Creeping motion in granular flow

The core of a quasi-two-dimensional rotating cylinder filled more than half full with glass beads rotates slightly faster than the cylinder itself and decreases in radius over time. Core precession depends linearly on the number of tumbler revolutions while core erosion varies logarithmically. Both processes serve to quantify the slow granular motion in the “fixed” bed and depend on the filling fraction and the tumbler rotation rate. A simple model, based on experimental observations of an exponential decrease in velocity parallel to the free surface, captures the primary features of the core dynamics.

Figure 1

Images of the tumbler after (a) 0, (b) 4, (c) 500, and (d) 1000 complete revolutions showing core development, precession, and erosion ($f=0.663$, $\nu =1\mathrm{rpm}$). The angular position of the tumbler is the same in all images, while the angular position of the core advances in the rotation direction (clockwise). A gray scale enhances contrast between the red (white) and blue (black) beads.

Figure 2

The rotating tumbler geometry with the coordinate systems and the system parameters.

Figure 3

(a) Evolution of the precession angle, $\theta$, for different fill fractions: ◇, $f=0.631$; ▵, $f=0.663$; ◻, $f=0.671$; 엯, $f=0.705$; *, $f=0.849$ $(\nu =1\mathrm{rpm})$. Data were taken every fifth revolution; values averaged over 45 rotations are shown. (b) The precession rate $m$ vs $f$ from (a); the solid curve is a fit to our model, which gives $y_0∕d=3.4$. (c) Dependence of $m$ on the tumbler rotation rate at $f=0.70$. The solid curve is a fit to our model, which also yields $y_0∕d=3.4$, and according to which the primary mechanism for the increase in $m$ with $\nu$ is the increase in the depth of the flowing layer $\delta$. The uncertainty in $f$ is $\pm 0.003$ in all cases.

Figure 4

(a) Decrease in scaled core radius, $(r_c-r_0)∕d$ vs $\ln N$ for various filling fractions $(\nu =1\mathrm{rpm})$: ◇, $f=0.631$; ▵, $f=0.663$; ◻, $f=0.671$; 엯, $f=0.705$; and *, $f=0.849$. (b) Dependence of $(r_c-r_0)∕d$ vs $\ln N$ on $\nu$ for various rotation rates $(f=0.700)$: ◇, $\nu =0.5\mathrm{rpm}$; ▵, $\nu =1\mathrm{rpm}$; ◻, $\nu =1.5\mathrm{rpm}$; 엯, $\nu =2\mathrm{rpm}$; and *, $\nu =2.5\mathrm{rpm}$. In both (a) and (b) the slope of the data is nearly constant and matches the predictions of our model [Eq. (2)]. The slopes of the solid lines are −1 and −2 and correspond to characteristic length scales from our model of $d$ and $2d$, respectively. Data were taken every fifth revolution; values averaged over fixed ratio intervals (1:1.2) are shown.

Figure 5

Velocity parallel to the free surface vs scaled depth at the middle of the free surface measured using PIV/PTV velocimetry $(f=0.67,\nu =1\mathrm{rpm})$. The slope of the solid line gives $\dot{\gamma }=27.5{\mathrm{s}}^{-1}$. The inset, with a logarithmic ordinate, shows the exponential velocity regime; the solid line is a fit to $u_0{e}^{-y∕y_0}$ with $y_0=1.48d$ and $u_0=101\mathrm{cm}∕\mathrm{s}$, so $u_t=\dot{\gamma }{y}_0=3.62\mathrm{cm}∕\mathrm{s}$ and $\delta =y_0\ln (u_0∕u_t)=4.93d$. Data are analyzed in the reference frame of the rotating cylinder.