Effect of pressure on segregation in granular shear flows

The effect of confining pressure (overburden) on segregation of granular material is studied in discrete element method (DEM) simulations of horizontal planar shear flow. To mitigate changes to the shear rate due to the changing overburden, a linear with depth variation in the streamwise velocity component is imposed using a simple feedback scheme. Under these conditions, both the rate of segregation and the ultimate degree of segregation in size bidisperse and density bidisperse granular flows decrease with increasing overburden pressure and scale with the overburden pressure normalized by the lithostatic pressure of the particle bed. At overburdens greater than approximately 20 times the lithostatic pressure at the bottom of the bed, the density segregation rate is zero while the size segregation rate is small but nonzero, suggesting that different physical mechanisms drive the two types of segregation. The segregation rate scales close to linearly with the inertial number for both size bidisperse and density bidisperse mixtures under various flow conditions, leading to a proposed pressure dependence term for existing segregation velocity correlations. Surprisingly, particle stiffness has only a minor effect on segregation, although it significantly affects the packing density.

Figure 1

Schematic (side view) of size bidisperse shear flow $0.5\mathrm{s}$ after the onset of shear but before significant segregation has occurred. Vertical dashed lines represent streamwise periodic boundary conditions; the domain is also periodic in the spanwise direction. The top and bottom walls are modeled as flat frictional planes. Top wall mass is varied to change the overburden pressure $P$, and the top wall is free to move vertically due to dilation or compaction of the bed particles. Particle sizes are $d_S=2\mathrm{mm}$ and $d_L=4\mathrm{mm}$ in the simulation shown. The particle bed shown here is truncated in the streamwise direction compared to actual simulations.

Figure 2

Streamwise velocities of individual particles in a size bidisperse simulation with species diameter ratio $d_L/d_S=2$, shear rate $\dot{\gamma }=5{\mathrm{s}}^{-1}$, and overburden pressure $P=340\mathrm{Pa}$. Particle sizes are $d_S=2\mathrm{mm}$ and $d_L=4\mathrm{mm}$. Data are recorded for all particles in the simulation at an instant 0.5 s after onset of shear, at which point the particles are still relatively well mixed.

Figure 3

Sideview images of simulation domain at steady state after 100 s of shear flow in $d_L/d_s=2$ size bidisperse (left) and ${\rho }_H/{\rho }_L=9$ density bidisperse (right) mixtures. For both types of mixtures, the steady-state degree of segregation decreases with increasing overburden pressure. Particle sizes are $d_S=2\mathrm{mm}$ and $d_L=4\mathrm{mm}$ for size bidisperse mixtures with particle density $\rho =2500\mathrm{kg}/{\mathrm{m}}^3$, and $d=3\mathrm{mm}$ for density bidisperse mixtures with particle densities ${\rho }_H=4500\mathrm{kg}/{\mathrm{m}}^3$ and ${\rho }_L=500\mathrm{kg}/{\mathrm{m}}^3$. The binary collision time is $t_c=1.25\times {10}^{-4}\mathrm{s}$, corresponding to particle normal stiffness $k_n=5909\mathrm{N}/\mathrm{m}$ for the size bidisperse mixtures and $k_n=4039\mathrm{N}/\mathrm{m}$ for the density bidisperse mixtures.

Figure 4

Time series of the rising particle species center of mass $y_R$ for a size bidisperse mixture ($d_L/d_S=2$) with binary collision time $t_c=1.25\times {10}^{-4}\mathrm{s}$ (particle stiffness $k_n=5909\mathrm{N}/\mathrm{m}$), overburden pressure $P=340\mathrm{Pa}$, and shear rate $\dot{\gamma }=5{\mathrm{s}}^{-1}$. $w_S$ characterizes the initial rate of segregation, and $y_{R,f}$ measures the final height of the rising particle species. Particle diameters are $d_S=2\mathrm{mm}$ and $d_L=4\mathrm{mm}$, and particle density is $\rho =2500\mathrm{kg}/{\mathrm{m}}^3$.

