How body torque and Strouhal number change with swimming speed and developmental stage in larval zebrafish

Small undulatory swimmers such as larval zebrafish experience both inertial and viscous forces, the relative importance of which is indicated by the Reynolds number (Re). Re is proportional to swimming speed (v_swim) and body length; faster swimming reduces the relative effect of viscous forces. Compared with adults, larval fish experience relatively high (mainly viscous) drag during cyclic swimming. To enhance thrust to an equally high level, they must employ a high product of tail-beat frequency and (peak-to-peak) amplitude fA_tail, resulting in a relatively high fA_tail/v_swim ratio (Strouhal number, St), and implying relatively high lateral momentum shedding and low propulsive efficiency. Using kinematic and inverse-dynamics analyses, we studied cyclic swimming of larval zebrafish aged 2–5 days post-fertilization (dpf). Larvae at 4–5 dpf reach higher f (95 Hz) and A_tail (2.4 mm) than at 2 dpf (80 Hz, 1.8 mm), increasing swimming speed and Re, indicating increasing muscle powers. As Re increases (60 → 1400), St (2.5 → 0.72) decreases nonlinearly towards values of large swimmers (0.2–0.6), indicating increased propulsive efficiency with v_swim and age. Swimming at high St is associated with high-amplitude body torques and rotations. Low propulsive efficiencies and large yawing amplitudes are unavoidable physical constraints for small undulatory swimmers.     

Magnetically Propelled Fish‐Like Nanoswimmers

The swimming locomotion of fish involves a complex interplay between a deformable body and induced flow in the surrounding fluid. While innovative robotic devices, inspired by physicomechanical designs evolved in fish, have been created for underwater propulsion of large swimmers, scaling such powerful locomotion into micro‐/nanoscale propulsion remains challenging. Here, a magnetically propelled fish‐like artificial nanoswimmer is demonstrated that emulates the body and caudal fin propulsion swimming mechanism displayed by fish. To mimic the deformable fish body for periodic shape changes, template‐electrosynthesized multisegment nanowire swimmers are used to construct the artificial nanofishes (diameter 200 nm; length 4.8 μm). The resulting nanofish consists a gold segment as the head, two nickel segments as the body, and one gold segment as the caudal fin, with three flexible porous silver hinges linking each segment. Under an oscillating magnetic field, the propulsive nickel elements bend the body and caudal fin periodically to generate travelling‐wave motions with speeds exceeding 30 μm s^?1. The propulsion dynamics is studied theoretically using the immersed boundary method. Such body‐deformable nanofishes exhibit a high swimming efficiency and can serve as promising biomimetic nanorobotic devices for nanoscale biomedical applications.     

The fluid dynamics of swimming by jumping in copepods

Copepods swim either continuously by vibrating their feeding appendages or erratically by repeatedly beating their swimming legs, resulting in a series of small jumps. The two swimming modes generate different hydrodynamic disturbances and therefore expose the swimmers differently to rheotactic predators. We developed an impulsive stresslet model to quantify the jump-imposed flow disturbance. The predicted flow consists of two counter-rotating viscous vortex rings of similar intensity, one in the wake and one around the body of the copepod. We showed that the entire jumping flow is spatially limited and temporally ephemeral owing to jump-impulsiveness and viscous decay. In contrast, continuous steady swimming generates two well-extended long-lasting momentum jets both in front of and behind the swimmer, as suggested by the well-known steady stresslet model. Based on the observed jump-swimming kinematics of a small copepod Oithona davisae, we further showed that jump-swimming produces a hydrodynamic disturbance with much smaller spatial extension and shorter temporal duration than that produced by a same-size copepod cruising steadily at the same average translating velocity. Hence, small copepods in jump-swimming are in general much less detectable by rheotactic predators. The present impulsive stresslet model improves a previously published impulsive Stokeslet model that applies only to the wake vortex.     

Fin sweep angle does not determine flapping propulsive performance

The importance of the leading-edge sweep angle of propulsive surfaces used by unsteady swimming and flying animals has been an issue of debate for many years, spurring studies in biology, engineering, and robotics with mixed conclusions. In this work, we provide results from three-dimensional simulations on single-planform finite foils undergoing tail-like (pitch-heave) and flipper-like (twist-roll) kinematics for a range of sweep angles covering a substantial portion of animals while carefully controlling all other parameters. Our primary finding is the negligible 0.043 maximum correlation between the sweep angle and the propulsive force and power for both tail-like and flipper-like motions. This indicates that fish tails and mammal flukes with similar range and size can have a large range of potential sweep angles without significant negative propulsive impact. Although there is a slight benefit to avoiding large sweep angles, this is easily compensated by adjusting the fin’s motion parameters such as flapping frequency, amplitude and maximum angle of attack to gain higher thrust and efficiency.     

