Hydrodynamic interaction between two swimmers at low Reynolds number

We investigate the hydrodynamic interactions between micro-organisms swimming at low Reynolds number. By considering simple model swimmers, and combining analytic and numerical approaches, we investigate the time-averaged flow field around a swimmer. At short distances the swimmer behaves like a pump. At large distances the velocity field depends on whether the swimming stroke is invariant under a combined time-reversal and parity transformation. We then consider two swimmers and find that the interaction between them consists of two parts: a passive term, independent of the motion of the second swimmer, and an active term resulting from the simultaneous swimming action of both swimmers. The swimmer-swimmer interaction is a complicated function of their relative displacement, orientation, and phase, leading to motion that can be attractive, repulsive, or oscillatory.     

Brownian dynamics of a microswimmer

We report on dynamic properties of a simple model microswimmer composed of three spheres and propelling itself in a viscous fluid by spinning motion of the spheres under zero net torque constraint. At a fixed temperature and increasing the spinning frequency, the swimmer demonstrates a transition from dissipation-dominated to a pumping-dominated motion regime characterized by negative effective friction coefficient. In the limit of high frequencies, the diffusion of the swimmer can be described by a model of an active particle with constant velocity.     

Chirality in microswimmer motion: From circle swimmers to active turbulence

In this minireview, recent progress in our understanding of the basic physical principles of microswimmers which perform a motion characterized by chirality is summarized. We discuss both the chiral motion of a single circle swimmer and the occurrence of bacterial turbulence where swirls of different chirality are formed spontaneously in an interacting ensemble of linear microswimmers. Some recent highlights in this context as obtained by theory, simulation and experiment are summarized and briefly discussed.     

Dynamics of Purcell's three-link microswimmer with a passive elastic tail

One of the few possible mechanisms for self-propulsion at low Reynolds number is undulations of a passive elastic tail, as proposed in the classical work of Purcell (1977). This effect is studied here by investigating a variant of Purcell's three-link swimmer model where the front joint angle is periodically actuated while the rear joint is driven by a passive torsional spring. The dynamic equations of motion are formulated and explicit expressions for the leading-order solution are derived by using perturbation expansion. The dependence of the motion on the actuation amplitude and frequency is analyzed, and optimization with respect to the swimmer's geometry is conducted.     

Mechanical response of a small swimmer driven by conformational transitions

A conformation space kinetic model is constructed to drive the deformation cycle of a three-sphere swimmer to achieve propulsion at low Reynolds number. We analyze the effect of an external load on the performance of this kinetic swimmer and show that it depends sensitively on where the force is exerted, so that there is no general force-velocity relation. We discuss how the conformational cycle of such swimmers should be designed to increase their performance in resisting forces applied at specific points.     

Dynamics of Purcell’s three-link microswimmer with a passive elastic tail

One of the few possible mechanisms for self-propulsion at low Reynolds number is undulations of a passive elastic tail, as proposed in the classical work of Purcell (1977). This effect is studied here by investigating a variant of Purcell??s three-link swimmer model where the front joint angle is periodically actuated while the rear joint is driven by a passive torsional spring. The dynamic equations of motion are formulated and explicit expressions for the leading-order solution are derived by using perturbation expansion. The dependence of the motion on the actuation amplitude and frequency is analyzed, and optimization with respect to the swimmer??s geometry is conducted.      

Speedup of self-propelled helical swimmers in a long cylindrical pipe

Pipe-like confinements are ubiquitously encountered by microswimmers.Here we systematically study the ratio of the speeds of a force-and torque-free microswimmer swimming in the center of a cylindrical pipe to its speed in an unbounded fluid(speed ratio).Inspired by E.coli,the model swimmer consists of a cylindrical head and a double-helical tail connected to the head by a rotating virtual motor.The numerical simulation shows that depending on swimmer geometry,confinements can enhance or hinder the swimming speed,which is verified by Reynolds number matched experiments.We further developed a reduced model.The model shows that the swimmer with a moderately long,slender head and a moderately long tail experiences the greatest speed enhancement,whereas the theoretical speed ratio has no upper limit.The properties of the virtual motor also affect the speed ratio,namely,the constant-frequency motor generates a greater speed ratio compared to the constant-torque motor.     

