ratcheting motion, and comparethe results of our experiments and simulations. The criti-cal insights provided by our simulations are discussed inSec. IV.

II. EXPERIMENTS

Experiments were performed using the set-up shownin Fig. 1. An electrodynamic shaker9 is used to verticallyvibrate a bath of silicon oil (density ρ = 949 kg m−3, sur-face tension σ = 20.6 × 10−3 N/m, and kinematic viscosityν = 20 cSt) with an acceleration of γ sin ωt. For this fluid,the sub-harmonic Faraday waves have a wavelength of λF ≈4.75 mm and the Faraday threshold is γF ≈ 4.3 g, where gdenotes gravitational acceleration. Droplets of a desired size,composed of the same fluid, are created using a piezoelectricdroplet generator.17 A ratcheting pair is formed by generatingtwo unequal drops, of radii R1 and R2, and by manually push-ing them together until they settle into a stable bound statewith an inter-drop distance d . As reported by Eddi et al.,18

there are a discrete set of stable separation distances. Fol-lowing the convention of Eddi et al.,18 we use the bindingnumbers n = 1, 1.5, 2, 2.5, . . . to denote the quantized stablebinding lengths. Integer and half-integer values of n denotethat the drops in a pair will bounce in- or out-of-phase,respectively, after they have undergone the period-doublingtransition from a (1, 1) to a (2, 2) mode.

An overhead camera is used to record the horizontaltrajectory of the droplets and a transparent lid is used to iso-late the bath from ambient air currents.19 Three submergedacrylic launchers are spaced evenly around the bath’s edgeand serve to direct ratcheting pairs toward the bath’s center.This geometric adaptation makes data collection more effi-cient, allowing the experiment to run relatively continuouslyby eliminating the need to remove the lid in order to reposi-tion the drops after they reach the bath’s edge. Following Eddiet al.,1 we define the ratcheting speed, vR, to be the speed ofthe pair’s center of mass along the radial direction between the

(a) (b) (c)

(d)

FIG. 1. (a) A schematic diagram of the experimental set-up. A detailed description of the shaker can be found in Harris and Bush.9 (b) A schematic illus-tration showing a view of the bath as seen by the overhead camera. Submerged launchers are used to direct ratcheting pairs towards the center of the bath.(c) A ratcheting pair consists of two unequally sized drops of radii R1 and R2 interacting at a distance d through their common wavefield. This image wascaptured using the visualization technique developed by Harris et al.10 (d) An overhead view of the wavefield generated by a ratcheting pair.

096112-3 Galeano-Rios et al. Chaos 28, 096112 (2018)

drops, and use the convention that vR > 0 when the smallerdrop follows the larger drop.

We considered droplet pairs with five different size dif-ferentials, denoted A through E in Table I. For each pair, wecharacterized the dependence of the inter-drop distance, d ,and the ratcheting speed, vR, on the driving acceleration ofthe bath, γ , and the pair’s binding number, n. For a givenpair size, the driving acceleration of the bath was first setto γ = 1.2 g, slightly above the bouncing threshold, γB, andan initial separation distance corresponding to binding num-ber n = 1, 1.5, 2 or 2.5 was chosen. Note that at γ = 1.2 g,all drops bounce in phase in a (1, 1) mode: the possibility ofthe drops being out of phase only arises when both drops areabove the period-doubling transition. No ratcheting motionwas observed for pairs with binding numbers n > 2.5: thewave force imparted during impact at such large separationdistances is evidently too weak to generate horizontal motionand the droplets bounce in place. For a given pair size andbinding number, n, γ is then slowly increased in incrementsof 0.1 g − 0.2 g. As γ is increased progressively, the pair firsttransitions from a static to a ratcheting state, and then theratcheting state eventually destabilizes into an orbiting state.At each increment of γ , an overhead video of the pair isrecorded at a frame rate of 5 fps, allowing us to track the hor-izontal motion of each drop. We note that for pair A, whichhas the largest size differential, ratcheting motion with bind-ing number n = 1 was observed all the way down to the valueof γ at which the larger drop coalesced with the bath. A videoshowing several examples of the ratcheting motion of pair Awith binding number n = 1 is included in the supplementarymaterial.

