Derivation of a hydrodynamic theory for mesoscale dynamics in microswimmer suspensions

TL;DR

This paper derives a comprehensive fourth-order continuum hydrodynamic theory for mesoscale turbulence in three-dimensional microswimmer suspensions, incorporating hydrodynamic and nematic interactions, extending previous two-dimensional models.

Contribution

It introduces a new three-dimensional hydrodynamic framework that generalizes earlier two-dimensional theories, including both hydrodynamic and nematic interactions for microswimmer suspensions.

Findings

The theory captures mesoscale turbulence behavior.

Effective microswimmer dynamics depend on flow and orientation coupling.

Simplified models are valid at high suspension viscosity.

Abstract

In this paper we systematically derive a fourth-order continuum theory capable of reproducing mesoscale turbulence in a three-dimensional suspension of microswimmers. We start from overdamped Langevin equations for a generic microscopic model (pushers or pullers), which include hydrodynamic interactions on both, small length scales (polar alignment of neighboring swimmers) and large length scales, where the solvent flow interacts with the order parameter field. The flow field is determined via the Stokes equation supplemented by an ansatz for the stress tensor. In addition to hydrodynamic interactions, we allow for nematic pair interactions stemming from excluded-volume effects. The results here substantially extend and generalize earlier findings [Phys. Rev. E 94, 020601(R) (2016)], in which we derived a two-dimensional hydrodynamic theory. From the corresponding mean-field Fokker-Planck equation combined with a self-consistent closure scheme, we derive nonlinear field equations for the polar and the nematic order parameter, involving gradient terms of up to fourth order. We find that the effective microswimmer dynamics depends on the coupling between solvent flow and orientational order. For very weak coupling corresponding to a high viscosity of the suspension, the dynamics of mesoscale turbulence can be described by a simplified model containing only an effective microswimmer velocity.