Decoupling translational and rotational effects on the phase synchronization of rotating helices

The locomotion of swimming microorganisms often relies on synchronized motions; examples include the bundling of flagella and metachronal coordination of cilia. It is now generally accepted that such behavior can result from hydrodynamic interactions alone. In this paper we consider the interactions between two side-by-side rigid helices driven by constant torques. We use the method of regularized Stokeslets to simulate an end-pinned model, in which restoring forces and torques are applied at one end of each helix. This allows us to decouple the respective effects of translation and rotation on phase synchronization. We find that while translational freedom leads to synchrony, rotational freedom can result in either synchrony or antisynchrony, depending on the stiffness of the system. In addition, we characterize the nature of the physical mechanisms driving these behaviors, focusing on the individual effects of each applied force and torque. For translational freedom, there is a single underlying mechanism in which the interaction forces indirectly influence the helix rotation rates. Multiple mechanisms are at play for rotational freedom: the interaction torques may exert either direct or indirect influence depending on stiffness. These characterizations are important to the future development of reduced-order models, which should capture not only the expected end behaviors (synchrony or antisynchrony), but also the nature of the driving mechanisms.

Figure 1

Schematic showing helix position and orientation (not to scale).

Figure 2

Schematic showing how helix phase is defined. The helices are shown in a perspective view with each helix axis parallel to the $x$ axis (out of the page). Helix 1 has a phase ${\theta }_1=0$ and helix 2 has a phase ${\theta }_2=\pi /2$.

Figure 3

Schematic of helix geometry used in simulations (to scale). The local origins are shown at their equilibrium positions and the helix phase difference is $\Delta \theta =\pi /2$.

Figure 4

Illustration of constrained helix motions. (a) Axial translation; (b) axial rotation; (c) transverse translation; (d) transverse rotation.

Figure 5

Time history of helix phase difference for $k_{\mathrm{tr}}=100$, showing convergence to a synchronized state. Time is normalized by the rotation period of an isolated helix.

Figure 6

Comparison of self- and interaction-force contributions to the change in phase difference for $k_{\mathrm{tr}}=100$. While self-forces drive a long-term decrease in phase difference, the interaction-force contribution oscillates around zero. (The contributions of self-torques and interaction torques, not shown, also oscillate around zero, with even smaller amplitude than that of the interaction force.)

Figure 7

Comparison of $y$ displacement for $k_{\mathrm{tr}}=100$. Helix 1 (the leading helix) experiences larger amplitude displacements than helix 2 (the lagging helix).

Figure 8

Self-force contributions ($y$ component only) to rotation rate for $k_{\mathrm{tr}}=100$. Shaded regions indicate times during which helix 1 (leading) slows down relative to helix 2 (lagging). (${\Omega }_i$ is parallel to the negative axis, so positive contributions slow down the helices.) The amplitude difference between the two curves results in net synchronization.

Figure 9

5% settling time from an initial phase difference of $\pi /2$, as a function of translational spring constant. Note that the variation is nonmonotonic.

Figure 10

Component contributions to $y$ velocity for helix 1: (a) $k_{\mathrm{tr}}=10$; (b) $k_{\mathrm{tr}}=100$; (c) $k_{\mathrm{tr}}=1000$. For small spring constants, the force contribution only modifies the torque contribution by a small amount. For large spring constants the modification is large but out of phase, resulting in small total velocities. For intermediate spring constants, we get both large modifications and large total velocities.

Figure 11

Time history of helix phase difference. Time is normalized by the rotation period of an isolated helix. We see that for transverse rotation, both antisynchronous and synchronous behaviors are possible.

Figure 12

5% settling time from an initial phase difference of $\pi /2$, as a function of translational spring constant. We observe both antisynchronous and synchronous behavior. Labels correspond to cases further illustrated in Fig. 13.

Figure 13

Comparison of dominant contributions to ${\Omega }_{x,1}$ for different spring constants. (a) $k_{\mathrm{rot}}=100$; (b) $k_{\mathrm{rot}}=10000$; (c) $k_{\mathrm{rot}}=50000$; (d) $k_{\mathrm{rot}}=100000$. Note that the component responsible for the phase behavior changes. For instance, axial self-torque is only important for $k_{\mathrm{rot}}=100$. Furthermore, the nature of the contribution can change. For $k_{\mathrm{rot}}=10000$ transverse self-torques drive the system toward antisynchrony while for $k_{\mathrm{rot}}=50000$ the opposite is true.

Figure 14

Normalized settling time for combined rotation and translation. Synchronous settling times are shown as negative. Antisynchronous settling times are shown as positive.

Figure 15

Settling time as a function of $k_{\mathrm{rot}}$, for various fixed values of $k_{\mathrm{tr}}$. For a stiff spring with $k_{\mathrm{tr}}=500$ the behavior resembles that for an infinitely stiff spring. In contrast, for a flexible spring with $k_{\mathrm{tr}}=100$ the behavior at small values of $k_{\mathrm{rot}}$ is fundamentally different.

Figure 16

Comparison of dominant contributions to phase difference, for different motion constraints. (a) Both transverse translation and transverse rotation ($k_{\mathrm{tr}}=100$, $k_{\mathrm{rot}}=10000$); (b) transverse translation only ($k_{\mathrm{tr}}=100$); (c) transverse rotation only ($k_{\mathrm{rot}}=10000$). The behavior for rotation combined with translation is not a simple superposition. When both translation and rotation are allowed, the contributions have different dynamics (e.g., growth rates, oscillation amplitudes).