It is well established that many flagellated bacteria, such as Escherichia coli, swim in clockwise circles above rigid surfaces. However, in a cylindrical microwell with asymmetric top-bottom boundary conditions, such that bacteria segregate into two populations of differing sizes at opposing flat boundaries, the smaller bacterial vortex has been observed to rotate in the opposite direction to that expected in the absence of the other population [K. Beppu, Z. Izri, T. Sato, Y. Yamanishi, Y. Sumino and Y. T. Maeda, Proc. Natl. Acad. Sci. U. S. A., 2021, 118, e2107461118]. Motivated by these observations, we employ flow singularities to investigate the motion of a population of chiral swimmers near one flat boundary of a cylindrical geometry, subject to the flows generated by a bacterial vortex at the opposing surface. We show numerically that, purely due to hydrodynamic interactions, the rotational direction of the bacterial population reverses in the presence of a sufficiently large vortex on the opposite boundary. Our numerical results are fully explained by an analytical theory in the continuum limit, which captures the essential hydrodynamic interactions governing the observed reversal.

Abstract Contact-free and object-agnostic three-dimensional (3D) rotation remains a challenge at both the micro and nanoscale, with broad relevance to advanced imaging, biology, microrobotics, and materials science. Specifically, precise 3D rotation is desirable in diffusion-suppressing environments, where conventional micromanipulation methods fail. Here we introduce an opto-thermoviscous strategy that scans a focused laser spot within a two-dimensional plane to robustly generate 3D helical thermoviscous flows (TVFs) within highly viscous media. We further report on the discovery of opto-hydrodynamic focusing that converges a spiral motion to a defined particle height. By exploiting symmetry relations, we use rational design to decouple out-of-plane rotation from lateral displacements, leading to stable spinning with positional fluctuations below 200 nm, and demonstrate compatibility with a broad range of microstructures, from nano-printed tiles to stained biological cells, and even perfectly round homogenous spheres. Finally, leveraging the kinematic nature of thermoviscous manipulations, we demonstrate how stepwise rotation, alternated with 3D volumetric microscopy, can be combined with established multi-view image fusion strategies to increase resolution in biological imaging. Conceptually, this elevates TVFs from planar transport to symmetry-engineered volumetric actuation, delivering robust, material-agnostic, out-of-plane rotational control and sheathless opto-hydrodynamic focusing for all-optical micromanipulations and augmented microscopy.

The passive rotation of rigid helical filaments is the propulsion strategy used by flagellated bacteria and some artificial microswimmers to navigate at low Reynolds numbers. In a classical 1976 paper, Lighthill calculated the 'optimal' resistance coefficients in a local (logarithmically accurate) resistive-force theory that best approximates predictions from the non-local (algebraically accurate) slender-body theory for force-free swimming of a rotating helix without an attached load (e.g. no cell body). These coefficients have since been widely applied, often beyond the conditions for which they were originally derived. Here, we revisit the problem for the case where a load is attached to the rotating filament, such as the cell body of a bacterium or the head of an artificial swimmer. We show that the optimal resistance coefficients depend in fact on the size of the load, and we quantify the increasing inaccuracy of Lighthill's coefficients as the load grows. Finally, we provide a physical explanation for the origin of this unexpected load-dependence.This article is part of the theme issue 'Biological fluid dynamics: emerging directions'.

High-precision micromanipulation techniques, including optical tweezers and hydrodynamic trapping, have garnered wide-spread interest. Recent advances in optofluidic multiplexed assembly and microrobotics demonstrate significant progress, particularly by iteratively applying laser-induced, localized flow fields to manipulate microparticles in viscous solutions. However, these approaches still face challenges such as undesired hydrodynamic coupling and instabilities when multiple particles are brought into close proximity. By leveraging an analytical model of thermoviscous flows, this work introduces a stochastic optimization approach that selects flow fields for precise particle arrangement without relying on rule-based heuristics. Through minimizing a comprehensive objective function, the method achieves sub-micrometer alignment accuracy even in a crowded setting, avoiding instabilities driven by undesired coupling or particle collisions. An autonomously emerging "action at a distance" strategy-placing the laser scan path farther from the manipulated particles over time-exploits the 1/r2 decay of thermoviscous flow to refine positioning. Overall, objective function-based model predictive control enhances the versatility of automated optofluidic manipulations, opening new avenues in assembly, micromanufacturing, robotics, and life sciences.

