Shear-induced segregation of particles by material density

Recently, shear rate gradients and associated gradients in velocity fluctuations (e.g., granular temperatures or kinetic stresses) have been shown to drive segregation of different-sized particles in a manner that reverses at relatively high solids fractions $\left(\langle f\rangle >0.50\right)$. Here we investigate these effects in mixtures of particles differing in material density through computational and theoretical studies of particles sheared in a vertical chute where we vary the solids fraction from $\langle f\rangle =0.2$ to 0.6. We find that in sparse flows, $\langle f\rangle =0.2$ to 0.4, the heavier (denser) particles segregate to lower shear rates similarly to the heavier (larger) particles in mixtures of particles differing only in size. However, there is no segregation reversal at high $f$ in mixtures of particles differing in density. At all solids fractions, heavier (denser) particles segregate to regions of lower shear rates and lower granular temperatures, in contrast with segregation of different-sized particles at high $f$, where the heavier (larger) particles segregate to the region of higher shear rates. Kinetic theory predicts well the segregation for both types of systems at low $f$ but breaks down at higher $f$'s. Our recently proposed mixture theory for high $f$ granular mixtures captures the segregation trends well via the independent partitioning of kinetic and contact stresses between the two species. In light of these results, we discuss possible directions forward for a model framework that encompasses segregation effects more broadly in these systems.

Figure 1

(a) Sketch of a vertical chute. [(b)–(d)] Time-averaged profiles of kinematic quantities for four mixtures at steady state, here $t=5$–6 s for $\langle f\rangle =$ 0.2 (green solid curve), $t=20$–30 s for $\langle f\rangle =$ 0.4 (blue dash-dotted curve), $t=30$–40 s for $\langle f\rangle =$ 0.5 (red dashed curve), and $t=300$–310 s for $\langle f\rangle =$ 0.6 (black dotted curve): (b) streamwise velocity $\overline{w}$ of the mixture, (c) kinematic granular temperature $\overline{T}=(\overline{u^{\prime }{u}^{\prime }}+\overline{v^{\prime }{v}^{\prime }}+\overline{w^{\prime }{w}^{\prime }})/3$ of the mixture, and (d) local solids fraction of the mixture $\overline{f}$.

Figure 2

Snapshots of three mixtures at the beginning and steady state of each simulation. (The steady-state time is determined using data plotted in Fig. 4.) (a) The beginning of the simulations ($t=0$ s). From left to right, $\langle f\rangle =0.2$, 0.4, and 0.6, respectively; (b) the steady state of the simulations. From left to right, $\langle f\rangle =0.2$ at $t=5$ s, $\langle f\rangle =0.4$ at $t=10$ s, and $\langle f\rangle =0.6$ at $t=300$ s, respectively. The different species are distinguishable by color: 2-mm denser particles, blue (dark); 2-mm less-dense particles, green (light).

Figure 3

Segregation kinematics of three systems with $\langle f\rangle$ as noted on top of each column for the mixture (m) [green (lighter line)] and dense (d) [red (darker line)] and less dense (l) [blue (bold dark line)] particles. Row 1: ${\overline{f}}_i$ at steady state (SS). Row 2: Segregation fluxes ${\overline{f}}_i\mathrm{\Delta }{\overline{v}}_i$ as defined in text averaged over $t=0-1$ s (row 2). Row 3: Kinematic granular temperature $T_i=(\overline{u_i^{\prime }{u}_i^{\prime }}+\overline{v_i^{\prime }{v}_i^{\prime }}+\overline{w_i^{\prime }{w}_i^{\prime }})/3$ at steady state. We note that the scales of the vertical axes in rows 2 and 3 vary for the different solids fractions.

Figure 4

Time dependence of average downstream velocity $\langle w\rangle$ and segregation index $S$: First row shows plots of the downstream velocity averaged across the width of the chute $\langle w\rangle ={\sum }_{j=0}^{N_{\mathrm{bin}}}{\overline{f}}_j{\overline{w}}_j/{\sum }_{j=0}^{N_{\mathrm{bin}}}{\overline{f}}_j$, where ${\overline{w}}_j$ and ${\overline{f}}_j$ are the average vertical velocity and average solids fraction of the mixture in bin $j$ and $N_{\mathrm{bin}}=2500$ is the number of bins in the $y$ direction and the second row shows plots of a measure of the segregation in the chute $S$ [see Eq. (3)]. Symbols are data measured from DEM simulations and solid lines are exponential fits to the data. For $\langle f\rangle =$ 0.2 and 0.4, the fit equations are $f\left(t\right)=A+B\exp (-t/\tau )$. For $\langle f\rangle =0.6$, when $t<100$ s (stage I), the fit equation is the same as those at $\langle f\rangle =$ 0.2 and 0.4, and when $t>100$ s (stage II), the fit equation is $f\left(t\right)=A+B\exp [-(t-t_0)/\tau ]$. Here $A,B,t_0$, and $\tau$ are fitting parameters: $A+B$ and $A$ represent the initial and final values for each variable, $\tau$ is the time scale, and $t_0$ is indicative of the effective start time of the exponential decay during stage II for $\langle f\rangle =0.6$. The fitting coefficients are shown in Table 3.

