An optimal microswimmer with a given geometry has a surface velocity profile that minimises energy dissipation for a given swimming speed. An axisymmetric swimmer can be puller-, pusher- or neutral-type depending on the sign of the stresslet strength. We numerically investigate the type of optimal surface-driven active microswimmers using a minimum dissipation theorem for optimum microswimmers. We examine the hydrodynamic resistance and stresslet strength with nonlinear dependence on various deformation modes. Optimum microswimmers with fore-and-aft symmetry exhibit neutral-type behaviour. Asymmetrical geometries exhibit pusher-, puller- or neutral-type behaviour, depending on the dominant deformation mode and the nonlinear dependence of the stresslet for an optimum microswimmer on deformation mode and amplitude.

Active fluids encompass a wide range of non-equilibrium fluids, in which the self-propulsion or rotation of their units can give rise to large-scale spontaneous flows. Despite the diversity of active fluids, they are commonly viscoelastic. Therefore, we develop a hydrodynamic model of isotropic active liquids by accounting for their viscoelasticity. Specifically, we incorporate an active stress term into a general viscoelastic liquid model to study the spontaneous flow states and their transitions in two-dimensional channel, annulus and disk geometries. We have discovered rich spontaneous flow states in a channel as a function of activity and Weissenberg number, including unidirectional flow, travelling-wave and vortex-roll states. The Weissenberg number acts against activity by suppressing the spontaneous flow. In an annulus confinement, we find that a net flow can be generated only if the aspect ratio of the annulus is not too large nor too small, akin to some three-dimensional active-flow phenomena. In a disk geometry, we observe a periodic chirality switching of a single vortex flow, resembling the bacteria-based active fluid experiments. The two phenomena reproduced in our model differ in Weissenberg number and frictional coefficient. As such, our active viscoelastic model offers a unified framework to elucidate diverse active liquids, uncover their connections and highlight the universality of dynamic active-flow patterns.

The encapsulation of active particles, such as bacteria or active colloids, inside a droplet gives rise to a non-trivial shape dynamics and droplet displacement. To understand this behaviour, we derive an asymptotic solution for the fluid flow about a deformable droplet containing an active particle, modelled as a Stokes-flow singularity, in the case of small shape distortions. We develop a general solution for any Stokes singularity and apply it to compute the flows and resulting droplet velocity due to common singularity representations of active particles, such as Stokeslets, rotlets and stresslets. The results show that offsetting of the active particle from the centre of the drop breaks symmetry and excites a large number of generally non-axisymmetric shape modes as well as particle and droplet motion. In the case of a swimming stresslet singularity, a run-and-tumble locomotion results in superdiffusive droplet displacement. The effect of interfacial properties is also investigated. Surfactants adsorbed at the droplet interface counteract the internal flow and arrest the droplet motion for all Stokes singularities except the Stokeslet. Our results highlight strategies to steer the flows of active particles and create autonomously navigating containers.

Poiseuille flow is a fundamental flow in fluid mechanics and is driven by a pressure gradient in a channel. Although the rheology of active particle suspensions has been investigated extensively, knowledge of the Poiseuille flow of such suspensions is lacking. In this study, dynamic simulations of a suspension of active particles in Poiseuille flow, situated between two parallel walls, were conducted by Stokesian dynamics assuming negligible inertia. Active particles were modelled as spherical squirmers. In the case of inert spheres in Poiseuille flow, the distribution of spheres between the walls was layered. In the case of non-bottom-heavy squirmers, on the other hand, the layers collapsed and the distribution became more uniform. This led to a much larger pressure drop for the squirmers than for the inert spheres. The effects of volume fraction, swimming mode, swimming speed and the wall separation on the pressure drop were investigated. When the squirmers were bottom heavy, they accumulated at the channel centre in downflow, whereas they accumulated near the walls in upflow, as observed in former experiments. The difference in squirmer configuration alters the hydrodynamic force on the wall and hence the pressure drop and effective viscosity. In upflow, pusher squirmers induced a considerably larger pressure drop, while neutral and puller squirmers could even generate negative pressure drops, i.e. spontaneous flow could occur. While previous studies have reported negative viscosity of pusher suspensions, this study shows that the effective viscosity of bottom-heavy puller suspensions can be negative for Poiseuille upflow, which is a new finding. The knowledge obtained is important for understanding channel flow of active suspensions.