Active flexible filaments form the classical continuum framework for modelling the locomotion of spermatozoa and algae driven by the periodic oscillation of flagella. This framework also applies to the locomotion of various artificial swimmers. Classical studies have quantified the relationship between internal forcing (localized or distributed internal moments or forces) and external output (filament shape and swimming speed). In this paper, we pose locomotion as a mathematical optimization problem and demonstrate that the swimming of an isolated active filament can be accurately described and optimized using a small number of eigenmodes, significantly reducing computational complexity. In particular, we reveal that the motion of a filament with monophasic forcing, relevant to recently proposed artificial swimmers, is governed by exactly four forcing eigenmodes, only two of which are independent. We further present optimizations of such swimmers under various constraints.This article is part of the theme issue 'Biological fluid dynamics: emerging directions'.

Recent microfluidic experiments have explored the precise positioning of micron-sized particles in liquid environments via laser-induced thermoviscous flow. From micro-robotics to biology at the subcellular scale, this versatile technique has found a broad range of applications. Through the interplay between thermal expansion and thermal viscosity changes, the repeated scanning of the laser along a scan path results in fluid flow and hence net transport of particles, without physical channels. Building on previous work focusing on two-dimensional microfluidic settings, we present an analytical, theoretical model for the thermoviscous and thermal expansion flows and net transport induced by a translating heat spot in three-dimensional, unconfined fluid. We first numerically solve for the temperature field due to a translating heat source in the experimentally relevant limit. Then, in our flow model, the small, localised temperature increase causes local changes in the mass density, shear viscosity and bulk viscosity of the fluid. We derive analytically the instantaneous flow generated during one scan and compute the net transport of passive tracers due to a full scan, up to quadratic order in the thermal expansion and thermal shear viscosity coefficients. We further show that the flow and transport are independent of bulk viscosity. In the far field, while the leading-order instantaneous flow is typically a three-dimensional source or sink, the leading-order average velocity of tracers is instead a source dipole, whose strength depends on the relative magnitudes of the thermal expansion and thermal shear viscosity coefficients. Our quantitative results reveal the potential for future three-dimensional net transport and manipulation of particles at the microscale.

We study slender, helical elastic rods subject to distributed forces and moments. Focussing on the case when the helix axis remains straight, we employ the method of multiple scales to systematically derive an 'equivalent-rod' theory from the Kirchhoff rod equations: the helical filament is described as a naturally-straight rod (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Importantly, our analysis is asymptotically exact in the limit of a 'highly-coiled' filament ( i.e. , when the helical wavelength is much smaller than the characteristic lengthscale over which the filament bends due to external loading) and is able to account for large, unsteady displacements. In addition, our analysis yields explicit conditions on the external loading that must be satisfied for a straight helix axis. In the small- deformation limit, we exactly recover the coupled wave equations used to describe the free vibrations of helical coil springs, thereby justifying previous equivalent-rod approximations in which linearised stiffness coefficients are assumed to apply locally and dynamically. We then illustrate our theory with two loading scenarios: (I) a heavy helical rod deforming under its own weight; and (II) the dynamics of axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. In both scenarios, we demonstrate excellent agreement with solutions of the full Kirchhoff rod equations, even beyond the formal limit of validity of the 'highly-coiled' assumption. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, and yields analytical insight into their elastic instabilities. In particular, our analysis indicates that tensile instabilities are a generic phenomenon when helical rods are subject to a combination of distributed forces and moments.

