Coarsening of axial segregation patterns of slurries in a horizontally rotating drum

Segregation structures of granular mixtures in rotating drums represent classical examples of pattern formation in granular material. We investigate the coarsening of axial segregation patterns of slurries in a long horizontally rotating cylinder. The dynamics and the three-dimensional geometry of the segregation structures are analyzed with optical methods and nuclear magnetic resonance imaging. Previous studies have mainly considered global statistical features of the pattern dynamics. In order to get insight into driving mechanisms for the coarsening process, we focus on the details of the dissolution of individual bands. We treat the coarsening as a consequence of interactions of adjacent bands in the pattern, which are determined by their geometrical relations. In addition to initially homogeneous mixtures, which evolve to spontaneously formed patterns, we study the evolution of specially prepared simple initial states. The role of the three-dimensional geometry of the axial core in the dissolution process of segregation bands is demonstrated. Relations between geometry and dynamic processes are established, which may help to find the correct microscopic models for the coarsening mechanism.

Figure 1

(Color online) Sketch of the experimental setup. The drum rests on two ball bearings on each side; it is driven by a dc motor. The rotation speed is monitored by a stroboscopic counter. The drum is illuminated from the top. A digital camera at the same height as the drum, positioned in a distance of $150\mathrm{cm}$, automatically takes photos at regular intervals. The width of the drum shown is $\approx 1\mathrm{m}$. For a typical experiment, the observation time ranges between one day and one week.

Figure 2

Axial segregation pattern of a 50%:50% mixture of glass beads, after $20\min$ at $15\mathrm{rpm}$. Dark areas represent bands of small particles; bright bands contain large particles at the surface. The width of the drum is $\approx 1\mathrm{m}$.

Figure 3

Spatiotemporal presentation of the stripe evolution for mixture $L$ and the $994\mathrm{mm}$ drum for three different rotation speeds; (a) $5\mathrm{rpm}$, (b) $15\mathrm{rpm}$, (c) $30\mathrm{rpm}$. Time runs from top to bottom.

Figure 4

(Color online) Number of stripes $N$ in a $994\mathrm{mm}$ long half-filled tube (mixture $L$) as a function of time for different rotation speeds of the drum. Plot (a) contains data for intermediate rotation rates $\left(10\text{to}22.5\mathrm{rpm}\right)$. The data follow a logarithmic decay, with few exceptions for high rotation rates. The plot (b) collects the data for high rotation rates; with increasing rotation rate the tendency to form a plateau (stationary pattern) is exhibited. There are data from 21 individual experimental runs in plot (a), and in addition 10 runs in plot (b).

Figure 5

Spatiotemporal plot of the evolution of segregation patterns in a bidisperse mixture of small and large glass beads (mixture $L$ at $20\mathrm{rpm}$). Note that in an array of stripes, usually the narrow dark stripes dissolve first, their material is distributed among the neighboring stripes. The box highlights a selection that is discussed in detail in the text. Some slight intensity fluctuations in the time domain are artifacts due to contrast variations between individual photos, which have not been corrected in this plot.

Figure 6

Spatiotemporal plot of the evolution of segregation patterns in a mixture of sand and large (diameter $2\mathrm{mm}$) glass beads, $15\mathrm{rpm}$. Both the dark and bright bands can dissolve in the coarsening process. The large particles contained in the bright stripes can penetrate the dark zones of polydisperse irregular small-grained particles.

Figure 7

Spatiotemporal plot of the evolution of single prepared stripes of fine beads surrounded by large beads. (a) Small stripe; the material dissolves and forms a core structure. (b) The core extends to the lateral end of the drum where the small beads are collected in a stripe again. The tube length is $662\mathrm{mm}$. The stripes have initial widths of $4\mathrm{cm}$ (a) and $12\mathrm{cm}$ (b), respectively, and their right edge is initially $20\mathrm{cm}$ from the tube end.

Figure 8

The image on the top shows a tube with two prepared stripes of small-sized beads of $105\mathrm{mm}$ and $140\mathrm{mm}$ width, respectively. During the first rotations, the axial core starts to form and the stripe widths shrink by a few percent. The tube length is $662\mathrm{mm}$, the images show the full length of the tube on the horizontal axis. The initial distance of the stripes is (a) $40\mathrm{mm}$, (b) $80\mathrm{mm}$, (c) $120\mathrm{mm}$, and (d) $160\mathrm{mm}$. In all these experiments, the small stripe disappears.