Hydrodynamics of the double-wave structure of insect spermatozoa flagella

In addition to conventional planar and helical flagellar waves, insect sperm flagella have also been observed to display a double-wave structure characterized by the presence of two superimposed helical waves. In this paper, we present a hydrodynamic investigation of the locomotion of insect spermatozoa exhibiting the double-wave structure, idealized here as superhelical waves. Resolving the hydrodynamic interactions with a non-local slender body theory, we predict the swimming kinematics of these superhelical swimmers based on experimentally collected geometric and kinematic data. Our consideration provides insight into the relative contributions of the major and minor helical waves to swimming; namely, propulsion is owing primarily to the minor wave, with negligible contribution from the major wave. We also explore the dependence of the propulsion speed on geometric and kinematic parameters, revealing counterintuitive results, particularly for the case when the minor and major helical structures are of opposite chirality.     

行波推进仿生机器鱼

介绍5种常见的鱼类游动模式,分析讨论仿生机器鱼的生物学原型。基于对裸臀鱼属（gymnarchus niloticus）与裸背鳗属（gymnotus carapo）活体鱼的实验研究,建立了长背鳍扭转行波动态曲面的数学模型。依靠长鳍行波实现推进与控制的新型仿生机器鱼,其基本结构是“刚性身体”与“柔性背鳍”的组合,这有利于改善现有摆动式机器鱼的技术性能。裸臀鱼属与裸背鳗属鱼类可作为发展特种机器鱼潜艇的生物学原型,以提高现行水下航行器的效率和机动性能。     

Speed limits on swimming of fishes and cetaceans.

Physical limits on swimming speed of lunate tail propelled aquatic animals are proposed. A hydrodynamic analysis, applying experimental data wherever possible, is used to show that small swimmers (roughly less than a metre long) are limited by the available power, while larger swimmers at a few metres below the water surface are limited by cavitation. Depending on the caudal fin cross-section, 10-15 m s(-1) is shown to be the maximum cavitation-free velocity for all swimmers at a shallow depth.     

Coordination of multiple appendages in drag-based swimming

Krill are aquatic crustaceans that engage in long distance migrations, either vertically in the water column or horizontally for 10 km (over 200 000 body lengths) per day. Hence efficient locomotory performance is crucial for their survival. We study the swimming kinematics of krill using a combination of experiment and analysis. We quantify the propulsor kinematics for tethered and freely swimming krill in experiments, and find kinematics that are very nearly metachronal. We then formulate a drag coefficient model which compares metachronal, synchronous and intermediate motions for a freely swimming body with two legs. With fixed leg velocity amplitude, metachronal kinematics give the highest average body speed for both linear and quadratic drag laws. The same result holds for five legs with the quadratic drag law. When metachronal kinematics is perturbed towards synchronous kinematics, an analysis shows that the velocity increase on the power stroke is outweighed by the velocity decrease on the recovery stroke. With fixed time-averaged work done by the legs, metachronal kinematics again gives the highest average body speed, although the advantage over synchronous kinematics is reduced.     

The management of fluid and wave resistances by whirligig beetles

Whirligig beetles (Coleoptera: Gyrinidae) are semi-aquatic insects with a morphology and propulsion system highly adapted to their life at the air–water interface. When swimming on the water surface, beetles are subject to both fluid resistance and wave resistance.The purpose of this study was to analyse swimming speed, leg kinematics and the capillarity waves produced by whirligig beetles on the water surface in a simple environment. Whirligig beetles of the species Gyrinus substriatus were filmed in a large container, with a high-speed camera. Resistance forces were also estimated.These beetles used three types of leg kinematics, differing in the sequence of leg strokes: two for swimming at low speed and one for swimming at high speed. Four main speed patterns were produced by different combinations of these types of leg kinematics, and the minimum speed for the production of surface waves (23 cm s⁻¹) corresponded to an upper limit when beetles used low-speed leg kinematics. Each type of leg kinematics produced characteristic capillarity waves, even if the beetles moved at a speed below 23 cm s⁻¹. Our results indicate that whirligig beetles use low- and high-speed leg kinematics to avoid maximum drag and swim at speed corresponding to low resistances.     

Propulsive performance of a fish-like travelling wavy wall

Summary. A numerical simulation is performed to investigate the viscous flow over a smooth wavy wall undergoing transverse motion in the form of a streamwise travelling wave, which is similar to the backbone undulation of swimming fish. The objective of this study is to elucidate hydrodynamic features of the flow structure over the travelling wavy wall and to get physical insights to the understanding of fish-like swimming mechanisms in terms of drag reduction and optimal propulsive efficiency. The effect of phase speed, amplitude and Reynolds number on the flow structure over the wavy wall, the drag force acting on the wall, and the power consumption required for the propulsive motion of the wall is investigated. The phase speed and the amplitude, which are two important parameters in this problem, predicted based on the optimal propulsive efficiency agree well with the available data obtained for the wave-like swimming motion of live fish in nature.     

