Derivation of a hydrodynamic theory for mesoscale dynamics in microswimmer suspensions

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 [S. Heidenreich et al., Phys. Rev. E 94, 020601 (2016)2470-004510.1103/PhysRevE.94.020601], 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.

Anisotropic mesoscale turbulence and pattern formation in microswimmer suspensions induced by orienting external fields

“…The effective velocity of the microswimmers is calculated as the sum v 0 P+u. In the limit of weak coupling between the solvent flow and the polar order, the dependence on u can be neglected and the dynamics is adequately described by one field [19]. In contrast to the phenomenological approach, the coefficients of the field equation are directly linked to the parameters of the microscopic Langevin model [19] (see also appendix A).…”

“…For the first part of this article, however, we will consider the full model consisting of both the dynamics of the polar order parameter P and the solvent flow u. As shown in [19], the dynamical equation for P(x, t) can be conveniently written in potential form…”

“…The anisotropic passive contribution is treated in detail in the theory of liquid crystals [73]. For the purpose of this study, however, we assume that the passive contribution pass s is not significant compared to the active stress, see [19] for a more detailed discussion. Performing an expansion of the active stress up to fifth order in gradients of the orientation [19]…”

“…To describe and understand the fascinating collective behavior from a theoretical point of view, models on different levels of detail have been applied, from studies simulating large numbers of individual particles [7,8] to phenomenological approaches [9][10][11][12][13][14] standing on the opposite side of the spectrum. To bridge the gap, many efforts have been made to derive coarse-grained hydrodynamic theories from microscopic models [15][16][17][18][19]. For a selection of recent reviews on the full scope of active matter see [20][21][22][23][24][25][26][27][28].…”

“…are rescaled using the microswimmer length ℓ as length scale, the self-swimming speed v 0 as characteristic velocity and ℓ/v 0 as time scale[19]. The coefficients can then be written as…”