Micromechanics of intruder motion in wet granular medium

We investigate the effective friction encountered by an intruder moving through a sedimented medium, which consists of transparent granular hydrogels immersed in water, and the resulting motion of the medium. We show that the effective friction ${\mu }_e$ on a spherical intruder is captured by the inertial number $I$ given by the ratio of the timescale over which the intruder moves and the inertial timescale of the granular medium set by the overburden pressure. Further, ${\mu }_e$ is described by the function ${\mu }_e\left(I\right)={\mu }_s+\alpha I^{\beta }$, where ${\mu }_s$ is the static friction, and $\alpha$ and $\beta$ are material-dependent constants which are independent of intruder depth and size. By measuring the mean flow of the granular component around the intruder, we find significant slip between the intruder and the granular medium. The motion of the medium is strongly confined near the intruder compared with a viscous Newtonian fluid and is of the order of the intruder size. The return flow of the medium occurs closer to the intruder as its depth is increased. Further, we study the reversible and irreversible displacement of the medium by not only following the medium as the intruder moves down but also while returning the intruder back up to its original depth. We find that the flow remains largely reversible in the quasistatic regime, as well as when ${\mu }_e$ increases rapidly over the range of $I$ probed.

Figure 1

(a) Schematic of the experimental system consisting of an intruder descending through sedimented granular hydrogels immersed in water. The depth $z$ of the intruder is measured from the top of the sedimented hydrogels denoted by $z_o$ to the bottom of the intruder. The height of the water column $H_w$ and the granular hydrogel medium $H_h$ in the container are also shown. (b) A transect of the sedimented medium and the intruder illuminated by a thin laser sheet. (c) The intruder depth $z$ as a function of time $t$ for a range of ${\xi }_i$, intruder density relative to the medium.

Figure 2

The yield stress ${\tau }_o$ as a function of overburden pressure $P_p$ for various combinations of intruder densities and size. The slope corresponds to the effective static friction ${\mu }_s$. The error bars are the same as the symbol size and thus not drawn separately.

Figure 3

The ratio of the relative magnitude of the inertial term and the gravitational terms becomes steadily smaller as the intruder slows down. The data correspond to depth vs time curve shown for ${\xi }_i=3.1\times {10}^{-3}$ in Fig. 1.

Figure 4

(a) The effective friction ${\mu }_e$ as a function of intruder speed $v_i$ is observed to approach a constant value at low speeds. Inset: Same plot in linear scale. (b) The effective friction ${\mu }_e$ as a function of inertial number $I$ along with Eq. (8). Inset: Same plot in linear scale shows that the data collapse onto a single curve both at low and high velocities as a function of $I$. The key is the same as in panel (a). (c) ${\mu }_e$ as a function of inertial number $I$ for $d_i=5\mathrm{cm}$ for various depths. ${\mu }_e$ is observed to collapse onto the same curve, irrespective of depth. The measurement errors are smaller than the marker size and not drawn for clarity.

Figure 5

Effective friction as a function of the viscous number $J$ is not observed to collapse onto a single curve. The observed scatter is far greater than measurement error, which is smaller than symbol size.

Figure 6

(a) Schematic of the experimental system used to measure the medium rearrangements. The boxes represent the displacements shown in panel (b) and velocity field in panel (c). (b) Tracers in a vertical plane are observed to follow a systematic trajectory as the intruder is moved from a depth $z_A$ (blue filled circle) to $z_B$ (empty circle) and back up to $z_A$ (green filled circle) as shown in the inset. The net displacement of the tracers after the cycle is shown by a red arrow and is observed to decrease with distance from the intruder. (c) The velocity field of the medium and its curl $\left(v_i=0.01\mathrm{mm}{\mathrm{s}}^{-1}\right)$. The arrows indicate the direction of the flow. The magnitude of the curl is given by the color bar.

Figure 7

(a) The vertical component of the medium velocity $v_z$ as a function of horizontal distance along the equatorial plane of the intruder as it moves down with various speed $v_i\left(z/d_i=3\right)$. The velocity of the medium is significantly lower compared to that of a sphere moving with the same speed in a viscous fluid (solid line). (b) $v_z$ as a function of horizontal distance along the equatorial plane of the intruder at various depths $(v_i={10}^{-3}$ m/s). Greater variation is observed with respect to changes in intruder depth compared with intruder velocity. The measurement errors are smaller than the marker size and not drawn for clarity.

Figure 8

(a) Vertical displacement of the medium normalized by the intruder diameter $\mathrm{\Delta }{z}_{AB}^m/d_i$ for various $I$ when the intruder moves down from $z_A$ to $z_B$. (b) The displacements at $r/d_i\approx 0.5$ [indicated by vertical dashed line in panel (a)] are plotted as a function of $I$ and observed to increase systematically in magnitude. (c) Vertical displacement of the medium normalized by the intruder diameter $\mathrm{\Delta }{z}_{AB}^m/d_i$ for various $I$ when the intruder moves down from $z_A$ to $z_B$ and then returns back up to original depth $z_A$. (d) The displacement near the intruder corresponding to $r/d_i\approx 0.5$ [the vertical dashed line in panel (c)] is plotted as a function of $I$ and is observed to decrease and change sign with $I$. The symbols shown in panels (a) and (c) also correspond to the velocity key in Fig. 7. The measurement errors are smaller than the marker size and not drawn for clarity.