Figure 5

(a) Normalized final height of large particle species ${\tilde{y}}_{R,f}$ and (b) normalized segregation rate $w_S/\left(\dot{\gamma }\overline{d}\right)$ vs overburden $P$ for size ratio $d_L/d_S=2,d_S=2\mathrm{mm},d_L=4\mathrm{mm}$, and shear rate $\dot{\gamma }=5{\mathrm{s}}^{-1}$. $t_c=1\times {10}^{-3}\mathrm{s},k_N=92\mathrm{N}/\mathrm{m}$ ($▵$); $t_c=5\times {10}^{-4}\mathrm{s},k_N=370\mathrm{N}/\mathrm{m}$ ($◊$); $t_c=2.5\times {10}^{-4}\mathrm{s},k_N=1477\mathrm{N}/\mathrm{m}$ ($\square$); $t_c=1.25\times {10}^{-4}\mathrm{s},k_N=5909\mathrm{N}/\mathrm{m}$ ($○$).

Figure 6

(a) Normalized final height of less dense particle species ${\tilde{y}}_{R,f}$ and (b) normalized segregation rate $w_S/\left(\dot{\gamma }\overline{d}\right)$ vs overburden $P$ for density ratio ${\rho }_H/{\rho }_L=9,d=3\mathrm{mm}$, and shear rate $\dot{\gamma }=5{\mathrm{s}}^{-1}$. $t_c=1\times {10}^{-3}\mathrm{s},k_N=63\mathrm{N}/\mathrm{m}$ ($▵$); $t_c=5\times {10}^{-4}\mathrm{s},k_N=252\mathrm{N}/\mathrm{m}$ ($◊$); $t_c=2.5\times {10}^{-4}\mathrm{s},k_N=1010\mathrm{N}/\mathrm{m}$ ($\square$); $t_c=1.25\times {10}^{-4}\mathrm{s},k_N=4039\mathrm{N}/\mathrm{m}$ ($○$).

Figure 7

Mean packing fraction $\varphi$ vs (a) overburden $P$ and (b) the inverse of effective particle stiffness $1/\kappa =P\overline{d}/k_n$ for size bidisperse simulations. Symbols as in Fig. 5.

Figure 8

Mean packing fraction $\varphi$ vs (a) overburden $P$ and (b) the inverse of effective particle stiffness $1/\kappa =P\overline{d}/k_n$ for density bidisperse simulations. Symbols as in Fig. 6.

Figure 9

Normalized final height of large particle species ${\tilde{y}}_{R,f}$ vs mean packing fraction $\varphi$ for (a) $d_L/d_S=2$ size bidisperse simulations and (b) ${\rho }_H/{\rho }_L=9$ density bidisperse simulations. Symbols as in Fig. 5 (size bidisperse data) and Fig. 6 (density bidisperse data).

Figure 10

Normalized final height of large particle species ${\tilde{y}}_{R,f}$ vs normalized overburden pressure $P/\left({\rho }_{B}gh\right)$ for (a) $d_L/d_S=2$ size bidisperse simulations and (b) ${\rho }_H/{\rho }_L=9$ density bidisperse simulations for the conditions described in Table 1; $t_c=1.25\times {10}^{-4}\mathrm{s}$.

Figure 11

Normalized segregation rate $w_S/\left(\dot{\gamma }\overline{d}\right)$ vs normalized overburden pressure $P/\left({\rho }_{B}gh\right)$ for (a) $d_L/d_S=2$ size bidisperse simulations and (b) ${\rho }_H/{\rho }_L=9$ density bidisperse simulations for the conditions described in Table 1; $t_c=1.25\times {10}^{-4}\mathrm{s}$. The same data are plotted on log-log axes in the insets.

Figure 12

Normalized segregation rate $w_S/\sqrt{g\overline{d}}$ vs inertial number $I=\dot{\gamma }\overline{d}/\sqrt{P/\overline{\rho }}$ for (a) $d_L/d_S=2$ size bidisperse simulations and (b) ${\rho }_H/{\rho }_L=9$ density bidisperse simulations for the conditions described in Table 1; $t_c=1.25\times {10}^{-4}\mathrm{s}$. (c) The same normalized segregation rate $w_S/\sqrt{gd}$ data for density bidisperse simulations vs inertial number accounting for a critical pressure cutoff $I^{*}=\dot{\gamma }\overline{d}/\sqrt{[P/(1-P/P_{\mathrm{crit}})]/\overline{\rho }}$, where $P_{\mathrm{crit}}=20{\rho }_{B}gh$.