Microswimming cells and robots exhibit diverse behaviours due to both their swimming and their environment. One key environmental feature is the presence of a background flow. While the influences of select flows, particularly steady shear flows, have been extensively investigated, these only represent special cases. Here, we examine inertialess swimmers in more general flows, specifically general linear planar flows that may possess rapid oscillations, and impose weak symmetry constraints on the swimmer (ensuring planarity, for instance). We focus on swimmers that are inefficient, in that the time scales of their movement are well separated from those associated with their motility-driving deformation. Exploiting this separation of scales in a multiple-time-scale analysis, we find that the behaviour of the swimmer is dictated by two effective parameter groupings, excluding mathematically precise edge cases. These systematically derived parameters measure balances between angular velocity and the rate of strain of the background flow. Remarkably, one parameter governs the orientational dynamics, whilst the other completely captures translational motion. Further, we find that the long-time translational dynamics is solely determined by properties of the flow, independent of the details of the swimmer. This illustrates the limited extent to which, and how, microswimmers may control their behaviours in planar linear flows.

Undulatory slender objects have been a central theme in the hydrodynamics of swimming at low Reynolds number, where the slender body is usually assumed to be inextensible, although some microorganisms and artificial microrobots largely deform with compression and extension. Here, we theoretically study the coupling between the bending and compression/extension shape modes, using a geometrical formulation of kinematic microswimmer hydrodynamics to deal with the non-commutative effects between translation and rotation. By means of a coarse-grained minimal model and systematic perturbation expansions for small bending and compression/extension, we analytically derive the swimming velocities and report three main findings. First, we revisit the role of anisotropy in the drag ratio of the resistive force theory, and generally demonstrate that no motion is possible for uniform compression with isotropic drag. We then find that the bending–compression/extension coupling generates lateral and rotational motion, which enhances the swimmer’s manoeuvrability, as well as changes in progressive velocity at a higher order of expansion, while the coupling effects depend on the phase difference between the two modes. Finally, we demonstrate the importance of often-overlooked Lie bracket contributions in computing net locomotion from a deformation gait. Our study sheds light on compression as a forgotten degree of freedom in swimmer locomotion, with important implications for microswimmer hydrodynamics, including understanding of biological locomotion mechanisms and design of microrobots.

We present numerical analysis of the lateral movement of a spherical capsule in the steady and pulsatile channel flow of a Newtonian fluid for a wide range of oscillatory frequencies. Each capsule membrane satisfying strain-hardening characteristics is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$. Our numerical results showed that capsules with high $Ca$ exhibit axial focusing at finite $Re$ similarly to the inertialess case. We observe that the speed of the axial focusing can be substantially accelerated by making the driving pressure gradient oscillate in time. We also confirm the existence of an optimal frequency that maximises the speed of axial focusing, which remains the same found in the absence of inertia. For relatively low $Ca$, however, the capsule exhibits off-centre focusing, resulting in various equilibrium radial positions depending on $Re$. Our numerical results further clarify the existence of a specific $Re$ for which the effect of the flow pulsation to the equilibrium radial position is maximum. The roles of channel size on the lateral movements of the capsule are also addressed. Throughout our analyses, we have quantified the radial position of the capsule in a tube based on an empirical expression. Given that the speed of inertial focusing can be controlled by the oscillatory frequency, the results obtained here can be used for label-free cell alignment/sorting/separation techniques, e.g. for circulating tumour cells in cancer patients or precious haematopoietic cells such as colony-forming cells.

In this paper, we consider the dynamics of a helicoidal object, which can be either a passive particle or an active swimmer, with an arbitrary shape in a linear background flow at low Reynolds number, and derive a generalized version of the Jeffery equations for the angular dynamics of the object, including a new constant from the chirality of the object as well as the Bretherton constant. The new constant appears from the inhomogeneous chirality distribution along the axis of the helicoidal symmetry, whereas the overall chirality of the object contributes to the drift velocity. Further investigations are made for an object in a simple shear flow, and it is found that the chirality effects generate non-closed trajectories of the director vector which will be stably directed parallel or anti-parallel to the background vorticity vector depending on the sign of the chirality. A bacterial swimmer is considered as an example of a helicoidal object, and we calculate the values of the constants in the generalized Jeffery equations for a typical morphology of Escherichia coli. Our results provide useful expressions for the studies of microparticles and biological fluids, and emphasize the significance of the symmetry of an object on its motion at low Reynolds number.

We have investigated the dynamics of a monotrichous bacteria cell near a wall boundary, taking elastic hook flexibility into consideration. Combining theoretical linear stability analysis and direct numerical computations via the boundary element method, we have found that the elastohydrodynamic coupling between the hook elasticity and cell rotational motion enables a stable vertical spinning behaviour like a low-Reynolds-number spinning top. The forwardly rotated flagellum, which generates the force exertion pushing towards the cell body, typically destabilizes the vertical upright position and leads to a boundary-following motion. In contrast, the backward rotation of the flagellum, generating a force pulling the cell body, contributes to stable upright behaviour in a large range of hook rigidity. Further numerical investigations have demonstrated that the non-spherical geometry of the cell body and boundary adhesive interactions affect the bacterial dynamics, leading to complex behaviours such as horizontal spinning and unstable vertical spinning motions, both of which are experimentally observed in Pseudomonas aeruginosa bacteria. These results highlight the rich diversity of bacterial surface motility emerging from mechanical boundary interactions coupled with the cell swimming and hook flexibility.

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

Induction of flow is commonly used to control the migration of a microswimmer in a confined system such as a microchannel. The motion of a swimmer, in general, is governed by nonlinear equations due to non-trivial hydrodynamic interactions between the flow and the swimmer near a wall. This paper derives analytical expressions for the equations of motion governing a circular treadmilling swimmer in simple shear near a no-slip wall by combining the reciprocal theorem for Stokes flow with an exact solution for the dragging problem of a cylinder near a wall. We demonstrate that the reduced dynamical system possesses a Hamiltonian structure, which we use to show that the swimmer cannot migrate stably at a constant distance from a wall but only exhibit periodic oscillatory motion along the wall, or to escape from it. A treadmilling swimmer with the lowest two treadmilling modes is investigated in detail by means of a bifurcation analysis of the reduced dynamical system. It is found that the swimming direction of oscillatory motion is clarified by the sign of the Hamiltonian in the absence of flow, and that the induction of the flow suppresses upstream migration but aligns swimmer orientations in downstream migration. These results could inform strategies for the transport and control of micro-organisms and micromachines.

The motion of a spherical squirmer in unsteady Stokes flow is investigated for a deeper understanding of unsteady inertial effects on swimming micro-organisms and differences of swimming strokes between a wave pattern and a flapping motion. An asymptotic analysis with respect to the small amplitude and the small inertia is performed, and the average swimming velocity after a long period of time under an assumption of a time-periodic stroke is obtained. This analysis shows that the scallop theorem still holds in a long-time asymptotic sense for tangential deformation, but that the time variation of the shape generates a net velocity even for a reciprocal swimmer. It is also found that the inertial effects on the swimming velocity are significant for a flapping swimmer, as contrasted with little influence on that of a swimmer with a wave pattern. The inertial effect is also illustrated with a simple squirmer, so that a reciprocal motion can be almost an optimal stroke under a constraint on energy consumption.