Theoretical framework for two-microswimmer hydrodynamic interactions

Abstract

Hydrodynamic interactions are crucial for determining the cooperative behavior of microswimmers at low Reynolds numbers. Here we provide a comprehensive analysis of the scaling laws and the strength of the interactions in the case of a pair of three-sphere swimmers. Both stroke-based and force-based elastic microswimmers are analyzed using an analytic perturbative approach, focusing on passive and active interactions. The former are governed by the cycle-averaged flow field of a single swimmer, which is dipolar at long range. However, at intermediate distances, with a cross-over at the order of 102 swimmer lengths, the quadrupolar field dominates which, notably, yields an increase of the swimming velocity compared to individual swimmers, even when the swimmers are one behind another. Furthermore, we find that active rotations resulting from the interplay of the time-resolved swimming stroke and the ambient flow fields and, even more prominently, active translations are model-dependent. A mapping between the stroke-based and force-based swimmers is only possible for the low driving frequency regime where the characteristic time scale is smaller than the viscous one. Finally, we find that the long-term behavior of the swimmers, while sensitive to the initial relative positioning, does not depend on the pusher or puller nature of the swimmer. These results clearly indicate that the behavior of swarms will depend on the swimmer model, which was hitherto not well appreciated.

Lattice Boltzmann Simulations of Two Linear Microswimmers Using the Immersed Boundary Method

“…While some swimmers, such as the squirmer, only experience passive effects [62], shape-changing swimmers such as the bead-spring swimmer experience both types. In particular, bead-spring swimmers will generally alter their swimming stroke as a consequence of the time-dependent flow field they find themselves in, giving rise to active effects [39]. These active effects have been reported to be particularly important as small swimmer separations [61], and thus are expected to play an important role in our simulations.…”

“…To compare the results of the lattice Boltzmann method, we make use of an analytical framework [39] where the hydrodynamic interactions between the particles are calculated using either the Oseen or Rotne-Prager approximation [58]. The equation of motion of the bead-spring system reads…”

“…As expected by symmetry, we find that the two swimmers always have the same velocity for all the distances we investigated among them, i.e., all investigated configurations are stationary on the time scales accessible to lattice Boltzmann simulations. However, theoretical work [39,61] has shown that parallel swimmers typically also experience sideways interaction as well as rotation, which will in general break stationarity.…”

“…Recently, the focus of theoretical research has shifted towards swimming in complex environments like channels [35,36] or near fluid interfaces [31,37], and the question of collective swimmer dynamics has become of major interest. The present work aims at understanding the collective dynamics of several microswimmers [38,39]. For this purpose, we consider a relatively simple situation of two linear microswimmers in different configurations.…”

Hydrodynamic particle interactions in linear and radial viscosity gradients

“…Hydrodynamic interactions between particles of approximately spherical shape are of great importance for many areas in research and industry, including colloid dynamics (Rex & Löwen 2008), polymers (Kirkwood & Riseman 1948), deposition processes (Dabros 1989), and active systems such as microswimmers (Najafi & Golestanian 2004; Ishikawa, Simmonds & Pedley 2006; Pooley, Alexander & Yeomans 2007; Pande & Smith 2015; Ziegler et al. 2021) and Janus particles (Sharifi-Mood, Mozaffari & Córdova-Figueroa 2016). Due to the small length scales involved, the corresponding hydrodynamic flows are dominated by viscous friction rather than by fluid inertia.…”