Self-propulsion of a helical swimmer in granular matter

The motion of helicoidal swimmers moving in a pool filled with a granular medium is studied experimentally. The horizontal displacement through granular beads is measured considering geometrical modifications of the swimmer, the size, and frictional properties of the media. We found three main parameters which affect the swimming performance: the diameter, the wavelength, and the angle of the helix. The swimming speed scales with the rotation speed, $\omega R$. The size of particles does not affect the swimming speed significantly; the swimming speed is reduced when the particle's angle of repose increases. It was found that a maximum swimming speed is achieved when the helix angle is close to ${55}^{\circ }$. The experimental data are compared with predictions of the granular resistive force theory, which was extended to be applicable for a swimmer with a rigid helical tail, leading to good agreement.

Figure 1

(a) The head, where a permanent magnet is enclosed, has a diameter and length $D_h$ and $L_h$, respectively. The helix can be characterized by its wavelength ($\lambda$), diameter ($2R$), helix angle ($\theta$), and the thickness the spring ($2a$). (b) Scheme of the experimental setup. The work space is $70\times 70\times 140 {\mathrm{mm}}^3$. (c) Configuration of the swimmer inside the granular matter with an acrylic plate on the top.

Figure 2

The schematic representation for the seven swimmers employed in all experiments: case (i), constant $2R$. Note that the wire length in the helix, $L_w=L_t/cos\theta$, is the same for all swimmers.

Figure 3

Photos of the particles used in this investigation as granular matter: (a) glass beads, 2.0 mm mean diameter; (b) mustard seeds, 2.2 mm mean diameter.

Figure 4

Swimming speed for case (i) with glass particles of 0.325 mm in diameter. (a) Swimming speed as a function of rotational speed, $\omega$. (b) Swimming speed as a function of helix angle, $\theta$ (also as a function of wavelength, $\lambda$). All data for swimmers with $2R$ constant.

Figure 5

Swimming speed as a function of rotational speed for different particle sizes. Each panel shows the results for different swimmers (see Table 1): (a) D11P6, (b) D11P20, (c) D11P41, and (d) D11P60.

Figure 6

Swimming speed for mustard seeds. On (a), the speed is plotted as a function of rotational speed; on (b), the swimming speed is shown as a function of helix angle. Case (i) swimmers are tested. The properties of the grains are shown in Table 2.

Figure 7

Normalized speed as a function of the angle for all swimmers in case (i). On the left [(a) and (c)] the speed is normalized with $\omega R$, while on the right [(b) and (d)] the normalization is done with $\omega \lambda$. The lines are the predictions from granular resistive force theory: the solid lines show the model with the head [Eq. (A2)]; the dashed lines show the predictions without the effect of the head [Eq. (4)]. The dotted line in (a) shows the model prediction from Ref. [14]. (a) and (b) Glass particles and (c) and (d) Mustard seeds.

Figure 8

Speed as a function of helix angle, $\theta$, and wavelength, $\lambda$, for different rotation speeds. The lines show the predictions from Eq. (A2), which does account for the presence of the head. Each color, both symbols and lines, corresponds to different values of the angular speed, $\omega$. (a) Glass particles and (b) Mustard seeds.

Figure 9

(a) Speed as function of helix angle, $\theta$, and (b) wavelength, $\lambda$. Predictions from the RFT model [Eqs. (4) and (A2)]. Predictions for two values of $\mathrm{\Gamma }$, low packing (dashed lines) and high packing (solid lines), and for three values of the angle of repose.

Figure 10

The schematic representation for the nine swimmers employed in all experiments. Case (ii), constant $\lambda$; case (iii), constant $\theta$. Note that the wire length is the tail; $L_w=L_t/cos\theta$, is constant for each case.

Figure 11

Swimming speed for cases (ii) and (iii) with glass particles of 0.325 mm in diameter. On the left and right columns, the swimming speed is shown as a function of the rotational speed, $\omega$, and the helix angle, $\theta$ (or alternatively as a function of wavelength $\lambda$ or helix size $2R$). (a) and (b) Case (ii) $\lambda$ is constant and (c) and (d) Case (iii) $\theta$ is constant.