Shear-driven size segregation of granular materials: Modeling and experiment

Granular materials segregate by size under shear, and the ability to quantitatively predict the time required to achieve complete segregation is a key test of our understanding of the segregation process. In this paper, we apply the Gray-Thornton model of segregation (developed for linear shear profiles) to a granular flow with an exponential shear profile, and evaluate its ability to describe the observed segregation dynamics. Our experiment is conducted in an annular Couette cell with a moving lower boundary. The granular material is initially prepared in an unstable configuration with a layer of small particles above a layer of large particles. Under shear, the sample mixes and then resegregates so that the large particles are located in the top half of the system in the final state. During this segregation process, we measure the velocity profile and use the resulting exponential fit as input parameters to the model. To make a direct comparison between the continuum model and the observed segregation dynamics, we map the local concentration (from the model) to changes in packing fraction; this provides a way to make a semiquantitative comparison with the measured global dilation. We observe that the resulting model successfully captures the presence of a fast mixing process and relatively slower resegregation process, but the model predicts a finite resegregation time, while in the experiment resegregation occurs only exponentially in time.

Figure 1

(Color online) Schematic of experimental apparatus (not to scale) showing initial configuration and coordinate system.

Figure 2

(Color online) (a) $H\left(t\right)$ for 11 experimental runs (gray lines), with thicker line representing the run for which we took movies and measured the velocity profile. The thick black line is the average of the 11 runs, used for comparison with the GGT model in Fig. 7. Positions of particles (○ indicates small, × indicates large) overlaid on images taken through the window in the outer wall (b) initially, $t=0\text{s}$, (c) in a mixed state, $t=127\text{s}$, and (d) in the re-segregated state, $t=1820\text{s}$.

Figure 3

(Color online) (a) Measured velocity profile $u\left(z_i\right)$ (●) for full cell height, with boundary layers of thickness $d_L$ above and below the dashed horizontal lines which bound the modeled region. Nondimensional height variable $z$ is scaled so that $z=0$ at the bottom dashed line and $z=1$ at the top line. (b) Dimensionless $u\left(z_i\right)$ with velocities scaled by $u=5.5\text{mm}/\text{s}$ so that $u\left(0\right)=1$. The dashed lines correspond to those in (a). The solid curve is the fit to Eq. (1). Bars represent width of the velocity distribution at half the height of the peak.

Figure 4

(Color online) (a) Dimensionless shear rate $\dot{\gamma }=\left|du/dz\right|$ (●) on logarithmic axis over full cell height, with dashed lines showing same boundary layers as in Fig. 3. (b) $\dot{\gamma }=\left|du/dz\right|$ within the region $z∊\left[0,1\right]$, on linear axes. Solid line is the fit from Fig. 3, plotted as Eq. (2).

Figure 5

Numerical solution of the initial value problem Eq. (5, 6, 7), with parameters $u_0=0.866,\lambda =0.205,z_0=\frac{1}{2}$. For this plot, we set the segregation parameter $s=16.2$ so that $t=t_f=1$ is the (nondimensional) time at the final, fully segregated state.

Figure 6

(Color online) (a) Concentration map specified by Eq. (13). (b) Contour plot of $H_{\text{min}}-{\tilde{H}}_{\text{min}}$ (solid lines) and $\Delta H-\Delta \tilde{H}$ (dashed). Symbol ● marks the intersection of the two zero contours at $V_p/\left(A{\rho }_{\text{min}}\right)=42.0\text{mm}$ and $\Delta \rho /{\rho }_{\text{min}}=0.072$.

Figure 7

(Color online) Comparison of the experimentally measured $H\left(t\right)$ (solid line, see Fig. 2) and the calculated proxy height $\tilde{H}$ (dashed line) for the parameters given in Fig. 6.