An approximation to the added mass matrix of an assembly of spheres is constructed on the basis of potential flow theory for situations where one sphere is much larger than the others. In the approximation the flow potential near a small sphere is assumed to be dipolar, but near the large sphere it involves all higher order multipoles. The analysis is based on an exact result for the potential of a magnetic dipole in the presence of a superconducting sphere. Subsequently, the approximate added mass hydrodynamic interactions are used in a calculation of the swimming velocity and rate of dissipation of linear chain structures consisting of a number of small spheres and a single large one, with account also of frictional hydrodynamic interactions. The results derived for periodic swimming on the basis of a kinematic approach are compared with bilinear theory, valid for small amplitude of stroke, and with the numerical solution of the approximate equations of motion. The calculations cover the whole range of scale number between the friction-dominated Stokes limit and the inertia-dominated regime.
Through billions of years of evolution, microorganisms mastered unique swimming behaviors to thrive in complex fluid environments. Limitations in nanofabrication have thus far hindered the ability to design and program synthetic swimmers with the same abilities. Here we encode multi-behavioral responses in artificial swimmers such as microscopic, self-propelled tori using nanoscale 3D printing. We show experimentally and theoretically that the tori continuously transition between two primary swimming modes in response to a magnetic field. The tori also manipulate and transport other artificial…
The swimming velocity and rate of dissipation of a linear chain consisting of two or three little spheres and a big sphere is studied on the basis of low Reynolds number hydrodynamics. The big sphere is treated as a passive cargo, driven by the tail of little spheres via hydrodynamic and direct elastic interaction. The fundamental solution of Stokes' equations in the presence of a sphere with no-slip boundary condition, as derived by Oseen, is used to model the hydrodynamic interactions between the big sphere and the little spheres.
The swimming of a sphere immersed in a viscous incompressible fluid with inertia is studied for surface modulations of small amplitude on the basis of the Navier-Stokes equations. The mean swimming velocity and the mean rate of dissipation are expressed as quadratic forms in term of the surface displacements. With a choice of a basis set of modes the quadratic forms correspond to two hermitian matrices. Optimization of the mean swimming velocity for given rate of dissipation requires the solution of a generalized eigenvalue problem involving the two matrices. It is found for surface modulation…
The swimming of a deformable planar slab in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations. A continuum of plane wave displacements, symmetric on both sides of the slab and characterized by a polarization angle, allows optimization of the swimming efficiency with respect to polarization. The mean swimming velocity and mean rate of dissipation are calculated to second order in the amplitude of the stroke. The optimum efficiency depends on the ratio of viscosity and mass density of the fluid. For high viscosity a stroke is found with significantly higher ef…
The swimming of a deformable uniform sphere is studied in second order perturbation theory in the amplitude of the stroke. The effect of the first order reaction force on the first order center of mass velocity is calculated in linear response theory by use of Newton's equation of motion. The response is characterized by a dipolar admittance, which is shown to be proportional to the translational admittance. As a consequence the mean swimming velocity, calculated in second order perturbation theory, depends on the added mass of the sphere. The mean swimming velocity and the mean rate of dissip…
Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function…
Translational and rotational swimming at small Reynolds number of a planar assembly of identical spheres immersed in an incompressible viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The translational and rotational swimming velocities of the assembly are deduced from momentum and angular momentum balance equations. The mean pow…
The swimming of a two-sphere system and of a three-sphere chain in an incompressible viscous fluid is studied on the basis of simplified equations of motion which take account of both Stokes friction and added mass effects. The analysis is based on an explicit expression for the asymptotic periodic swimming velocity and a corresponding evaluation of the mean rate of dissipation. The mean swimming velocity of the two-sphere system is found to be non-vanishing provided that the two spheres are not identical. The swimming of a comparable chain of three identical spheres is much more efficient.
