Two-dimensional motion of Brownian swimmers in linear flows

The motion of viruses and bacteria and even synthetic microswimmers can be affected by thermal fluctuations and by external flows. In this work, we study the effect of linear external flows and thermal fluctuations on the diffusion of those swimmers modeled as spherical active (self-propelled) particles moving in two dimensions. General formulae for their mean-square displacement under a general linear flow are presented. We also provide, at short and long times, explicit expressions for the mean-square displacement of a swimmer immersed in three canonical flows, namely, solid-body rotation, shear and extensional flows. These expressions can now be used to estimate the effect of external flows on the displacement of Brownian microswimmers. Finally, our theoretical results are validated by using Brownian dynamics simulations.     

Self-propelled micro-swimmers in a Brinkman fluid

We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system.     

Effects of Geometric Parameters on Swimming of Micro Organisms with Single Helical Flagellum in Circular Channels

We present a computational fluid dynamics (CFD) model for the swimming of micro organisms with a single helical flagellum in circular channels. The CFD model is developed to obtain numerical solutions of Stokes equations in three dimensions, validated with experiments reported in literature, and used to analyze the effects of geometric parameters, such as the helical radius, wavelength, radii of the channel and the tail and the tail length on forward and lateral swimming velocities, rotation rates, and the efficiency of the swimmer. Optimal shapes for the speed and the power efficiency are reported. Effects of Brownian motion and electrostatic interactions are excluded to emphasize the role of hydrodynamic forces on lateral velocities and rotations on the trajectory of swimmers. For thin flagella, as the channel radius decreases, forward velocity and the power efficiency of the swimmer decreases as well; however, for thick flagella, there is an optimal radius of the channel that maximizes the velocity and the efficiency depending on other geometric parameters. Lateral motion of the swimmer is suppressed as the channel is constricted below a critical radius, for which the magnitude of the lateral velocity reaches a maximum. Results contribute significantly to the understanding of the swimming of bacteria in micro channels and capillary tubes.     

Effects of Geometric Parameters on Swimming of Micro Organisms with Single Helical Flagellum in Circular Channels

We present a computational fluid dynamics (CFD) model for the swimming of micro organisms with a single helical flagellum in circular channels. The CFD model is developed to obtain numerical solutions of Stokes equations in three dimensions, validated with experiments reported in literature, and used to analyze the effects of geometric parameters, such as the helical radius, wavelength, radii of the channel and the tail and the tail length on forward and lateral swimming velocities, rotation rates, and the efficiency of the swimmer. Optimal shapes for the speed and the power efficiency are reported. Effects of Brownian motion and electrostatic interactions are excluded to emphasize the role of hydrodynamic forces on lateral velocities and rotations on the trajectory of swimmers. For thin flagella, as the channel radius decreases, forward velocity and the power efficiency of the swimmer decreases as well; however, for thick flagella, there is an optimal radius of the channel that maximizes the velocity and the efficiency depending on other geometric parameters. Lateral motion of the swimmer is suppressed as the channel is constricted below a critical radius, for which the magnitude of the lateral velocity reaches a maximum. Results contribute significantly to the understanding of the swimming of bacteria in micro channels and capillary tubes.     

Effect of body roll amplitude and arm rotation speed on propulsion of arm amputee swimmers

Only a limited amount of research has gone into evaluating the contribution made by the upper arm to the propulsion of elite swimmers with an amputation at elbow level. With assistance of computational fluid dynamics (CFD) modelling, the swimming technique of competitive arm amputee swimmers can be assessed through numerical simulations which test the effect of various parameters on the effectiveness of the swimming propulsion.This numerical study investigates the effect of body roll amplitude and of upper arm rotation speed on the propulsion of an arm amputee swimmer, at different mean swimming speeds. Various test cases are simulated resulting in a thorough analysis of the complex body/fluid interaction with a detailed quantitative assessment of the effect of the variation of each parameter on the arm propulsion. It is found that a body roll movement with an amplitude of 45° enhances greatly the propulsive contribution from the upper arm with an increase of about 70% in the propulsive force compared to the no roll condition. An increase in the angular velocity of the upper arm also leads to a concomitant increase in the propulsive forces produced by the arm.Such results have direct implications for competitive arm amputee front crawl swimmers and for those who coach them. One important message that emerges in this present work is that there exists, for any given swimming speed, a minimum angular velocity at which the upper arm must be rotated to generate effective propulsion. Below this velocity, the upper arm will experience a net resistive drag force which adversely affects swimming performance.     

