Phase Transitions in Shear-Induced Segregation of Granular Materials

We computationally study shear-induced segregation of different-sized particles in vertical chute flow. We find that, for low solid fractions, large particles segregate toward regions of low shear rates where the granular temperature (velocity variance) is low. As the solid fraction increases, this trend reverses, and large particles segregate toward regions of high shear rates and temperatures. We find that this is a global phenomenon: local segregation trends reverse at high system solid fractions even where local solid fractions are small. The reversal corresponds to the growth of a single enduring cluster of 30%–60% of the particles that we propose changes the segregation dynamics.

Figure 1

(a) Sketch of a vertical chute. (b)–(d) Profiles of kinematic quantities for four mixtures once steady state is reached ($t=50\mathrm{s}$ for $⟨\overline{f}⟩=0.21$ and 0.34 and $t=400\mathrm{s}$ for $⟨\overline{f}⟩=0.47$ and 0.60). (b) Average streamwise velocity $\overline{w}$, (c) average granular temperature $\overline{T}$, and (d) average solid fraction of the mixture $\overline{f}$. (e)–(h) Snapshots of two mixtures at $t=0\mathrm{s}$ and steady state. (e) $⟨\overline{f}⟩=0.21$ at $t=0\mathrm{s}$, (f) $⟨\overline{f}⟩=0.21$ at $t=50\mathrm{s}$, (g) $⟨\overline{f}⟩=0.60$ at $t=0\mathrm{s}$, and (h) $⟨\overline{f}⟩=0.60$ at $t=400\mathrm{s}$. Two millimeter particles, blue (dark); 3 mm particles, green (light).

Figure 2

Segregation kinematics of three systems with $⟨\overline{f}⟩$’s as noted for the mixture (M) [green (light gray line)] and large (L) [red (thin dark gray line)] and small (S) [blue (thick dark gray line)] particles. Row 1: ${\overline{f}}_i(y)$ at steady state (SS). Rows 2 and 3: normalized segregation fluxes ${\overline{f}}_i\Delta {\tilde{v}}_i(y)$ as defined in text averaged over $t=0.5–1\mathrm{s}$ (row 2) and 1–10 s (row 3). Row 4: normalized net outward average segregation fluxes for large particles $⟨{\overline{f}}_L\Delta {\tilde{v}}_L{⟩}_o$ as a function of normalized time $\tilde{t}$.

Figure 3

Average segregation velocity $\delta v={\overline{v}}_S-{\overline{v}}_L$ at 0.5–1 s from simulation results and from kinetic theory [11, 13]. Derivation summarized in Ref. 16 for (a) $⟨\overline{f}⟩=0.21$, (b) $⟨\overline{f}⟩=0.34$, (c) $⟨\overline{f}⟩=0.55$, (d) $⟨\overline{f}⟩=0.60$.

Figure 4

(a) Sketch illustrating 4 clusters with 2 singletons. (b)–(c) Probability distribution function of $N_c/N$ for (b) $⟨\overline{f}⟩=0.21$ and 0.37, and (c) $⟨\overline{f}⟩=0.47$ and 0.60. (d)–(e) Time dependence of the maximum cluster size for (d) $⟨\overline{f}⟩=0.21$ and 0.37, and (e)–(f) $⟨\overline{f}⟩=0.47$ and 0.60.

Figure 5

$⟨{\overline{f}}_L\Delta {\tilde{v}}_L{⟩}_o$ ($○$), and $N_{\mathrm{cm}}/N$ ($\times$) vs $⟨\overline{f}⟩$ calculated for (a) $t=0.5–1\mathrm{s}$ for $⟨{\overline{f}}_L\Delta {\tilde{v}}_L{⟩}_o$ and $t=0.75\mathrm{s}$ for $N_{\mathrm{cm}}/N$ and (b) $t=1–10\mathrm{s}$ for $⟨{\overline{f}}_L\Delta {\tilde{v}}_L{⟩}_o$ and $t=5\mathrm{s}$ for $N_{\mathrm{cm}}/N$. The dashed line indicates $⟨{\overline{f}}_L\Delta {\tilde{v}}_L{⟩}_o=0$, and the dash-dotted line indicates $N_{\mathrm{cm}}/N=0.34$, the value for $⟨\overline{f}⟩=0.47$ at $\tilde{t}\approx 50$, the segregation reversal time indicated in Fig. 2, row 4.