Showing papers by "John W. M. Bush published in 2018"

Walking droplets interacting with single and double slits

Abstract: Couder & Fort (Phys. Rev. Lett., vol. 97, 2006, 154101) demonstrated that when a droplet walking on the surface of a vibrating bath passes through a single or a double slit, it is deflected due to the distortion of its guiding wave field. Moreover, they suggested the build-up of statistical diffraction and interference patterns similar to those arising for quantum particles. Recently, these results have been revisited (Andersen et al., Phys. Rev. E, vol. 92 (1), 2015, 013006; Batelaan et al., J. Phys.: Conf. Ser., vol. 701 (1), 2016, 012007) and contested (Andersen et al. 2015; Bohr, Andersen & Lautrup, Recent Advances in Fluid Dynamics with Environmental Applications, 2016, Springer, pp. 335–349). We revisit these experiments with a refined experimental set-up that allows us to systematically characterize the dependence of the dynamical and statistical behaviour on the system parameters. The system behaviour is shown to depend strongly on the amplitude of the vibrational forcing: as this forcing increases, a transition from repeatable to unpredictable trajectories arises. In all cases considered, the system behaviour is dominated by a wall effect, specifically the tendency for a drop to walk along a path that makes a fixed angle relative to the plane of the slits. While the three dominant central peaks apparent in the histograms of the deflection angle reported by Couder & Fort (2006) are evident in some of the parameter regimes considered in our study, the Fraunhofer-like dependence of the number of peaks on the slit width is not recovered. In the double-slit geometry, the droplet is influenced by both slits by virtue of the spatial extent of its guiding wave field. The experimental behaviour is well captured by a recently developed theoretical model that allows for a robust treatment of walking droplets interacting with boundaries. Our study underscores the importance of experimental precision in obtaining reproducible data.

Introduction to focus issue on hydrodynamic quantum analogs.

Abstract: Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protiere, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269–292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is “path memory” [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433–463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics, a field that attempts to answer the following simple but provocative question: Might deterministic chaotic pilot-wave dynamics underlie quantum statistics?Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protiere, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269–292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is “path memory” [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433–463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is c...

A review of the theoretical modeling of walking droplets: Toward a generalized pilot-wave framework.

Abstract: The walking droplet system has extended the range of classical systems to include several features previously thought to be exclusive to quantum systems. We review the hierarchy of analytic models that have been developed, on the basis of various simplifying assumptions, to describe droplets walking on a vibrating fluid bath. Particular attention is given to detailing their successes and failures in various settings. Finally, we present a theoretical model that may be adopted to explore a more generalized pilot-wave framework capable of further extending the phenomenological range of classical pilot-wave systems beyond that achievable in the laboratory.

Walking droplets in a circular corral: Quantisation and chaos

Abstract: A millimetric liquid droplet may walk across the surface of a vibrating liquid bath through a resonant interaction with its self-generated wavefield. Such walking droplets, or “walkers,” have attracted considerable recent interest because they exhibit certain features previously believed to be exclusive to the microscopic, quantum realm. In particular, the intricate motion of a walker confined to a closed geometry is known to give rise to a coherent wave-like statistical behavior similar to that of electrons confined to quantum corrals. Here, we examine experimentally the dynamics of a walker inside a circular corral. We first illustrate the emergence of a variety of stable dynamical states for relatively low vibrational accelerations, which lead to a double quantisation in angular momentum and orbital radius. We then characterise the system’s transition to chaos for increasing vibrational acceleration and illustrate the resulting breakdown of the double quantisation. Finally, we discuss the similarities and differences between the dynamics and statistics of a walker inside a circular corral and that of a walker subject to a simple harmonic potential.A millimetric liquid droplet may walk across the surface of a vibrating liquid bath through a resonant interaction with its self-generated wavefield. Such walking droplets, or “walkers,” have attracted considerable recent interest because they exhibit certain features previously believed to be exclusive to the microscopic, quantum realm. In particular, the intricate motion of a walker confined to a closed geometry is known to give rise to a coherent wave-like statistical behavior similar to that of electrons confined to quantum corrals. Here, we examine experimentally the dynamics of a walker inside a circular corral. We first illustrate the emergence of a variety of stable dynamical states for relatively low vibrational accelerations, which lead to a double quantisation in angular momentum and orbital radius. We then characterise the system’s transition to chaos for increasing vibrational acceleration and illustrate the resulting breakdown of the double quantisation. Finally, we discuss the similarities ...

