Reciprocal movement cannot be used for locomotion at low-Reynolds number in an infinite fluid or near a rigid surface. Here we show that this limitation is relaxed for a body performing reciprocal motions near a deformable interface. Using physical arguments and scaling relationships, we show that the nonlinearities arising from reciprocal flow-induced interfacial deformation rectify the periodic motion of the swimmer, leading to locomotion. Such a strategy can be used to move toward, away from, and parallel to any deformable interface as long as the length scales involved are smaller than intrinsic scales, which we identify. A macro-scale experiment of flapping motion near a free surface illustrates this new result.
The motion of a rotating helical body in a viscoelastic fluid is considered. In the case of force-free swimming, the introduction of viscoelasticity can either enhance or retard the swimming speed and locomotive efficiency, depending on the body geometry, fluid properties, and the body rotation rate. Numerical solutions of the Oldroyd-B equations show how previous theoretical predictions break down with increasing helical radius or with decreasing filament thickness. Helices of large pitch angle show an increase in swimming speed to a local maximum at a Deborah number of order unity. The numer…
Actuating periodically an elastic filament in a viscous liquid generally breaks the constraints of Purcell's scallop theorem, resulting in the generation of a net propulsive force. This observation suggests a method to design simple swimming devices - which we call "elastic swimmers" - where the actuation mechanism is embedded in a solid body and the resulting swimmer is free to move. In this paper, we study theoretically the kinematics of elastic swimming. After discussing the basic physical picture of the phenomenon and the expected scaling relationships, we derive analytically the elastic s…
Swimming microorganisms often self propel in fluids with complex rheology. While past theoretical work indicates that fluid viscoelasticity should hinder their locomotion, recent experiments on waving swimmers suggest a possible non-Newtonian enhancement of locomotion. We suggest a physical mechanism, based on fluid-structure interaction, leading to swimming in a viscoelastic fluid at a higher speed than in a Newtonian one. Using Taylor's two-dimensional swimming sheet model, we solve for the shape of an active swimmer as a balance between the external fluid stresses, the internal driving mome…
Microorganisms are rarely found in Nature swimming freely in an unbounded fluid. Instead, they typically encounter other organisms, hard walls, or deformable boundaries such as free interfaces or membranes. Hydrodynamic interactions between the swimmer and nearby objects lead to many interesting phenomena, such as changes in swimming speed, tendencies to accumulate or turn, and coordinated flagellar beating. Inspired by this class of problems, we investigate locomotion of microorganisms near deformable boundaries. We calculate the speed of an infinitely long swimmer close to a flexible surface…
Swimming cells often have to self-propel through fluids displaying non-Newtonian rheology. While past theoretical work seems to indicate that stresses arising from complex fluids should systematically hinder low-Reynolds number locomotion, experimental observations suggest that locomotion enhancement is possible. In this paper we propose a physical mechanism for locomotion enhancement of microscopic swimmers in a complex fluid. It is based on the fact that micro-structured fluids will generically phase-separate near surfaces, leading to the presence of low-viscosity layers which promote slip a…
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…
The propulsion of mammalian spermatozoa relies on the spontaneous periodic oscillation of their flagella. These oscillations are driven internally by the coordinated action of ATP-powered dynein motors that exert sliding forces between microtubule doublets, resulting in bending waves that propagate along the flagellum and enable locomotion. We present an integrated chemomechanical model of a freely swimming spermatozoon that uses a sliding-control model of the axoneme capturing the two-way feedback between motor kinetics and elastic deformations while accounting for detailed fluid mechanics ar…
A variety of swimming microorganisms, called ciliates, exploit the bending of a large number of small and densely-packed organelles, termed cilia, in order to propel themselves in a viscous fluid. We consider a spherical envelope model for such ciliary locomotion where the dynamics of the individual cilia are replaced by that of a continuous overlaying surface allowed to deform tangentially to itself. Employing a variational approach, we determine numerically the time-periodic deformation of such surface which leads to low-Reynolds locomotion with minimum rate of energy dissipation (maximum ef…
Biological organisms swimming at low Reynolds number are often influenced by the presence of rigid boundaries and soft interfaces. In this paper we present an analysis of locomotion near a free surface with surface tension. Using a simplified two-dimensional singularity model, and combining a complex variable approach with conformal mapping techniques, we demonstrate that the deformation of a free surface can be harnessed to produce steady locomotion parallel to the interface. The crucial physical ingredient lies in the nonlinear hydrodynamic coupling between the disturbance flow created by th…
