Figure 1

(a) Schematic side-view of our split-bottomed Couette cell, showing the stationary bottom disk and inner cylinder (both gray), and the rotating bottom ring (striped) and outer cylinder. A thin layer of felt between the rings assures smooth rotation. The inner and outer cylinder radii are fixed at 65 and 105 mm, respectively, while $R_s$ can be varied. (b) Setup in disk geometry. (c) Top view of the filled set up in the Couette geometry, where the rectangle indicates the area recorded by the video camera.Reuse & Permissions

Figure 2

Main features of the normalized angular surface velocity in the Couette geometry for 0.3 mm glass beads (mixture I) and $R_s=85\mathrm{m}\mathrm{m}$. (a) $\omega (r)$ for a range of equidistant heights $h=3,8,\dots ,53\mathrm{m}\mathrm{m}$ (right to left). (b) Contour plots of $\omega (r)$, where the symbols correspond to, from left to right, $\omega =0.1,0.25,0.5,0.75$, and $0.9$. The curve indicates the strain rate maximum and shows the rapid qualitative change of the profiles when the inner cylinder is approached.Reuse & Permissions

Figure 3

(a) Collapse of all 35 bulk profiles obtained for $H=4,5,6,\dots ,38\mathrm{m}\mathrm{m}$ (mixture I, $R_s=85\mathrm{m}\mathrm{m}$, as in Fig. 2) when plotted as function of the dimensionless coordinate $\lambda ≔(r-R_c)/W$. The gray curve is an error function. (b) Differences $\xi$ between bulk profiles and fits to Eq. (1). (c) Differences $\xi$ between bulk profiles and fits to $1/2+1/2\tanh (l\lambda )$, where $l\approx 1.20274$ compensates for the difference in scale between error function and hyperbolic tangent. The gray curve shows $[\mathrm{e}\mathrm{r}\mathrm{f}(\lambda )-\tanh (l\lambda )]/2$. (d) Left tail of the velocity profile in log-lin scale, compared to the best fit to either a hyperbolic tangent and error function (gray curves). Dots, crosses, and plusses are obtained for time lags 0.3, 3, and 30 s, respectively.Reuse & Permissions

Figure 4

Shear zone positions $R_c$ versus height, where $H$ is restricted to the universal regime detected by a ${\chi }^2$ test. (a) Comparison between $R_c(H)$ for glass beads I-IV (closed symbols), nonspherical particles V-VII (open symbols) and Eq. (2) (curve). For all runs, $R_s$ is 95 mm. (b) $R_c(H)$ for mixture III and $R_s=95$, 85, and 75 mm in the Couette geometry (open symbols) and $R_s=95$, 65, and 45 mm in the disk geometry (closed symbols), compared to Eq. (2) (curves).Reuse & Permissions

Figure 5

Width of the shear zones versus $H$. (a) Linear and (b) log-log plots of $W$ for spherical glass beads of increasing sizes for $R_s=95\mathrm{m}\mathrm{m}$ (mixture I-IV). The width grows with particle size. The straight lines in (b) have slopes $1/2$, $2/3$, and $1$, showing that $W$ grows faster than $\sqrt{H}$ but slower than $H$. (c) Irregular particle shapes diminish $W$ substantially, as can be seen by comparing the widths for the plastic flakes (V/pluses), aluminum oxide beads (VI/open circles), and coarse sand (VII/crosses) to those of the glass beads shown in panel (a). (d) $W$ for mixture III and $R_s=95$, 85, and 75 mm (Couette geometry; open symbols) and $R_s=95$, 65, and 45 mm (disk geometry; closed symbols). The strong upward deviations observed in the disk geometry coincide with the symmetry breaking of the velocity profiles (see text). Apart from this, the setup geometry does not affect $W$.Reuse & Permissions

Figure 6

Measurements of $\omega (r)$ inside the material for the disk geometry and $R_s=95\mathrm{m}\mathrm{m}$. (a) Initial pattern at $H_b=31\mathrm{m}\mathrm{m}$. (b) The same pattern after adding material up to $H=49\mathrm{m}\mathrm{m}$, rotating, and removing the upper layers. (c) Velocity profiles for $H=49\mathrm{m}\mathrm{m}$, and $H_b=7,13,19,\dots ,49\mathrm{m}\mathrm{m}$ (from right to left). (d) $R_c(H_b)$ for $H=25\mathrm{m}\mathrm{m}$ (plusses), $H=37\mathrm{m}\mathrm{m}$ (crosses), and $H=49\mathrm{m}\mathrm{m}$ (circles) compared to $R_c(H)$ given by Eq. (2) (curve). (e) Corresponding $W(H_b)$.Reuse & Permissions