Canonical orbits for rapidly deforming planar microswimmers in shear flow

Classically, the rotation of ellipsoids in shear Stokes flow is captured by Jeffery's orbits. Here we demonstrate that Jeffery's orbits also describe high-frequency shape-deforming swimmers moving in the plane of a shear flow, employing only basic properties of Stokes flow and a multiple-scales asymptotic analysis. In doing so, we support the use of these simple models for capturing shape-changing swimmer dynamics in studies of active matter and highlight the ubiquity of ellipsoid-like dynamics in complex systems. This result is robust to weakly confounding effects, such as distant boundaries, and also applies in the low-frequency limit.

Figure 1

Swimmer and laboratory frames. A generic swimmer with time-dependent boundary $\partial \mathrm{\Omega }$ is shown along with the basis $\{{\mathit{e}}_x,{\mathit{e}}_y,{\mathit{e}}_z\}$ of a swimmer-fixed frame. The swimmer frame is instantaneously at an angle $\theta$ to the laboratory frame $\{{\mathit{e}}_X,{\mathit{e}}_Y,{\mathit{e}}_Z\}$, which corresponds to rotation about the shared ${\mathit{e}}_z={\mathit{e}}_Z$ axis in the ${\mathit{e}}_X{\mathit{e}}_Y$ plane of swimmer motion.