Deformable microswimmer in an external force field

External forces, such as gravity, play significant role in the swimming properties of autonomous biological microswimmers as well as artificial swimming robots. Here we have studied the influence of the external forces on the transport characteristics of the triangular bead-spring microswimmers. The microswimmer, formed by connecting three beads using three springs in an equilateral triangular arrangement, is capable of performing autonomous translational (“mover”) and rotational (“rotor”) motions. We show that for a mover triangle the application of a small external force results in the alignment of swimming direction with that of the external force, a phenomenon known as “gravitaxis.” We demonstrate that this gravitactic nature of the active triangle is purely due to the hydrodynamic interaction among the beads. Under large external force, however, the gravitactic nature is lost. This transition from gravitactic to nongravitactic motion of the microswimmer is characterized by a saddle node or pitchfork bifurcations (depending on nature of active forces), where the strength of the critical external force scales linearly with the active force amplitude, $f_e^c\sim f_a$. However, for the rotor triangle only saddle node bifurcation is observed, which results in a vanishing angular velocity as the strength of the external force is increased. The critical value of the external force for the rotor, however, scales as $f_e^c\sim f_a^{2/3}$. These findings will provide insights into the nature of biological swimming under gravity, especially the gravitactic microorganisms such as Chlamydomonas, as well as help in the design of underwater vehicles.

Figure 1

Schematic of passive (left) and active (right) triangular bead-spring assemblies placed in an external force field ${\mathbf{f}}_e$. The swimmer orientation $\theta$ is characterized by the angle made by the vector joining the triangle center with one of the beads and the direction of external force ${\mathbf{f}}_e$. The unit vector ${\widehat{\mathbf{e}}}_m$ marks the direction of motion for a mover triangle.

Figure 2

Schematics of (a) stable and (b) unstable steady state configurations of a passive triangle in an external force field ${\mathbf{f}}_e$.

Figure 3

Bifurcation diagrams for the gravitactic swimming of a mover with (a) $A>0$ and (b) $A<0$ as obtained from the numerical simulation of Eq. (1). For $f_e<f_e^c$, the swimming direction for a pusher ($\alpha >0$) is the same as that of external force. For a puller swimmer ($\alpha <0$), however, the swimming direction depends on the sign of $A$ (also see Fig. 4). The solid and dashed lines represent the stable and unstable steady states, respectively. Values of other parameters are $f_a/k=0.03$, $l/a=100$, $\alpha =\pi /2,-\pi /4$, and $\lambda =0.8$, 0.3. (c) Phase diagram of the mover triangle from the numerical simulation of Eq. (1) for different values of active and external forces. The critical value of the external force separating the gravitactic (shaded region) and nongravitactic behaviors scales linearly with the active force amplitude. The triangles represent the steady state behavior of the microswimmer in the presence of ${\mathbf{f}}_e$, with the arrows in the triangles denoting the respective direction of autonomous propulsion.

Figure 4

Phase diagram showing swimming direction of puller ($-\pi <\alpha <0$) and pusher ($0<\alpha <\pi$) swimmers in the gravitactic regime as obtained from numerical simulation of the mover triangle. The triangles in the three regions represent the steady state swimmer configuration with the arrows denoting respective swimming directions. Values of other parameters are $l/a=100$, $f_a/k=0.01$, and $f_e/k=0.5$.

Figure 5

(a) Bifurcation diagram for the rotor triangle from the numerical simulation of the rotor triangle. For $f_e<f_e^c$, the rotor triangle has a nonzero angular velocity. The solid and dashed lines represent the stable and unstable steady states, respectively. Values of other parameters are $f_a/k=0.03$, $l/a=100$, and $\lambda =0.8$. (b) Phase diagram of the rotor triangle as obtained from numerical simulations. The critical value of the external force separating the rotating (shaded region) and nonrotating states scales as $f_e^c\sim f_a^{2/3}$.

Figure 6

(a) Velocity field ${\mathbf{u}}_{\mathrm{FD}}$ due to the force-dipoles induced by the passive bead-spring triangle in external force field. Parameter values are $l/a=100$, $f_e/k=1$. (b) Schematic showing the forces acting on the three beads of the passive triangle in the reference frame comoving with the triangle.

Figure 7

Velocity fields (averaged over one cycle of activity) due to a mover triangle with (a) $f_a/k=0.01$, $f_e/k=0$, (b) $f_a/k=0.02$, $f_e/k=0.01$, and a rotor triangle with (c) $f_a/k=0.01$, $f_e/k=0$, and (d) $f_a/k=0.02$, $f_e/k=0.01$ in the reference frame comoving with the swimmer. Other parameters are $l/a=100$, $\lambda =0.8$, (a), (b) ${\alpha }_{12}={\alpha }_{13}=0$, ${\alpha }_{23}=\pi /2$, and (c), (d) ${\alpha }_{12}=-{\alpha }_{13}=-2\pi /3$, ${\alpha }_{23}=0$. Corresponding lower panels show the schematics with the singularity representations where straight and curved arrows mark the average stokeslets and rotlets, respectively.