Figure 9

Space-time plots of experiments in a $662\mathrm{mm}$ tube, initial stripe distance of $40\mathrm{mm}$ in all experiments, and different initial stripe widths: (a) $60\mathrm{mm}$ and $80\mathrm{mm}$, mixture $L$ (1:3), $15\mathrm{rpm}$; (b) $60\mathrm{mm}$ and $80\mathrm{mm}$, mixture $L$ (1:3), $30\mathrm{rpm}$; (c) $80\mathrm{mm}$ and $60\mathrm{mm}$, mixture $S$ (1:2), $20\mathrm{rpm}$; (d) $93\mathrm{mm}$ and $47\mathrm{mm}$, mixture $L$ (1:3), $15\mathrm{rpm}$.

Figure 10

Prepared stripe pair in a $662\mathrm{mm}$ tube, stripe distance $120\mathrm{mm}$; the initial stripe widths are $105\mathrm{mm}$ and $140\mathrm{mm}$, mixture $L$ (1:3), $15\mathrm{rpm}$. The image on the left shows the stripe dissolution; the image on the right presents the position of the inner edge of the left-hand stripe as a function of the number of rotations. The asymptotic dissolution velocity is taken from the tangent to the slope (straight line).

Figure 11

Velocity $v$ of the dissolution of the stripes as a function of the stripe distance $d$. Data are from stripe pairs with initial width ratios of approximately 3:4 (solid spheres) and 1:2 (solid squares); one datum point (open triangle) is from an experiment where the broader stripe dissolves; all other data originate from the dissolution of the narrower stripes.

Figure 12

Inverse transport rate $1∕v$ in two-stripe systems as a function of the stripe distance $d$ for different initial stripe widths. The solid line suggests a $v\propto 1∕d$ dependence, which obviously holds independent of which stripe dissolves.

Figure 13

(a) Experiment with a prepared array of seven equidistant identical stripes (mixture $S$, radius ratio 1:2). (b) Visualization of tracer particles by image processing (see the text). The tracers stay almost exclusively in the stripe where they have been deposited. Exchange with neighbor stripes is negligible unless a stripe dissolves.

Figure 14

MR images (sagittal sections) of a prepared two-stripe pattern (for details, see text), (a) before rotation, (b) after 225 rotations, (c) after 4275 rotations, and (d) after 4785 rotations. The right-hand stripe has disappeared optically after 4740 rotations. The bottom picture sketches the position of the MR image relative to the prepared pattern.

Figure 15

MR images (sagittal cross section) of a prepared two-stripe pattern after 225 rotations (a) and after 4785 rotations (b), in the region to the right of the right-hand stripe.

Figure 16

MR image (sagittal cross section) of a prepared two-stripe pattern after 5500 rotations (at the same position as in Fig. 14). The right-hand stripe has disappeared.

Figure 17

MR images (axial sections) of a prepared stripe pattern consisting of two bands of small beads ($8\mathrm{cm}$ and $6\mathrm{cm}$), separated by a $4\mathrm{cm}$ band of large beads. The three axial slices are measured in the band of small beads (left), near the edge of the two bands (middle), and in the center of the band with large beads (right). The white lines in (b) and (c) mark the regions, which consist mainly of small-sized beads. The bottom image sketches the position of the three slices.

Figure 18

Relative area covered by the small-sized beads in the axial MRI slices along the axis (related to the drum cross-section area). A coverage of 50% means that there are only small beads in the respective cross section; the rest is granulate-free (water). The development of the size and position of the two stripes in this experiment is shown in Fig. 19 where the instants at which MRI data have been sampled are marked by vertical lines.

Figure 19

Position of the bands in the prepared two-band pattern for the MRI measurements as seen from the surface, and their relation to the MRI data of Fig. 18. Open and solid circles symbolize the edges of the bands as determined optically. Shaded areas represent the regions of small particles (dark) and large particles (bright); they are meant merely to guide the eye. Vertical lines mark times where the cross sections of Fig. 18 have been taken.