A mechanical model of swimming and flying in an incompressible viscous fluid in the absence of gravity is studied on the basis of assumed equations of motion. The system is modeled as an assembly of rigid spheres subject to elastic direct interactions and to periodic actuating forces which sum to zero. Hydrodynamic interactions are taken into account in the virtual mass matrix and in the friction matrix of the assembly. An equation of motion is derived for the velocity of the geometric center of the assembly. The mean power is calculated as the mean rate of dissipation. The full range of visco…
The swimming of a bead-spring chain in a viscous incompressible fluid as a model of a sperm is studied in the framework of low Reynolds number hydrodynamics. The optimal mode in the class of planar flagellar strokes of small amplitude is determined on the basis of a generalized eigenvalue problem involving two matrices which can be evaluated from the mobility matrix of the set of spheres constituting the chain. For an elastic chain with a cargo constraint for its spherical head, the actuating forces yielding a nearly optimal stroke can be determined. These can be used in a Stokesian dynamics s…
We propose a reciprocal, self-propelled model swimmer at intermediate Reynolds numbers ($Re$). Our swimmer consists of two unequal spheres that oscillate in antiphase generating nonlinear steady streaming (SS) flows. We show computationally that the SS flows enable the swimmer to propel itself, and also switch direction as $Re$ increases. We quantify the transition in the swimming direction by collapsing our data on a critical $Re$ and show that the transition in swimming directions corresponds to the reversal of the SS flows. Based on our findings, we propose that SS can be an important physi…
In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers Re ~ 1-1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case-specific. Here, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillati…
We study swimming of small spherical particles who regulate fluid flow on their surface by applying tangential squirming strokes. We derive translational and rotational velocities for any given stroke which is not restricted by axial symmetry as assumed usually. The formulation includes inertia of both the fluid and the swimmer, motivated by inertia's relevance for large Volvox colonies. We show that inertial contribution to mean speed comes from dynamic coupling between translation and rotation, which occurs only for strokes that break axial symmetry. Remarkably, this effect enables overcomin…
The swimming of a two-sphere system oscillating in a viscous fluidis studied on the basis of simplified equations of motion which take account of both friction and inertial effects. In the model the friction follows from an Oseen approximation to the mobility matrix, and the inertial effects follow from a dipole approximation to the added mass matrix. The resulting mean swimming velocity is evaluated analytically in a first harmonics approximation. For specific choices of the parameters this is compared with the exact result following from a numerical calculation including higher harmonics. Th…
We analyze the dynamics of a microswimmer in pressure-driven Poiseuille flow, where fluid inertia is small but non-negligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a `swimming lift', which we evaluate for neutral and pusher/puller-type swimmers. Accounting for fluid inertia engenders a rich spectrum of novel complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer's hydrodynami…
In this paper, we address a crucial point regarding the description of moderate to high Reynolds numbers aquatic swimmers. For decades, swimming animals have been classified in two different families of propulsive mechanisms based on the Reynolds number: the "resistive" swimmers, using local friction to produce the necessary thrust force for locomotion at low Reynolds number and the "reactive" swimmers, lying in the high Reynolds range, and using added mass acceleration (described by perfect fluid theory). However, inertial swimmers are also systems that dissipate energy, due to their finite s…
A body immersed in a highly viscous fluid can locomote by drawing in and expelling fluid through pores at its surface. We consider this mechanism of jet propulsion without inertia in the case of spheroidal bodies, and derive both the swimming velocity and the hydrodynamic efficiency. Elementary examples are presented, and exact axisymmetric solutions for spherical, prolate spheroidal, and oblate spheroidal body shapes are provided. In each case, entirely and partially porous (i.e. jetting) surfaces are considered, and the optimal jetting flow profiles at the surface for maximizing the hydrodyn…
By synergistically combining modeling, simulation and experiments, we show that there exists a regime of self-propulsion in which the inertia in the fluid dynamics can be separated from that of the swimmer. This is demonstrated by the motion of an asymmetric dumbbell that, despite deforming in a reciprocal fashion, self-propagates in a fluid due to a non-reciprocal Stokesian flow field. The latter arises from the difference in the coasting times of the two constitutive beads. This asymmetry acts as a second degree of freedom, recovering the scallop theorem at the mesoscopic scale.
Locomotion in Stokes flow is an intensively-studied problem because it describes important biological phenomena such as the motility of many species' sperm, bacteria, algae and protozoa. Numerical computations can be challenging, particularly in three dimensions, due to the presence of moving boundaries and complex geometries; methods which combine ease-of-implementation and computational efficiency are therefore needed. A recently-proposed method to discretise the regularised Stokeslet boundary integral equation without the need for a connected 'mesh' is applied to the inertialess locomotion …
Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection, and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming (tens of microns and below). The focus is on the fundamental flow physics phenomena occurring in this inertia-less realm, and the emphasis is on the simple physical picture. We review the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming, such as resistance matrices fo…
Micro-scale swimming robots have been envisaged for many medical applications such as targeted drug delivery, where the microrobot will be expected to navigate in a fluid through channels carrying a payload. Alternatively, in many cases, such a payload does not have to be physically bound to the swimmer, but may be instead manipulated and steered through the channel by the microrobot. We investigate this problem of contactless manipulation of a microparticle by mobile microswimmer in a fluid at low Reynolds number. We consider a model of a magnetically actuated artificial microswimmer, whose l…
Fluid-suspended microorganisms have evolved different swimming and feeding strategies in order to cope with an environment dominated by viscous effects. For instance ciliated organisms rely on the collective motion of flexible appendices to move and feed. By performing a non-reciprocal motion, flexible filaments can produce a net propulsive force, or pump fluid, in the absence of inertia. Inspired by such fundamental concept, we propose a strategy to produce macroscopic pumping and mixing in creeping flow. We measure experimentally the net motion of a Newtonian viscous fluid induced by the rec…
Cilia and flagella are actively bending slender organelles, performing functions such as motility, feeding and embryonic symmetry breaking. We review the mechanics of viscous-dominated microscale flow, including time-reversal symmetry, drag anisotropy of slender bodies, and wall effects. We focus on the fundamental force singularity, higher order multipoles, and the method of images, providing physical insight and forming a basis for computational approaches. Two biological problems are then considered in more detail: (1) left-right symmetry breaking flow in the node, a microscopic structure i…
