Passive swimming of a microcapsule in vertical fluid oscillation

The artificial microswimmer is a cutting-edge technology with applications in drug delivery and micro-total-analysis systems. The flow field around a microswimmer can be regarded as Stokes flow, in which reciprocal body deformation cannot induce migration. In this study, we propose a microcapsule swimmer that undergoes amoeboidlike shape deformations under fluid oscillation conditions. This is a study on the propulsion principle using a capsule with a solid membrane, and one of only a few studies using fluid oscillation. The microswimmer consists of an elastic capsule containing fluid and a rigid sphere. Opposing forces are generated when fluid oscillations are applied, because the densities of the internal fluid and sphere are different. The opposing forces induce nonreciprocal body deformation, which leads to migration of the microswimmer under Stokes flow conditions. Using numerical simulations, we found that the microswimmer propels itself in one of two modes, i.e., stroke swimming or drag swimming. We discuss the feasibility of the proposed microswimmer and show that the most efficient swimmer can migrate tens of micrometers per second. These findings pave the way for future artificial microswimmer designs.

Figure 1

Basic structure of the microcapsule and the problem settings. The dotted line indicates the area under which the spring forces are considered.

Figure 2

Upward propulsion of the stroke swimmer (with $a_1/a_2=3,A=20,f^{*}=2,\mathrm{and}\mathrm{Bo}=1$). (a) Initial reference shape with $a_1/a_2=3$. The material point P is located $0.525a_c$ from the vertical body axis. (b) Time change of $z$ component of the volume center $r_{g,z}^{*}$. Periodic motion can be observed when $t^{*}\ge 5$. (c) Periodic deformation of the capsule during one period of oscillation, where $T$ is the period and $n$ is an integer. The upper graph shows the time change of flow acceleration during one period of oscillation. (See also Movie 1 in the Supplemental Material [58].) (d) Time change of the active stress (red outward, blue inward directed force).

Figure 3

Strokes generated by material points on the membrane during one period of oscillation (with $a_1/a_2=3,A=20,f^{*}=2,\mathrm{Bo}=1$). (a) Trajectories of three pairs of material points on the membrane. Insets (i)–(iii) are the magnified plots of the trajectories. Red points and arrows indicate trajectories with positive flow acceleration, while blue points and arrows indicate trajectories with negative flow acceleration. (b) Distribution of $\lambda$, defined by Eq. (14), on the membrane. Although $\lambda$ is a time-averaged quantity, the color contour is plotted on the capsule surface at $t=nT$ for convenience, where $n$ is an integer.

Figure 4

Effect of the amplitude of the fluid oscillation $A$ (with $a_1/a_2=3,f^{*}=2,$ and $\mathrm{Bo}=1$). (a) Effect of $A$ on the time-average upward swimming velocity $\overline{u_z^{*}}$. (b) Trajectories of the material point P, specified in Fig. 2, with $A=2,10$, and 20. Red points and arrows indicate trajectories with positive flow acceleration, while blue points and arrows indicate trajectories with negative flow acceleration. The same notations are used in Figs. 5, 6, 7, and 9 as well.

Figure 5

Effect of Bond number $\mathrm{Bo}$ (with $a_1/a_2=3,A=20$, and $f^{*}=2$). (a) Effect of $\mathrm{Bo}$ on the time-averaged upward swimming velocity $\overline{u_z^{*}}$. (b) Trajectories of the material point P, specified in Fig. 2, with $\mathrm{Bo}=0.1,0.25$, and $0.6$ (left panel) and with $\mathrm{Bo}=0.6,1.25$, and $2.5$ (right panel).

Figure 6

Effect of the oscillation frequency $f^{*}$ (with $a_1/a_2=3,A=15$, and $\mathrm{Bo}=1$). (a) Effect of $f^{*}$ on the time-averaged upward swimming velocity $\overline{u_z^{*}}$. (b) Trajectories of the material point P, specified in Fig. 2, with $f^{*}=0.5,1.25$, and $5.0$.

Figure 7

Effect of the aspect ratio $a_1/a_2$ (with $A=15,f^{*}=2$, and $\mathrm{Bo}=1$). (a) Effect of $a_1/a_2$ on the time-averaged upward swimming velocity $\overline{u_z^{*}}$. (b) Trajectories of the material point P, specified in Fig. 2, with $a_1/a_2=1.5$ and $3.0$. (c) Distribution of $\lambda$, defined by Eq. (14), on the membrane with $a_1/a_2=1.5$. Although $\lambda$ is a time-averaged quantity, the color contour is plotted on the capsule surface at $t=nT$ for convenience, where $n$ is an integer. (d) Effect of $a_1/a_2$ on the surface-averaged $\lambda$.

Figure 8

Propulsion of the drag swimmer. (a) Direction of forces induced by the fluid oscillation on the rigid sphere (black arrows) and the fluid inside the capsule (white arrow). The left figure represents the stroke swimmer with ${\rho }_c<{\rho }_{\infty }<{\rho }_r$, while the right figure represents the drag swimmer with ${\rho }_{\infty }<{\rho }_c<{\rho }_r$. (b) Time change of the $z$ component of the volume center $r_{g,z}^{*}$ with $A=0$ and 20 ($a_1/a_2=3,f^{*}=2,$ and $\mathrm{Bo}=1$).

Figure 9

Propulsion mechanism of the drag swimmer (with $a_1/a_2=3,A=20,f^{*}=2,\mathrm{and}\mathrm{Bo}=1$). (a) Trajectory of the material point P, specified in Figs. 2 and 9. (b) Distribution of $\lambda$, defined by Eq (14), on the membrane. (c) Periodic deformation of the capsule at $t/T=n+1/4$, when the upward flow acceleration is at a maximum, and at $t/T=n+3/4$, when the downward flow acceleration is at a maximum, where $T$ is the period and $n$ is an integer. (d) Time change of the vertical drag coefficient $C_z$, which is calculated by holding the shape fixed at each instance. The red and blue broken lines indicate the average values of $C_z$ in the regions $\frac{t}{T}=0-0.5$ and $\frac{t}{T}=0.5-1$, respectively.

Figure 10

Comparison between the drag swimmer and the stroke swimmer. The effects of the amplitude of the fluid oscillation $A$ and the aspect ratio $a_1/a_2$ on the time-averaged upward swimming velocity $\overline{u_z^{*}}$, relative to that with $A=0$ are plotted. Although the density conditions are different between the drag swimmer $({\rho }_{\infty }<{\rho }_c<{\rho }_r)$ and the stroke swimmer $({\rho }_c<{\rho }_{\infty }<{\rho }_r)$, both swimmers have the common parameters $f^{*}=2$ and $\mathrm{Bo}=1$. (a) Effect of $A$ (with $a_1/a_2=3$). (b) Effect of $a_1/a_2$ (with $A=20$).