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Surface topography measurements of the bouncing droplet experiment

DOI:10.1007/s00348-016-2251-4
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Adam P. Damiano
Adam P. Damiano
Adam P. Damiano
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Pt Brun at Princeton University
Pt Brun
Daniel M. Harris at Brown University
Daniel M. Harris
Carlos Galeano Rios at University of Birmingham
Carlos Galeano Rios
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Abstract and Figures

A free-surface synthetic Schlieren (Moisy et al. in Exp Fluids 46:1021–1036, 2009; Eddi et al. in J Fluid Mech 674:433–463, 2011) technique has been implemented in order to measure the surface topography generated by a droplet bouncing on a vibrating fluid bath. This method was used to capture the wave fields of bouncers, walkers, and walkers interacting with boundaries. These wave profiles are compared with existing theoretical models and simulations and will prove valuable in guiding their future development. Specifically, the method provides insight into what type of boundary conditions apply to the wave field when a bouncing droplet approaches a submerged obstacle.
Layout of our experimental setup (Harris and Bush 2015) comprising an electromagnetic shaker (1) interfaced to the bath via a rod (2) and a linear air bearing (3). The air bearing carriage is fixed to an optical table (4) via a leveled platform (5). The bath acceleration is monitored via two accelerometers (6) and is surmounted by a lid (7). The bath (8) comprises a pattern printed on paper inserted below a transparent lower plate so that the pattern may be used as an imaging background
Layout of our experimental setup (Harris and Bush 2015) comprising an electromagnetic shaker (1) interfaced to the bath via a rod (2) and a linear air bearing (3). The air bearing carriage is fixed to an optical table (4) via a leveled platform (5). The bath acceleration is monitored via two accelerometers (6) and is surmounted by a lid (7). The bath (8) comprises a pattern printed on paper inserted below a transparent lower plate so that the pattern may be used as an imaging background
… 
The FSSS method (Moisy et al. 2009): a A sloped interface causes an apparent displacement of the points on a background image, u(x, y). The refraction angle is denoted by ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document} and the interface slope by ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}. The background image is a random dot pattern (b) and the apparent displacement of the pattern creates a gradient field (c) that is used to reconstruct the surface topography (d)
The FSSS method (Moisy et al. 2009): a A sloped interface causes an apparent displacement of the points on a background image, u(x, y). The refraction angle is denoted by ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document} and the interface slope by ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}. The background image is a random dot pattern (b) and the apparent displacement of the pattern creates a gradient field (c) that is used to reconstruct the surface topography (d)
… 
Regime diagram [see Molácek and Bush (2013b) for its derivation] displaying the drop’s bouncing or walking mode as a function of the vibration number ω0/σ/ρR3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0/\sqrt{\sigma /\rho R^3}$$\end{document} and driving acceleration γ/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /g$$\end{document}. A drop in the (m,n)i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m, n)^i$$\end{document} mode bounces n times in m forcing periods, with the integer i ordering multiple (m, n) states according to their total mechanical energy, with i = 1 being the lowest. Here, we have ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,{\mathrm{Hz}}$$\end{document}, ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,{\mathrm{cSt}}$$\end{document} and fixed drop radius R = 0.38 mm. The red symbols indicate the experiments performed in our study
Regime diagram [see Molácek and Bush (2013b) for its derivation] displaying the drop’s bouncing or walking mode as a function of the vibration number ω0/σ/ρR3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0/\sqrt{\sigma /\rho R^3}$$\end{document} and driving acceleration γ/g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /g$$\end{document}. A drop in the (m,n)i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m, n)^i$$\end{document} mode bounces n times in m forcing periods, with the integer i ordering multiple (m, n) states according to their total mechanical energy, with i = 1 being the lowest. Here, we have ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,{\mathrm{Hz}}$$\end{document}, ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,{\mathrm{cSt}}$$\end{document} and fixed drop radius R = 0.38 mm. The red symbols indicate the experiments performed in our study
… 
Temporal evolution of the wave field excited by a period-doubled bouncing droplet of radius 0.38 mm. The acceleration of the 20 cSt oil bath is γ=0.77γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.77\,\gamma _\mathrm{F}$$\end{document} and ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\mathrm{Hz}$$\end{document} (see the red star in Fig. 3). The wave field is shown at phases a 0°, b 150°, c 300°, d 450° and e 600°. Phase 0° corresponds to the bath at its maximum vertical displacement. Black lines show a projection of the crosssection of the wave field passing through the droplet’s center. The gray cylinders indicate the location of the droplet’s projected shadow, where the wave field cannot be determined.
Temporal evolution of the wave field excited by a period-doubled bouncing droplet of radius 0.38 mm. The acceleration of the 20 cSt oil bath is γ=0.77γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0.77\,\gamma _\mathrm{F}$$\end{document} and ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\mathrm{Hz}$$\end{document} (see the red star in Fig. 3). The wave field is shown at phases a 0°, b 150°, c 300°, d 450° and e 600°. Phase 0° corresponds to the bath at its maximum vertical displacement. Black lines show a projection of the crosssection of the wave field passing through the droplet’s center. The gray cylinders indicate the location of the droplet’s projected shadow, where the wave field cannot be determined.