Microorganisms often encounter anisotropy, for example in mucus and biofilms. We study how anisotropy and elasticity of the ambient fluid affects the speed of a swimming microorganism with a prescribed stroke. Motivated by recent experiments on swimming bacteria in anisotropic environments, we extend a classical model for swimming microorganisms, the Taylor swimming sheet, actuated either by transverse or longitudinal traveling waves in a three-dimensional nematic liquid crystal without twist. We calculate the swimming speed and entrained volumetric flux as a function of the swimmer's stroke p…
We study self-propulsion (or locomotion) of a robot (or an underwater vehicle) in an inviscid incompressible fluid. The robot's body is rigid, while its locomotion ability is due to an internal actuator, which can perform controlled translational and rotational oscillations. Our attention is focused on two classes of the plane analytic exact solutions, describing rectilinear locomotion. Solutions of the first class describe the \emph{tumbling locomotion}, while the second class corresponds to the \emph{zigzag locomotion} without tumbling. We show, that tumbling locomotion is more efficient. Bo…
Inspired by recent experiments using synthetic microswimmers to manipulate droplets, we investigate the low-Reynolds-number locomotion of a model swimmer (a spherical squirmer) encapsulated inside a droplet of comparable size in another viscous fluid. Meditated solely by hydrodynamic interactions, the encaged swimmer is seen to be able to propel the droplet, and in some situations both remain in a stable co-swimming state. The problem is tackled using both an exact analytical theory and a numerical implementation based on boundary element method, with a particular focus on the kinematics of th…
The locomotion of swimming bacteria in simple Newtonian fluids can successfully be described within the framework of low Reynolds number hydrodynamics. The presence of polymers in biofluids generally increases the viscosity, which is expected to lead to slower swimming for a constant bacterial motor torque. Surprisingly, however, several experiments have shown that bacterial speeds increase in polymeric fluids, and there is no clear understanding why. Therefore we perform extensive coarse-grained simulations of a bacterium swimming in explicitly modeled solutions of macromolecular polymers of …
Locomotion by shape changes (spermatozoon swimming, snake slithering, bird flapping) or gas expulsion (rocket firing) is assumed to require environmental interaction, due to conservation of momentum. As first noted in (Wisdom, 2003) and later in (Guéron, 2009) and (Avron et al, 2006), in curved space or spacetime the non-commutativity of translations permits translation without momentum exchange, just as falling cats and lizards can self-deform to reorient in flat space without environmental interaction. Translation in curved space can occur not only in gravitationally induced curved spacetime…
The swimming behavior of bacteria and other microorganisms is sensitive to the physical properties of the fluid in which they swim. Mucus, biofilms, and artificial liquid-crystalline solutions are all examples of fluids with some degree of anisotropy that are also commonly encountered by bacteria. In this article, we study how liquid-crystalline order affects the swimming behavior of a model swimmer. The swimmer is a one-dimensional version of G. I. Taylor's swimming sheet: an infinite line undulating with small-amplitude transverse or longitudinal traveling waves. The fluid is a two-dimension…
Propulsion through fluids is a key component in the life cycle of many microbes, be it in development, infection, or simply finding nutrients. In systems of biomedical relevance, this propulsion is often through polymer suspensions that endow the fluid with complex non-Newtonian properties, such as shear-thinning and viscoelastic behaviour. Due to the complexity of 3D non-Newtonian modelling, 2D undulatory propulsion has recently been extensively-studied as a means of garnering physical intuition for these systems. However, while streamlines, swimming speeds, and swimmer trajectories are strik…
Active matter comprised of self-propelled interacting units holds a major promise for extraction of useful work from its seemingly chaotic out-of-equilibrium dynamics. Streamlining active matter to produce work is especially important at microscale, where the viscous forces prevail over inertia and the useful modes of transport require very specific non-reciprocal type of motion. Here we report that microscopic active droplets representing aqueous dispersions of swimming bacteria Bacillus subtilis show unidirectional propulsion when placed in an inactive nematic medium. Random motion of bacter…
