Spontaneous Orbiting of Two Spheres Levitated in a Vibrated Liquid

In the absence of gravity, particles can form a suspension in a liquid irrespective of the difference in density between the solid and the liquid. If such a suspension is subjected to vibration, there is relative motion between the particles and the fluid which can lead to self-organization and pattern formation. Here, we describe experiments carried out to investigate the behavior of two identical spheres suspended magnetically in a fluid, mimicking weightless conditions. Under vibration, the spheres mutually attract and, for sufficiently large vibration amplitudes, the spheres are observed to spontaneously orbit each other. The collapse of the experimental data onto a single curve indicates that the instability occurs at a critical value of the streaming Reynolds number. Simulations reproduce the observed behavior qualitatively and quantitatively, and are used to identify the features of the flow that are responsible for this instability.

Figure 1

The main panel shows the collapse of the onset data in terms of the dimensionless variables $A_r^{*}/d$ and $\delta /d$. The kinematic viscosity is in the range $1.13–2.43{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ for 1 mm glass spheres; $2.76–8.37{\mathrm{mm}}^2{\mathrm{s}}^{-1}$, 2 mm glass; $7.14–14.9{\mathrm{mm}}^2{\mathrm{s}}^{-1}$, 3 mm glass; and $6.46–15.6{\mathrm{mm}}^2{\mathrm{s}}^{-1}$, 2.8 mm bismuth shot. The inset shows the dependence of the scaled orbital angular velocity $\Omega /\omega$ on the scaled relative amplitude $A_r/\delta$ for representative data sets. The points at $\Omega /\omega =0$ are extrapolations; onsets determined from these points are shown by filled symbols on the main panel.

Figure 2

An image of the spheres taken from above when the fluid is vibrated but there is no orbital motion. Color is used to enhance contrast. The vibration is applied along the direction perpendicular to the plane of this image. The red and green streak lines indicate the steady streaming flow in the plane of the spheres. There is a strong outward flow from the contact point of the spheres and an inward flow elsewhere, as illustrated by the white arrows. The superimposed black lines show the corresponding streak lines obtained from simulation.

Figure 3

A representation, obtained from simulation, of the time-averaged vortex structure around the two spheres vibrated under conditions for which they are not rotating. The closed loops represent the curl of the velocity field and are schematic vortex lines as described in the text. The direction of the arrows indicates the sense of rotation, i.e., the direction of the curl of the velocity field. The double-headed arrow indicates the direction of vibration.

Figure 4

Data from simulations showing the instantaneous flow field and vorticity map in the equatorial plane of the spheres under conditions for which they orbit anticlockwise. The direction of vibration is normal to the plane of the figure. The small white arrows indicate the instantaneous flow and the background color map indicates the strength and sense of the vorticity. Large arrows indicate the direction of the orbital motion.