Figure 2 shows the dependence of the inter-drop dis-tance, d , on the dimensionless driving acceleration, γ /γF ,for ratcheting pairs A through E for each accessible bindingnumber n. We note that pair E, which has the smallest sizedifferential, exhibited no ratcheting motion beyond a bind-ing number of n = 1.5, and pair A, which has the largest sizedifferential, was the only pair to exhibit ratcheting motion atn = 2.5. Figure 2 suggests that the shift in d with increasingγ may be attributed to the shift in the wavelength of a drop’swavefield as the drop transitions through various bouncingmodes to reach the (2, 1) mode. We observe that this shift ismost pronounced for pairs with binding number n = 1. Notethat below the period-doubling transition, all drops bounce inphase, while above it, the drops bounce either in phase or outof phase according to their separation distance.

Figure 3 shows the dependence of the ratcheting speed,vR, on γ /γF for pairs A through E for each accessible bind-ing number n. The largest ratcheting speeds are seen to occur

TABLE I. The radius of the larger drop, R1, and the smaller drop, R2, for thefive ratcheting pairs considered in our experiments. The pairs are labeled inorder of the largest (pair A) to the smallest (pair E) difference in drop size.All drop radii have an experimental uncertainty of approximately ±0.01 mm.

Pair A B C D E

R1 (mm) 0.444 0.406 0.400 0.383 0.406R2 (mm) 0.353 0.354 0.361 0.365 0.393

0 0.5 1 1.5 2 2.5 30

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FIG. 2. The dependence of the dimensionless inter-drop distance, d/λF , onthe dimensionless driving acceleration of the bath, γ /γF , for ratcheting pairsof size A through E and binding numbers n = 1 to n = 2.5. In each dataset,the bottommost point corresponds to the value of γ /γF at which the pairfirst starts ratcheting (or below which coalescence occurs for pair A, n = 1)and the topmost point indicates when the ratcheting pair destabilizes into anorbiting pair. The experimental uncertainty on d is approximately ±0.1 mm,as determined by the resolution of the overhead camera. Smooth curves havebeen drawn through the data for the sake of clarity, and the correspondingratcheting speeds are shown in Fig. 3. The relative vertical bouncing phaseof the drops for each pair (in or out of phase) at the beginning and endof the ratcheting regime is indicated. For the sake of comparison, Besselfunctions J0

[k(1,1)d

]and J0

[k(2,1)d

]are shown, which are the wave fields

produced, respectively, by (1, 1) and (2, 1) bouncers, located at d = 0; as pre-dicted by the stroboscopic model.11 Note that k(1,1) = 2.20 mm−1 and k(2,1) =1.32 mm−1 are the wavenumbers obtained from the water-wave dispersionrelation using forcing frequencies of 80 Hz and 40 Hz, respectively.

when the drops are closest together and when the size differ-ence between the drops is the largest. Multiple reversals indirection are apparent as γ is increased. The number of rever-sals and the values of γ /γF at which they occur are weaklydependent on the pair’s size and strongly dependent on thebinding number n.

III. MODELING

We simulate the behavior of ratcheting pairs using thetheoretical model developed by Milewski et al.5 This modelrelies on a linear quasi-potential approximation of the freesurface flow,20 in which the effects of impacting dropletsare included by means of a time-dependent moving pressurefield. The free surface elevation, η, and the velocity potential,φ(x, y, z, t), of the fluid bath are governed by the equations

0 = φ, (1)

φt = −g[1 + γ sin(ωt)]η + σ

ρHη + 2ν∗Hφ − P

ρ, (2)

ηt = 2ν∗Hη + φz, (3)

where = ∂xx + ∂yy + ∂zz, H = ∂xx + ∂yy, and P = P(x, y, t)is the forcing term that models the effect of the impactingdroplets, and the constants σ , ρ, and g denote, respectively,the surface tension, liquid density, and gravitational accel-eration. Equation (1) is valid in the semi-space z < 0, andEqs. (2) and (3) in the xy-plane.