Active suspensions encompass a wide range of complex fluids containing microscale energy-injecting particles, such as cells, bacteria or artificially powered active colloids. Because they are intrinsically non-equilibrium, active suspensions can display a number of fascinating phenomena, including turbulent-like large-scale coherent motion and enhanced diffusion. Here, using a recently developed active fast Stokesian dynamics method, we present a detailed numerical study of the hydrodynamic diffusion in apolar active suspensions of squirmers. Specifically, we simulate suspensions of active but non-self-propelling spherical squirmers (or ‘shakers’), of either puller type or pusher type, at volume fractions from 0.5 % to 55 %. Our results show little difference between pulling and pushing shakers in their instantaneous and long-time dynamics, where the translational dynamics varies non-monotonically with the volume fraction, with a peak diffusivity at around 10 % to 20 %, in stark contrast to suspensions of self-propelling particles. On the other hand, the rotational dynamics tends to increase with the volume fraction as is the case for self-propelling particles. To explain these dynamics, we provide detailed scaling and statistical analyses based on the activity-induced hydrodynamic interactions and the observed microstructural correlations, which display a weak local order. Overall, these results elucidate and highlight the different effects of particle activity versus motility on the collective dynamics and transport phenomena in active fluids.

A model is formulated of a two-dimensional migrating, or swimming, inviscid bubble in a viscous fluid whose unsteady displacement is caused by the spreading over its surface of an initial distribution of insoluble surfactant. Assuming small capillary and Reynolds numbers, and a linear equation of state giving the surface tension as a function of surfactant concentration, the quasi-steady Stokes flow around the bubble is found analytically and explicit formulas are determined for the time-dependent bubble speed and its final overall displacement. At infinite surface Péclet number this is done using a complex version of the method of characteristics to solve a complex partial differential equation of Burgers type. For a finite non-zero surface Péclet number, the problem is shown to be linearizable by a complex variant of the classical Cole–Hopf transformation. The formulation allows general statements to be made on the bubble speed and its total net displacement in terms of the initial surfactant distribution. A weak finite-time singularity in the surface activity associated with an isolated clean point on the bubble surface is also identified and studied in detail.

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.

Understanding the effect of intricate surface wettability conditions on microswimmers is crucial for precisely navigating them across narrow microcirculatory networks. Here, we adopt the spherical squirmer model and Navier slip condition to delineate the microswimmer locomotion under a Poiseuille flow in a slit microchannel. Through a combined analytical–numerical approach utilizing bispherical coordinates and the superposition technique, we resolve the slip-modulated simultaneous hydrodynamic interaction with substrate boundaries. Phase portraits reveal that slip significantly alters propulsion mechanisms, destabilizing centreline stable oscillations of pullers beyond a threshold slip length. Superhydrophobic surfaces suppress near-wall rheotaxis states but preserve centreline focusing, facilitating slip-assisted directed transport without surface accumulation. Under strong background flows, subcritical Hopf bifurcation emerges for pullers at a critical slip length, transitioning dynamics from coexisting stable and unstable states to purely unstable behaviour. Contrastingly, for pushers, slip causes a transition from unstable to either stable or fixed-amplitude oscillations. Increased slip length reduces hydrodynamic repulsion on pullers from the walls by enhancing rotational velocity near the walls, whereas it counteracts the torque that causes unstable oscillations of pushers. Three-dimensional analysis of the trajectories reveals the significant role of the out-of-plane orientation of the microswimmer in its transitions between different swimming states. The presented regime maps offer parametric combinations for specific motion behaviours, guiding the development of smart microfluidic drug delivery systems and preventing biofilm deposition in biomedical devices.