Figure 5

Spatiotemporal profiles of concentration of denser particles $\left({\varphi }_d=f_d/f\right)$ for (a) $\langle f\rangle =0.2$ at $t=0$–5 s, (b) $\langle f\rangle =0.4$ at $t=0$–20 s, (c) $\langle f\rangle =0.6$ at $t=0$–100 s, and (d) $\langle f\rangle =$ 0.6 at $t=0$–300 s. The legend indicates the shade of gray that corresponds to particular fraction of denser particles. For example, ${\varphi }_d=1$ for white pixels and ${\varphi }_d=0$ for black pixels.

Figure 6

Profiles of the relative diffusion velocities ${\overline{v}}_l-{\overline{v}}_d$ between less dense and denser particles in the $y$ direction averaged across the width of the chute and over the first 1 s for three different systems with $\langle f\rangle$ indicated in the figure. Solid lines denote ${\overline{v}}_l-{\overline{v}}_d=0$ to guide eye.

Figure 7

The $y$ component of total and partial normal stresses in the $y$ direction for the mixture and the two species at a quasi steady state ($t=50$–100 s). (a) Total, kinetic, and contact stresses for the mixture; (b) contact stresses for the two species; (c) kinetic stresses for the two species; (d) species concentrations. The dashed lines in (a) are exponential fits for the kinetic and contact stresses of the mixtures. For kinetic stresses: ${\sigma }_{yy}^k\left(y\right)=A\exp \left(\right|By\left|\right)$ based on a linearized least-squares fit, where $A=3.6\times {10}^{-2}$ N/${\mathrm{m}}^2$ and $B=0.25{\mathrm{mm}}^{-1}$; for contact stresses: ${\sigma }_{yy}^c\left(y\right)=C-A\exp \left(\right|By\left|\right)$, where $A=1.6\times {10}^{-2}$ N/${\mathrm{m}}^2,B=0.24{\mathrm{mm}}^{-1}$, and $C=27.05$ N/${\mathrm{m}}^2$.

Figure 8

Profiles of partial stress coefficients $R_i^c$ (a) and $R_i^k$ (b) averaged at $t=50$–100 s. (c) Profiles of $R_i^c-R_i^k$. (d) A parametric plot from the data shown in (c) and Fig. 7 for $R_L^c-R_L^k$ vs ${\varphi }_D$. The dashed lines in (a)–(c) are used to indicate the case where $R_i^{k,c}=1$ for both components, i.e., indicating the values on the plot for which the stresses would be equally partitioned between the components. The solid line in (d) is a linear least-squared fit for $R_L^c-R_L^k=B{\varphi }_D$; $B\approx 0.8$.

Figure 9

Theoretical predictions of spatiotemporal profiles of the concentration of dense particles compared to those in Figs. 5 and 5. (a) $t=0$–100 s; (b) $t=0$–300 s. The legend indicates the shade of gray that corresponds to particular fraction of denser particles. For example, ${\varphi }_d=1$ for white pixels and ${\varphi }_d=0$ for black pixels.

Figure 10

Comparison of profiles of segregation velocity $v_l-v_d$ vs $y$ between theoretical predictions from Eq. (12) and DEM simulations at (a) the first second and (b) $t=300$–310 s for $\langle f\rangle =0.6$.

Figure 11

Profiles of dynamic granular temperature and dynamic pressure for two species in the $y$ direction averaged across the width of the chute and over the first 1 s for three systems of different $\langle f\rangle$'s as indicated in the figure. First row: Plots of dynamic granular temperature $T_D$; second row: Plots of dynamic pressure $P$.

Figure 12

Profiles of driving forces in the $y$ direction averaged across the width of the chute and over the first 1 s for three $\langle f\rangle$'s as indicated in the figure. The three different diffusion forces are $d_T,d_p$, and $d_n$ vs $y$.