The constant ν∗ is the corrected value of the kinematicviscosity, ν, required to match the Faraday threshold, γF ,observed in experiments.5 For the experimental parametersused here, the effective viscosity ν∗ = 0.8025ν. The equa-tions of the model are formulated in the frame of reference of

096112-4 Galeano-Rios et al. Chaos 28, 096112 (2018)

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FIG. 3. The dependence of the ratch-eting speed, vR, on the dimensionlessdriving acceleration of the bath, γ /γF ,for ratcheting pairs of size A throughE and binding numbers (a) n = 1, (b)n = 1.5, (c) n = 2, and (d) n = 2.5. Weuse the convention that vR is positivewhen the smaller drop follows the largerdrop. Each reported vR is obtained byaveraging the ratcheting speeds mea-sured in two separate trials and theassociated experimental uncertainty isapproximately ±10%. For each curve,the leftmost point corresponds to thevalue of γ /γF at which the pair firststarts ratcheting (or below which coales-cence occurs for pair A, n = 1) and therightmost point indicates when the ratch-eting pair destabilizes into an orbitingpair. Smooth curves are drawn throughthe data for the sake of clarity.

the shaker, which requires the use of a time dependent grav-ity field, G(t) = g (1 + γ sin ωt). A thorough derivation ofthese equations is presented in Galeano-Rioset et al.6 Periodicboundary conditions are imposed in x and y, allowing the useof spectral methods. Spectral decomposition also diagonal-izes the Dirichlet-to-Neumann map, defined as φ(x, y, 0, t) �→φz(x, y, 0, t), which is necessary to reduce Eqs. (2) and (3) to atwo-dimensional problem in the plane z = 0.

In flight, a droplet’s motion is described by

d2zi

dt2= −G(t), (4)

mid2xi

dt2= −6πRiμair

dxi

dt, (5)

where xi = (xi, yi), (xi, yi, zi) are the coordinates of the low-ermost point (“south pole”) of the i-th droplet and Ri, mi,and μair denote, respectively, the radius and mass of thei-th droplet and the dynamic viscosity of air. While air dragis significant in the drop’s horizontal motion, it is negligiblein its vertical motion.21

A droplet’s impact is defined as the period of time duringwhich the south pole of the droplet is predicted to be below thelevel of the free surface, the latter being calculated without theforcing due to the ongoing impact pressure. During impact,the vertical motion of a droplet is calculated using the non-linear spring model of Molácek and Bush,3 namely,

⎛

⎝1 + c3

ln2∣∣∣ c1Ri

zi−ηi

∣∣∣

⎞

⎠ mid2zi

dt2+ 4πνρRic2

3ln2∣∣∣ c1Ri

zi−ηi

∣∣∣

d

dt(zi − ηi)

+ 2πσ

ln∣∣∣ c1Ri

zi−ηi

∣∣∣(zi − ηi) = −miG(t), (6)

where the values of the constants c1, c2, and c3 are as given inMilewski et al.5 The variable ηi is the estimate of the free sur-face elevation at xi, as would arise in the absence of the dropimpact in question. It is calculated from a separate solution ofthe free surface flow, which is not subject to the forcing of thecurrent impact of the i-th droplet. The variable ηi is required ifa spring model is used to account for the interaction betweenthe droplet and the bath, as (zi − ηi) prescribes the intru-sion depth of the impacting droplet. Since we here have twodroplets to consider, at times we need to calculate up to threefree surfaces; namely, that resulting from the impact of bothdroplets and those arising from the individual impacts. Thevertical force that the bath exerts on the i-th droplet is givenby Fi(t) = max

[0, mi(d2zi/dt2) + miG(t)

]. We note that suc-

tion forces, as may be significant during droplet rebound,21

are not considered. The forces Fi(t) are used to calculate thepressure field

P(x, y, t) =∑

i

Pi(x, y, t) (7)

in Eq. (2), where

Pi = Fi(t)

πr2i (t)

χ{[(x, y), |(x, y) − (xi, yi)| < ri(t)]} , (8)

with ri(t) = min(√

2|zi − ηi|Ri, Ri/3), and χ is the character-istic function of the pressed area. These modeling choices aremade following those of Milewski et al.5

The horizontal motion of the droplet during impact isdetermined by

mid2xi

dt2+ D (t)

dxi

dt= −Fi (t) ∇ ηi|xi

, (9)

D(t) = c4

√ρRi

σF(t) + 6πRiμair, (10)

