Segregation of large particles in dense granular flows suggests a granular Saffman effect

We report on the scaling between the lift force and the velocity lag experienced by a single particle of different size in a monodisperse dense granular chute flow. The similarity of this scaling to the Saffman lift force in (micro-) fluids, suggests an inertial origin for the lift force responsible for segregation of (isolated, large) intruders in dense granular flows. We also observe an anisotropic pressure field surrounding the particle, which potentially lies at the origin of the velocity lag. These findings are relevant for modeling and theoretical predictions of particle-size segregation. At the same time, the suggested interplay between polydispersity and inertial effects in dense granular flows with stress and strain gradients, implies striking new parallels between fluids, suspensions, and granular flows with wide application perspectives.

Figure 1

Schematic of the simulations: 3D monodisperse granular flow down an incline, with angle $\theta ={22}^{\circ }$. Only base (white) and surface (blue) particles are shown, as well as three bulk particles. The flow contains three intruder particles that are held with springs around three different $z$ positions $z_I$ (intruder positions in the schematic are to scale) but move freely in the $x\text{−}y$ plane.

Figure 2

(a) The velocity lag ${\lambda }_x$ of the intruder particle as a function of size ratio $S$, for $z_I=15$ and $z_I=23$. (b) Velocity lag as a function of the average vertical position $\langle z_i\rangle$ of an intruder for $S=2.4$. The dashed lines in (a) and (b) are fits of Eq. (2), with $a=0.24$. The circles indicate the outliers. (c) The data from (a) and (b) are plotted here as $\eta {\lambda }_x$ versus $S$. The yellow circles are the data from (b), with the black circles indicating the outliers. (d) The data from (a) and (b) are plotted here as $\eta {\lambda }_x$ versus $a(1/S-1)$. The solid black line has a slope of 1.0. The yellow circles are again the data from (b), with the black circles indicating the outliers.

Figure 3

(a) Cross sections $P(x,0,z)$ around the intruder, centered at the origin, for an intruder at $z_I=15$. The blue circle (diameter $d_i$) corresponds to the intruder. The edge of the white circle (diameter $d_i+d_b$) corresponds to the position of the first layer of bulk particles. (b) Cross sections $P_L(x,0,z)$, where $P_L=P-P_H$, around the intruder at $z_I=15$.

Figure 4

(a) Local intruder solids fraction ${\nu }_i$ versus $S$. Different (almost collapsing) symbols correspond to intruders with ${\mu }_{bi}=0.5$, ${\mu }_{bi}=0$, $z_I=15$, $z_I=23$, $\theta ={22}^{\circ },{23}^{\circ },{24}^{\circ },{25}^{\circ }$, and ${26}^{\circ }$, $k=20$ and $k=\infty$. Solid line corresponds to Eq. (5) with $c=-1.2$ and $\nu =0.577$. The schematic depicts the Voronoi volume $\widetilde{V_i}$ (dotted octagon) of the intruder (dashed circle). (b) The measured forces $F_b$, $F_L$, and $F_L+F_b$, normalized by $F_{g_z}$, for $z_I=23$, as well as a fit of $F_L$ with Eq. (8) (solid red line), with $a=0.24$ and $b=130.0$. The value of $a$ is obtained from the fit in Fig. 2. The buoyancy force $F_b$ (blue circles) corresponds to Eq. (4) with ${\nu }_i$ from (a). (c) The measured forces $F_b$, $F_L$, $F_{n_z}$, and $F_{t_z}$, normalized by $F_{g_z}$, for $S=2.4$, ${\mu }_{bi}=0$ and 0.5, at $z_I=15$.

Figure 5

(a) The measured lift force $F_L$ (red triangles) and the buoyancy force $F_b$ (blue squares) as a function of the chute inclination angle $\theta$, for $S=2.4$. The data are normalized by the vertical component of the gravity force on the intruder $F_{g_z}$. (b) The measured net contact force $F_c$ (blue circles) and the lift force $F_L$ (red triangles) on an intruder, as a function of depth, for $S=2.4$, normalized by the gravity force. The solid black line is a fit of Eq. (8) using the functional form for the lag Eq. (2), with $a=0.24$ and $b=130.0$, while the dashed line uses the raw velocity lag data from Fig. 2. Near the bed ($z_i<8$) a boundary effect occurs, likely due to layering [41].