Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction

Matthieu Labousse, Stéphane Perrard, Yves Couder, and Emmanuel Fort

Phys. Rev. E 94, 042224 – Published 27 October 2016

Abstract

The back-reaction of a radiated wave on the emitting source is a general problem. In the most general case, back-reaction on moving wave sources depends on their whole history. Here we study a model system in which a pointlike source is piloted by its own memory-endowed wave field. Such a situation is implemented experimentally using a self-propelled droplet bouncing on a vertically vibrated liquid bath and driven by the waves it generates along its trajectory. The droplet and its associated wave field form an entity having an intrinsic dual particle-wave character. The wave field encodes in its interference structure the past trajectory of the droplet. In the present article we show that this object can self-organize into a spinning state in which the droplet possesses an orbiting motion without any external interaction. The rotation is driven by the wave-mediated attractive interaction of the droplet with its own past. The resulting “memory force” is investigated and characterized experimentally, numerically, and theoretically. Orbiting with a radius of curvature close to half a wavelength is shown to be a memory-induced dynamical attractor for the droplet's motion.

Received 1 June 2016

DOI:https://doi.org/10.1103/PhysRevE.94.042224

Physics Subject Headings (PhySH)

Biological self-organization

Nonlinear waves

Chaos & nonlinear dynamics Density functional theory development

Fluid Dynamics

Authors & Affiliations

Matthieu Labousse^1,2,*, Stéphane Perrard², Yves Couder², and Emmanuel Fort^1,†

¹Institut Langevin, ESPCI Paris, PSL Research University, CNRS, 1 rue Jussieu, 75005 Paris, France
²Matière et Systèmes Complexes, Université Paris Diderot, CNRS, Sorbonne Paris Cité, Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France

^*Present address: Matériaux et Phénomènes Quantiques, Université Paris Diderot—CNRS, Sorbonne Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France.
^†Corresponding author: emmanuel.fort@espci.fr

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Vol. 94, Iss. 4 — October 2016

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Images

Figure 1
Experimental realization of self-orbiting modes. (a) Scheme of the core of the experimental setup. The whole bath is submitted to a global magnetic field generated by two coils that are not shown here. A magnetic field maximum is created by the concentration of flux lines due to a sharp rod of pure iron. (b) Top view of the walker in a self-orbiting mode in a long memory regime $(Me \approx 140)$ . The droplet has a diameter $D \approx 0.7 mm$ and a velocity $V \approx 7 mm s^{- 1}$ . The magnetic field that initially trapped the walker, had been turned off 3 s before this picture was taken. The recent trajectory is superimposed. The nondimensional radius of the free orbit is $R_{c} / λ_{F} = 0.385 \pm 0.01$ .
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Figure 2
Typical trajectories observed when the confinement is turned off for two values of the memory parameters. (a) Short memory $(Me \approx 10)$ . (b) Long memory $(Me \approx 140)$ (see Movie S1 [36]). (c,d) Temporal evolution of the normalized trajectory radius $R_{c} / λ_{F}$ along these two trajectories (black: magnetic field on, green: transition time, and red: no central force.
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Figure 3
Numerical investigation of the stability. (a) Temporal evolution of the normalized radius of curvature $R_{c} / λ_{F}$ along three trajectories (pink, $Me = 11$ ; blue, $Me = 15$ ; black $Me = 27$ . The central force is turned off at $t = 0 s$ . Insets (b,c) associated trajectories for $Me = 11$ , and $Me = 27$ . (d) Temporal evolution of the normalized radius of curvature $R_{c} / λ_{F}$ for a fixed value of the memory $(Me = 27)$ and several values of noise amplitude expressed in radius unit $ɛ = σ / R$ , with $R$ the instantaneous radius: pink, $ɛ = 2 \times 10^{- 2}$ ; blue $ɛ = 1.6 \times 10^{- 2}$ ; red $ɛ = 1.2 \times 10^{- 2}$ ; black $ɛ = 4 \times 10^{- 3}$ . (e) Probability of capture $p$ in the self-orbiting mode (color code on the right) when varying the noise amplitude ε and the memory parameter $Me$ . Each probability is calculated over 20 realizations.
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Figure 4
Theoretical characterization of the forces acting in self-orbits. (a) The computed orbit and wave field at long memory $(Me = 100)$ . The dotted lines are the zeros of $J_{0}$ . (b) The wave-induced radial forces and the centrifugal pseudoforce $F_{inert}$ (dashed red line) as a function of the normalized orbit radius $R / λ_{F}$ . The estimation wave-induced force is shown for several values of the memory parameters (solid line, from light gray to black: $Me = 11$ , 15, 19, 23, 27) with $\frac{C h_{0} k_{F}}{m_{eff}} = 8.05 \times 10^{- 2} N / kg$ and $m_{eff} = 3.82 \times 10^{- 7} kg$ . Equilibrium is possible when the two curves cross. Vertical lines indicate the first zeros of the $J_{0}$ Bessel function $R_{0}$ and $R_{1}$ . The arrows indicate the stable orbits in the vicinity of $R_{0}$ and $R_{1}$ . (c) The possible orbits (black lines: stable solutions; gray: unstable solutions) (d) The evolution with memory of the contributions of the first Bessel modes to the tangential force $F_{Me}^{/ /}$ with an equilibrium speed $V = 10.9 mm / s$ which enables to estimate $γ = 7.8 \times 10^{- 6} kg / s$ . (e) The evolution with memory of the contributions of the first Bessel modes to the radial force $F_{Me}^{⊥}$ . The dashed lines represent the contribution of the first three centered Bessel modes. The vertical line indicates the critical memory parameter $M e_{c}$ ; the horizontal line shows the asymptotic value of $F_{Me}^{⊥} = F_{inert}$ when $J_{0} (k_{F} R) = 0$ .
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Figure 5
Energy stored in the global wave field and its modes. (a) The evolution of the global wave field energy of a walker with the memory. While it diverges for a free walker walking linearly (in red), it reaches a finite value for a self-orbiting mode (black line). The contribution of the first three centered Bessel modes are shown in dashed lines. The vertical line indicates the critical memory parameter $M e_{c}$ . The horizontal line indicates the energy of the self-orbit mode for long memory. (b) The surface profile at approximately the time of impact of the droplet with the surface. The colors distinguish the protrusions in red from the troughs in blue. (c,d,e) show the contributions to the surface height of the first three centered Bessel wave modes $J_{0}, J_{1}$ , and $J_{2}$ , respectively. The radius of the orbit (black line) is slightly larger than that of the first nodal line of $J_{0}$ (dashed black line). As shown in Fig. 4 the centripetal force exerted on the drop is essentially due to $J_{0}$ while the tangential force is due to $J_{1}$ and $J_{2}$ .
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Figure 6
Spontaneous transient self-orbiting loops. At very long memory, the trajectory of a confined walker exhibits generically the spontaneous formation of multiple transient self-orbiting loops. (a) Trajectory in a wide potential well at $Me = 500$ . (b) Photograph of the wave field structure in this regime showing, in the vicinity of the drop, the self-organized structure of a small self-orbit. (c) The same transient self-orbits observed when a long memory walker is confined in a circular corral.
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