Motivated by the aim of understanding the effect of media heterogeneity on the swimming dynamics of flagellated bacteria, we study the rotation and swimming of rigid helices in dilute suspensions experimentally and theoretically. We first measure the torque experienced by, and thrust force generated by, helices rotating without translating in suspensions of neutrally buoyant particles with varying concentrations and sizes. Using the ratio of thrust to drag forces FF as an empirical proxy for propulsion efficiency, our experiments indicate that FF increases with the concentration of particles in the fluid, with the enhancement depending strongly on the geometric parameters of the helix. To rationalize these experimental results, we then develop a dilute theoretical approach that accounts for the additional hydrodynamic stress generated by freely suspended spheres around the helical tail. We predict similar enhancements in the drag coefficient ratio and propulsion at a given angular speed in a suspension and study its dependence on the helix geometry and the spatial distribution of the suspended spheres. These results are further reinforced by experiments on freely swimming artificial swimmers, which propel faster in dilute suspensions, with speed increases over 60% for optimal geometries. Our findings quantify how biological swimmers might benefit from the presence of suspended particles, and could inform the design of artificial self-propelled devices for biomedical applications.

Slip effects on solid boundaries are common in complex fluids. Boundary depletion layers in polymer solutions can create apparent slip effects, which can in turn significantly impact the dynamics of moving bodies. Motivated by microswimmer locomotion in such environments, we derive a series of slip slender-body theories for filamentous bodies experiencing slip-like boundary conditions. Using Navier’s slip model, we derive three slip slender-body theories, linking the body’s velocity to the distribution of hydrodynamic forces. The models are shown to be consistent with each other and with existing numerical computations. As the slip length increases, we show that the drag parallel to the body decreases towards zero while the perpendicular drag remains finite, in a manner which we quantify. This reduction in drag ratio is shown to be inversely related to microswimmer mobility in two simple swimmer models. This increase could help rationalise empirically observed enhanced swimming in complex fluids.

The mobility of particles in fluid membranes is a fundamental aspect of many biological and physical processes. In a 1975 paper (Saffman PG, Delbr & uuml;ck M. 1975 Brownian Motion in Biological Membranes. Proc. Natl Acad. Sci. USA 72, 3111-3113. (doi:10.1073/pnas.72.8.3111)), Saffman and Delbr & uuml;ck demonstrated how the presence of external Stokesian solvents is crucial in regularizing the apparently singular flow within an infinite flat membrane. In the present paper, we extend this classical work and compute the rotational mobility of a rigid finite-sized particle located inside a spherical membrane embedded in Stokesian solvents. Treating the particle as a spherical cap, we solve for the flow semi-analytically as a function of the Saffman-Delbr & uuml;ck (SD) length (ratio of membrane to solvent viscosity) and the solid angle formed by the particle. We study the dependence of the mobility and flow on inclusion size and SD length, recovering the flat-space mobility as a special case. Our results will be applicable to a range of biological problems including rotational Brownian motion, the dynamics of lipid rafts, and the motion of aquaporin channels in response to water flow. Our method will provide a novel way of measuring a membrane's viscosity from the rotational diffusion of large inclusions, for which the commonly used planar Saffman-Delbr & uuml;ck theory does not apply.

The propulsion of many eukaryotic cells is generated by flagella, flexible slender filaments that are actively oscillating in space and time. The dynamics of these biological appendages have inspired the design of many types of artificial microswimmers. The magnitude of the filament's viscous propulsion depends on the time-varying shape of the filament, and that shape depends in turn on the spatial distribution of the bending rigidity of the filament. In this work, we rigorously determine the relationship between the mechanical (bending) properties of the filament and the viscous thrust it produces using mathematical optimisation. Specifically, by considering a model system (a slender elastic filament with an oscillating slope at its base), we derive the optimal bending rigidity function along the filament that maximises the time-averaged thrust produced by the actuated filament. Instead of prescribing a specific functional form, we use functional optimisation and adjoint-based variational calculus to formally establish the link between the distribution of bending rigidity and propulsion. The optimal rigidities are found to be stiff near the base, and soft near the distal end, with a spatial distribution that depends critically on the constraints used in the optimisation procedure. These findings may guide the optimal design of future artificial swimmers.