096112-5 Galeano-Rios et al. Chaos 28, 096112 (2018)

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0.08 FIG. 4. Examples of the simulated verticalmotion of droplets in ratcheting pair A with bind-ing number n = 1. The solid blue and red linesdenote the heights of the south poles of the largerand smaller drops, respectively. The dashed blueand red lines show the heights of the free surfacebeneath each drop. Panel (a) shows the initial tran-sient at the start of the simulation as the dropletssettle into (1, 1) bouncing modes at γ /γF = 0.2.Panel (b) shows both droplets bouncing in a (2, 2)

mode at γ /γF = 0.62.

as was developed by Molácek and Bush.3 We use a value ofc4 = 0.13 for the horizontal drag coefficient.5 We note thatEq. (5), used during flight, is simply Eq. (10) for the caseF(t) = 0.

A. Simulations

We chose to focus our simulations on pairs A and B withbinding numbers n = 1, 1.5, and 2, as these pairs exhibited therichest behavior in our experiments. Each simulation is initial-ized by releasing a pair of droplets, separated by a distance d ,from a height of 1 mm onto an undisturbed fluid surface attime t = 0, with γ /γF = 0.2. The initial inter-drop distance,d , is varied to obtain pairs of different binding numbers, n.After an initial transient, the droplets quickly settle into the(1, 1) bouncing mode and bounce with similar impact phases[Fig. 4(a)]. As shown in Fig. 4, our simulations allow us totrack the vertical trajectory of the south pole of each drop,zi(t), and the positions of the fluid interface below each drop,ηi(t). We note that with our model, there can be a slight mis-match between zi and ηi when the drop is in contact with thebath, owing to our method for approximating the waveform.5

After the initial transient, the droplets begin to movehorizontally as a ratcheting pair, with the smaller droplet fol-lowing the larger one [Fig. 5(a)]. We define the x-coordinateto be along the ratcheting direction, oriented to point fromthe smaller drop to the larger drop. Each simulation is runat a fixed driving acceleration, γ , until the horizontal speedof both droplets, averaged over a bouncing period of thepair, converges to a steady value, at which point the ratch-eting speed, vR, is recorded. We then progressively increase

γ in increments of 0.1g, as was done in the experiments.After each increment, the pair is allowed to settle into its newequilibrium state. With each increase in γ , we observe a cor-responding change in the impact phase of the droplets, whichaffects both the accompanying wavefields generated at impactand the ratcheting speed, vR. Figure 5 gives examples of vR

changing with specific increments of γ . Eventually, a value ofγ is reached at which the drops begin to execute long-periodoscillations along their line of centers, and the simulation isstopped. In certain cases, an overall motion along the line ofthe centers persists accompanied by a periodic oscillation ofthe separation distance. This behavior was not apparent in theexperiments.

Figure 6 shows the dependence of the inter-drop separa-tion distance, d , on the driving acceleration, γ , for the pairsconsidered in our simulations, compared directly to the exper-imental data. Our simulations are able to capture adequatelythe experimentally observed shifts in d as γ is increased,including the sharp jump observed at binding number n = 1.For pair B, this sharp jump in d is due to a droplet in thepair changing bouncing mode: the larger drop undergoes a(4, 3) → (2, 1) transition. For pair A, the cause is more subtleas the jump in d occurs due to a change in the vertical dynam-ics of both drops within the (4, 3) mode. Specifically, the jumpoccurs when the drops change from taking one long bounce(of duration 2TF) and two short bounces (each of duration TF)in synchrony to the large/small drop taking the long bouncewhile the small/large drop takes two short bounces.

As seen in Fig. 6, for binding numbers n = 1.5 and n = 2,the model consistently yields a slight overprediction for theinter-drop distance, d , likely due to small errors introduced by

0 100 200 300-0.4

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FIG. 5. Examples of the simulated horizon-tal motion of droplets in ratcheting pair Awith binding number n = 1. The blue andred lines indicate the horizontal positions ofthe larger and smaller droplets, respectively.The dashed vertical lines indicate the times atwhich the driving acceleration was increasedby γ = 0.1 g. Panel (a) shows the initial tran-sient at the start of a simulation as the pairsettles into the stable binding length corre-sponding to n = 1. In this panel, we can alsosee the first reversal takes place at γ /γF =0.22, beyond which the droplets start to moveat an extremely slow speed in the oppositedirection. Panel (b) shows examples of howthe ratcheting speed changes with successiveincrements in γ .

