Dynamics of granular band formation: Long-term behavior in slurries, parameter space, and tilted cylinders

Band formation (axial segregation) and subsequent coarsening of bidisperse mixtures in long circular tumblers is well documented for the case where the cylinder is at a single fill level and the interstitial fluid is air. However, little information is available for a range of fill levels, nor is the effect of rotational speed on segregation clear. Moreover just a handful of studies have focused on slurry systems, where the interstitial fluid is a liquid. This is precisely the parameter space covered in this study. Experiments are conducted using a 2:1 mixture of 882 and $272\mu \mathrm{m}$ glass beads with water as the interstitial fluid. Several different phenomena are uncovered. Results indicate that bands are less likely to form at low rotational speeds and low fill levels. As the fill level and rotational speed increase, more bands form and they form more quickly. However, at fill levels near 50% and high rotational speeds the bands contain a mixture of particles rather than being relatively pure. Furthermore, the evolution of the core of small beads that forms deep in the bed depends on the fill level and the rotational speed. For certain fill levels and rotational speeds, the core remains prominent as bands form, while in other cases the core disappears entirely between bands. Finally, when the tumbler is tilted so that the fill level varies from 14% at one end to slightly more than half full at the other end, the bands and core that form locally qualitatively correspond with those that would form for the corresponding fill level in a horizontal cylinder.

Figure 1

Image of an experimental run cropped so that only the granular bed is shown. Smaller particles (dark) form a core as well as bands between bands of larger particles (white and gray). Illumination is from the bottom of the tumbler.

Figure 2

Example of a space-time series (2000 revolutions from top to bottom). Cropped images are appended atop each other to show the band dynamics of an entire experimental run, as seen at the surface of the granular bed.

Figure 3

Phase portrait of space-time plots as a function of the angular velocity and the fill level.

Figure 4

Experimental space-time series for $26\mathrm{rpm}$ and 45% fill level for the first 400 revolutions. The “fuzzy” interfaces between dark and white bands are “wavy” or “unstable.”

Figure 5

Top: The effect of rotation rate on the area of bands rich in smaller particles (dark bands in the experiments) after 2000 revolutions. The lines are least squares fits of the data for various fill levels. Bottom: The evolution of the area of bands rich in smaller particles over time for a fill level of 40%.

Figure 6

The effect of fill level and rotation rate on the number of dark bands (rich in smaller particles). The upper plot averages the number of bands over all rotation rates after 2000 revolutions (error bars correspond to plus or minus one standard deviation). The lower plot shows that lower fill levels result in few bands and higher fill levels result in many. The lines are two linear fits at different ranges of rotation rates for each fill level.

Figure 7

Top three images are of experiments showing the three types of core behavior. Top image: Radial segregation with no banding. Second image: Banding with continuous central core connecting the bands. Third image: Banding with no core between bands (completely separated bands). The bottom three images are axially cropped and thresholded images of the top three images to emphasize where a core exists and does not exist. The arrow in the second to last image indicates the flow of small particles when a band disappears by merging into an adjacent band.

Figure 8

Comparison of tilted cylinder experiments with a rotation rate of $8\mathrm{rpm}$ to horizontal experiments at the same rotation rate. The left end of the cylinder has the lowest fill level. Top: Space-time series for the horizontal experiments at various fill levels compared to the inclined experiment. Bottom: Images of horizontal experiments for the same fill levels compared to the inclined experiment.

Figure 9

Space-time series for tilted cylinder experiments with rotation rates of $16\mathrm{rpm}$ (left) and $24\mathrm{rpm}$ (right). The left end of the cylinder has the lowest fill level.