Segregation patterns in three-dimensional granular flows

Flow of size-bidisperse particle mixtures in a spherical tumbler rotating alternately about two perpendicular axes produces segregation patterns that track the location of nonmixing islands predicted by a dynamical systems approach. To better understand the paradoxical accumulation of large particles in regions defined by barriers to transport, we perform discrete element method (DEM) simulations to visualize the three-dimensional structure of the segregation patterns and track individual particles. Our DEM simulations and modeling results indicate that segregation pattern formation in the biaxial spherical tumbler is due to the interaction of size-driven radial segregation with the weak spanwise component of the advective surface flow. Specifically, we find that after large particles segregate to the surface, slow axial drift in the flowing layer, which is inherent to spherical tumblers, is sufficient to drive large particles across nominal transport barriers and into nonmixing islands predicted by an advective flow model in the absence of axial drift. Axial drift alters the periodic dynamics of nonmixing islands, turning them into “sinks” where large particles accumulate even in the presence of collisional diffusion. Overall, our results indicate that weak perturbation of chaotic flow has the potential to alter key dynamical system features (e.g., transport barriers), which ultimately can result in unexpected physical phenomena.

Figure 1

Flow and segregation structures for the (${57}^{\circ },{57}^{\circ }$) protocol. (a) Poincaré section from a continuum model (CM) simulation of a half-full spherical tumbler at $r=0.95$. (b) 3D nonmixing KAM tubes visualized by transport barriers from CM for $0.55\le r\le 0.95$ in radial increments of 0.05. The six conelike KAM tubes (A1–A3 and B1–B3) are period-3. The dashed white line indicates an intersecting plane used for analysis in Sec. 2b. (c) Segregation experiment in a half-full 7 cm radius spherical tumbler with fraction $f=0.05$ large blue (shaded gray) particles and $1-f=0.95$ small red (gray) particles by volume after $N=30$ iterations of the (${57}^{\circ },{57}^{\circ }$) protocol with angular velocity $\omega =2.6$ rpm. All images show the mixture-filled half of the tumbler viewed from below. Adapted with permission from [32].

Figure 2

Comparison of segregation patterns in (left) experiment [32] and (right) DEM simulation at $N=30$ iterations for the (${57}^{\circ },{57}^{\circ }$) protocol using mixtures of 4 mm blue (shaded gray) and 2 mm red (gray) particles in a half-filled spherical tumbler of radius $R_o=7$ cm for three large-particle fractions $f$ (rows) viewed from below.

Figure 3

Large-particle movement in a DEM simulation during the $N=31$ iteration of the (${57}^{\circ },{57}^{\circ }$) protocol for an initially uniform mixture with large-particle fraction $f=0.05$. The free surface corresponds to the flat portion of the hemisphere, which is offset from the repose angle to horizontal for ease of visualization. Arrows indicate streamwise velocity in the flowing layer. Labels identify two groups of nonmixing clusters A1–A3 and B1–B3.

Figure 4

Local large-particle concentration, $c_{\mathrm{L}},$ from DEM simulations in steady state averaged over iterations $25\le N\le 35$ and tumbler radius ($0.9<r<1$) under the protocol (${57}^{\circ },{57}^{\circ }$) for increasing (top to bottom) large-particle fraction $f$ as indicated. Concentration fields are visualized from below using a Lambert azimuthal projection, which preserves the area between the 3D hemisphere and the 2D plane. Boundaries of nonmixing islands predicted by the CM at $r=0.95$ are indicated by red (gray) curves.

Figure 5

Steady-state ($N=30$) particle distributions from DEM simulations for three large-particle fractions $f$ in the intersecting plane indicated by the dashed white line in Fig. 1 under the (${57}^{\circ },{57}^{\circ }$) protocol. Dotted white curves indicate boundaries of the conelike nonmixing regions (KAM tubes) predicted by the CM. Solid curves indicate radii $r$.

Figure 6

Net angular displacement, $\mathrm{\Delta }\psi$, of large particles (top row) and small particles (bottom row) from DEM simulations with $0.8\le r\le 1$ over two flow periods (six iterations) during steady-state iterations ($13\le N\le 35$) of protocol (${57}^{\circ },{57}^{\circ }$) for mixtures with large-particle fraction $f$ varied from 0 to 1 as indicated. Grid cells with no large particles are shaded gray. Results are shown in bottom views of the spherical tumbler presented in Lambert azimuthal projections.

Figure 7

Net angular displacement, $\mathrm{\Delta }\psi$, of large particles (top row) and small particles (bottom row) from DEM simulations with $0.8\le r\le 1$ over six flow periods (18 iterations) during steady-state iterations ($13\le N\le 35$) of protocol (${57}^{\circ },{57}^{\circ }$) for mixtures with large-particle fraction $f$ between 0 and 1. Grid cells with no large particle are gray. Results are shown in bottom views of the spherical tumbler presented in Lambert azimuthal projections.

