Dynamics of a helical swimmer crossing an interface between two immiscible fluids

In this study, we experimentally investigate the mechanical process of a self-propelled helical swimmer to move across an interface between two immiscible fluids. This configuration is aimed to emulate some aspects of the process used by bacteria to trespass mucus layers or epithelial cell membranes to cause infections. We consider two configurations: head-first and tail-first. We find that, in both cases, the head of the swimmer deforms the interface generating a meniscus that induces a significant reduction of the swimming speed, that lasts until the interface is pierced. In both cases, the dynamics of penetration is complex leading to significant variations of the swimming speed during the process. We observed interesting differences in the penetration dynamics for the two cases; we argue that the differences arise from the significantly different shape and direction of the menisci that forms during the penetration process. A model that accounts for thrust, drag, buoyant, and capillary forces is used to rationalize the results.

Figure 1

The experimental setup. (a) The helical swimmer. The dimensions of the device are: $D_h=3\mathrm{mm}$; $L_h=16\mathrm{mm}$; $L_t=16\mathrm{mm}$; $\lambda =5.3\mathrm{mm}$; $2R=3\mathrm{mm}$; and the pitch angle $\theta ={45}^{\circ }$. The thickness of the wire, $d$, was 0.9 mm. (b) The upper and (c) the lower fluids. The swimmer with (d) head and (e) tail firsts.

Figure 2

The helical swimmer velocity without interface in (a) dense fluid, $U_o$ and (b) light fluid, $U_1$. The physical properties of the fluids are shown in Table 1. The error bars indicate the measurement uncertainty.

Figure 3

Time sequence of the swimmer crossing the interface. (a)–(h) depict different relevant instants of the process, described in the text; these instants are also shown in the plot in Fig. 4. Images (i) and (j) show close-ups of the swimmer during the formation of the upper and the lower meniscus, respectively. A thin line was drawn at the interface location for clarity. The swimming speed for this case was 6.34 mm/s, which corresponds to a $\mathrm{Re}=9.9\times {10}^{-3}$ and $\mathrm{Ca}=0.29$.

Figure 4

Normalized speed of the swimmer $U/U_o$ as a function of position, $z/D_h$. $U_o$ is the speed of the swimmer in lower fluid without interface. The blue line indicates the location of the interface. The letters correspond to the time instants depicted in Fig. 3. The black-continuous and red-dashed curves correspond to swimmer speeds $U_o=6.34\mathrm{mm}/\mathrm{s}$ and $U_o=4.22\mathrm{mm}/\mathrm{s}$, respectively. The green line (lower right side) indicates the dimensionless swimming speed in oil (single phase).

Figure 5

(a) The minimum speed $U_{\text{min}}/U_0$ as a function of $C_a$ during penetration. (b) The piercing speed $U_{\text{imp}}/U_0$, after meniscus rupture. (c) The maximum speed $U_{\text{max}}/U_0$ during penetration. The solid and the dashed lines show the predictions from Eq. (14) in panel (a) or Eq. (15) in panel (c), considering the maximum and the minimum value of the radius of dimensionless curvature of the meniscus, $\kappa$ [Eq. (16)], respectively. Each point on the graph is the average of five measurements.

Figure 6

Time sequence of the swimmer crossing the interface. (a)–(h) depict different relevant instants of the process, described in the text. Panels (i) and (j) are close-ups of the formation of the lower and upper meniscus, respectively. The inverse meniscus is formed under the small orange region close to the tail. A thin line was drawn at the interface location for clarity. The swimming speed for this case was 5.77 mm/s, which corresponds to a $\mathrm{Re}=9.0\times {10}^{-3}$ and $\mathrm{Ca}=0.27$.

Figure 7

Normalized speed of the swimmer $U/U_0$ as a function of position, $z/D_h$. The blue line indicates the location of the interface. The letters correspond to the time instants depicted in Fig. 6. The black and red curves correspond to swimmer speeds $U_o=5.77\mathrm{mm}/\mathrm{s}$ and $U_o=4.85\mathrm{mm}/\mathrm{s}$, respectively. The green line (lower right side) indicates the dimensionless swimming speed in oil (single phase).

Figure 8

(a) The maximum speed $W_{\text{max}}/U_0$ (b) the minimum $W_{\text{min}}/U_0$ and (c) the adhesion speed $W_{\text{ad}}/U_0$ during the penetration as a function of the capillary number, $\mathrm{Ca}$. The solid and the dashed line show the predictions from Eq. (14), considering the maximum and the minimum value of the normalized radius of curvature of the meniscus, $\kappa$ [Eq. (16)], respectively. Each point on the graph is the average of five measurements.