Promenading pairs of walking droplets: Dynamics and stability

Abstract: An integrated experimental and theoretical investigation of the promenade mode, a bound state formed by a pair of droplets walking side by side on the surface of a vibrating fluid bath, highlights the role of bouncing phase adaptation in stabilizing the promenade mode.

Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets.

Abstract: A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal “walking” motion is induced by repeated impacts with its accompanying Faraday wave field For ergodic long-time dynamics, we derive the relationship between the droplet’s stationary statistical distribution and its mean wave field in a very general setting We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line By analyzing the system’s periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force We demonstrate that as the vibrational forcing is increased progressively, the periodic oscillations become chaotic via the Ruelle-Takens-Newhouse route We rationalize the role of the local pilot-wave structure on the resulting droplet motion, which is akin to a random walk We characterize the emergence of wavelike statistics influenced by the effective potential that is induced by the mean Faraday wave fieldA millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal “walking” motion is induced by repeated impacts with its accompanying Faraday wave field For ergodic long-time dynamics, we derive the relationship between the droplet’s stationary statistical distribution and its mean wave field in a very general setting We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line By analyzing the system’s periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force We demonstrate that as the vibrational forcing is increased progressively, the perio

Hydrodynamic spin states.

Abstract: 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.

Ratcheting droplet pairs

Abstract: 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 findingsMillimetric 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, allowi

The interaction of a walking droplet and a submerged pillar: From scattering to the logarithmic spiral

Abstract: Millimetric droplets may walk across the surface of a vibrating fluid bath, propelled forward by their own guiding or “pilot” wave field. We here consider the interaction of such walking droplets with a submerged circular pillar. While simple scattering events are the norm, as the waves become more pronounced, the drop departs the pillar along a path corresponding to a logarithmic spiral. The system behavior is explored both experimentally and theoretically, using a reduced numerical model in which the pillar is simply treated as a region of decreased wave speed. A trajectory equation valid in the limit of weak droplet acceleration is used to infer an effective force due to the presence of the pillar, which is found to be a lift force proportional to the product of the drop’s walking speed and its instantaneous angular speed around the post. This system presents a macroscopic example of pilot-wave-mediated forces giving rise to apparent action at a distance.

Spin lattices of walking droplets

Abstract: This paper is associated with a video winner of a 2017 APS/DFD Gallery of Fluid Motion Award for work presented at the DFD Gallery of Fluid Motion. The original video is available from the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2017.GFM.V0018

Exploring orbital dynamics and trapping with a generalized pilot-wave framework

Abstract: We explore the effects of an imposed potential with both oscillatory and quadratic components on the dynamics of walking droplets. We first conduct an experimental investigation of droplets walking on a bath with a central circular well. The well acts as a source of Faraday waves, which may trap walking droplets on circular orbits. The observed orbits are stable and quantized, with preferred radii aligning with the extrema of the well-induced Faraday wave pattern. We use the stroboscopic model of Oza et al. [J. Fluid Mech. 737, 552–570 (2013)] with an added potential to examine the interaction of the droplet with the underlying well-induced wavefield. We show that all quantized orbits are stable for low vibrational accelerations. Smaller orbits may become unstable at higher forcing accelerations and transition to chaos through a path reminiscent of the Ruelle-Takens-Newhouse scenario. We proceed by considering a generalized pilot-wave system in which the relative magnitudes of the pilot-wave force and drop inertia may be tuned. When the drop inertia is dominated by the pilot-wave force, all circular orbits may become unstable, with the drop chaotically switching between them. In this chaotic regime, the statistically stationary probability distribution of the drop’s position reflects the relative instability of the unstable circular orbits. We compute the mean wavefield from a chaotic trajectory and confirm its predicted relationship with the particle’s probability density function.

Bouncing droplet dynamics above the Faraday threshold.

Abstract: We present the results of an experimental investigation of the dynamics of droplets bouncing on a vibrating fluid bath for forcing accelerations above the Faraday threshold Two distinct fluid viscosity and vibrational frequency combinations (20 cS-80 Hz and 50 cS-50 Hz) are considered, and the dependence of the system behavior on drop size and vibrational acceleration is characterized A number of new dynamical regimes are reported, including meandering, zig-zagging, erratic bouncing, coalescing, and trapped regimes Particular attention is given to the regime in which droplets change direction erratically and exhibit a dynamics akin to Brownian motion We demonstrate that the effective diffusivity increases with vibrational acceleration and decreases with drop size, as suggested by simple scaling arguments