Hydrodynamic drag constrains head enlargement for mouthbrooding in cichlids

Presumably as an adaptation for mouthbrooding, many cichlid fish species have evolved a prominent sexual dimorphism in the adult head. Since the head of fishes serves as a bow during locomotion, an evolutionary increase in head volume to brood more eggs can trade-off with the hydrodynamic efficiency of swimming. Here, the differences between males and females in three-dimensional shape and size of the external head surfaces and the effect thereof on drag force during locomotion was analysed for the Nile tilapia (Oreochromis niloticus), a maternal mouthbrooder. To do so, three-dimensional body surface reconstructions from laser scans and computational fluid dynamics simulations were performed. After scaling the scanned specimens to post-cranial body volume, in order to theoretically equalize propulsive power, the external volume of the head of females was 27% larger than that of males (head length + 14%; head width + 9%). These differences resulted in an approximate 15% increase in drag force. Yet, hydrodynamics imposed important constraints on the adaptation for mouthbrooding as a much more drastic drop in swimming efficiency seems avoided by mainly enlarging the head along the swimming direction.     

A microcontinuum analysis of the self propulsion of the spermatozoa in the cervical canal

The hydrodynamics of the self propulsion of a spermatozoa, swimming through the mucus filling the cervical channel, is investigated. The mucus is modeled as a micropolar fluid and the spermatozoa as a 2-dimensional sheet swimming at low Reynolds number between two rigid walls. The wavelengths of the propulsive waves passing down the sheet are assumed to be very large compared to the channel spacing, but the amplitude of the propulsive waves is arbitrary. Expressions for the propulsive velocity and the energy expended by the swimming sheet are obtained in terms of various parameters involved. The results are elaborated through graphs. It is found that both the propulsive velocity and the rate of working by the sheet increase as the value of the micropolar parameters $\text{N}$ increases and that of $\text{L}$ decreases.     

Generalized squirming motion of a sphere

A number of swimming microorganisms, such as ciliates (Opalina) and multicellular colonies of flagellates (Volvox), are approximately spherical in shape and swim using beating arrays of cilia or short flagella covering their surfaces. Their physical actuation on the fluid may be mathematically modeled as the generation of surface velocities on a continuous spherical surface—a model known in the literature as squirming, which has been used to address various aspects of the biological physics of locomotion. Previous analyses of squirming assumed axisymmetric fluid motion and hence required all swimming kinematics to take place along a line. In this paper we generalize squirming to three spatial dimensions. We derive analytically the flow field surrounding a spherical squirmer with arbitrary surface motion and use it to derive its three-dimensional translational and rotational swimming kinematics. We then use our results to physically interpret the flow field induced by the swimmer in terms of fundamental flow singularities up to terms decaying spatially as \({\sim } 1/r^3\) . Our results will make it possible to develop new models in biological physics, in particular in the area of hydrodynamic interactions and collective locomotion.     

Development of a multi-pathway probabilistic health risk assessment model for swimmers exposed to chloroform in indoor swimming pools

For swimmers, exposure to chloroform, a probable carcinogen, in indoor swimming pools can be through different pathways such as ingestion, dermal absorption, inhalation during swimming, and inhalation during resting. In order to evaluate health risk results from excessive exposure to chloroform, concentrations of chloroform in pool water were first collected and analyzed. Then, a two-layer model is used, which is capable of estimating the concentrations of chloroform in the boundary layer adjacent to the water surface and the concentrations of chloroform in indoor swimming pool air. The use of stratification model is important for estimating the risks for swimmers since they are exposed to these kinds of situations while performing swimming and resting in indoor swimming pools environment. The incremental lifetime cancer risk (ILCR) was then estimated using the multi-pathway exposure model. The results showed that the 95th percentile of ILCRs calculated for male and female swimmers were 2.80 × 10(-4) and 2.47 × 10(-4), respectively. The major exposure routes were found to be inhalation during swimming which contributes to more than 99% of the total health risk. Our study suggested that to protect swimmers from excessive exposure to chloroform, alternative methods or processes of disinfection should be considered for swimming pool managers.     

Snakes mimic earthworms: propulsion using rectilinear travelling waves

In rectilinear locomotion, snakes propel themselves using unidirectional travelling waves of muscular contraction, in a style similar to earthworms. In this combined experimental and theoretical study, we film rectilinear locomotion of three species of snakes, including red-tailed boa constrictors, Dumeril''s boas and Gaboon vipers. The kinematics of a snake''s extension–contraction travelling wave are characterized by wave frequency, amplitude and speed. We find wave frequency increases with increasing body size, an opposite trend than that for legged animals. We predict body speed with 73–97% accuracy using a mathematical model of a one-dimensional n-linked crawler that uses friction as the dominant propulsive force. We apply our model to show snakes have optimal wave frequencies: higher values increase Froude number causing the snake to slip; smaller values decrease thrust and so body speed. Other choices of kinematic variables, such as wave amplitude, are suboptimal and appear to be limited by anatomical constraints. Our model also shows that local body lifting increases a snake''s speed by 31 per cent, demonstrating that rectilinear locomotion benefits from vertical motion similar to walking.     

