Promenading pairs of walking droplets: Dynamics and stability

Arbelaiz, Juncal; Oza, Anand U.; Bush, John W. M.

doi:10.1103/physrevfluids.3.013604

Cited by 36 publications

(57 citation statements)

References 32 publications

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“…A future direction would be to compare our results with those of a mathematical stability analysis of spin state solutions to an iterated map. 29 It has been shown that phase adaptation, wherein the walker's impact phase sin varies with both the forcing acceleration and the instantaneous wave height h, is crucial to stabilizing the orbital 21 and promenade 22 modes for interacting pairs of walkers. We thus expect that spin states may be stabilized by phase adaptation.…”

Section: Discussionmentioning

confidence: 99%

“…For the sake of simplicity, we neglect the effect of spatial damping, 19,20 which is the experimentally observed exponential decay of the surface waves in the far field. We also neglect the dependence of the impact phase sin on both the forcing acceleration γ and the instantaneous wave height h[x p (t), t], an effect that has been shown to stabilize both the orbital 21 and promenade 22 modes executed by pairs of walkers in the absence of an external force. We first nondimensionalize the trajectory equation using the Faraday wavelength and memory time, and so let x → k F x, t → t/T M , and → 2m /D in Eq.…”

Section: Generalized Pilot-wave Modelmentioning

confidence: 99%

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Hydrodynamic spin states

Oza

Rosales

Bush

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We present the results of a theoretical investigation of hydrodynamic spin states, wherein a droplet walking on a vertically vibrating fluid bath executes orbital motion despite the absence of an applied external field. In this regime, the walker's self-generated wave force is sufficiently strong to confine the walker to a circular orbit. We use an integro-differential trajectory equation for the droplet's horizontal motion to specify the parameter regimes for which the innermost spin state can be stabilized. Stable spin states are shown to exhibit an analog of the Zeeman effect from quantum mechanics when they are placed in a rotating frame.

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Section: Discussionmentioning

confidence: 99%

Section: Generalized Pilot-wave Modelmentioning

confidence: 99%

Hydrodynamic spin states

Oza

Rosales

Bush

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…These are reminiscent of cascades from oscillating walkers to orbits that have recently been observed in experiments. 16…”

Section: B Discrete-turning Walkersmentioning

confidence: 99%

“…In walker-walker interactions, neglecting the transient wave and assuming a simplified wave structure are reasonable within two Faraday wavelengths, but the assumption of constant phase breaks down and thus far this has only been addressed using an empirical fix for the particular system being considered. 15,16 Here we take the Oza-Rosales-Bush description 2 as a theoretical pilot-wave model and explore the behaviors predicted for a simple extension to dynamics of two identical, in-phase bouncing droplets. We find parallel walkers and promenading pairs as well as a rich array of more exotic dynamics such as regularly and chaotically switching walkers, wandering walkers and intriguing closed-loop trajectories in regions of parameter space where wave forcing and/or inertia play a significant role.…”

Section: Introductionmentioning

confidence: 99%

Pilot-wave dynamics of two identical, in-phase bouncing droplets

Valani

Slim

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza-Rosales-Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, κ, and the ratio of wave forcing to drag, β. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate towards and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.A droplet of liquid can bounce on the surface of a bath of the same liquid indefinitely if the bath is experiencing vertical vibrations in an appropriate range of frequencies. At each bounce, the droplet generates a decaying surface wave. The slope of the wave at the point of the next bounce can impart a horizontal force on the droplet, leading to the droplet walking. Since its discovery 1 in 2005, this experimental system has excited significant research because it can exhibit quantum-like behavior. The droplet is a particle that interacts with its environment through the wave it generates, i.e., it is a pilot-wave system. In this paper we use a previously-published model 2 to explore a two-droplet pilot-wave system. We find a remarkably wide range of behaviors, which we explore in detail.

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“…14 The neglect of the traveling fronts in the stroboscopic models also suggests their limitations in rationalizing the relative stability of various dynamical states through consideration of the system's global energetics. For example, in order for the binding energy of promenading pairs 15,24 to be a meaningful system diagnostic, it must capture the influence of the moving wave fronts.…”

Section: Discussionmentioning

confidence: 99%

Ratcheting droplet pairs

Galeano-Rios

Couchman

Bush

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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 placed nearby, they interact through their common wavefield in such a way as to self-propel through a ratcheting mechanism. We present the results of an integrated experimental and theoretical investigation of such ratcheting pairs. Particular attention is given to characterizing the dependence of the ratcheting behavior on the droplet sizes and vibrational acceleration. Our experiments demonstrate that the quantized inter-drop distances of a ratcheting pair depend on the vibrational acceleration, and that as this acceleration is increased progressively, the direction of the ratcheting motion may reverse up to four times. Our simulations highlight the critical role of both the vertical bouncing dynamics of the individual drops and the traveling wave fronts generated during impact on the ratcheting motion, allowing us to rationalize the majority of our experimental findings.

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Promenading pairs of walking droplets: Dynamics and stability

Cited by 36 publications

References 32 publications

Hydrodynamic spin states

Hydrodynamic spin states

Pilot-wave dynamics of two identical, in-phase bouncing droplets

Ratcheting droplet pairs

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