096112-6 Galeano-Rios et al. Chaos 28, 096112 (2018)

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FIG. 6. A comparison of the experimental (circles) and theoretical (solid curves) dependence of the dimensionless separation distance, d/λF , on the dimen-sionless driving acceleration of the bath, γ /γF , for ratcheting pairs A and B in modes n = 1, n = 1.5, and n = 2. Vertical dashed lines indicate the changes inbouncing modes apparent in the simulations. The bouncing modes of the larger and smaller drops in the simulation are indicated in blue and red, respectively.

our modeling of the waves. Specifically, the decay rate of thewaves is prescribed by the effective viscosity, ν∗, as was cho-sen to match the Faraday threshold. The model is thus unlikelyto predict correctly the decay rate for driving accelerationsfar from the Faraday threshold. We expect the resulting errorto increase with n, as the wave fronts produced by the dropstake longer to arrive at their companions, so the effects of theanomalous decay rate will be more pronounced. Both trendsare apparent in Fig. 6.

Figure 7 shows the predicted dependence of the ratchet-ing speed, vR, on the driving acceleration γ , compared directlyto our experimental data. The net horizontal wave force act-ing on the pair, averaged over a period of the pair’s verticalmotion, determines the direction of the ratcheting motion andis shown in green for reference. We note that the sharp jumps

in separation distances observed in Figs. 6(a) and 6(d) cor-respond to sharp corners in the velocity curve in Fig. 7. Thecorner evident at γ /γF ≈ 0.66 in Fig. 7(b) is also due to a rear-rangement of the two (4, 3) modes of the two droplets. Whilesimilar to that described for pair A at n = 1, in this case it isfor a pair that is out of phase.

In the majority of cases, our simulations were able to cap-ture all of the observed reversals in the ratcheting directionand the qualitative agreement between the experimental andtheoretical ratcheting speeds is satisfactory. However, thereare some notable discrepancies apparent for pair B at largevalues of γ /γF , where the simulations predict a reversal inthe ratcheting direction that was not observed in the experi-ments. We shall attempt to rationalize these discrepancies inwhat follows.

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FIG. 7. A comparison of the exper-imental (circles) and theoretical (solidcurves) dependence of the dimension-less ratcheting speed, vR/CP, on thedimensionless driving acceleration ofthe bath, γ /γF , for ratcheting pairsA and B in modes n = 1, n = 1.5,and n = 2. CP is the phase speedof a wave with the Faraday wave-length, defined as CP = λFfF , whereλF and fF are the Faraday wavelengthand frequency, respectively. Specifically,CP = 190 mm/s in our experiments andCP = 188 mm/s in our simulations. Ver-tical dashed lines indicate the changesin bouncing modes apparent in the sim-ulations. The bouncing modes of thelarger and smaller drops in the sim-ulation are indicated in blue and red,respectively. The dashed green curvesindicate the dimensionless, net horizon-tal wave force acting on the pair aver-aged over one bouncing period,

⟨FP

W

⟩ =⟨FW ,1/m1g + FW ,2/m2g

⟩/2, as calculated

from our simulations.

B. Details of the ratcheting mechanism

Eddi et al.1 attributed the two reversals in the ratchetingdirection apparent in their experiments to the successive tran-sition of the small and large drops from a (1, 1) to a (2, 2)

bouncing mode. However, in our simulations, this period-doubling transition always occurs simultaneously for bothdroplets and is not associated with any reversal in the ratch-eting direction. We also note that if the drops were bothemitting standing waves, one would expect the ratchetingpairs to align themselves in the extrema of the Bessel wave-forms generated by their partner. Figure 2 makes clear thatwhile such is a fair approximation, it is not precisely correct.In order to better understand what determines the inter-dropspacing, speed, and direction of a ratcheting pair, we pro-ceed by examining the coupling of each droplet with thebath.

