Size and shape affect swimming of a triangular bead-spring microswimmer

We investigate analytically the transport characteristics of the triangular bead-spring microswimmer and its dependence on the sizes of the beads as well as on the relative bead configurations. The microswimmer is composed of three beads connected by linear springs in an isosceles triangular arrangement. The bead at the apex of the isosceles triangle is taken to be of arbitrary size, while the other two beads are of equal and fixed size. This arrangement of the beads and springs undergoes autonomous propulsion thanks to the time-dependent active forces which act along the connecting springs. For small active force amplitudes we obtain an explicit expression for the average velocity of the microswimmer as a function of the shape of the isosceles triangle and the size of the central bead. This enables us to identify the conditions on the shape configuration and the relative bead sizes that yield the fastest motion. We find that the magnitude as well as the direction of motion critically depend on these parameters. In the limit where the central bead is large enough, the swimming direction becomes insensitive to the shape of the microswimmer. This limit is appropriate for modeling Chlamydomonas reinhardtii, an alga which moves with the help of its two flagella. Finally, we estimate the efficiency of the microswimmer and identify the model parameters that produce efficient swimming. These findings will help us understand the nature of biological swimming mechanisms and aid in the designing and tuning of rapid and efficient microswimmers capable of moving along arbitrary paths.

Figure 1

Schematic of a triangular bead-spring microswimmer.

Figure 2

The steady-state trajectories of the individual beads relative to microswimmer center of mass for different $\beta =a/R$ and ${\theta }_0$ with $\alpha =-\pi /2$. The bead sizes and the trajectory amplitudes have been exaggerated in the plot for better visualization.

Figure 3

Dependence of the microswimmer velocity on the radius of the central bead for (a) equilateral triangular swimmer (${\theta }_0=\pi /6$) and for (b) different equilibrium swimmer configurations characterized by different values of the angle ${\theta }_0$.

Figure 4

Dependence of microswimmer velocity on its shape for (a) identical three beads ($\beta =1$) and (b) different radii of the central bead.

Figure 5

The far field averaged velocity field over one cycle of active force in the microswimmer reference frame for small active force amplitude. Here $f={10}^{-3}k, \beta =1$, and ${\theta }_0=\pi /6$.

Figure 6

The averaged velocity fields close to the microswimmer in the microswimmer reference frame for active force amplitude $f=0.1k$.

Figure 7

Speed diagram showing the dependence of the velocity on the radius of the central bead and the microswimmer shape. The white curve corresponds to zero swimmer velocity.

Figure 8

Radius of central bead at fixed ${\theta }_0$ and swimmer shape for fixed $\beta$ for motion with maximum speed and highest efficiencies.

Figure 9

Dependence of average (a) power input $\mathcal{P}$ and the transport efficiencies (b) ${\mathrm{\Pi }}_1$ and (c) ${\mathrm{\Pi }}_2$ in a cycle of active force on shape of microswimmer and central bead radius.