Using a combination of theory, experiments, and numerical simulations, we investigate the stability of coherent structures in a suspension of strongly aligned active swimmers. We show that a dilute jet of pullers undergoes a pearling instability, while a jet of pushers exhibits a helical (or, in two dimensions, zigzag) instability. We further characterize the nonlinear evolution of these instabilities, deriving exact and approximate solutions for the spreading and mutual attraction of puller clusters, as well as the wavelength coarsening of the helical instability. Our theoretical predictions closely match the experimentally observed wavelengths, timescales, and flow fields in suspensions of photophobic algae, as well as results from direct numerical simulations. These findings reveal the intrinsic instability mechanisms of aligned active suspensions and demonstrate that coherent structures can be destabilized by the flows they generate.

In many biological systems, from sinuses to airways, ciliary arrays are essential for transporting particles and bacteria. Challenging long-held assumptions, we show that the classical Eulerian view of cilia-induced flow overlooks a key feature of time-averaged transport. Using theoretical modeling and numerical simulations, we demonstrate that the drift generated by ciliary motion is not uniform, as previously predicted, but varies with distance from the ciliary surface in unexpected ways. This variation reveals drift stagnation points, offering a potential means to distinguish healthy from impaired ciliary beating and refines our understanding of fluid-driven transport.

Muscle contraction, both in skeletal and cardiac tissue, is driven by sarcomeres, the microscopic units inside muscle cells where thick myosin and thin actin filaments slide past each other. During contraction and relaxation, the sarcomere's volume changes, causing sarcoplasm (intra-sarcomeric fluid) to flow out during contraction and back in as the sarcomere relaxes. We present a quantitative model of this sarcoplasmic flow, treating the sarcomere as an anisotropic porous medium with regions defined by the presence and absence of thick and thin filaments. Using semi-analytic methods, we solve for axial and lateral fluid flow within the filament lattice, calculating the permeabilities of this porous structure. We then apply these permeabilities within a Darcy model to determine the flow field generated during contraction. The predictions of our continuum model show excellent agreement with finite element simulations, reducing computational time by several orders of magnitude while maintaining accuracy in modelling the biophysical flow dynamics.

Inspired by Richard Feynman's 1959 lecture and the 1966 film Fantastic Voyage, the field of micro/nanorobots has evolved from science fiction to reality, with significant advancements in biomedical and environmental applications. Despite the rapid progress, the deployment of functional micro/nanorobots remains limited. This review of the technology roadmap identifies key challenges hindering their widespread use, focusing on propulsion mechanisms, fundamental theoretical aspects, collective behavior, material design, and embodied intelligence. We explore the current state of micro/nanorobot technology, with an emphasis on applications in biomedicine, environmental remediation, analytical sensing, and other industrial technological aspects. Additionally, we analyze issues related to scaling up production, commercialization, and regulatory frameworks that are crucial for transitioning from research to practical applications. We also emphasize the need for interdisciplinary collaboration to address both technical and nontechnical challenges, such as sustainability, ethics, and business considerations. Finally, we propose a roadmap for future research to accelerate the development of micro/nanorobots, positioning them as essential tools for addressing grand challenges and enhancing the quality of life.

Active fluids can develop spontaneous flow instabilities and complex patterns. However, spatiotemporal control of active particles has remained challenging, despite its relevance in biological and applied contexts. Here, we harnessed phototaxis to steer millions of swimming "puller" Chlamydomonas reinhardtii algae to create active jets and control both pearling and buckling instabilities through the preferential orientation of the cells. Our experiments, supported by a full analytical model and simulations, confirm long-standing predictions that self-generated flows can lead to jet destabilization. Our results further indicate that pullers can behave analogously to pushers when their orientation is properly tuned, and demonstrate how light enables the efficient control of active fluids.