Comparing flow thresholds and dynamics for oscillating and inclined granular layers

The onset and dynamics of flow in shallow horizontally oscillating granular layers are studied as a function of the depth of the layer and imposed acceleration. Measurements of the flow velocity made from the top and side are presented in the frame of reference of the container. As is also found for avalanches of inclined layers, the thresholds for starting and stopping of flow are slightly different. The variation with depth of the starting acceleration ${\Gamma }_{\mathrm{start}}$ for the oscillating layer is similar to the corresponding variation of the tangent of the starting angle $\tan \left({\theta }_{\mathrm{start}}\right)$ for avalanches in the same container at low frequencies, but deviates as the frequency is increased. However, the threshold behavior depends significantly on the measurement protocol. Just above ${\Gamma }_{\mathrm{start}}$, the motion decays with time as the material reorganizes over a minute or so, causing the apparent threshold to increase. Furthermore, the rms velocity as a function of acceleration rises more sharply above the starting threshold if the first minute or so of excitation is discarded. Once excited, the rheology of the material is found to vary in time during the cycle in surprising ways. If the maximum inertial force (proportional to the container acceleration amplitude) is slightly higher than that required to produce flow, the flow velocity grows as soon as the inertial force exceeds zero in each cycle, but jamming occurs long before the inertial force returns to zero. At higher $\Gamma$, the motion is fluidlike over the entire cycle. However, the fraction of the cycle during which the layer is mobile is typically far higher than what one would predict from static considerations or the behavior of the inclined layer. Finally, we consider the flow profiles as a function of both the transverse distance across the cell at the free surface and also as a function of the vertical coordinate in the boundary layer near the sidewall. These profiles have time-dependent shapes and are therefore significantly different from profiles previously measured for avalanche flows.

Figure 1

(Color online) (a) Sketch of the horizontally oscillated experiment. A plexiglass cell containing the granular layer is attached to an electromagnetic shaker via a steel shaft confined by a low friction roll-bearing. A CCD camera is used to track the grain motion from the top and the side. (b) Sketch of the avalanche experiment performed in the same cell with the inclination angle $\theta$. The coordinate axes are shown, with motion always along $x$.

Figure 2

Acceleration dependence of the rms $x$ velocity at the free surface of the grain layer for depth of $7d$, as a function of the dimensionless acceleration $\Gamma$. All velocity measurements in this paper are in the box frame of reference. The solid upward pointing triangles are obtained when the acceleration is increased. The open downward pointing triangles are obtained when the acceleration is decreased. (a) Method A (see text). Dashed lines show the best linear fit to the rms velocity emerging from the background noise (full line). (b) Method B, in which measurements are made after the transient decay $\left(1\min \right)$. Here, the rise is sharper, and the hysteresis less. In both cases the nondimensional frequency $\eta =0.21$ $(f_o=3.3\mathrm{Hz})$.

Figure 3

(Color online) Transient decay of the surface $x$-velocity amplitude (or maximum surface velocity) for three accelerations near threshold, as a result of rapid reorganization of the grains for $\eta =0.31$ $(f_o=5.0\mathrm{Hz})$. Velocities are normalized by $f_{o}d$, so the displacements are many particle diameters. For sufficiently large $\Gamma$, no decay is observed. In method B, thresholds are determined after the decay (if present).

Figure 4

(Color online) Dependence of the thresholds ${\Gamma }_{\mathrm{start}}$ and ${\Gamma }_{\mathrm{stop}}$ on the layer depth. In all cases, solid symbols denote upward transitions and open symbols denote downward transitions. (a) Comparison of the two experimental procedures, for excitation at $\eta =0.22$ ($f_o=3.5\mathrm{Hz}$; triangles, method A; squares, method B). (b) Comparison of the oscillating layer to the avalanche experiment. Triangles: Oscillating driving (method A, $\eta =0.22$). Circles: Tangent of the critical avalanche inclination angle ${\theta }_{\mathrm{start}}$. The dashed line is the fit proposed in [13]. (c) Frequency dependence, method B (stars, $\eta =0.31$; squares, $\eta =0.22$). The measured oscillatory thresholds vary somewhat with the measurement method, but are fairly close to the avalanche case at the lower frequency.

Figure 5

(Color online) Temporal trace of the spatially averaged $x$ velocity $⟨V⟩$ along the direction of excitation (full line) at the surface of the layer. Velocities are made nondimensional by $f_{o}d$. The layer depth is $9d$, and $\eta =0.21$. (a) $\Gamma =0.39$, (b) $\Gamma =0.44$, (c) $\Gamma =0.46$, and (d) $\Gamma =0.48$. Dashed lines show the container position (which is approximately proportional to the effective force). For intermediate accelerations, the rheology changes during the cycle.

Figure 6

(Color online) Nondimensional spatially averaged $x$ velocity $⟨V⟩$ as a function of the approximate instantaneous non-dimensional acceleration $x\left(t\right){\omega }^2∕g$, for the four accelerations given in Fig. 5. Dashed lines show the stopping thresholds, where $x\left(t\right){\omega }^2∕g=\pm {\Gamma }_{\mathrm{stop}}$. Arrows show the circulation of the trajectory. In (b) and (c), the material starts to move without threshold, but becomes solid-like as the acceleration falls.

Figure 7

(Color online) Mobility cycle fraction of the surface layer obtained with method A by decreasing $\Gamma$. The fraction of time for which the surface particles move is plotted as function of the ratio $\Gamma ∕{\Gamma }_{\mathrm{start}}$ for seven layer depths. The dotted curve shows the theoretical fraction of time for which $\Gamma$ would be larger than ${\Gamma }_{\mathrm{stop}}$ once ${\Gamma }_{\mathrm{start}}$ has been exceeded.

Figure 8

(Color online) Time dependence of the velocity standard deviation ${\sigma }_V$ (including both components), compared to the mean particle velocity (dashed). The fluctuations are largest close to (but slightly later than) the instant of the maximum velocity along the direction of excitation.

Figure 9

(Color online) Temporal traces of the $x$ velocities at the surface for different distances to the sidewall, $V_y\left(t\right)$ (full lines). (a) $\Gamma =0.45$ (intermittent regime) and (b) $\Gamma =0.49$ (fully mobilized at the top). In both cases $h=9d$ and $\eta =0.21$. The distance between two successive curves is about $2d$. The dashed line is the spatially averaged velocity $⟨V{⟩}_t$. The vertical cuts through these curves show the instantaneous velocity profiles $V_t\left(y\right)$ at different times, represented in the inset by the same symbols. The profiles are normalized by the spatially averaged velocity at the corresponding time. The dashed line in the inset is the profile of the rms (time-averaged) surface velocity normalized by its space averaged value. The cross-cell profile of the horizontal velocity is clearly time-dependent in the intermittent regime.

Figure 10

(Color online) Temporal traces of the $x$ velocity at the sidewall $V_z\left(t\right)$ (full lines) for different distances below the surface. (a) $\Gamma =0.45$ (intermittent regime) and (b) $\Gamma =0.49$ (fully mobilized at the top). In both cases $h=9d$ and $\eta =0.21$. The vertical separation between two successive curves is about one particle diameter. Vertical cuts through these curves show the instantaneous velocity profiles $V_t\left(z\right)$ at different times, represented in the inset by the same symbols. The profiles are normalized by the velocity of the top layer. The dashed line in the inset is the profile of the rms (time-averaged) surface velocity normalized by the rms surface velocity. The vertical profile of the horizontal velocity is clearly time-dependent.