Three-dimensional non-rotating odd viscous liquids give rise to Taylor columns and support axisymmetric inertial-like waves (J. Fluid Mech., vol. 973, 2023, A30). When an odd viscous liquid is subjected to rigid-body rotation however, there arise in addition a plethora of other phenomena that need to be clarified. In this paper, we show that three-dimensional incompressible or two-dimensional compressible odd viscous liquids, rotating rigidly with angular velocity $\varOmega$, give rise to both oscillatory and evanescent inertial-like waves or a combination thereof (which we call of mixed type) that can be non-axisymmetric. By evanescent, we mean that along the radial direction, typically when moving away from a solid boundary, the velocity field decreases exponentially. These waves precess in a prograde or retrograde manner with respect to the rotating frame. The oscillatory and evanescent waves resemble respectively the body and wall-modes observed in (non-odd) rotating Rayleigh–Bénard convection (J. Fluid Mech., vol. 248, 1993, pp. 583–604). We show that the three types of waves (wall, body or mixed) can be classified with respect to pairs of planar wavenumbers $\kappa$ which are complex, real or a combination, respectively. Experimentally, by observing the precession rate of the patterns, it would be possible to determine the largely unknown values of the odd viscosity coefficients. This formulation recovers as special cases recent studies of equatorial or topological waves in two-dimensional odd viscous liquids which provided examples of the bulk–interface correspondence at frequencies $\omega <2\varOmega$. We finally point out that the two- and three-dimensional problems are formally equivalent. Their difference then lies in the way data propagate along characteristic rays in three dimensions, which we demonstrate by classifying the resulting Poincaré–Cartan equations.

Communities of swimming microorganisms often thrive near liquid–air interfaces. We study how such ‘active carpets’ shape their aquatic environment by driving biogenic transport in the water column beneath them. The hydrodynamic stirring that active carpets generate leads to diffusive upward fluxes of nutrients from deeper water layers, and downward fluxes of oxygen and carbon. Combining analytical theory and simulations, we examine the biogenic transport by studying fundamental metrics, including the single and pair diffusivity, the first passage time for particle pair encounters and the rate of particle aggregation. Our findings reveal that the hydrodynamic fluctuations driven by active carpets have a region of influence that reaches orders of magnitude further in distance than the size of the organisms. These non-equilibrium fluctuations lead to a strongly enhanced diffusion of particles, which is anisotropic and space dependent. Fluctuations also facilitate encounters of particle pairs, which we quantify by analysing their velocity pair correlation functions as a function of distance between the particles. We found that the size of the particles plays a crucial role in their encounter rates, with larger particles situated near the active carpet being more favourable for aggregation. Overall, this research broadens our comprehension of aquatic systems out of equilibrium and how biologically driven fluctuations contribute to the transport of fundamental elements in biogeochemical cycles.

Understanding the propulsion of a swimmer in a large group of individuals holds the key to unravelling the intriguing dynamics of active matter collective motion. Here, we develop a two-dimensional (2-D) self-assembled rotor, powered by bacterial flagella. At a water–air interface, the average direction of rotation of a rotor is fixed. When the chiral rotor is put into a 2-D bacterial suspension, we examine the average and fluctuation of the angular velocity of the rotor. Remarkably, the average angular velocity of a rotor is found to increase up to 3 times when the density of surrounding bacterial suspension increases and the increase is nonlinear. In a dense suspension of bacteria, the existence of a rotor disrupts vortices in the surrounding active turbulence, and the acceleration of the rotor is independent of the activity level of the surrounding free bacteria. The nonlinear acceleration thus results from hydrodynamic interaction with surrounding crowdedness that can be quantitatively explained by hydrodynamic simulation. The simultaneity between the acceleration of rotor and free bacteria in active turbulence suggests that crowding-induced acceleration may promote the onset of instability. The result will inspire new active-matter-based microfluidic devices with improved transport properties.

The mobility of externally driven phoretic propulsion of particles is evaluated by simultaneously solving the solute conservation equation, interaction potential equation and the modified Stokes equation. While accurate, this approach is cumbersome, especially when the interaction potential decays slowly compared with the particle size. In contrast to external phoresis, the motion of self-phoretic particles is typically estimated by relating the translation and rotation velocities with the local slip velocity. While this approach is convenient and thus widely used, it is only valid when the interaction decay length is significantly smaller than the particle size. Here, by taking inspiration from Brady (J. Fluid Mech., vol. 922, 2021, A10), which combines the benefits of two approaches, we reproduce their unified mobility expressions with arbitrary interaction potentials and show that these expressions can conveniently recover the well-known mobility relationships of external electrophoresis and diffusiophoresis for arbitrary double-layer thickness. Additionally, we show that for a spherical microswimmer, the derived expressions relax to the slip velocity calculations in the limit of the thin interaction length scales. We also employ the derived mobility expressions to calculate the velocities of an autophoretic Janus particle. We find that there is significant dampening in the translation velocity even when the interaction length is an order of magnitude larger than the particle size. Finally, we study the motion of a catalytically self-propelled particle, while it also propels due to external concentration gradients, and demonstrate how the two propulsion modes compete with each other.

