Darcy-Reynolds forces during intrusion into granular-fluid beds

We experimentally study intrusion into fluid-saturated granular beds by a free-falling sphere, varying particle size and fluid viscosity. We test our results against Darcy-Reynolds theory, where the deceleration of the sphere is controlled by Reynolds dilatancy and the Darcy flow resistance. We find the observed intruder dynamics are consistent with Darcy-Reynolds theory for varied particle size. We also find that our experimental results for varied viscosity are consistent with Darcy-Reynolds theory, but only for a limited range of the viscosity. For large viscosities, observed forces begin to decrease with increasing viscosity, in contrast with the theoretical prediction. We suggest that a dynamic lubrication mechanism may be responsible for the observed discrepancy.

Figure 1

(a) Sketch of the experiment. The magnet releases the sphere, which is connected to the accelerometer (accel.) via a threaded rod. The magnet is released at height $H$ onto the submersed settled particle bed (sample). (b) Sketch of the particle bed slowing down the spherical intruder impacting the bed of particles with diameter $d$ and packing fraction ${\varphi }_0$. The penetration of the intruder with speed $V$, diameter $D$, and mass $m$ will make the particles bulge out from the bed surface (long dashed line). At penetration depth $z=\delta$ the particles have been sheared in a region with size $L$. This shear required dilation of the packing and thus forces the local absorption of fluid with viscosity ${\eta }_f\left(B\right)$ whose properties may depend on the applied magnetic field strength $B$. (c) SEM image of the glass beads used, showing slight polydispersity and a mild roughness (d) SEM image of dried ferrofluid on the glass beads. The individual ferrofluid nanoparticles are too small to observe, yet the leftovers from capillary bridges are clearly visible. Image width in (c) and (d) is $200\mu \text{m}$.

Figure 2

Experimental data for impacts into mixtures of water, ${\eta }_f=1$ cp, and glass beads of varying diameter $d$, using an intruder of diameter $D=20$ mm and mass $m=0.047$ kg. (a) Acceleration $a=-\ddot{z}$ as a function of time for impacts with similar impact speed, $V\approx 1$ m/s, for each of the five values of $d$. (b) Dimensionless acceleration $\tilde{a}$ as a function of dimensionless time $\tilde{t}$, as defined in Eq. (7). Smaller grains collapse well, where the assumptions leading up to this equation are valid. (c) Three representative curves of acceleration vs time for the largest beads (600 to $850\mu \mathrm{m}$), where the assumptions for Eq. (7) are not valid. These larger grains are better captured by Eq. (11).

Figure 3

Experimental data for impacts into mixtures of water, ${\eta }_f=1$ cp, and glass beads of varying diameter $d$, using an intruder of diameter $D=20$ mm and mass $m=0.047$ kg. (a) $a_{\max }$ vs $V$ for all five values of $d$. Smaller beads obey the scaling law in Eq. (9), as expected, since $z\ll D$ for these impacts. Larger beads obey $a_{\max }\propto V$ from Eq. (11), as expected, since $z\sim D$ for these impacts. (b, c) We perform linear fits to the logarithmic data in panel (a) to obtain the best fit for the function $a_{\max }=B{V}^{\beta }$. DRT predictions are shown: $B\propto d^{-0.8}$ for panel (b); $\beta =1.6$ for small beads and $\beta =1$ for large beads in panel (c).

Figure 4

Acceleration vs time for different impacts into beds with $d=90\mu \mathrm{m}$ and varying fluid viscosity using an intruder of diameter $D=20$ mm and mass $m=0.047$ kg. (a) Impacts with $V\approx 1$ where ${\eta }_f$ is varied by adding glycerol to water. (b) Impacts with $V\approx 2.5$ where ${\eta }_f$ is varied by using a ferrofluid and changing the external magnetic field. A zero-field ferrofluid impact result is also shown in panel (a) for the ferrofluid (dash-dotted line).

Figure 5

Experimental data for impacts into mixtures with varied ${\eta }_f$ and $d=90\mu \mathrm{m}$ using an intruder of diameter $D=20$ mm and mass $m=0.047$ kg. (a) $a_{\max }$ vs $V$ for all values of ${\eta }_f$. (b), (c) We perform linear fits to the logarithmic data in panel (a) to obtain the best fit for the function $a_{\max }=B{V}^{\beta }$. DRT predictions are shown: $B\propto {\eta }_f^{0.4}$ for panel (b); $\beta =1.6$ in panel (c). Similar to the data from Fig. 4, we find that DRT breaks down for ${\eta }_f>10$ cp, and forces begin to decrease with increasing viscosity. The horizontal error bar represents the uncertainty in the viscosity for the 100% glycerol, as discussed in Sec. 3a.