Shear and extensional rheology of cellulose/ionic liquid solutions

In this study, we characterize the shear and extensional rheology of dilute to semidilute solutions of cellulose in the ionic liquid 1-ethyl-3-methylimidazolium acetate (EMIAc). In steady shear flow, the semidilute solutions exhibit shear thinning, and the high-frequency complex modulus measured in small amplitude oscillatory shear flow exhibits the characteristic scaling expected for solutions of semiflexible chains. Flow curves of the steady shear viscosity plotted against shear rate closely follow the frequency dependence of the complex viscosity acquired using oscillatory shear, thus satisfying the empirical Cox-Merz rule. We use capillary thinning rheometry (CaBER) to characterize the relaxation times and apparent extensional viscosities of the semidilute cellulose solutions in a uniaxial extensional flow that mimics the dynamics encountered in the spin-line during fiber spinning processes. The apparent extensional viscosity and characteristic relaxation times of the semidilute cellulose/EMIAc solutions increase dramatically as the solutions enter the entangled concentration regime at which fiber spinning becomes viable.     

QUANTITATIVE BEHAVIORAL STUDY OF BOTTLENOSE DOLPHINS IN SWIM-WITH-DOLPHIN PROGRAMS IN THE UNITED STATES

The behavior of dolphins in four Swim-With-Dolphin programs was compared by type of Swim encounter, defined by the presence ("Controlled") or absence ("Not-Controlled") of explicit trainer regulation of interactions between dolphins and human swimmers. Dolphin-swimmer interactions involving aggressive, submissive, or sexual behavior were designated as "high-risk" in the Swim context; sexual behavior was included as high-risk based on analyses that demonstrated co-occurrence of sexual and agonistic behaviors. High-risk activity comprised a substantial proportion of dolphin-swimmer social activity during Not-Controlled Swims. In contrast, high-risk activity rarely occurred during Controlled Swims, even though agonistic and sexual behaviors were normal components of the same dolphins' free-time social repertoire. These results indicated that direct trainer control of dolphin-swimmer interactions virtually eliminated high-risk activity from the Swim context, and thereby diminished the potential for dolphin distress, swimmer injury, and rejection of dolphins from Swim programs due to swimmer injury. This study illustrates effective use of quantitative behavioral sampling techniques for evaluation of captive management concerns and promotes broader use of these techniques for a better understanding of cetacean behavior.     

Helical swimming can provide robust upwards transport for gravitactic single-cell algae; a mechanistic model

In still fluid, many phytoplankton swim in helical paths with an average upwards motion. A new mechanistic model for gravitactic algae subject to an intrinsic torque is developed here, based on Heterosigma akashiwo, which results in upwards helical trajectories in still fluid. The resultant upwards swimming speed is calculated as a function of the gravitactic and intrinsic torques. Helical swimmers have a reduced upwards speed in still fluid compared to cells which swim straight upwards. However a novel result is obtained when the effect of fluid shear is considered. For intermediate values of shear and intrinsic torque, a new stable equilibrium solution for swimming direction is obtained for helical swimmers. This results in positive upwards transport in vertical shear flow, in contrast to the stable equilibrium solution for straight swimmers which results in downwards transport in vertical shear flow. Furthermore, for strong intrinsic torque, when there is no longer a stable orientation equilibrium, we show that the average downwards transport of helical swimmers in vertical shear flow is greatly suppressed compared to straight swimmers. We hypothesise that helical swimming provides robustness for upwards transport in the presence of fluid shearing motions.     

