Stirring by squirmers

TL;DR

This paper models the mixing effects of microswimmers using a viscous squirmer model, highlighting the importance of direction changes and stresslet contributions, validated by simulations and extended to finite Reynolds numbers.

Contribution

It introduces a viscous squirmer model for mixing, emphasizing the role of direction changes and stresslet effects, with validation through numerical simulations and analysis of finite Reynolds number corrections.

Findings

Particle displacements are dominated by random direction changes.

Effective diffusivity is influenced mainly by the stresslet term.

Displacement PDFs have exponential tails and show superdiffusive behavior at short times.

Abstract

We analyse a simple 'Stokesian squirmer' model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress [Physics Letters A, 374, 3487 (2010), arXiv:0911.5511], where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that for the viscous case the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate nonzero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly-swimming squirmers exhibit PDFs with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to 'sticking' near the stagnation points on the squirmer's surface.