Motility and rotation of multi-timescale microswimmers in linear background flows

TL;DR

This work develops a two-timescale framework to analyze inertialess, inefficient microswimmers swimming in general planar linear background flows, including oscillatory components. By applying a multiple-scales expansion with fast gait dynamics ($T=\omega t$) and slow net motion, the authors derive slow equations for the averaged orientation $\overline{\theta}_0$ and averaged position $\overline{\mathbf{x}}_0$, revealing a simple two-parameter classification: the angular-forcing ratio $a^2/(b^2+c^2)$ and the flow ratio $\overline{\Omega^*}^2/(\overline{E^{11}}^2+\overline{E^{12}}^2)$. The angular dynamics separate into tumbling versus asymptoting behavior, while translational dynamics fall into exponential, linear, or oscillatory regimes depending on flow shear and rotation balances; special symmetric swimmer cases recover Jeffery-like or Bretherton-type dynamics and demonstrate possible violations of Purcell’s scallop theorem via flow-swimmer interactions. The results provide a concise framework to predict swimmer trajectories (e.g., hyperbolic in irrotational flow, parabolic in oscillatory shear) and offer insight into navigation and control of microrobots in general linear background flows, with implications for collective modelling and rheotaxis. The study also emphasizes the nontrivial role of flow-swimmer interactions and the need for careful averaging when linking fast actuation to slow motility.

Abstract

Microswimming cells and robots exhibit diverse behaviours due to both their swimming and their environment. One of the core environmental features impacting inertialess swimming is background flows. While the influence of select flows, particularly shear flows, have been extensively investigated, these are special cases. Here, we examine inertialess swimmers in more general flows, specifically general linear planar flows that may also possess rapid oscillations. Relatively weak symmetry constraints are imposed on the swimmer to ensure planarity and to reduce complexity. A further constraint reflecting common observation is imposed, namely that the swimmer is inefficient, which we suitably define. This introduces two separate timescales: a fast timescale associated with swimmer actuation, and a second timescale associated with net swimmer movement, with inefficiency dictating that this latter timescale is much slower, allowing for a multiple timescale simplification of the governing equations. With the exception of mathematically precise edge cases, we find that the behaviour of the swimmer is dictated by two parameter groupings, both of which measure balances between the angular velocity and rate of strain of the background flow. While the measures of flow angular velocity and strain rates that primarily govern the rotational dynamics are modulated by swimmer properties, the primary features of the translational motion are determined solely by a ratio of flow angular velocity to strain rate. Hence, a simple classification of the swimmer dynamics emerges. For example, this illustrates the limited extent to which, and how, microswimmers may control their orientations and trajectories in flows.

Figures (3)

• Figure 1: We illustrate a model swimmer in a planar, uni-directional, linear flow in the $x$-direction ($\bm{e}_1)$ and varying in the $y$-direction ($\bm{e}_2$), with the swimmer moving in the $xy$-plane. The swimmer orientation in the plane is captured via the unit vector $\ehat{1}$, which makes an angle $\theta$ with the $\bm{e}_1$ axis.
• Figure 2: Systematically averaged dynamics of a reciprocal swimmer in stationary shear flow. Shown both as dynamics on a cylinder and in the plane, we illustrate the semi-periodic phase space that corresponds to a reciprocal swimmer in stationary shear flow. We showcase three qualitatively distinct regimes: in (a,d), we have $(\overline{U_I B}, \overline{B}) = (0,0.5)$, leading to no motion at leading order; in (b,e), we have $(\overline{U_I B}, \overline{B}) = (1,0.5)$ and long-time periodic motion; in (c,f), we have $(\overline{U_I B}, \overline{B}) = (1,1.5)$ and progression. In (f), the dashed lines correspond to stable states of the angular dynamics.
• Figure 3: Behaviours of reciprocal swimmers in unsteady shear flow with zero average shear rate. Note in (a) that the long-time trajectory has a gradient of magnitude unity, a universal prediction for swimmers with sufficient symmetry (such as maintaining a shape that is always a body of revolution with fore-aft symmetry).