Non-Newtonian rheology of leukemic blood and plasma: are n and k parameters of power law model diagnostic?

When discussing the rheological properties of normal and leukemic blood it must be considered that blood is a suspension of cells in aqueous solution which is also known as plasma. Whole blood viscosity and plasma viscosity were determined by Rheometer LS30 which allows measuring whole blood and plasma viscosity in the middle and low shear rate ranges. The measurements of the viscosity showed that whole blood and plasma behave as non-Newtonian power law fluid. The values of n (non-Newtonian index) and k (consistency index) of power law fluid were calculated for both leukemic blood and plasma samples. The importance of this phenomenon for the micro-circulation is discussed.     

Flow Loading Induces Oscillatory Trajectories in a Bloodstream Parasite

The dynamics of isolated microswimmers are studied in bounded flow using the African trypanosome, a unicellular parasite, as the model organism. With the help of a microfluidics platform, cells are subjected to flow and found to follow an oscillatory path that is well fit by a sine wave. The frequency and amplitudes of the oscillatory trajectories are dependent on the flow velocity and cell orientation. When traveling in such a manner, trypanosomes orient upstream while downstream-facing cells tumble within the same streamline. A comparison with immotile trypanosomes demonstrates that self-propulsion is essential to the trajectories of trypanosomes even at flow velocities up to ～40 times higher than their own swimming speed. These studies reveal important swimming dynamics that may be generally pertinent to the transport of microswimmers in flow and may be relevant to microbial pathogenesis.     

Viscosity study of interactions between sodium alginate and CTAB in dilute solutions at different pH values

Interactions between anionic polyelectrolyte sodium alginate and the cationic surfactant cetytrimethylammonium bromide (CTAB) have been investigated by viscosity measurement techniques. The polymer–surfactant interactions are observed between alginate and CTAB at different pH in dilute solution. The results show that the rheological response of alginate dilute solutions is sensitive to a change of pH in the low pH range. The steady shear and intrinsic viscosity measurements reveal that the strong association between alginate and CTAB by electrostatic attraction above pH 5.0. However, as the pH value of solution decrease from 5.0 to 3.0, the strong association between alginate and CTAB is affected by not only electrostatic attraction but also hydrophobic interaction.     

An estimation of drag in front crawl swimming

Propulsive arm forces of twelve elite male swimmers during a front crawl swimming-like activity were measured. The swimmers pushed off against grips which are attached to a 23 m tube at 0.8 m under the water surface. The tube was fixed to a force transducer. Since at constant speed, mean propulsive force equals mean drag force this method also provides the mean active drag on a moving swimmer. The mean propulsive force at a speed of v = 1.48 m s-1 appeared to be 53.2 +/- 5.8 N which is two to three times smaller than what is reported by other authors for active drag but which is in agreement with values reported for passive drag on a (towed) swimmer who is not moving. Discrepancies with indirect active drag measurements are discussed.     

Theoretical and experimental analysis of the sedimentation kinetics of concentrated red cell suspensions in a centrifugal field: determination of the aggregation and deformation of RBC by flux density and viscosity functions

