Research ArticlePHYSICS

A hydrodynamic analog of Friedel oscillations

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Science Advances  15 May 2020:
Vol. 6, no. 20, eaay9234
DOI: 10.1126/sciadv.aay9234
  • Fig. 1 Walker dynamics in the vicinity of a circular well.

    (A) Oblique view of a walking drop passing over a submerged circular well (dashed line) (see movie S1). (B) Plan view and schematic cross section of the experimental setup. (C) Walker trajectories with γ/γF = 0.990 and free-walking speed v0 = 7.1 mm s−1. The arrows denote the direction of motion and trajectories are color coded according to drop speed. A total of 449 trajectories were collected, culminating in speed modulations shown in (iii) (see movie S2). (D) Experimental trajectories colored according to their impact parameter yi. These trajectories are obtained by rotating those shown in (C, iii) until the drop’s initial direction is parallel to the x axis. α then denotes the scattering angle. (E) Scattering angle α versus impact parameters yi for experimental (yellow dots) and simulated (solid lines) trajectories with walkers of different size and speed at γ/γF = 0.990. The dotted magenta line corresponds to the same drop as the blue solid line but at a lower memory, γ/γF = 0.970.

  • Fig. 2 Emergent statistical behavior.

    Top view illustrating the experimental (A) incoming and (B) outgoing drop trajectories, color coded according to speed v. Trajectories are obtained by splitting those shown in Fig. 1G at the point nearest the well center (dashed line). Concentric speed modulations appear in the outgoing phase. (C) Faraday waves observed above the well at threshold γ = γF. Note the spatial correspondence between the Faraday wave extrema and the speed modulations evident in (B). (D) Incoming and (E) outgoing trajectories, color coded according to speed, corresponding to the slowest walker with γ/γF = 0.990 in Fig. 1F. Red arrows identify the outermost trajectory crossing the well. White arrows indicate the trajectory with a radial approach. Dependence of the normalized speed v on radius r for (F) incoming and (G) outgoing walkers of different size and speed. The gray area denotes the well’s extent. (H and I) Histograms of the drop’s radial position corresponding to the data shown in (F) and (G). The bin size is λFh/13. The histograms have been normalized by their respective height at r/λFh=2, corresponding to the first speed minimum outside the well observed in (G). (J and K) Two-dimensional histograms (normalized by the histogram height at the center of the well) resulting from the experimental (B) and simulated (E) outgoing trajectories. Asterisks denote normalized quantities.

  • Fig. 3 Wave-mediated interaction.

    (A) Experimental and (B) simulated walker wave field η during its interaction with a submerged well (solid circle). Snapshots illustrate the walker (i) approaching the well in straight-line motion, (ii) spiraling inward, (iii) exciting localized large-amplitude waves as it crosses the well, and (iv and v) exiting the well along a straight trajectory. (C) Well-induced wave perturbation ξ=ηη̄ (normalized by the instantaneous maximum wave amplitude ηmax = max ∣η∣) obtained by subtracting from the wave field η shown in (B) the computed wave field η̄ of a drop following the same trajectory in the absence of the well. A sliding beam-like wave mode emerges as the drop spirals inward (ii). Along the outgoing trajectory, the drop crosses a spatially fixed wave field centered on the well (iv). The resulting speed variations give rise to the wavelike statistics evident in Fig. 2. The simulated walker corresponds to γ/γF = 0.990, v0 = 4.4 mm s−1, and yi/λFh=6. See movies S3 to S5.

  • Fig. 4 Well-induced wave perturbation deduced from simulations.

    3D visualization of the well-induced wave perturbation as the drop (A) spirals inward, (B) crosses the well, and (C) exits in a rectilinear fashion. Solid blue lines projected onto the vertical (xz and yz) planes illustrate the evolution of the perturbation amplitude beneath the drop. The drop approaches the well in an ever-deepening trough and then exits radially across a field of waves centered on the well. The simulated walker corresponds to that shown in Fig. 3 (B and C). See movie S5.

  • Table 1 Range of drops considered in our study.

    The values reported for γ/γF = 0.990 are from experimental observations with h = 1.6 mm. The case with γ/γF = 0.970 corresponds to our simulations in which the impact phase φ was adjusted to match v0, and so drop inertia, for the smallest drop. This adjustment allowed us to assess the role of memory on the range of interaction, independent of drop inertia (see Fig. 1E).

    R (mm)γ/γFv0 (mm/s)φ/2π
    0.3550.9904.40.259
    0.3755.80.266
    0.3907.10.269
    0.4007.60.277
    0.42510.00.286
    0.3550.9704.40.268

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