Figure 8(a) shows the time-dependence of the horizon-tal component of the wave-force, FW = −Fi (t) ∇ ηi|xi

, actingon each droplet in pair A with binding number n = 1 atγ /γF = 0.2, over a bouncing period. Figure 8(a) shows thatthe wave force acting on each drop is approximately sinu-soidal, and reverses sign approximately halfway through the

time of impact of the drop. The origin of this reversal inthe wave force is shown in Figs. 8(b)–8(f). Upon impact, thewavefield produced by the previous impacts causes both dropsto be attracted to each other. During impact, each droplet trig-gers a traveling front that propagates outward. As this frontapproaches the partner drop, it reverses the surface gradient,thus repelling the partner drop. If the net wave force onthe pair averaged over a bouncing period,

⟨FP

W

⟩, is non-zero,

then the pair will ratchet. The ratcheting speed is determinedby the balance between the averaged net wave force andthe drag force, and is equal to the speeds of the individ-ual droplets averaged over a bouncing period. The ratchetingspeed, ∼1 mm/s, is much slower than the typical walkingspeed of an individual drop, ∼ 10 mm/s, due to the small netwave force resulting from the reversal in surface gradient dur-ing contact. We note that although in our experiments thedroplets look to move at a steady speed with a fixed inter-drop distance, our simulations have highlighted that the speedof each droplet is actually fluctuating over the timescale ofcontact with the bath [Fig. 8(a)].

The reversal of the horizontal wave force on each dropletwas observed for all of the ratcheting pairs considered in oursimulations, some examples of which are shown in Fig. 9.

096112-8 Galeano-Rios et al. Chaos 28, 096112 (2018)

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FIG. 8. (a) Time dependence of the horizontal wave force, FW , acting on each drop (solid lines) and the speed, v, of each drop (dashed lines) in ratcheting pairA with a binding number of n = 1 at γ /γF = 0.2. (b)–(f) Snapshots of the wave field at the times marked in panel (a). The larger drop and its correspondingaverage surface deflection η1 are shown in blue and the corresponding curves for the smaller droplet in red. In panel (b), the impact of the larger droplet triggersthe traveling front that moves toward the smaller droplet. In (c), the impacts of the smaller droplet triggers the second moving front. In (d), the horizontal forceis still inward for both droplets as the fronts have not yet moved enough to reverse the surface gradient. In (e), the surface gradient is reversed by the travelingfronts. In (f), the fronts continue to move past the droplets, reversing the forces until the end of contact.

These reversals in wave force are expected when the timebetween successive wave crests at a given point is compa-rable to a droplet’s contact time. Whether the wave forceaverages to a positive or negative value is highly sensitiveto the phase of impact of each droplet with the bath andexplains why many reversals in direction occurred withina given bouncing mode. For example, Fig. 9(a) shows thewave force acting on pair A in the (1, 1) bouncing modewith a binding number of n = 1 at γ /γF = 0.2. In this case,the wave forces on each drop average to a positive valueand the small drop follows the large. However, as shown inFig. 9(b), if the driving acceleration is increased to γ /γF =0.25, although the drops remain in a (1, 1) bouncing mode,the phase of impact has changed sufficiently that the net

wave forces imparted to each drop assume negative values,so the large drop follows the small. We note that changesin the bouncing mode that result in sharp differences in thebouncing phase [e.g., the (4, 3) → (2, 1)1 transition] causesudden variations in the ratcheting velocity. All instances inwhich the velocity curves in Fig. 7 have a discontinuity inslope arise in the vicinity of discrete changes in the bouncingmode.

The reversal of the wave force during impact also allowsus to rationalize why ratcheting pairs have a discrete set ofinter-drop distances. As shown in Fig. 10, a series of mov-ing wave fronts travel away from the location of each dropletimpact. In order to be horizontally stable, the drops in a pairmust be separated by a distance such that, during the impact

096112-9 Galeano-Rios et al. Chaos 28, 096112 (2018)

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FIG. 9. The time dependence of the horizontal wave force, FW , acting on each drop (solid lines) and the speed, v, of each drop (dashed lines) for pair A with bind-ing number n and driving acceleration γ /γF as follows: (a) n = 1, γ /γF = 0.2, (b) n = 1, γ /γF = 0.25, (c) n = 1, γ /γF = 0.44, and (d) n = 1.5, γ /γF = 0.51.In panels (a) and (b) both drops are in (1, 1) bouncing modes. In panels (c) and (d) both drops are in (2, 2) bouncing modes.

