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Active suspensions, which consist of suspended self-propelling particles such as swimming microorganisms, often exhibit non-trivial transport properties. Continuum models are frequently employed to elucidate phenomena in active suspensions, such as shear trapping of bacteria, bacterial turbulence, and bioconvection patterns in suspensions of algae. Yet, these models are often empirically derived and may not always agree with the individual-based description of active particles. Here we establish a more rigorous foundation to fully develop a continuum model based on the respective microscopic dynamics through coarse-graining. All the assumptions needed to reach popular continuum models from a multi-particle Fokker-Planck equation, which governs the probability of the full configuration space, are explicitly presented. In the dilute limit, this approach leads to the mean-field model (a.k.a. Doi-Saintillan-Shelley model), which can be further reduced to a continuum equation for particle density. Moreover, we review the limitations and highlight the challenges related to continuum descriptions, including significant issues in implementing physical boundary conditions and the possible emergence of singular solutions., Comment: 14 pages

Bioconvection is the prototypical active matter system for hydrodynamic instabilities and pattern formation in suspensions of biased swimming microorganisms, particularly at the dilute end of the concentration spectrum where cell-cell interactions typically are neglected. Confinement is an inherent characteristic of such systems, including those that are naturally-occurring or industrially-exploited, so it is important to understand the impact of boundaries on the hydrodynamic instabilities. Despite recent interest in this area we note that commonly-adopted symmetry assumptions in the literature, such as for a vertical channel or pipe, are uncorroborated and potentially unjustified. Therefore, by employing a combination of analytical and numerical techniques, we investigate whether confinement itself can drive asymmetric plume formation in a suspension of bottom-heavy swimming microorganisms (gyrotactic cells). For a class of solutions in a vertical channel we establish the existence of a first integral of motion, and reveal that asymptotic asymmetry is plausible. Furthermore, numerical simulations from both Lagrangian and Eulerian perspectives demonstrate with remarkable agreement that asymmetric solutions can indeed be more stable than symmetric; asymmetric solutions are in fact dominant for a large, practically-important region of parameter space. In addition, we verify the presence of blip and varicose instabilities for an experimentally accessible parameter range. Finally, we extend our study to a vertical Hele-Shaw geometry to explore whether a simple linear drag approximation can be justified. We find that although two-dimensional bioconvective structures and associated bulk properties have some similarities with experimental observations, approximating near wall physics in even the simplest confined systems remains challenging.

Flagella allow eukaryotic cells to move and pump fluid. We present the first three-dimensional, time-resolved imaging of the flagellar waveform of Chlamydomonas reinhardtii, a model alga found in fresh water. During the power stroke, we find that the flagella show rotational symmetry about the cell's centreline, but during the recovery stroke they display mirror symmetry about the same axis. We use our three-dimensional imaging data to test the applicability of resistive force theory when a force-free configuration is approximated using a cell on a micropipette. The inferred values of cells' swimming speeds and rotation rates show good agreement with experimental results on freely swimming cells., Comment: 7 pages, 4 figures