Cooperative motion of intrinsic and actuated semiflexible swimmers

We examine the phenomenon of hydrodynamic-induced cooperativity for pairs of flagellated micro-organism swimmers, of which spermatozoa cells are an example. We consider semiflexible swimmers, where inextensible filaments are driven by an internal intrinsic force and torque-free mechanism (intrinsic swimmers). The velocity gain for swimming cooperatively, which depends on both the geometry and the driving, develops as a result of the near-field coupling of bending and hydrodynamic stresses. We identify the regimes where hydrodynamic cooperativity is advantageous and quantify the change in efficiency. When the filaments' axes are parallel, hydrodynamic interaction induces a directional instability that causes semiflexible swimmers that profit from swimming together to move apart from each other. Biologically, this implies that flagella need to select different synchronized collective states and to compensate for directional instabilities (e.g., by binding) in order to profit from swimming together. By analyzing the cooperative motion of pairs of externally actuated filaments, we assess the impact that stress distribution along the filaments has on their collective displacements.

Figure 1

Swimming velocity of a driven semiflexible filament as a function of Sp for the two different driving mechanisms: (a) actuated swimmer and (b) intrinsic swimmer. For an actuated swimmer, the speed is normalized by its asymptotic value at large Sp, whereas for an intrinsic swimmer the apparent speed of the quadrupole wave $c^{*}=c{L}^{*}/L$ is introduced, where $L^{*}$ is the apparent length, i.e., the projection of the filament length $L$ on the propulsion direction; this magnitude decreases for larger Sp.

Figure 2

Sketch of the geometry of the three initial semiflexible swimmer configurations analyzed here: (a) transverse swimmers (T), (b) coplanar swimmers (C), and (c) frontal swimmers (F). In the simulations, the filaments are prepared by swimming separately until they reach a steady state. In configurations T and C the initial swimming velocity of the two swimmers is the same, $v$, while the initial velocities of frontal swimmers do not have to coincide.

Figure 3

Velocity (top panels) and amplitude (bottom panels) as a function of filament separation $d$ (rescaled by filament length), for $\text{Sp}=2.26$, when intrinsic swimmers move in parallel planes (geometry T, left) and in the same plane (geometry C, right). In the legends, the labels T and C stand for the geometries described in Fig. 2 and the labels A and P for antiphase and in-phase, respectively. For the parameters used in the plot, $A(d\to \infty )=0.01L$, but the results are equivalent for higher amplitudes.

Figure 4

Characteristic hydrodynamic interaction length for nearby semiflexible swimmers, $\beta$ (in units of bead size), as a function of Sp, showing a different dependence for intrinsic and actuated swimmers. Here $A(d\to \infty )=0.01L$.

Figure 5

Relative velocity $v_r$ of frontal intrinsic microswimmers as a function of the minimum distance between them. As $v_r>0$, there is repulsion at initial times in this configuration. Here $d$ is the minimum distance between beads of different filaments, $A(d\to \infty )=0.01L$, and $\text{Sp}=2.26$. In the inset we depict the same on the log-log scale, showing the algebraic decay at large distances $v_r\sim d^{-3}$.

Figure 6

Turning angular velocity $\omega$ in the short-time regime as a function of distance for intrinsic swimmers (a) transversal in phase (TP) and (b) coplanar in phase (CP), for different Sp numbers, at $t={\tau }_c$, where ${\tau }_c=L{\xi }_{\perp }/T_0$ is the turning time. The angular velocity has been normalized by the propulsive velocity $v_0$ divided by the length of the filament $L$. The amplitude for this parameter value is $A(d\to \infty )=0.01L$. The case of antiphase beating (not shown) follows the same behavior.

Figure 7

Long-time turning regime: turning angle $\theta$ as a function of time for transverse bending swimmers for in-phase beating, in coplanar (CP) and transverse (TP) swimmer configurations for $\text{Sp}=2.26$ and $A(d\to \infty )=0.01L$, where ${\tau }_c=L{\xi }_{\perp }/T_0$ is the turning time. In the two cases we subtract the intrinsic rotation when computing $\theta$.

Figure 8

(a) Velocity and (b) amplitude as a function of $d$, for $\text{Sp}=2.26$ and $A(d\to \infty )=0.01L$, when actuated swimmers move in the same plane and in parallel planes and they are driven in phase or in antiphase.

Figure 9

(a) Total efficiency of coplanar and transversal intrinsic swimmers, at small-amplitude driving $A(d\to \infty )=0.01L$, for $\text{Sp}=2.26$. The inset shows the energy consumed after moving a distance $L$ for both geometries. (b) Dissipative efficiency of coplanar semiflexible swimmers when driven at small amplitudes $A(d\to \infty )=0.01L$, for $\text{Sp}=2.26$. The inset shows the energy dissipated after moving a distance $L$ for coplanar swimmers.