Figure 8

Development of surface segregation patterns as shown by tracking large particles through barriers to mixing over $7\le N\le 34$ iterations of (${57}^{\circ },{57}^{\circ }$) protocol in DEM simulations viewed from below. “Persistent” particles [orange (light gray)] start and end in nonmixing regions, “captured” particles [blue (gray)] start in the chaotic region and end in nonmixing regions, and “dispersed” particles [green (dark gray)] start in nonmixing regions and end in the chaotic region. All other large particles start and end in the chaotic region (small black dots). Transport barriers (closed black curves) are determined from the CM.

Figure 9

Continuum model (CM, $v_{\text{ax}}=0, {\sigma }_{\text{ax}}=0$) and modified continuum model (MCM) results for (${57}^{\circ },{57}^{\circ }$) protocol in Lambert azimuthal projection viewed from below: (a) without axial drift or diffusion (CM); (b) with random-walk diffusion of ${\sigma }_{\text{ax}}$ only; (c) with axial velocity but no diffusion; (d) with axial velocity and diffusion ${\sigma }_{\text{ax}}$; (e) with axial velocity and increased diffusion $20{\sigma }_{\text{ax}}$. Top row: Initial random distribution of tracer points. Middle row: Tracer point distribution at $N$ = 30 with obvious clustering in (c), (d). Bottom row: Cumulative tracer point positions for $30\le N\le 45$ and nonmixing island perimeters [red (gray) curves] determined by the CM, which does not include drift or diffusion. Only tracer points in the bulk are plotted.

Figure 10

Segregation pattern under the (${90}^{\circ },{90}^{\circ }$) protocol. (a) Poincaré section from CM at $r=0.95$. (b) Lambert projection of particles ($0.9\le r\le 1$) in the DEM simulation of a half-filled spherical tumbler with a mixture of $f=0.15$ large particles at $N=30$ of the same protocol. (c) Average concentration of large particles ($0.9\le r\le 1$) in the DEM simulation for $15<N<30$ (color map as in Fig. 4). (d) MCM with axial velocity and diffusion ${\sigma }_{\text{ax}}$ for an initial random distribution of tracer points at $N=30$. Boundaries of nonmixing islands predicted by the CM are indicated by red (gray) curves in [(c),(d)]. All results are viewed from below.

Figure 11

Absence of segregation patterns for the (${68}^{\circ },{68}^{\circ }$) protocol. (a) No nonmixing islands in the Poincaré section of the CM. (b) Bottom view of DEM simulation with initially mixed small red (gray) particles and large blue (dark gray) particles ($f=0.15$) at $N=40$. (c) Average concentration in 2D Lambert projection for $26\le N\le 40$ from DEM simulation data with $0.9<r<1$ (color map as in Fig. 4).

Figure 12

Global mixing barrier spanning the tumbler for the (${45}^{\circ },{45}^{\circ }$) protocol (bottom view). (a) Poincaré section from the CM shows large interpenetrating fingerlike structures. Stable manifolds of the CM flow are indicated by green (light gray) curves. (b) Large blue (dark gray) particles with $f=0.15$ in DEM simulations remain primarily in the half of the domain in which they are initially placed after $N=30$ iterations. (c) Average concentration of large particles ($0.9\le r\le 1$) in the DEM simulation for $15\le N\le 30$ (color map as in Fig. 4). (d) Large-particle distribution from the MCM ($v_{\text{ax}}\ne 0,{\sigma }_{\text{ax}}$) at $N=30$ for the same initial conditions as in (b).

Figure 13

Segregation of density-bidisperse mixtures of lighter [blue (dark gray)] and heavier [(red gray)] 2-mm-diam particles in DEM simulations for the (${57}^{\circ },{57}^{\circ }$) protocol at density ratios (a),(b) $R_{\rho }=3$ and (c),(d) $R_{\rho }=8$. (a),(c) Bottom view of DEM simulations at $N=30$, and (b),(d) average concentration of lighter particles for $25\le N\le 35$ for $0.97\le r\le 1$ (one-particle-diameter-thick layer adjacent to the tumbler wall) indicates weak pattern formation (color map as in Fig. 4, except for lighter particles).

Figure 14

DEM simulation of a size-bidisperse mixture in a spherical tumbler rotating about the $z$-axis with angular speed $\omega =3$ rpm. Particles cross the free surface in a thin flowing layer in the direction indicated by the white arrow and slowly drift toward the poles due to a weak axial velocity (see the main text).