Drag force in immersed granular materials

We investigate the drag forces acting on objects moving through a granular packing immersed in water. In this aim, we conducted uplift experiments involving pulling out horizontal plates at a prescribed velocity vertically. During these tests, we observed that the drag force reaches to peak at a low displacement and then decays. Results show that the peak drag force strongly increases with the velocity and depends on the plate size and grain diameter. We identify empirical scaling laws for these properties and introduce a Darcy-flow mechanism that can explain them. Furthermore, we conducted tests involving suddenly stopping the motion of the plate, which evidenced a progressive relaxation of the drag force in time. We discuss how a visco-elasto-plastic mechanical analog can reproduce these dynamics. These results and analyses highlight fundamental differences in drag force between dry and immersed granular materials.

Figure 1

Experimental uplift tests. (a) Experimental setup showing the loading frame, load cell, shaft, and the container filled with glass beads and water. (b) Photo of the glass beads used (top) $d$ = 0.3 mm; (middle) $d$=0.6 mm; (bottom) $d$= 1 mm; the scale is the same on all three photos. (c) Illustration of a typical force (F) versus plate displacement (P) response obtained when pulling the plate vertically at a constant velocity.

Figure 2

Drag force $F$ as a function of the plate displacement $P$ during uplift tests ($d=$ 0.3 mm, $H=$ 120 mm, and $B=$ 40 mm). (a) Tests performed in an immersed packing at different velocities; the inset focuses on small displacements. (b) Comparison between tests performed in a dry and in an immersed packing using the same plate and the same grains.

Figure 3

Effect of the uplift velocity $v$ on the peak drag force $F_0$. (a) and (b) Peak drag forces measured at different velocities with different grain sizes and plate diameters (see legends): (a) $d=$ 0.3 mm, (b) $B=$ 40 mm. Dashed lines correspond the best linear fits of the data by Eq. (1) using $F_0^{qs}$ and $m$ as fitting parameters. (c) and (d) Values of the parameter $m$ that best fit data in (a) and (b), respectively. Error bars represent the confidence intervals for the parameter $m$ associated with the linear regression. Dashed lines show power laws with an exponent 3 (c) and $-2$ (d) for a visual reference.

Figure 4

Origin of the peak drag force in dry (left) and immersed (right) granular packings. In dry condition, $F_0$ corresponds to the weight of the grains comprising the truncated cone (patterned area). In fully immersed conditions, this weight accounting for buoyancy corresponds to the quasistatic component of $F_0$. We propose a Darcy-flow mechanism to explain the rate-dependent component of the drag force, whereby water recirculates around the plate through the granular matrix on a typical pathway length ${\alpha }_{l}B$.

Figure 5

Comparison of the parameter $m_{\exp }$ [measured experimentally by fitting the data $F_0\left(v\right)$ by Eq. (1)] with the value $m_{\mathrm{theory}}$ [predicted by the proposed model in Eq. (14) using ${\alpha }_l=0.62]$. The dashed line represents the function $m_{\exp }=m_{\mathrm{theory}}$ for visual reference.

Figure 6

Post-peak behavior measured from data shown in Fig. 2 ($B=$ 40 mm). (a) Displacement of the plate $P_0$ when the peak drag force is reached. The dashed line represents the best linear fit using Eq. (15), obtained with $P_0^{qs}\approx$ 0.67; 0.6; 0.47 mm and ${\lambda }_0\approx$ 0.24; 0.18; 0.17 s for grain size of 0.3; 0.6; 1 mm, respectively. (b) Post-peak relative drag force drop $\mathrm{\Delta }F/F_0$ [see Fig. 1].

Figure 7

Examples of mechanical analog: (a) spring, (b) dashpot, and (c) slider elements; (d) elasto-plastic, (e) visco-plastic, and (f) elasto-visco-plastic analogs.

Figure 8

Relaxation tests performed in 0.3 mm grains. (a) and (b) Displacement $P$ and measured drag force during a test where the plate (B= 40 mm) is uplifted at a constant velocity $v=$ 0.4 mm/s during $t_0$= 4 s and then stopped. (c) Semilog plot of the drag force evolution during relaxation [same data as in (b)]. The dashed line represents the best fit using the exponential decay in Eq. (17), using $\lambda$ as sole fitting parameter. (d) Values of relaxation time $\lambda$ obtained by similarly fitting tests conducted with different velocities $v$.

Figure 9

Proposed mechanical analog for the drag force in immersed granular packings. The force $F$ is distributed between the elasto-plastic element (left) and the visco-elastic element (right): $F=F^{ep}+F^{ve}$. Both elements undergo the same deformation $P$ and deformation rate $\dot{P}$ at any point in time. The dashed red line indicates that large slider deformations may lead to lowering the dashpot viscous parameter $\xi$; this could explain the post-peak drop in drag force observed in Fig. 2.

Figure 10

Drag force relaxation dynamics. (a) Three experimental plate displacement where a constant velocity $v$ is applied before the plate is stopped. (b)–(d) Drag force evolution corresponding to tests on (a): v = 10 mm/min (b), 400 mm/min (c), and 1000 mm/min (d). (b)–(d) Compare the experimental drag force measurements ($F_{\exp }$) to the predictions of the proposed visco-elasto-plastic model in Fig. 9 ($F_{\mathrm{sim}}$, together with its elasto-plastic and visco-elastic components). The model parameter used are summarized on Table 3.