Faraday-Wave Contact-Line Shear Gradient Induces Streaming and Tracer Self-Organization: From Vortical to Hedgehoglike Patterns

In this Letter, we experimentally demonstrate self-organization of small tracers under the action of longitudinal Faraday waves in a narrow container. We observe a steady current formation dividing the interface in small cells given by Faraday-wave symmetries. These streaming currents rotate in each cell, and their circulation increases with wave amplitude. This streaming flow drives the tracers to form patterns, whose shapes depend on the Faraday-wave amplitude: From low to high amplitudes, we find tracers dispersed on vortices, narrow rotating rings, and a hedgehoglike pattern. We first describe the main pattern features and characterize the wave and tracers’ motion. We then show experimentally that the main source of the streaming flow is the spatiotemporal-dependent shear at the wall contact line created by the Faraday wave itself. We end by presenting a 2D compressible advection model that considers the minimal ingredients present in the Faraday experiment, namely, the stationary circulation, the stretching component due to the oscillatory wave, and a steady converging field, which combined produce the observed self-organized patterns.

Figure 1

Experimental setup and hedgehoglike pattern. (a) An electromechanical shaker vibrates vertically the acrylic container with a water-surfactant liquid mixture and a small amount of silver-coated hollow glass microspheres. Side (b) and top (c) views of the patterns observed on a single Faraday wavelength. Solid (dashed) lines indicate antinodes (nodes). (c) Arrows depict streaming flows.

Figure 2

Elementary hedgehoglike pattern. (a) Pattern at a given phase for various wave amplitudes $A$. (b) Breathing motion during a single oscillation (inverse gray scale); the formation of a new spike is highlighted (purple).

Figure 3

(a) Tracer velocities around a unit-cell rotating ring, for $A=2.1\mathrm{mm}$. Streaming velocities, shown as arrows, are measured at eight different locations around the ring, being much larger close to the wall. (b) Streaming speed averaged over the three near-wall points as a function of wave amplitude. Solid points correspond to measurements with well-developed spikes along the ring. The solid line is a quadratic fit for $A<2.4\mathrm{mm}$. Inset: Examples of instantaneous velocity components at a given point around the ring.

Figure 4

(a) Schematic illustration of the rotating disk setup. (b) Top view of the observed streaming pattern, $A=R\theta =2.67\mathrm{mm}$ and $f=2\mathrm{Hz}$. The dashed-line rectangle shows the PTV window. (c) Streaming speed as a function of disk oscillation amplitude $A$, for $f=2\mathrm{Hz}$ (Down pointing triangle) and $f=4\mathrm{Hz}$ (Circle). Inset: $v_s$ linear frequency dependence for fixed $A$. (d) $v_s$ as a function of $C\omega A^2/L$, with $L=R$ and $C=0.35$ for the disk and $L=\lambda$ and $C=2.5$ for Faraday data. The disk data collapse on a single curve, saturating at higher values. For both experiments, an $A^2$ scaling is observed for the lower-amplitude regime, shown as straight lines.

Figure 5

(a) Steady circulation, (b) periodic stretching, and (c) a steady converging field featured by a ring are the basic ingredients to generate hedgehoglike structures. The superposition of the velocity fields allows agglomeration of particles (d) and similar dynamics, including rotation, breathing, and formation of spikes in every cycle (e) as those observed on Faraday waves.