FIG. 10. The positions of droplet impact in relation to the moving extrema of the wavefield for ratcheting pair A at γ /γF = 0.2. Maxima and minima of thewavefield are indicated in black and gray, respectively, and the impact positions of the smaller and larger droplets are denoted by red and blue, respectively.Panels (a) and (b) show the extrema of η1 and η2, respectively, for n = 1 and panels (c) and (d) show the extrema of η1 and η2, respectively, for n = 2.

096112-10 Galeano-Rios et al. Chaos 28, 096112 (2018)

of each drop, a minimum in the traveling wave front emit-ted by its partner sweeps beneath it, causing a reversal in thewave force. For example, Figs. 10(a) and 10(c) show that forbinding numbers n = 1 and n = 2, the larger (blue) droplet inpair A always impacts at a position such that a minimum pro-duced by the smaller (red) droplet sweeps beneath it duringcontact. Similarly, Figs. 10(b) and 10(d) show that the mini-mum produced by the larger (blue) droplet sweeps beneath thesmaller (red) droplet during impact. Figure 10 also highlightsthat an accurate picture of the wave field must include travel-ing fronts, consideration of which are necessary to rationalizethe changes in surface slope that take place during impact.

In order to understand why the quantized inter-drop dis-tances are stable, consider the standing Bessel wavefieldsshown in Fig. 2, which represent a snapshot, at the time ofimpact, of the moving fronts. Due to the finite contact time ofeach drop with the bath, each drop must first impact slightlyto the right of a minimum produced by the partner drop, sothat the inter-drop distance initially starts to decrease. Theminimum then sweeps outward beneath the drop, causingthe surface gradient to reverse and the inter-drop distanceto increase again, resulting in a net change in the inter-dropdistance of zero. This is a stable position because if the inter-drop distance d is slightly larger than the stable position, thetraveling fronts take longer to arrive and so they reverse thesurface gradient at a later stage of the contact, leading to anet attraction. Similarly, if d is decreased, the traveling frontsarrive earlier, causing a net repulsion. One may thus rational-ize why the standing wave-model (see Fig. 2) always slightlyunderestimates the stable separation distances; specifically, itneglects the interaction of the drops with the traveling wavefronts during their finite contact time.

Finally, we address the discrepancies previously men-tioned between our experimental and theoretical results. InFig. 7, the simulated velocity curves sometimes have kinksthat are not present in the experimental data [such as betweenγ /γF ≈ 0.6–0.7 in Fig. 7(b)]. For pair B, with binding num-bers of n = 1.5 and n = 2 [Figs. 7(e) and 7(f), respectively],the simulations also predict an additional reversal in directionthat was not observed in our experiments. The most likelycause for these discrepancies is differences in the simulatedand experimental bouncing modes. Although the wave-modelof Milewski et al.5 has been verified against experimentaldata,16 coupling it to the logarithmic spring model can resultin spurious bouncing modes not seen in the experiment.5

As the ratcheting motion is highly sensitive to the bounc-ing phase of the droplets, if the model predicts a differentbouncing mode than that seen in the experiment, we shouldexpect spurious results. For example, in Fig. 7(b), if pair Ain the n = 1.5 mode smoothly transitioned from a (2, 2) to a(2, 1) mode instead of passing through the (8, 7) and (4, 3)

modes deduced from the simulations, the erroneous abruptchanges in the ratcheting speed not seen in experiments mighthave been averted. We note that the shortcomings of the cur-rent simulations might be eliminated through the applicationof the most recent model of Galeano-Rios, Milewski, andVanden-Broeck,6 which more accurately models the droplet-bath interaction and has been shown to predict bouncingmodes more accurately.

