Hydrodynamic interactions between rotating helices

Escherichia coli bacteria use rotating helical flagella to swim. At this scale, viscous effects dominate inertia, and there are significant hydrodynamic interactions between nearby helices. These interactions cause the flagella to bundle during the “runs” of bacterial chemotaxis. Here we use slender-body theory to solve for the flow fields generated by rigid helices rotated by stationary motors. We determine how the hydrodynamic forces and torques depend on phase and phase difference, show that rigid helices driven at constant torque do not synchronize, and solve for the flows. We also use symmetry arguments based on kinematic reversibility to show that for two rigid helices rotating with zero phase difference, there is no time-averaged attractive or repulsive force between the helices.

Figure 1

Two identical rotating helices.

Figure 2

Line segment approximation of a helix. Only two turns are shown.

Figure 3

(a) Reference configuration. (b) The two helices after rotation about the $z$ axis by $\pi$ and translation along $y$ by $h$, but before reversing the rotation speeds.

Figure 4

Torque versus phase difference for $h=3b$. The symbols in the legend correspond to direct numerical calculation. The solid flat line is the torque on a single isolated helix turning at speed $\omega$; the other solid lines correspond to the analytic forms for the ${\overline{A}}_{\mu \nu }$ described in the text.

Figure 5

(a) Time dependence of $F_{1x}$ and $F_{2x}$ for ${\omega }_1=1.5{\omega }_2$. (b) Time-averaged axial components of the force.

Figure 6

Flow fields for two left-handed helices rotating with the same angular velocity $\omega =\omega \widehat{\mathbf{x}}$.