Propulsion by reciprocal motion into granular media

The force experienced by an object moving in a granular medium is a necessary input to describe locomotion problems in such a context. Towards this objective, resistive force theory (RFT) has been developed for granular flows inspired by previous developments in the case of viscous flows. In viscous fluids, a reciprocal motion does not lead to a net propulsion due to the kinematics reversibility of viscous flows that is included in the RFT. We show that, in a granular medium, a reciprocal motion allows for propulsion. We investigate the specific mechanisms underlying this propulsion and discuss how they differ from that of viscous flows. Understanding these phenomena permit both the accurate description of locomotion in sand and the refinement of the RFT for granular materials. These achievements provide opportunities towards the development of robots that are able to progress inside granular media and are suitable for controlling industrial processes.

Figure 1

(a) The experimental setup. A rod with a length of $L=60$ mm and a diameter of $D=6$ mm oscillates according $\theta \left(t\right)={\theta }_{m}sin\left(2\pi \nu t\right)$ with respect to the $x$-direction. (b) Position of the rod's head along the $x$-direction as a function of time in glycerine (blue solid line) and in a granular medium (orange solid line) for an angular amplitude ${\theta }_m={45}^{\circ }$, frequency $\nu =0.26$ Hz, and depth $h=40$ mm.

Figure 2

Mean speed of the rod's head along the $x$-axis as a function of the experimental parameters: angular velocity $L\nu$ (a), angular amplitude ${\theta }_m$ (b), and depth $h$ (inset). Other parameters are maintained constants: $L=60$ mm, $D=6$ mm, $\nu =0.26$ Hz, $h=40$ mm, ${\theta }_m={45}^{\circ }$. In (b) the solid lines show the predictions of the numerical simulation introduced in Sec. 3 for $\alpha =0.28, {\theta }_t=8^{\circ }, {\theta }_i={40}^{\circ }, C_n/C_t=3$, and $F_d/C_{n}L=0.055h/L$.

Figure 3

Torque $T$ experienced by a rod ($L=60$ mm and $D=6$ mm) in a granular medium as a function of its angular position for a constant depth $h=40$ mm and ${\theta }_m={90}^{\circ }$. The blue dashed line sketches the torque for the same cycle in a viscous fluid.

Figure 4

(a) Normalized torque $T/T_0$ experienced by a rod as a function of its angular position $\theta$ after a change in direction ($D=6$ mm and $h=40$ mm). The blue, green, orange, and red solid lines correspond to rod lengths $L=15,30,45,60$ mm, respectively. The dark solid line indicates the empirical relation $T=T_0(1-e^{-(\theta +90)/{\theta }_t})$ with ${\theta }_t=8^{\circ }$. (b) Force along the $x$-axis as a function of the normalized time for a rod with a length of $L=60$ mm and diameter of $D=6$ mm oscillating at $\nu =0.26$ Hz with an angular amplitude ${\theta }_m={90}^{\circ }$ into a granular medium at a depth $h=40$ mm (orange solid line). The theoretical force experienced by the same rod moving in glycerine is indicated by the blue solid line as a reference and magnified by a factor of 10.

Figure 5

Successive positions of the rod and illustrations of the three mechanisms that govern its propulsion into a granular medium. A quarter of the cycle is highlighted in color. Color codes the norm of the drag: decreasing drag from red to yellow. Arrows indicate the sense of rod motion. (a) The head moves forward while the rod explores mainly unvisited zones. (b) The head moves backward in a less extend while the rod explores unvisited zones.

Figure 6

(a) Normalized displacement of the rod's head, $x/L$, as a function of $\nu t$ resulting from the numerical integration of Eq. (1) with the three rules to account for structuring effects, transient regimes, and long-range interactions. The parameters are ${\theta }_m={90}^{\circ }, C_n/C_t=3, F_d/L{C}_n=0.08, \alpha =0.28, {\theta }_t=8^{\circ }$, and ${\theta }_i={40}^{\circ }$. Inset: visited (yellow) and nonvisited (orange) areas in the $(x,y)$ plane after three oscillations of the rod (black solid line). (b) Normalized head position $x/L$ as a function of the normalized time $\nu t$. The red solid line corresponds to the numerical integration when the three effects are taken into account, the orange solid line in the absence of long-range interactions (${\theta }_i=0^{\circ }$), the green solid line in the absence of memory effect ($\alpha =0$), and the blue solid line in the absence of transient regime (${\theta }_t=0^{\circ }$).

Figure 7

Torque $T$ experienced by the rod as a function of time for two different packing fractions of the granular medium $\varphi =0.58$ (orange solid line) and $\varphi =0.61$ (red solid line).

Figure 8

Notations used in the model