… 
Dependence of the wave profile of a bouncing droplet on the dimensionless forcing accelerations γ/γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /\gamma _\mathrm{F}$$\end{document} where γF denotes the Faraday threshold. The drop radius is 0.38 mm, while ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,\mathrm{Hz}$$\end{document} and ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,\mathrm{cSt}$$\end{document}. Each experiment is represented by a dot in Fig. 3. The wave profile is captured when the bath is at its minimum vertical displacement. The dark vertical band in the vicinity of r = 0 represents the droplets projected shadow
+4
Dependence of the wave profile of a bouncing droplet on the dimensionless forcing accelerations γ/γF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma /\gamma _\mathrm{F}$$\end{document} where γF denotes the Faraday threshold. The drop radius is 0.38 mm, while ω0=80Hz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _0=80\,\mathrm{Hz}$$\end{document} and ν=20cSt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =20\,\mathrm{cSt}$$\end{document}. Each experiment is represented by a dot in Fig. 3. The wave profile is captured when the bath is at its minimum vertical displacement. The dark vertical band in the vicinity of r = 0 represents the droplets projected shadow
… 
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1 3
Exp Fluids (2016) 57:163
DOI 10.1007/s00348-016-2251-4
RESEARCH ARTICLE
Surface topography measurements of the bouncing droplet
experiment
Adam P. Damiano1 · P.‑T. Brun1 · Daniel M. Harris1 · Carlos A. Galeano‑Rios2,3 ·
John W. M. Bush1
Received: 15 June 2016 / Revised: 6 September 2016 / Accepted: 10 September 2016 / Published online: 11 October 2016
© Springer-Verlag Berlin Heidelberg 2016
unstable to Faraday waves. Couder et al. (2005) and Protière
et al. (2006) discovered that in certain experimental regimes,
the droplets may self-propel along the surface of the bath due
to interactions with their own wave fields. These walking
droplets, henceforth walkers, are spatially extended objects
that exhibit several phenomena reminiscent of quantum sys-
tems (Couder and Fort 2006; Eddi et al. 2009, 2011; Bush
2010, 2015a, b; Fort et al. 2010; Harris et al. 2013; Perrard
et al. 2014a, b; Oza et al. 2014; Harris and Bush 2014).
While the droplet, of typical radius 0.4 mm, is read-
ily discerned by eye, the waves excited by the droplet,
of typical amplitude of 1–20 μm, are relatively diffi-
cult to observe and quantify. Various theoretical models
have been developed to describe the waves created by a
bouncing droplet (Eddi et al. 2011; Molácek and Bush
2013a, b; Oza et al. 2013; Labousse 2014; Milewski
et al. 2015; Gilet 2016; Blanchette 2016), on the basis
of which much headway has been made in rationalizing
the behavior of the walkers in a variety of settings [see
Bush (2015a, b) for reviews]. Nevertheless, theoretical
developments would benefit from quantitative measure-
ments of the wave field. In particular, walker-boundary
interactions as arise in a number of key quantum ana-
logues (Couder and Fort 2006; Eddi et al. 2009; Harris
et al. 2013; Harris 2015) remain poorly characterized
and understood. Specifically, some theoretical models
of walkers near boundaries apply a zero-wave-amplitude
boundary condition (Gilet 2016; Blanchette 2016) while
others apply a zero slope boundary condition (Duber-
trand et al. 2016).
We here report the results of an experimental effort to
measure the surface topography in the walking drop system
using the surface synthetic Schlieren technique originally
developed by Moisy et al. (2009), as was applied by Eddi
et al. (2009, 2011). Specifically, we utilize the refracted
Abstract A free-surface synthetic Schlieren (Moisy et al.
in Exp Fluids 46:1021–1036, 2009; Eddi et al. in J Fluid
Mech 674:433–463, 2011) technique has been implemented
in order to measure the surface topography generated by a
droplet bouncing on a vibrating fluid bath. This method
was used to capture the wave fields of bouncers, walkers,
and walkers interacting with boundaries. These wave pro-
files are compared with existing theoretical models and
simulations and will prove valuable in guiding their future
development. Specifically, the method provides insight into
what type of boundary conditions apply to the wave field
when a bouncing droplet approaches a submerged obstacle.
1 Introduction
A millimetric drop placed onto a vibrating liquid bath can
bounce indefinitely on the fluid surface due to a thin film of
air that prevents coalescence and is replenished with each
bounce (Walker 1978; Couder et al. 2005; Terwagne et al.
2007; Vandewalle et al. 2006). The drop dynamics depends
critically on the forcing acceleration of the bath γ relative
to the critical threshold γF, at which the interface becomes
Electronic supplementary material The online version of this
article (doi:10.1007/s00348-016-2251-4) contains supplementary
material, which is available to authorized users.