The flow properties of blood are mostly determined using various viscometric approaches, and described in terms of a shear rate or shear stress dependent apparent viscosity. The interpretation of results are rather difficult, especially at low shear rates when particle sedimentation and migration within the viscometer gap are significant. By contrast, analysing the separation process in concentrated RBC suspensions in a centrifugal field also yields information about the viscosity function, including particle-particle interaction and deformation parameters. In this paper, the sedimentation process is approached by means of the theory of kinematic waves and theoretically described by solving the corresponding one-dimensional quasi-linear partial differential equation based on viscosity/flow function as a function of volume concentration. The sedimentation kinetics of rigid spherical RBC suspended in saline and normal RBC suspended in Dx-saline solutions were investigated by means of a separation analyser (LUMiFuge 114). The instrument detects the light transmission over the total length of the cell containing the suspension. During centrifugation the analyser automatically determines the position of the particle free fluid/suspension interface or the sediment by means of a special algorithm. The data obtained with sedimentation of rigid spherical RBC at different volume concentrations demonstrate that, in the case of suspensions rotated in containers of constant cross section, there is good agreement between the theory of kinematic waves developed by Anestis and Schneider (1983) and the results of the experiments. Such good agreement was obtained even though a restrictive one-dimensional model was used to obtain the theoretically derived sedimentation time course. In addition, we describe an algorithm enabling the experimental determination of the viscosity and related flux density function to be made for any suspension. Through this approach, we investigated in detail the rheological behavior of suspended rigid spheres at low Reynolds numbers ranging from 10(-6) to 10(-3). The method here introduced also enabled us to investigate RBC suspensions with respect to the deformability and interactions of the cells by means of the separation analysis. Normal, rigid as well as aggregating RBC exhibited marked differences in the sedimentation kinetics, which were quantified by means of the flux and viscosity functions based on the theory of kinematic waves.     

Reduction of erythrocyte deformability in Krushinskiĭ-Molodkina rats in acute hemorrhagic cerebrovascular circulation disorders

The ability of erythrocytes to change their shapes in the shear flow under acute strokes of hemorrhagic type in rats of the Krushinsky-Molodkina line was studied. The rigidity of membranes and the internal viscosity of erythrocytes were investigated by the laser diffraction method. The method consists in obtaining diffraction images from a thin layer of a dilute suspension of erythrocytes moving in the shear flow and subsequent computer processing of these images. It was shown that strokes of hemorrhagical type in rats of the Krushinsky-Molodkina line cause a reduction in the ability of erythrocytes to change theirs shapes.     

Viscometric flow for a many-body dissipative particle dynamics (MDPD) fluid with Lees–Edwards boundary condition

Viscometric properties of polymer are explored by the many-body dissipative particle dynamics (MDPD) using Lees–Edwards boundary conditions. The equation of state for the MDPD system is modified by fitting the density correction to different values of the cut-off radius. Due to the many-body interactions in MDPD, the viscosity contributed from the conservative force increases considerably with increasing repulsive coefficient, density and cut-off radius, and cannot be ignored compared to the ‘standard’ DPD case. The influence of these parameters on the MDPD viscosity is investigated, and we propose an equation to predict the viscosity in MDPD model. Additionally, the dumbbell polymer suspension model is investigated in the MDPD fluid, and the relations concerning first normal stress difference and shear rate, the relaxation time and spring constant, are consistent to theoretical works. We conclude that the MDPD model can be used to investigate the dynamics of non-Newtonian droplets.     

The Long-Time Dynamics of Two Hydrodynamically-Coupled Swimming Cells

Swimming microorganisms such as bacteria or spermatozoa are typically found in dense suspensions, and exhibit collective modes of locomotion qualitatively different from that displayed by isolated cells. In the dilute limit where fluid-mediated interactions can be treated rigorously, the long-time hydrodynamics of a collection of cells result from interactions with many other cells, and as such typically eludes an analytical approach. Here, we consider the only case where such problem can be treated rigorously analytically, namely when the cells have spatially confined trajectories, such as the spermatozoa of some marine invertebrates. We consider two spherical cells swimming, when isolated, with arbitrary circular trajectories, and derive the long-time kinematics of their relative locomotion. We show that in the dilute limit where the cells are much further away than their size, and the size of their circular motion, a separation of time scale occurs between a fast (intrinsic) swimming time, and a slow time where hydrodynamic interactions lead to change in the relative position and orientation of the swimmers. We perform a multiple-scale analysis and derive the effective dynamical system—of dimension two—describing the long-time behavior of the pair of cells. We show that the system displays one type of equilibrium, and two types of rotational equilibrium, all of which are found to be unstable. A detailed mathematical analysis of the dynamical systems further allows us to show that only two cell-cell behaviors are possible in the limit of t→∞, either the cells are attracted to each other (possibly monotonically), or they are repelled (possibly monotonically as well), which we confirm with numerical computations. Our analysis shows therefore that, even in the dilute limit, hydrodynamic interactions lead to new modes of cell-cell locomotion.     

