Coexistence and Transition between Shear Zones in Slow Granular Flows

We report experiments on slow granular flows in a split-bottom Couette cell that show novel strain localization features. Nontrivial flow profiles have been observed which are shown to be the consequence of simultaneous formation of shear zones in the bulk and at the boundaries. The fluctuating band model based on a minimization principle can be fitted to the experiments over a large variation of morphology and filling height with one single fit parameter, the relative friction coefficient ${\mu }_{\mathrm{rel}}$ between wall and bulk. The possibility of multiple shear zone formation is controlled by ${\mu }_{\mathrm{rel}}$. Moreover, we observe that the symmetry of an initial state, with coexisting shear zones at both side walls, breaks spontaneously below a threshold value of the shear velocity. A dynamical transition between two asymmetric flow states happens over a characteristic time scale which depends on the shear strength.

Figure 1

(a) The experimental setup and its top view (inset). (b) Schematic of the side view. $R_i=57\mathrm{mm}$ and $R_o=99\mathrm{mm}$. (c) Velocity profiles at different values of $H$ and $d_{\mathrm{bulk}}$ (in mm), and $\eta$. Symbols are experimental data, dashed curves are obtained from the variational approach [Eq. (1)], and solid curves are the fits with Eq. (3). The gray color denotes the localized profiles at the two extreme limits of $H$.

Figure 2

(a) Radial dependence of the strain rate. The same symbols and lines as in Fig. 1c are used. (b) ${\mu }_{\mathrm{rel}}$ obtained from the best fit of Eq. (1) to the data at different $H$. The horizontal solid lines indicate the mean values, and the dashed lines are the best linear fits. (c) The rate of energy dissipation $\chi$, scaled by the maximum dissipation of the bulk profile ${\chi }_{\mathrm{ref}}$, versus $H$. The dissipation of the wide shear zone (dashed line) is compared to that of the localized shear zone at the outer cylinder at different values of ${\mu }_{\mathrm{rel}}$ (solid lines).

Figure 3

Phase diagram of shear zone coexistence for $d_{\mathrm{bulk}}=2\mathrm{mm}$. The squares, circles, and stars denote the shear zone at the outer and inner cylinders and in the bulk, respectively. The gray shaded regions denote the values of (${\mu }_{\mathrm{rel}}$, $H$) for which the experiments are performed. Insets: Typical velocity fields (sketched with arrows) and the corresponding strain rates (red curves) (both corrected for the radial dependence).

Figure 4

(a),(b) Evolution of the decay exponents $b_o$ and $b_i$ with $H$. The lines indicate (a) exponential fits and (b) fits given by Eq. (4). (c) ${\mu }_{\mathrm{rel}}$ (averaged over all filling heights) versus the wall roughness $\eta$. The curves are exponential saturation fits as guides to the eye.

Figure 5

The surface flow profile at $\Omega =0.1\mathrm{rad}/\mathrm{s}$ (solid curve) and two typical profiles at $\Omega =4\times {10}^{-3}\mathrm{rad}/\mathrm{s}$ (dashed curves), and the corresponding strain rates (inset).