* P.-T. Brun
pierrethomas.brun@gmail.com
1 MIT, Cambridge, MA, USA
2 IMPA/National Institute of Pure and Applied Mathematics,
Est. D. Castorina, 110, Rio de Janeiro RJ 22460-320, Brazil
3 Department of Mathematical Sciences, University of Bath,
Bath BA2 7AY, UK
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... For instance, Zhang and Cox (1994) extracted the surface topography based on a system of color-coded collimated beams that emerge below the surface. Other approaches, including the present study, relate the surface topography to the optical deformation of a known pattern when it is imaged through a deformed surface (Fouras et al. 2008;Gomit et al. 2013;Moisy et al. 2009;Ng et al. 2011;Murase 1992;Damiano et al. 2016;Morris and Kutulakos 2011;Li et al. 2021). In this approach, the imaged patterns are either random-speckle or regular (e.g., grid) patterns, and they can either be fixed in time (e.g., patterns printed on boundaries of test sections) or change with time (e.g., patterns employed to simultaneously perform velocimetry). ...
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We aim to describe a droplet bouncing on a vibrating bath. Due to Faraday instability a surface wave is created at each bounce and serves as a pilot wave of the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by {\it Couder et al} [Phys. Rev. Lett. {\bf 97}, 154101 (2006)] there have been many attempts to accurately reproduce the experimental results. Here we present a simple and highly versatile model inspired from quantum mechanics. We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to Helmholtz equation with Neumann boundary conditions on the obstacle(s) and outgoing conditions at infinity. For a single slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It stands for a promising candidate to account for the presence of any boundaries in the walkers'dynamics.
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Quantumlike statistics of deterministic wave-particle interactions in a circular cavity
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A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faraday waves confined to a circular cavity. Its solutions are investigated theoretically and numerically. The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk and the action per unit mass h/m of a quantum particle. The statistics of droplet position and speed are shaped by the cavity eigenmodes, in remarkable agreement with the solution of Schrödinger equation for a quantum particle in a similar potential well.
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The new wave of pilot-wave theory
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  • Aug 2015
Small drops bouncing across a vibrating liquid bath display many features reminiscent of quantum systems.
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Faraday pilot-wave dynamics: Modelling and computation
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A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet's bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.
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Investigation of a path-memory dynamics: a theoretical trial
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  • Dec 2014
Waves and particles are distinct objects at a macroscopic scale. The existence of walkers, drops bouncing on a vertically vibrated fluid bath is a surprising case of dual objects at our scale. The drop is self-propelled, piloted by the standing surface waves generated by its previous rebounds. These objects exhibit a rich dynamics relying on the concept of path memory. Indeed, the wave field results from the position of the past impacts left all along the walker trajectory. The memory is tunable at will by simply changing the vertical acceleration of the bath. A series of experiments have revealed the surprising dynamical behaviors of this dual drop-wave entity. In this PhD, we give a theoretical understanding of the temporal non local structure of walkers. We explore the dynamics of numerical walkers in a two-dimensional harmonic potential. We observe that the system only reaches a relatively limited set of stable attractors, quantized in both extension and mean angular momentum, in excellent agreement with the experimental results. We investigate how the different time scales are intertwined, which decouples the short-time acting propulsion from the build-up of coherent wave structures at much longer time scales. We analyze the non-local mechanism revealing the internal symmetries of the walker which drives the convergence of the dynamics to a set of low-dimensional eigenstates.
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A model for Faraday pilot waves over variable topography
Article
  • Jan 2017
Couder et al. ( Nature , vol. 437 (7056), 2005, p. 208) discovered that droplets walking on a vibrating bath possess certain features previously thought to be exclusive to quantum systems. These millimetric droplets synchronize with their Faraday wavefield, creating a macroscopic pilot-wave system. In this paper we exploit the fact that the waves generated are nearly monochromatic and propose a hydrodynamic model capable of quantitatively capturing the interaction between bouncing drops and a variable topography. We show that our reduced model is able to reproduce some important experiments involving the drop–topography interaction, such as non-specular reflection and single-slit diffraction.
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Non-specular reflection of walking droplets
Article
  • Oct 2016
Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.
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Modeling the vertical motion of drops bouncing on a bounded fluid reservoir
Article
  • Mar 2016
We present a first-principles model of drops bouncing on a liquid reservoir. We consider a nearly inviscid liquid reservoir and track the waves that develop in a bounded domain. Bouncing drops are modeled as vertical linear springs. We obtain an expression for the contact force between drop and liquid surface and a model where the only adjustable parameter is an effective viscosity used to describe the waves on the reservoir’s surface. With no adjustable parameters associated to the drop, we recover experimental bouncing times and restitution coefficients. We use our model to describe the effect of the Bond, Ohnesorge, and Weber numbers on drops bouncing on a stationary reservoir. We also use our model to describe drops bouncing on an oscillated reservoir, describing various bouncing modes and a walking threshold.
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