Feedback Control of an Achiral Robotic Microswimmer

Magnetic microswimmers are useful for navigating and performing tasks at small scales.To demonstrate effective control over such microswimmers,we implemented feedback control of the three-bead achiral microswimmers in both simulation and experiment.The achiral microswimmers with the ability to swim in bulk fluid are controlled wirelessly using magnetic fields generated from electromagnetic coils.The achirality of the microswimmers introduces unknown handedness resulting in uncertainty in swimming direction.We use a combination of rotating and static magnetic fields generated from an approximate Helmholtz coil system to overcome such uncertainty.There are also movement uncertainties due to environmental factors such as unsteady flow conditions.A kinematic model based feedback controller was created based on data fitting of experimental data.However,the controller was unable to yield satisfactory performance due to uncertainties from environmental factors;i.e.,the time to reach target pose under adverse flow condition is too long.Following the implementation of an integral controller to control the microswimmers' swimming velocity,the microswimmers were able to reach the target in roughly half the time.Through simulation and experiments,we show that the feedback control law can move an achiral microswimmer from any initial conditions to a target pose.     

Radial and spiral stream formation in Proteus mirabilis colonies

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form a variety of spatial patterns in colonies. Colony formation involves two distinct cell types: swarmer cells that dominate near the surface and the leading edge, and swimmer cells that prefer a less viscous medium, but the mechanisms underlying pattern formation are not understood. New experimental investigations reported here show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to previously-reported concentric rings. These new observations suggest that swimmers are motile and that indirect interactions between them are essential in the pattern formation. To explain these observations we develop a hybrid model comprising cell-based and continuum components that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams results from a swimming bias of the cells near the surface of the substrate. The spatial patterns generated from the model are in qualitative agreement with the experimental observations.     

Viscoelasticity of rigid macromolecules with irregular shapes in the limit of overwhelming Brownian motion

We consider viscoelastic properties of complex rigid macromolecules in fluids undergoing steady or sinusoidal linear shearing. An arbitrary body is hydrodynamically described by six tensors to allow for irregular shapes that couple rotational and translational motions to each other and to the shear field. The viscosity increment ν is obtained for infinitely dilute suspensions in the limit of overwhelming Brownian motion by balancing drag forces and torques with entropic forces and torques. For sinusoidal shearing, ν is a complex number that exhibits five resonances at frequencies matching the j = 2 eigenvalues of the rotational diffusion equation for a stationary fluid. The resonance amplitudes are conveniently expressed in terms of a second-rank body-fixed tensor χ that characterizes alignment by the shear field, and special cases of symmetry are considered which reduce the number of contributing terms. Steadyshear methods, which assume a body matches the translational and rotational motions of the fluid element it replaces, are shown to slightly overestimate ν when complex shapes are involved. An algebraic criterion is found to locate the center of viscosity needed in these other methods although our treatment is independent of the choice of position in the body used for calculations. Bead-model expressions are derived in order to provide numerical treatments of complicated structures. As an example, we examine a long bent rod.     

A simulation study of the dynamics of a driven filament in an Aristotelian fluid

We describe a method, based on techniques used in molecular dynamics, for simulating the inertialess dynamics of an elastic filament immersed in a fluid. The model is used to study the "one-armed swimmer". That is, a flexible appendage externally perturbed at one extremity. For small-amplitude motion our simulations confirm theoretical predictions that, for a filament of given length and stiffness, there is a driving frequency that is optimal for both speed and efficiency. However, we find that to calculate absolute values of the swimming speed we need to slightly modify existing theoretical approaches. For the more relevant case of large-amplitude motion we find that while the basic picture remains the same, the dependence of the swimming speed on both frequency and amplitude is substantially modified. For large-amplitudes we show that the one-armed swimmer is comparatively neither inefficient nor slow. This begs the question, why are there little or no one-armed swimmers in nature?