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A collection of microswimmers immersed in an incompressible fluid ischaracterised by strong interactions due to the long-range nature of thehydrodynamic fields generated by individual organisms. As a result, suspensionsof rear-actuated `pusher' swimmers such as bacteria exhibit a collective motionstate often referred to as `bacterial turbulence', characterised by large-scalechaotic flows. The onset of collective motion in pusher suspensions isclassically understood within the framework of mean-field kinetic theories fordipolar swimmers. In bulk 2-D and 3-D, the theory predicts that the instabilityleading to bacterial turbulence is due to mutual swimmer reorientation and setsin at the largest length scale available to the suspension. Here, we constructa similar kinetic theory for the case of a dipolar microswimmer suspensionrestricted to a two-dimensional plane embedded in a three-dimensionalincompressible fluid. This setting qualitatively mimics the effect of swimmingclose to a two-dimensional interface. We show that the in-plane flow fields areeffectively compressible in spite of the incompressibility of the 3-D bulkfluid, and that microswimmers on average act as sources (pushers) or sinks(pullers). We analyse stability of the homogeneous and isotropic state, andfind two types of instability that are qualitatively different from the bulk,three-dimensional case: First, we show that the analogue of the orientationalpusher instability leading to bacterial turbulence in bulk systems insteadoccurs at the smallest length-scale available to the system. Second, aninstability associated with density variations arises in puller suspensions asa generic consequence of the effective in-plane compressibility. We concludethat confinement can have a crucial role in determining the collectivebehaviour of microswimmer suspensions.

Hydrodynamic instabilities in a 2-D sheet of microswimmers embedded in a 3-D fluid