Collinear Swimming of a Squirmer Pair in Newtonian and Shear-Thinning Fluids

TL;DR

This study provides analytical and numerical insights into the hydrodynamic interactions of two microswimmers in Newtonian and shear-thinning fluids, highlighting co-swimming behaviors and effects of non-Newtonian rheology.

Contribution

The paper derives an exact solution for two interacting squirmers in Newtonian fluids and explores how shear-thinning fluids alter their propulsion, offering benchmarks for future research.

Findings

Identified co-swimming configurations with identical velocities.

Validated numerical simulations with analytical solutions.

Demonstrated influence of shear-thinning fluids on swimmer propulsion.

Abstract

Pairwise hydrodynamic interactions of microswimmers form the fundamental building blocks for understanding their more complex collective behaviors. In this work, we revisit the canonical problem of two interacting squirmers swimming along their common line of centers in both Newtonian and shear-thinning fluids. For the Newtonian case, we first derive an exact, closed-form solution for the axisymmetric Stokes flow generated by the interacting pair, thereby complementing prior analyses based on the reciprocal theorem approach by providing direct access to the detailed knowledge of the flow around the swimmers. The analytical solution is then used to cross-validate numerical simulations based on the finite element method. The combined theoretical and numerical investigation reveals co-swimming configurations in which the two squirmers develop identical velocities over a range of separations. We rationalize these behaviors through symmetry arguments and quantify their propulsion performance in terms of the speed and energetic cost of swimming. Furthermore, motivated by the prevalence of shear-thinning biological fluids encountered by microswimmers, we examine how this ubiquitous non-Newtonian rheological behavior modifies the propulsion characteristics of these co-swimming pairs. Taken together, our results establish quantitative benchmarks for interacting squirmers in both Newtonian and shear-thinning fluids, laying the groundwork for future studies of many-body dynamics of microswimmers in complex fluid environments.