Rotating polygonal depression soliton clusters on the inner surface of a liquid ring

We report an experimental observation of rotating depression soliton sets on the inner surface of a viscous liquid ring, carrying background waves. These occur within a rotating shallow layer of oil inside a stationary cylindrical container. The solitons are organized either in single, two, or regular polygonal (triangle and hexagon) clusters; they travel in unison at a higher speed than the background traveling waves. The spectral power density reveals a possible energy exchange between the soliton clusters and the background mixed radial-azimuthal modulations through wave radiation.

Figure 1

Schematic of the experimental setup: The present tests were conducted using Spindle Mobil Velocite Oil No. 6. The height of the (quiescent) oil layer $h_0$ was 10 mm, and the disk and tank radii were $R_d=141\mathrm{mm}$ and $R_t=142\mathrm{mm}$, respectively.

Figure 2

Formation of the oil ring and the evolution of its inner free surface into a polygonal pattern through traveling waves with increasing disk speed: (a) disk speed = 63 rpm, (b) disk speed = 99 rpm, (c) disk speed = 126 rpm, and (d) disk speed = 155 rpm. Initial oil height $h_0=10\mathrm{mm}$.

Figure 3

Cluster of solitary waves observed at $h_0=10\mathrm{mm}$: (a) one solitary (disk speed = 114 rpm) wave, (b) two solitary waves (disk speed = 117 rpm), (c) three solitary waves (disk speed = 126 rpm), and (d) six solitary waves (disk speed = 126 rpm).

Figure 4

Solitary wave profiles: (a) one solitary wave, (b) two solitary waves, (c) three solitary waves, and (d) six solitary waves.

Figure 5

Unwound surface contours depicted in the space-time ($\theta ,t$) plane. Soliton clusters (brighter) on background traveling waves: (a) one solitary wave, (b) two solitary waves, (c) three solitary waves, and (d) six solitary waves.

Figure 6

Power spectra $S_{\eta }(k,\omega$) of the inner interface of the fluid ring, $\eta (x,t)$: (a) one solitary wave, (b) two solitary waves, (c) three solitary waves, and (d) six solitary waves.

Figure 7

Spatiotemporal wave dynamics of the inner surface of the ring: (a) one solitary wave, (b) two solitary waves, (c) three solitary waves, and (d) six solitary waves.

Figure 8

Curve fitting of the typical experimental soliton by ${\mathrm{sech}}^2$ profiles: (a) in time and (b) in space.