Figure 1

Our model for flagellar phase locking: Two infinite, parallel, 2D sheets propagate periodic waves at a speed $c$ leading to swimming in the opposite direction. The top sheet is allowed to move with velocity $U_{\Delta }$ with respect to the bottom sheet and is out of phase by an angle $\varphi$.Reuse & Permissions

Figure 2

Swimmers with reflection symmetries by the horizontal and vertical axes cannot phase lock: If a relative force exists in (a), we obtain the forces in (b) and (c) by reflection symmetries ($R$ planes). Applying kinematic reversibility to (c) (KR line) leads to a force in (d) which is minus the one in (b), indicating that they both must be zero.Reuse & Permissions

Figure 3

Top: Evolution of the phase angle, $\varphi$, from an initial value of $\pi /2$ for swimmers with three different waveforms [upper inset]: sine wave (black solid line), sine waves shifted forward and backwards by $3\pi /10$ (blue dotted and red dashed lines, respectively). As predicted by the theory, the perfect sine wave shows no phase evolution while the skewed sine waves evolve to phase-locked positions (in-phase and opposite-phase, respectively). The energy dissipated for the three waveforms is almost identical [lower inset] (see text). Bottom: For all initial conditions, the phase angle converge to a phase-locked state at the only stable fixed point, $\varphi =0$ (same parameters as in top figure, dotted line).Reuse & Permissions