We study the coupled hydrodynamics between a motile slender microswimmer and a deformable interface that separates two Newtonian fluid regions. From the disturbance field generated by the swimming motion, we quantitatively characterize the interface deformation and the manner in which the coupling modifies the microswimmer translation itself. We treat the role of the swimmer type (pushers and pullers), size and model an interface that can deform due to both surface tension and bending elasticity. Our analysis reveals a strong dependence of the hydrodynamics on the swimmer orientation and posit…
In many biological systems, microorganisms swim through complex polymeric fluids, and usually deform the medium at a rate faster than the inverse fluid relaxation time. We address the basic properties of such life at high Deborah number analytically by considering the small-amplitude swimming of a body in an arbitrary complex fluid. Using asymptotic analysis and differential geometry, we show that for a given swimming gait, the time-averaged leading-order swimming kinematics of the body can be expressed as an integral equation on the solution to a series of simpler Newtonian problems. We then …
The development of multifunctional and biocompatible microrobots for biomedical applications relies on achieving locomotion through viscous fluids. Here, we describe a framework for swimming in homogeneous magnetoelastic membranes composed of superparamagnetic particles. By solving the equations of motion, we find the dynamical modes of circular membranes in precessing magnetic fields, which are found to actuate in or out of synchronization with a magnetic field precessing above or below a critical precession frequency, $ω_c$, respectively. For frequencies larger than $ω_c$, synchronized rotat…
Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and alga…
When tiny soft ferromagnetic particles are placed along a liquid interface and exposed to a vertical magnetic field, the balance between capillary attraction and magnetic repulsion leads to self-organization into well-defined patterns. Here, we demonstrate experimentally that precessing magnetic fields induce metachronal waves on the periphery of these assemblies, similar to the ones observed in ciliates and some arthropods. The outermost layer of particles behaves like an array of cilia or legs whose sequential movement causes a net and controllable locomotion. This bioinspired many-particle …
Due to the kinematic reversibility of Stokes flow, a body executing a reciprocal motion (a motion in which the sequence of body configurations remains identical under time reversal) cannot propel itself in a viscous fluid in the limit of negligible inertia; this result is known as Purcell's scallop theorem. In this limit, the Reynolds numbers based on the fluid inertia and on the body inertia are all zero. Previous studies characterized the breakdown of the scallop theorem with fluid inertia. In this paper we show that, even in the absence of fluid inertia, certain dense bodies undergoing reci…
Physics governing the locomotion of microorganisms and other microsystems is dominated by viscous damping. An effective swimming strategy involves the non-reciprocal and periodic deformations of the considered body. Here, we show that a magnetocapillary-driven self-assembly, composed of three soft ferromagnetic beads, is able to swim along a liquid-air interface when powered by an external magnetic field. More importantly, we demonstrate that trajectories can be fully controlled, opening ways to explore low Reynolds number swimming. This magnetocapillary system spontaneously forms by self-asse…
Compared to agile legged animals, wheeled and tracked vehicles often suffer large performance loss on granular surfaces like sand and gravel. Understanding the mechanics of legged locomotion on granular media can aid the development of legged robots with improved mobility on granular surfaces; however, no general force model yet exists for granular media to predict ground reaction forces during complex limb intrusions. Inspired by a recent study of sand-swimming, we develop a resistive force model in the vertical plane for legged locomotion on granular media. We divide an intruder of complex m…
In nature, a variety of limbless locomotion patterns flourish from the small or basic life form (Escherichia coli, the amoeba, etc.) to the large or intelligent creatures (e.g., slugs, starfishes, earthworms, octopuses, jellyfishes, and snakes). Many bioinspired soft robots based on locomotion have been developed in the past decades. In this work, based on the kinematics and dynamics of two representative locomotion modes (i.e., worm-like crawling and snake-like slithering), we propose a broad set of innovative designs for soft mobile robots through simple mechanical principles. Inspired by an…
Interactions between swimming cells and surfaces are essential to many microbiological processes, from bacterial biofilm formation to human fertilization. However, in spite of their fundamental importance, relatively little is known about the physical mechanisms that govern the scattering of flagellated or ciliated cells from solid surfaces. A more detailed understanding of these interactions promises not only new biological insights into structure and dynamics of flagella and cilia, but may also lead to new microfluidic techniques for controlling cell motility and microbial locomotion, with p…
