Emergent rheotaxis of shape-changing swimmers in Poiseuille flow

Abstract

A simple model for the motion of shape-changing swimmers in Poiseuille flow was recently proposed and numerically explored by Omori 𝘦𝘵 𝘢𝘭. (𝘑. 𝘍𝘭𝘶𝘪𝘥 𝘔𝘦𝘤𝘩., vol. 930, 2022, A30). These explorations hinted that a small number of interacting mechanics can drive long-time behaviours in this model, cast in the context of the well-studied alga 𝘊𝘩𝘭𝘢𝘮𝘺𝘥𝘰𝘮𝘰𝘯𝘢𝘴 and its rheotactic behaviours in such flows. Here, we explore this model analytically via a multiple-scale asymptotic analysis, seeking to formally identify the causal factors that shape the behaviour of these swimmers in Poiseuille flow. By capturing the evolution of a Hamiltonian-like quantity, we reveal the origins of the long-term drift in a single swimmer-dependent constant, whose sign determines the eventual behaviour of the swimmer. This constant captures the nonlinear interaction between the oscillatory speed and effective hydrodynamic shape of deforming swimmers, driving drift either towards or away from rheotaxis.