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Surface topography measurements of the bouncing droplet experiment

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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.
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
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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.
... (b) Dependence of a bouncing droplet's touch-down (red) and take-off (blue) times on the bath's vibrational acceleration γ/γ F . The kinematic-match model of the droplet's vertical dynamics (curves) [193] (see section 4.4) is compared with experiments (shaded regions) [194]. The vertical lines indicate the transitions between different bouncing modes in experiment (solid) and simulations (dashed). ...
... (a), (b) Reprinted by permission from Springer Nature Customer Service GmbH: Experiments in Fluids. [194] © 2016. (c), (d) Reproduced with permission from [44]. ...
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... Despite its simplicity, Moisy et al. (2009) reported a precision of 1 μm for a view field of 10 cm . This method has been successfully used in the case of Faraday waves (Eddi et al. 2011;Lau et al. 2020), wave-droplet interactions (Eddi et al. 2009;Damiano et al. 2016) and floating objects (Poty et al. 2014;Metzmacher et al. 2017;Vandewalle et al. 2020). ...
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... The displacement field u(x, y) was determined using the MATLAB-based FCD code provided at [24] and subsequently ∇h(x, y) was inverted using the intgrad2 function in MATLAB [25]. Finally, the surface height profile h(x, y) was processed using a spatial band-pass filter, primarily to remove any nonphysical large wavelength distortions predicted by the method as described in [26]. ...
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... The displacement field u(x, y) was determined using the MATLAB-based FCD code provided at [21] and subsequently ∇h(x, y) was inverted using the intgrad2 function in MATLAB [22]. Finally, the surface height profile h(x, y) was processed using a spatial band-pass filter, primarily to remove any nonphysical large wavelength distortions predicted by the method as described in [23]. ...
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... This functional form serves a good approximation to the droplet-generated waves in the vicinity of the droplet. 35,58,59 Choosing a Bessel function wave form W(x) = J 0 (x) results in the wave form gradient f(x) = J 1 (x). Substituting in Eq. (5) gives ...
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