Showing papers by "John W. M. Bush published in 2013"

Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory

Abstract: We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

Wavelike statistics from pilot-wave dynamics in a circular corral

Abstract: Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.

Drops bouncing on a vibrating bath

Abstract: We present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically the drop size, driving acceleration and driving frequency. As the driving acceleration is increased, depending on drop size, we observe the transition from coalescing to vibrating or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the forcing frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence–bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and vibrating, for identical system parameters. In the former state, the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against experimental data. We first model the drop–bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

A trajectory equation for walking droplets : hydrodynamic pilot-wave theory

Abstract: We present the results of a theoretical investigation of droplets bouncing on a vertically vibrating fluid bath. An integro-differential equation describing the horizontal motion of the drop is developed by approximating the drop as a continuous moving source of standing waves. Our model indicates that, as the forcing acceleration is increased, the bouncing state destabilizes into steady horizontal motion along a straight line, a walking state, via a supercritical pitchfork bifurcation. Predictions for the dependence of the walking threshold and drop speed on the system parameters compare favourably with experimental data. By considering the stability of the walking state, we show that the drop is stable to perturbations in the direction of motion and neutrally stable to lateral perturbations. This result lends insight into the possibility of chaotic dynamics emerging when droplets walk in complex geometries.

Exotic states of bouncing and walking droplets

Abstract: We present the results of an integrated experimental and theoretical investigation of droplets bouncing on a vibrating fluid bath. A comprehensive series of experiments provides the most detailed characterisation to date of the system's dependence on fluid properties, droplet size, and vibrational forcing. A number of new bouncing and walking states are reported, including complex periodic and aperiodic motions. Particular attention is given to the first characterisation of the different gaits arising within the walking regime. In addition to complex periodic walkers and limping droplets, we highlight a previously unreported mixed state, in which the droplet switches periodically between two distinct walking modes. Our experiments are complemented by a theoretical study based on our previous developments [J. Molacek and J. W. M. Bush, J. Fluid Mech.727, 582–611 (Year: 2013);10.1017/jfm.2013.279J. Molacek and J. W. M. Bush, J. Fluid Mech.727, 612–647 (Year: 2013)]10.1017/jfm.2013.280, which provide a basis for rationalising all observed bouncing and walking states.

Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization

Abstract: We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath’s rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.

The pilot-wave dynamics of walking droplets

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Wavelike statistics from pilot-wave dynamics in a circular corral

Abstract: Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.

Optimal concentrations in transport systems.

Abstract: Many biological and man-made systems rely on transport systems for the distribution of material, for example matter and energy. Material transfer in these systems is determined by the flow rate and the concentration of material. While the most concentrated solutions offer the greatest potential in terms of material transfer, impedance typically increases with concentration, thus making them the most difficult to transport. We develop a general framework for describing systems for which impedance increases with concentration, and consider material flow in four different natural systems: blood flow in vertebrates, sugar transport in vascular plants and two modes of nectar drinking in birds and insects. The model provides a simple method for determining the optimum concentration copt in these systems. The model further suggests that the impedance at the optimum concentration μopt may be expressed in terms of the impedance of the pure (c = 0) carrier medium μ0 as μopt 2(α)μ0, where the power α is prescribed by the specific flow constraints, for example constant pressure for blood flow (α = 1) or constant work rate for certain nectar-drinking insects (α = 6). Comparing the model predictions with experimental data from more than 100 animal and plant species, we find that the simple model rationalizes the observed concentrations and impedances. The model provides a universal framework for studying flows impeded by concentration, and yields insight into optimization in engineered systems, such as traffic flow.

Biomimicry and the culinary arts

Abstract: We present the results of a recent collaboration between scientists, engineers and chefs. Two particular devices are developed, both inspired by natural phenomena reliant on surface tension. The cocktail boat is a drink accessory, a self-propelled edible boat powered by alcohol-induced surface tension gradients, whose propulsion mechanism is analogous to that employed by a class of water-walking insects. The floral pipette is a novel means of serving small volumes of fluid in an elegant fashion, an example of capillary origami modeled after a class of floating flowers. The biological inspiration and mechanics of these two devices are detailed, along with the process that led to their development and deployment.

The hydraulic bump: The surface signature of a plunging jet

Abstract: When a falling jet of fluid strikes a horizontal fluid layer, a hydraulic jump arises downstream of the point of impact, provided a critical flow rate is exceeded We here examine a phenomenon that arises below this jump threshold, a circular deflection of relatively small amplitude on the free surface that we call the hydraulic bump The form of the circular bump can be simply understood in terms of the underlying vortex structure and its height simply deduced with Bernoulli arguments As the incoming flux increases, a breaking of axial symmetry leads to polygonal hydraulic bumps The relation between this polygonal instability and that arising in the hydraulic jump is discussed The coexistence of hydraulic jumps and bumps can give rise to striking nested structures with polygonal jumps bound within polygonal bumps The absence of a pronounced surface signature on the hydraulic bump indicates the dominant influence of the subsurface vorticity on its instability

The aerodynamics of the beautiful game

Abstract: We consider the aerodynamics of football, specifically, the interaction between a ball in flight and the ambient air. Doing so allows one to account for the characteristic range and trajectories of balls in flight, as well as their anomalous deflections as may be induced by striking the ball either with or without spin. The dynamics of viscous boundary layers is briefly reviewed, its critical importance on the ball trajectories highlighted. The Magnus effect responsible for the anomalous curvature of spinning balls is seen to depend critically on the surface roughness of the ball, the sign of the Magnus force reversing for smooth balls. The origins of the fluttering of balls struck with nearly no spin is also discussed. Particular attention is given to categorizing and providing aerodynamic rationale for the various free kick styles.

Flexibility increases lift on passive fluttering wings

Abstract: Abstract We examine experimentally the influence of flexibility on the side-to-side fluttering motion of passive wings settling under the influence of gravity. Our results demonstrate the existence of an optimal flexibility that allows flexible wings to remain airborne twice as long as their rigid counterparts of identical mass and size. Flow visualization and measurements allow us to elucidate the role of flexibility in generating increased lift and wing circulation by shedding additional vorticity at the turning point. Theoretical scalings are derived from a reduced model of the flight dynamics and yield quantitative agreement with experiments. These scalings rationalize the strong positive correlation between flexibility and flight time. Our experimental results and theoretical scalings represent an ideal system for the validation of computational approaches to model biologically inspired fluid–structure interaction problems.