Chiral fluids – such as fluids under rotation or a magnetic field as well as synthetic and biological active fluids – flow in a different way than ordinary ones. Due to symmetries broken at the microscopic level, chiral fluids may have asymmetric stress and viscosity tensors, for example giving rise to a hydrostatic torque or non-dissipative (odd) and parity-violating viscosities. In this article, we investigate the motion of rigid bodies in such an anisotropic fluid in the incompressible Stokes regime through the mobility matrix, which encodes the response of a solid body to forces and torques. We demonstrate how the form of the mobility matrix, which is usually determined by particle geometry, can be analogously controlled by the symmetries of the fluid. By computing the mobility matrix for simple shapes in a three-dimensional (3-D) anisotropic chiral fluid, we predict counterintuitive phenomena such as motion at an angle to the direction of applied forces and spinning under the force of gravity.

Microbes play a primary role in wide-ranging biogeochemical and physiological processes, where ambient fluid flows are responsible for cell dispersal as well as mixing of dissolved resources, signalling molecules and biochemical products. Determining the simultaneous (and often coupled) transport properties of actively swimming cells together with passive scalars is key to understanding and ultimately predicting these complex processes. In recent work, Ran & Arratia (J. Fluid Mech., vol. 988, 2024, A25) present the striking observation that dilute concentrations of swimming bacteria severely hinder scalar transport through Lagrangian vortex boundaries in a chaotic flow. Analysis of rotation-dominated regions suggests that local accumulation of bacteria enhances the strength of transport barriers and highlights the role of understudied elliptical Lagrangian coherent structures in bacterial and multicomponent transport.

We investigate the effects of bacterial activity on the mixing and transport properties of a passive scalar in time-periodic flows in experiments and in a simple model. We focus on the interactions between swimming Escherichia coli and the Lagrangian coherent structures (LCSs) of the flow, which are computed from experimentally measured velocity fields. Experiments show that such interactions are non-trivial and can lead to transport barriers through which the scalar flux is significantly reduced. Using the Poincaré map, we show that these transport barriers coincide with the outermost members of elliptic LCSs known as Lagrangian vortex boundaries. Numerical simulations further show that elliptic LCSs can repel elongated swimmers and lead to swimmer depletion within Lagrangian coherent vortices. A simple mechanism shows that such depletion is due to the preferential alignment of elongated swimmers with the tangents of elliptic LCSs. Our results provide insights into understanding the transport of micro-organisms in complex flows with dynamical topological features from a Lagrangian viewpoint.

We use the entropy method to analyse the nonlinear dynamics and stability of a continuum kinetic model of an active nematic suspension. From the time evolution of the relative entropy, an energy-like quantity in the kinetic model, we derive a variational bound on relative entropy fluctuations that can be expressed in terms of orientational order parameters. From this bound we show isotropic suspensions are nonlinearly stable for sufficiently low activity, and derive upper bounds on spatiotemporal averages in the unstable regime that are consistent with fully nonlinear simulations. This work highlights the self-organising role of activity in particle suspensions, and places limits on how organised such systems can be.

Microorganisms are ubiquitous in nature and technology. They inhabit diverse environments, ranging from small river tributaries and lakes, to oceans, as well as wastewater treatment plants and food manufacturing. In many of these environments, microorganisms coexist with settling particles. Here, we investigate the effects of microbial activity (swimming E. coli) on the settling dynamics of passive colloidal particles using particle tracking methods. Our results reveal the existence of two distinct regimes in the correlation length scale ($L_u$) and the effective diffusivity of the colloidal particles ($D_{eff}$), with increasing bacterial concentration ($\phi _b$). At low $\phi _b$, the parameters $L_u$ and $D_{eff}$ increase monotonically with increasing $\phi _b$. Beyond critical $\phi _b$, a second regime is found where both $D_{eff}$ and $L_u$ are independent of $\phi _b$. We demonstrate that the transition between these regimes is characterized by the emergence of bioconvection. We use experimentally measured particle-scale quantities $L_u$ and $D_{eff}$ to predict the critical bacterial concentration for the diffusion–bioconvection transition.