Ferromagnetic microswimmers

We propose a model for a novel artificial low Reynolds number swimmer, based on the magnetic interactions of a pair of ferromagnetic particles: one with hard and the other with soft magnetic properties, connected by a linear spring. Using a computational model, we analyze the behavior of the system and demonstrate that for realistic values of the parameters involved, the swimmer is capable of self-propelling with average speeds of the order of hundreds of micrometers per second.     

Motion and mixing for multiple ferromagnetic microswimmers

This paper concerns the interaction of several ferromagnetic microswimmers, their motion and the resulting fluid mixing. Each swimmer consists of two ferromagnetic beads joined by an elastic link, and is driven by an external, time-dependent magnetic field. The external field provides a torque on a swimmer and, together with the varying attraction between the magnetic beads, generates a time-irreversible motion leading to persistent swimming in a low Reynolds number environment. The aim of the present paper is to consider the interactions between several swimmers. A regime is considered in which identical swimmers move in the same overall direction, and their motion is synchronised because of driving by the external field. It is found that two swimmers tend to encircle one another while three undergo more complicated motion that may involve the braiding of swimmer trajectories. By means of approximations it is established that the interaction between pairs of swimmers gives circulatory motion which falls off with an inverse square law and is linked to their overall speed of motion through the fluid. As groups of two or more swimmers move through the fluid they process fluid, leaving behind a trail of fluid that has undergone mixing: this is investigated by following streak lines numerically.     

Optimal stroke patterns for Purcell's three-link swimmer

Stroke patterns for Purcell's three-link swimmer are optimized. We model the swimmer as a jointed chain of three slender rods moving in an inertialess flow. The swimmer is optimized for efficiency and speed. We were able to attain swimmer designs significantly more efficient than those previously suggested by authors who only consider geometric design rather than kinematic criteria. The influence of slenderness on optimality is considered as well.     

How walls affect the dynamics of self-phoretic microswimmers

We study the effect of a nearby planar wall on the propulsion of a spherical phoretic micro-swimmer driven by reactions on its surface. An asymmetric coverage of catalysts on its surface which absorb reactants and generate products gives rise to an anisotropic interfacial flow that propels the swimmer. We analyse the near-wall dynamics of such a self-phoretic swimmer as a function of the asymmetric catalytic coverage of the surface. By an analysis of the fundamental singularities of the flow and concentration or electrostatic potential gradients generated we are able to obtain and rationalise a phase diagram of behaviours as a function of the characteristics of the swimmer surface. We find a variety of possible behaviours, from “bound states” where the swimmer remains near the wall to “scattering” or repulsive trajectories in which the swimmer ends far from the wall. The formation of some of the bound states is a purely wall-phoretic effect and cannot be obtained by simply mapping a phoretic swimmer to a hydrodynamic one.     

Purcell’s “rotator”: mechanical rotation at low Reynolds number

An object consisting of three spheres, linked like the spokes on a wheel, can undergo a net rotational movement when the relative positions of the spheres proceed through a four-step cycle. This rotational motion is the analogue of the two-hinged swimmer originally proposed by Purcell (1977), which has served as a prototype for mechanical implementations of swimming. We also note that the rotational motion analysed here may be useful in the design of micromachines and has similarities to molecular-scale rotational motors that have been identified recently.     

Microswimmers near surfaces

Both, in their natural environment and in a controlled experimental setup, microswimmers regularly interact with surfaces. These surfaces provide a steric boundary, both for the swimming motion and the hydrodynamic flow pattern. These effects typically imply a strong accumulation of microswimmers near surfaces. While some generic features can be derived, details of the swimmer shape and propulsion mechanism matter, which give rise to a broad range of adhesion phenomena and have to be taken into account to predict the surface accumulation for a given swimmer. We show in this minireview how numerical simulations and analytic theory can be used to predict the accumulation statistics for different systems, with an emphasis on swimmer shape, hydrodynamics interactions, and type of noisy dynamics.     