IV. DISCUSSION

We have reported the results of an experimental study ofratcheting pairs. Five distinct pairs were considered, in up tofour binding lengths, and their behavior characterized as γ

was increased progressively. Eddi et al.1 observed that theinter-drop distance was quantized, owing to dynamic con-straints imposed on the droplets by their shared wavefield,and observed up to two reversals in the ratcheting direction.We extended the work of Eddi et al.1 to characterize how thestable inter-drop distances shift with increasing γ and demon-strated that up to four reversals in the direction could occurfor pairs with a sufficiently large drop size differential anda binding number of n = 1. The majority of these reversalswere not correlated with a bouncing mode transition of asingle droplet,4,21,22 as suggested by Eddi et al.,1 promptinga detailed theoretical investigation of the mechanism throughwhich the droplets interact.

To understand the complex, wave-mediated couplingbetween the two droplets in a ratcheting pair, we adoptedthe theoretical model of Milewski et al.,5 wherein both thedroplet’s vertical dynamics and the traveling wave frontsproduced by a droplet impact are modeled explicitly. Our sim-ulations were able to closely reproduce the shift in the stableinter-drop distances with increasing γ , and provided ade-quate agreement with the experimentally observed ratchetingspeeds. Importantly, the simulations provided critical insightinto the mechanism governing the ratcheting dynamics. Atevery impact, each droplet produces a Bessel shaped wave-field whose extrema travel radially outwards. The partner dropthen interacts with one of these traveling fronts during impact.In order to remain in a stable position, the droplet must bouncein a position such that as a traveling front sweeps past dur-ing impact, the gradient of the wavefield beneath the dropreverses direction so that the net wave force on the dropletduring contact is zero. This explains why all of the experi-mentally observed inter-drop distances lie slightly to the rightof an extremum anticipated on the basis of a standing wavemodel (Fig. 2). The drop must first land slightly to the right ofan extremum in the wavefield produced by the partner drop sothat it first encounters an attractive wave force. Then, as theextremum sweeps by the drop during contact, the wave forcebecomes repulsive, resulting in a net wave force of zero.

If the droplet lands slightly earlier or later than the criti-cal impact time that results in a zero net force, it will cause theratcheting motion. The ratcheting speed is much slower thanthat of an individual walker because the reversal in the waveforce during the contact time leads to a small net force. Ourstudy also rationalizes how a reversal in direction can occurwithin a given bouncing mode. For example, Fig. 8(a) showsthe time evolution of the wave force during contact on pair Awith a binding number n = 1 at γ /γF = 0.2. In this case, thenet wave force on both drops is positive and the small dropfollows the larger. As γ is increased, however, the bouncingphase of the droplets slowly changes, the net force changesfrom positive to negative, and the ratcheting pair reversesdirection.

We note that while the stroboscopic model11 accuratelycaptures the shape of the wavefield at an instant in time,16

096112-11 Galeano-Rios et al. Chaos 28, 096112 (2018)

it does not capture the outward radial expansion of thefronts. Therefore, the stroboscopic model should be usedwith caution when the contact time between the drop andthe bath is sufficiently large that the wave force during con-tact changes appreciably, as is the case in our study. Onecan now better understand the shortcomings of the strobo-scopic models3,11,23 in rationalizing the observed behavior oforbiting14 and promenading15 pairs. For example, the inclu-sion of traveling wave fronts could account for the stabilityof orbiting pairs with binding number n = 0.5, which havebeen shown to be stable in the experiment, but cannot becaptured with the stroboscopic model.14 The neglect of thetraveling fronts in the stroboscopic models also suggests theirlimitations in rationalizing the relative stability of variousdynamical states through consideration of the system’s globalenergetics. For example, in order for the binding energy ofpromenading pairs15,24 to be a meaningful system diagnostic,it must capture the influence of the moving wave fronts.

SUPPLEMENTARY MATERIAL

See supplementary material for a video that shows fourexamples of the ratcheting motion of pair A with bindingnumber n = 1.

ACKNOWLEDGMENTS

We thank Professor Paul A. Milewski and ProfessorAndré Nachbin for fruitful discussions on the modeling ofvertical dynamics for multiple bouncing droplets. C.A.G.-R. gratefully acknowledges the support of EPSRC ProjectNo. EP/N018176/1. M.M.P.C. gratefully acknowledges thefinancial support of the NSERC through Grant No. 502891.P.C. was partially supported by the MIT-France program.J.W.M.B. gratefully acknowledges the financial support of theNSF through Grant Nos. DMS-1614043 and CMMI-1727565.

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