Biaxial fluid oscillations can propel a microcapsule swimmer in an arbitrary direction

Due to their potential usefulness in engineering and medical applications, various artificial microswimmers have been proposed. In our previous paper et al. [T. Morita, T. Omori, and T. Ishikawa, Phys. Rev. E 98, 023108 (2018)], we introduced a microcapsule swimmer that underwent amoeboidlike shape deformation under vertical fluid oscillation and showed the advantages of using a solid membrane and fluid oscillation in terms of swimmer controllability. Although the microcapsule was capable of migrating in the Stokes flow regime, the propulsion direction was limited to vertically upward or downward. In this paper, therefore, we attempted to control the propulsion of a microcapsule in an arbitrary direction by imposing biaxial fluid oscillations, which is a major step toward future applications. Numerical results showed that the microcapsule could migrate in the horizontal and vertical directions by imposing biaxial fluid oscillations in the vertical and horizontal planes, respectively. The horizontal propulsion can be understood in terms of effective and recovery strokes, i.e., a stroke swimmer, whereas vertical propulsion is similar to rigid body motion induced by a torque, i.e., a torque swimmer. By sequentially imposing three types of fluid oscillations, we successfully controlled the microswimmer to draw a Π-shaped trajectory. Thus, these results illustrate that the position and trajectory of the microswimmer can be controlled arbitrarily in three dimensions. The knowledge presented in this paper is important for future artificial microswimmer designs.

Figure 1

Basic structure of the microswimmer and the problem settings. Acceleration vector $\underline{\alpha }$ of the fluid oscillations is given as a function of time. The gravitational acceleration $\underline{\mathcal{g}}$ is imposed in the $-z$ direction. The dotted line on the capsule indicates the area under which spring forces are considered.

Figure 2

Horizontal propulsion of the stroke swimmer ($a_1/a_2=3,A_x=10,A_z=20,A_y=0,{\varphi }_x=0,\mathrm{and}{\varphi }_z=-\pi /2$). (a) Initial reference shape. The volumetric centers of the microswimmer and the rigid sphere are ${\underline{r_{cc}}}^{*}$ and ${\underline{r_{rc}}}^{*}$, respectively. (b) Periodic deformation of the capsule during one period of oscillation, where $T$ is the period and $n$ is an integer (see also movie 1 in Supplemental Material [39]). The upper graph shows the time dependence of flow acceleration during one oscillation period in the $x$ and $z$ directions. (c) Time dependence of the $x$ and $z$ components of ${\underline{r_{cc}}}^{*}$. Periodic motion is observed when $t^{*}\ge 5$.

Figure 3

Horizontal migration mechanism, where the orange color indicates the effective stroke and the green color indicates the recovery stroke ($a_1/a_2=3,A_x=20,{\varphi }_x=0,\mathrm{and}{\varphi }_z=-\pi /2)$. (a) Relative trajectory of ${\underline{r_{rc}}}^{*}$ from ${\underline{r_{cc}}}^{*}$ during one oscillation period, plotted for the capsule shape at $t/T=n$. $\theta$ is the angle between the $-z$ direction and the time-averaged vector of ${\underline{r_{rc}}}^{*}-{\underline{r_{cc}}}^{*}$. (b) Time dependence of ${\lambda }_x^{*}$ and the $x$ component of the volumetric center of the microswimmer $x_{cc}^{*}$ during one oscillation period.

Figure 4

Effect of parameters $a_1/a_2,A_x,\mathrm{and}{\varphi }_x-{\varphi }_x$ on the time-averaged propulsion velocity $\overline{{\underline{u}}^{*}}$ (a, c, e) and the function ${\underline{\lambda }}_{\mathrm{ave}}^{*}$ (b, d, f). The effects of (a, b) the amplitude of fluid oscillation $A_x$ ($a_1/a_2=3\mathrm{and}{\varphi }_x-{\varphi }_z=\pi /2),$ (c, d) the aspect ratio $a_1/a_2$ ($A_x=10\mathrm{and}{\varphi }_x-{\varphi }_z=\pi /2)$, and (e, f) the phase difference between ${\varphi }_x-{\varphi }_z$ ($A_x=10\mathrm{and}{a}_1/a_2=3)$. In (e), the attitude angle $\theta$ is also plotted.

Figure 5

Downward propulsion of the torque swimmer ($a_1/a_2=3, A_x=A_y=10,A_z=0,{\varphi }_x=0,\mathrm{and}{\varphi }_y=\pi /2$). (a) Periodic deformation of the capsule during one oscillation period, where $T$ is the period and $n$ is an integer (see also movie 2 in Supplemental Material [39]). The upper graph shows the time dependence of flow acceleration during one oscillation period in the $x$ and $y$ directions. (b) Time dependence of the $x, y$, and $z$ components of ${\underline{r_{cc}}}^{*}$. Periodic motion is observed when $t^{*}\ge 5$.

Figure 6

Propulsion mechanism of the torque swimmer. (a) Shapes of the microswimmer observed in the reference frame rotating with the acceleration field ($a_1/a_2=3, A_x=A_y=10, {\varphi }_x=0$, and ${\varphi }_y=-\pi /2$). The coordinate axes of each figure are shown at the top. (b) Schematic diagram of the forces exerted on the rigid sphere ${\underline{F}}_r$ and the fluid inside the capsule ${\underline{F}}_c$, as well as the resultant torque that acts at ${\underline{r}}_T^{*}$. (c) Another coordinate system $\eta -\zeta -\xi$ with an origin at ${\underline{r_{rc}}}^{*}$. The $\eta$ axis is taken in the direction to $\underline{r_{cc}}$ from $\underline{r_{rc}}$. The $\xi$ axis is defined as $\underline{\xi }=\underline{\eta }\times \frac{\left(\underline{\eta }\times \underline{z}\right)}{|\underline{\eta }\times \underline{z}|}$, where $\underline{z}$ is the unit vector of the $z$ axis. $\zeta$ is defined as $\underline{\zeta }=\underline{\xi }\times \underline{\eta }$.

Figure 7

Effect of the aspect ratio $a_1/a_2$ on the vertical velocities. (a) The vertical swimming velocity $\overline{u_z^{*}}$. $Biaxial$ indicates the present results under biaxial fluid oscillation conditions. $Uniaxial$ indicates the previous results [31] under uniaxial vertical fluid oscillation conditions. (b) The vertical net propulsion velocity $V_z^{*}$ obtained by assuming rigid body motion.

Figure 8

Controlling the microswimmer with $a_1/a_2=3$ to draw a Π-shaped trajectory in the x-z plane (see also movie 3 in Supplemental Material [39]). The microswimmer is initially placed at the origin, and the red line indicates the upward propulsion in uniaxial vertical fluid oscillations with $A_z=20$ and ${\varphi }_z=-\pi /2$ for $0\le t^{*}\le 100$, i.e., for 200 oscillation periods. The green line indicates the horizontal propulsion in biaxial fluid oscillations in the vertical plane with $A_x=10, A_z=10, {\varphi }_x=0$, and ${\varphi }_z=-\pi /2$ for $100\le t^{*}\le 150$, i.e., for 100 oscillation periods. The blue line indicates the results of biaxial fluid oscillations in the horizontal plane with $A_x=10, A_y=10, {\varphi }_x=0$, and ${\varphi }_y=-\pi /2$ for $150\le t^{*}\le 212$, i.e., for 123 oscillation periods.