Lévy flights in inhomogeneous environments

We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place. To this end we employ two model patterns of dynamical behavior: Langevin-driven and (Lévy-Schrödinger) semigroup-driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to Lévy oscillators), with respect to both dynamical mechanisms. We discuss the links of the Lévy semigroup motion scenario with that of random searches in spatially inhomogeneous media.     

A Charged Rotating Cylindrical Shell

We give an example of a spacetime having an infinite thin rotating cylindrical shell constituted by a charged perfect fluid as a source. As the interior of the shell the Bonnor–Melvin universe is considered, while its exterior is represented by the Datta–Raychaudhuri spacetime. We discuss the energy conditions and we show that our spacetime contains closed timelike curves. Trajectories of charged test particles both inside and outside the cylinder are also examined. An expression for the angular velocity of a circular motion inside the cylinder is given.     

Snell's law of refraction observed in thermal frontal polymerization

We demonstrate that Snell's law of refraction can be applied to thermal fronts propagating through a boundary between regions that support distinct frontal velocities. We use the free-radical frontal polymerization of a triacrylate with clay filler that allows for two domains containing two different concentrations of a peroxide initiator to be molded together. Because the polymerization reaction rates depend on the initiator concentration, the propagation speed is different in each domain. We study fronts propagating in two parallel strips in which the incident angle is 90 degrees. Our data fit Snell's law v(r)/v(i)=sin theta(r)/sin theta(i), where v(r) is the refracted velocity, v(i) is the incident velocity, theta(r) is the angle of refraction, and theta(i) is the incident angle. Further, we study circular fronts propagating radially from an initiation point in a high-velocity region into a low-velocity region (and vice versa). We demonstrate the close resemblance between the numerically simulated and experimentally observed thermal reaction fronts. By measuring the normal velocity and the angle of refraction of both simulated and experimental fronts, we establish that Snell's law holds for thermal frontal polymerization in our experimental system. Finally we discuss the regimes in which Snell's law may not be valid.     

Motion and collision of particles near Plebanski-Demianski black hole: Shadow and gravitational lensing

Here we consider accelerating and rotating charged Plebanski-Demianski (PD) class of black hole metric as a particle accelerator. We obtain the geodesic motions (timelike, null and spacelike) of particles in a non-equatorial plane around the PD black hole. We find the effective potential, energy, angular momentum, impact parameters, and discuss the circular orbit. We study the center of mass energy of two neutral particles falling from infinity to near the non-extremal horizons (event and Cauchy horizons), extremal horizon, accelerating horizons, and near the center of the PD black hole. Also, we study the collision of a particle and a massless photon. Then we find the center of mass energy due to the collision of two massless photons in the PD black hole background. We compute the redshift and blueshift of the emitted photons by massive particles while light signal travels along null geodesics towards the observer located far away from the source. We study the Penrose process, which occurs within the ergosphere, and examines the particle’s motion with its implications. Here, we analyze the PD black hole shadow’s apparent shape, which forms far away from the black hole. We study the possible visibility of the PD black hole through photon’s shadow and energy emission rate. We also investigate the effect on the shadow of the PD black hole in plasma for a distant observer. We study the strong gravitational lensing by PD black hole. Finally, we analyze the deflection angle, lens equation, position, magnification, Einstein ring and observables by taking the supermassive PD black hole in the Galaxy’s center.     

Interface dynamics under nonequilibrium conditions: From a self-propelled droplet to dynamic pattern evolution

In this article, we describe the instability of a contact line under nonequilibrium conditions mainly based on the results of our recent studies. Two experimental examples are presented: the self-propelled motion of a liquid droplet and spontaneous dynamic pattern formation. For the self-propelled motion of a droplet, we introduce an experiment in which a droplet of aniline sitting on an aqueous layer moves spontaneously at an air-water interface. The spontaneous symmetry breaking of Marangoni-driven spreading causes regular motion. In a circular Petri dish, the droplet exhibits either beeline motion or circular motion. On the other hand, we show the emergence of a dynamic labyrinthine pattern caused by dewetting of a metastable thin film from the air-water interface. The contact line between the organic phase and the aqueous phase forms a unique spatio-temporal pattern characterized as a dynamic labyrinth. Motion of the contact line is controlled by diffusion processes. We propose a theoretical model to interpret essential aspects of the observed dynamic behavior.