Dynamics of a grain-scale intruder in a two-dimensional granular medium with and without basal friction

We report on a series of experiments in which a grain-sized intruder is pushed by a spring through a two-dimensional granular material composed of photoelastic disks in a Couette geometry. We study the intruder dynamics as a function of packing fraction for two types of supporting substrates: A frictional glass plate and a layer of water for which basal friction forces are negligible. We observe two dynamical regimes: Intermittent flow, in which the intruder moves freely most of the time but occasionally gets stuck, and stick-slip dynamics, in which the intruder advances via a sequence of distinct, rapid events. When basal friction is present, we observe a smooth crossover between the two regimes as a function of packing fraction, and we find that reducing the interparticle friction coefficient causes the stick-slip regime to shift to higher packing fractions. When basal friction is eliminated, we observe intermittent flow at all accessible packing fractions. For all cases, we present results for the statistics of stick events, the intruder velocity, and the force exerted on the intruder by the grains. Our results indicate the qualitative importance of basal friction at high packing fractions and suggest a possible connection between intruder dynamics in a static material and clogging dynamics in granular flows.

Figure 1

Range of packing fractions $\varphi$ explored in each of the three experiments: BF-H: With basal friction and high interparticle friction; BF-L: With basal friction and low interparticle friction; NBF-H: No basal friction and high interparticle friction. Dark represents intermittent flow dynamics and light represents pure stick-slip dynamics.

Figure 2

Schematic of apparatus: (a) Top-down (camera) view of apparatus: I = Intruder, PA = Pusher Arm, FS = Force Sensors. The inset shows a closeup image of the ribbed rubber boundary next to a small grain. The angular position $\theta$ of the intruder is measured relative to the start position in a given experimental run. (b) Side view: CP = Circular Polarizer, SM = Stepper Motor. The zoom-in feature shows the approximate height above the table of floating particles. The dashed line (red) just above the CP is the cross-sectional plane with the torque spring, shown in Fig. 3.

Figure 3

Cross-sectional schematic view of Fig. 2 in the plane marked by the red dotted line: i is rotated at constant angular velocity by the stepper motor, and ii is rigidly fixed to the pusher arm. The timescale of the spring's relaxation ($\mathrm{\Delta }{t}_2\sim 0.4\text{s}$) is shorter than the timescale of typical stick events ($\mathrm{\Delta }{t}_1$) when pure stick-slip dynamics are observed (light blue in Fig. 1).

Figure 4

Sample snapshot of an experiment. The intruder is being driven counterclockwise. The black curve is a cable above the particles. The torque spring schematic from Fig. 3 is shown in the center.

Figure 5

Sample time series of the intruder's angular position $\theta$ (upper curve, black) and intruder velocity $\dot{\theta }$ (lower curve, blue) as a function of the cumulative drive angle of the stepper motor. The dotted line represents the drive velocity ${\dot{\theta }}_d$. (a) Sample from the maximum packing fraction $\varphi$ explored with basal friction. (b) Sample from the maximum $\varphi$ explored without basal friction. Inset: Zoom in of time series within dotted (red) box.

Figure 6

Sample time series for intruder velocity (lower curve, blue) and force (upper curve, red) at the lowest (${\varphi }_{\text{min}}$) (a, c) and highest (${\varphi }_{\text{max}}$) (b, d) packing fractions from the experiments with and without basal friction with high interparticle friction. Thick black segments are detected stick events and the gray dashed line is the drive velocity ${\dot{\theta }}_d$. Inset in (a): Detected stick event; an upper threshold of 0.04 rad/s (dash-dot, green) is used to define stick events.

Figure 7

Velocity distributions (bin count normalized to total number of data points) with varying packing fraction, $\varphi$. Vertical black dashed lines indicate ${\dot{\theta }}_d$. The dotted (gray) distribution is noise measured by driving the intruder around the annulus without any particles. (a) Basal friction, high interparticle friction. (b) Basal friction, low interparticle friction. (c) No basal friction, high interparticle friction. Inset of (b): Zoom in of plot to highlight the increase in the peak height at 0 rad/s with increasing $\varphi$; this trend is observed for all three experiments.

Figure 8

Force distributions with varying packing fraction, $\varphi$. The dotted (gray) distribution is noise measured by driving the intruder around the annulus without any particles. (a) Basal friction, high interparticle friction. (b) Basal friction, low interparticle friction. (c) No basal friction, high interparticle friction.

Figure 9

(a) The average force of the force time series as a function of packing fraction, $\varphi$, for each experimental condition. Each shaded region around a point ranges from the lower 10% cutoff to the upper 90% cutoff of that point's force distribution. Inset: Zoom in to highlight the case with no basal friction. (b) The average creep velocity of the intruder during detected stick events as a function of $\varphi$ (the average speeds associated with the thick black lines in Fig. 6).

Figure 10

(a) The complementary cumulative distribution function (CDF) of the distribution of waiting times, ${\tau }_w$, between stick events for all three experiments. The inset is a zoomed in plot to highlight the trend of the complementary CDFs at low ${\tau }_w$. No basal friction: Increasing thickness here denotes increasing packing fraction $\varphi$: $0.781,0.783,0.784,0.786$. (b) The average waiting time $\tau \equiv \langle {\tau }_w\rangle$ with error bars given by the standard error of the average.

Figure 11

Sample photoelastic images of representative stick events at $\varphi =0.762$ (a) and $\varphi =0.780$ (b) and a measured force of $\sim 0.8\text{N}$ for the experiments with basal friction and without basal friction at high interparticle friction. The intruder is marked with a circle, and the arrow pointing to the intruder indicates the direction of force applied to the intruder by the torque spring.