Dense, layered, inclined flows of spheres

We consider dense, inclined flows of spheres in which the particles translate in layers, whose existence may be promoted by the presence of a rigid base and/or sidewalls. We imagine that in such flows a sphere of a layer is forced up the back of a sphere of the layer below, lifting a column of spheres above it, and then falls down the front of the lower sphere, until it bumps against the preceding sphere of the lower layer. We calculate the forces and rate of momentum transfer associated with this process of rub, lift, fall, and bump and determine a relation between the ratio of shear stress to normal stress and the rate of strain that may be integrated to obtain the velocity profile. The fall of a sphere and that of the column above it results in a linear increase in the magnitude of the velocity fluctuations with distance from the base of the flow. We compare the predictions of the model with measured profiles of velocity and granular temperature in several different dense, inclined flows.

Figure 1

Sketch of 2D flow of spherical beads down a bumpy incline, comprising spheres (black) fixed to the bed of the flume.

Figure 2

The penetration of a sphere into the layer below showing the largest angle $\beta$ between the line of centers and the normal to the flow. The angle of the flow to the horizontal is $\theta$.

Figure 3

The rise of a sphere above the layer below, with the line by centers rotated by an angle $\beta -\theta$ between the two.

Figure 4

Trajectory of a particle (red, dashed line) as a result of flying and falling; the red star indicates the coordinate of the center at the moment of bumping against a particle of the layer below. The effective angle at contact is ${\beta }_{\mathrm{eff}}<\beta$.

Figure 5

Comparison of the predicted velocity with [solid blue line, Eq. (18)] and without friction [dashed red line, Eq. (15)] and temperature [solid blue line, Eq. (20)] profiles and those measured in the experiments in Ref. [14] (black circles) shown in Fig. 12 there. The parameters employed are $d=3\mathrm{mm},\theta ={21}^{\circ },\mu =0.1,\beta =\pi /6$.

Figure 6

Comparison of the predicted velocity with [solid blue line, Eq. (18)] and without friction [dashed red line, Eq. (15)] and temperature [solid blue line, Eq. (20)] profiles and those measured in the experiments in Ref. [14] (black circles) shown in Fig. 3 there. The parameters employed are $d=3\mathrm{mm},\theta ={23}^{\circ },\mu =0.1,\beta =\pi /6$.

Figure 7

Comparison of the predicted velocity with [solid blue line, Eq. (18)] and without friction [dashed red line, Eq. (15)] and temperature [solid blue line, Eq. (20)] profiles and those measured in the experiments in Ref. [58] (black circles). The parameters employed are $d=8\mathrm{mm},\theta ={34}^{\circ },\mu =0.2,\beta =\pi /5.5$.

Figure 8

Comparison of the predicted velocity with [solid blue line, Eq. (18)] and without friction [dashed red line, Eq. (15)] and temperature [solid blue line, Eq. (20)] profiles and those measured in the experiments in Ref. [2] at a slope angle of $21.2^{\circ }$ (black circles) and $23.4^{\circ }$ (black squares). The parameters employed are $d=3\mathrm{mm},\theta =23.4^{\circ },\mu =0.1,\beta =\pi /6$.

Figure 9

Comparison of the predicted velocity with [solid blue line, Eq. (18)] and without friction [dashed red line, Eq. (15)] and temperature [solid blue line, Eq. (20)] profiles and those measured in the experiments in Ref. [46] (black circles) shown in Fig. 21 there. The parameters employed are $d=2\mathrm{mm},\theta =35.7^{\circ },\mu =0.09,\beta =\pi /6,\omega =6$ r.p.m.

Figure 10

Comparison of the predicted velocity with [solid blue line, Eq. (18)] and without friction [dashed red line, Eq. (15)] and temperature [solid blue line, Eq. (20)] profiles and those measured in the experiments in Ref. [46] shown in Fig. 24 there for two drum widths: $W/d=3.33$ (black circles) and $W/d=6.66$ (black crosses). The other parameters employed are $d=3\mathrm{mm},\theta =26.3^{\circ },\mu =0.09,\beta =\pi /6,\omega =3$ r.p.m.

Figure 11

Comparison of the predicted velocity [Eq. (18)] and temperature [Eq. (20)] profiles for three different particle diameters (solid blue lines for $d_1=1$ mm, dashed red lines for $d_2=2$ mm, and black dash-dotted lines for $d_3=3$ mm) and those measured in the experiments in Ref. [46] shown in Fig. 23 there (blue o for $d_1=1$ mm, red x for $d_2=2$ mm, and black + for $d_3=3$ mm). The other parameters employed are ${\theta }_1=32.0^{\circ },{\theta }_2=29.1^{\circ },{\theta }_3=26.3^{\circ },\mu =0.09,\beta =\pi /6,\omega =3$ r.p.m.

Figure 12

Comparison of the predicted velocity [Eq. (18)] and temperature [Eq. (20)] profiles for three different rotation speeds of the cylinder (solid blue lines for ${\omega }_1=3$ r.p.m., dashed red lines for ${\omega }_2=6$ r.p.m. and black dash-dotted lines for ${\omega }_3=9$ r.p.m.) and those measured in the experiments in Ref. [46] shown in Fig. 22 there (blue o for ${\omega }_1=3$ r.p.m., red x for ${\omega }_2=6$ r.p.m., and black + for ${\omega }_3=9$ r.p.m.). The other parameters employed are ${\theta }_1=29.1^{\circ },{\theta }_2=35.7^{\circ },{\theta }_3=42.8^{\circ },\mu =0.09,\beta =\pi /6,d=2$ mm.