The hydrodynamic analysis of fish propulsion performance and its morphological adaptation

The hydrodynamic performance of fish undulatory swimming has been studied by means of the linear three-dimensional and the non-linear two-dimensional waving-plate theories developed recently by the present authors. The morphological adaptation of the anguilliform mode, carangiform mode, and lunate-tail swimming propulsion in the biological evolution process of fishes has been understood tentatively from the point of view of hydrodynamic analysis. This research work was supported by the National Natural Science Foundation of China.     

Controlled Propulsion of Two‐Dimensional Microswimmers in a Precessing Magnetic Field

Magnetically actuated micro‐/nanoswimmers can potentially be used in noninvasive biomedical applications, such as targeted drug delivery and micromanipulation. Herein, two‐dimensional (2D) rigid ferromagnetic microstructures are shown to be capable of propelling themselves in three dimensions at low Reynolds numbers in a precessing field. Importantly, the above propulsion relies neither on soft structure deformation nor on the geometrical chirality of swimmers, but is rather driven by the dynamic chirality generated by field precession, which allows an almost unconstrained choice of materials and fabrication methods. Therefore, the swimming performance is systematically investigated as a function of precession angle and geometric design. One disadvantage of the described propulsion method is that the fabricated 2D swimmers are achiral, which means that the forward/backward swimming direction cannot be controlled. However, it has been found that asymmetric 2D swimmers always propel themselves toward their longer arm, which implies that dynamic chirality can be constrained to be either right‐handed or left‐handed by permanent magnetization. Thus, the simplicity of fabrication and possibility of dynamic chirality control make the developed method ideal for applications and fundamental studies that require a large number of swimmers.     

Embodied linearity of speed control in Drosophila melanogaster

Fruitflies regulate flight speed by adjusting their body angle. To understand how low-level posture control serves an overall linear visual speed control strategy, we visually induced free-flight acceleration responses in a wind tunnel and measured the body kinematics using high-speed videography. Subsequently, we reverse engineered the transfer function mapping body pitch angle onto flight speed. A linear model is able to reproduce the behavioural data with good accuracy. Our results show that linearity in speed control is realized already at the level of body posture-mediated speed control and is therefore embodied at the level of the complex aerodynamic mechanisms of body and wings. Together with previous results, this study reveals the existence of a linear hierarchical control strategy, which can provide relevant control principles for biomimetic implementations, such as autonomous flying micro air vehicles.     

Swimming of the semi-infinite strip revisited

Idealized mathematical models have been devised over the years for study of the fundamentals of the swimming of fishes. The two-dimensional flexible strip propelled by execution of transverse traveling-wave undulation is one of the most well-studied of the simple models. This model is redeveloped here, with the finding that higher propulsive efficiencies are theoretically available within the undulatory swimming mode than have been previously exposed. This is by configuring the displacement wave-form for continuously zero circulation over the body length with time, and thereby avoiding the shedding of a vortex wake and its attendant induced drag. The thrust is reactive, via acceleration processes, rather than inductive via relative velocity and lift. As in most of the classical work on fish propulsion, the analysis assumes high Reynolds number and a thin boundary layer, which provides the use of ideal-flow theory. The advance speed is assumed constant and the analysis is initially linearized, but both nonlinear and linear transient analysis are provided in supporting the basic “wakeless swimming” possibility.     

Unsteady motion: escape jumps in planktonic copepods,their kinematics and energetics

We describe the kinematics of escape jumps in three species of 0.3–3.0 mm-sized planktonic copepods. We find similar kinematics between species with periodically alternating power strokes and passive coasting and a resulting highly fluctuating escape velocity. By direct numerical simulations, we estimate the force and power output needed to accelerate and overcome drag. Both are very high compared with those of other organisms, as are the escape velocities in comparison to startle velocities of other aquatic animals. Thus, the maximum weight-specific force, which for muscle motors of other animals has been found to be near constant at 57 N (kg muscle)⁻¹, is more than an order of magnitude higher for the escaping copepods. We argue that this is feasible because most copepods have different systems for steady propulsion (feeding appendages) and intensive escapes (swimming legs), with the muscular arrangement of the latter probably adapted for high force production during short-lasting bursts. The resulting escape velocities scale with body length to power 0.65, different from the size-scaling of both similar sized and larger animals moving at constant velocity, but similar to that found for startle velocities in other aquatic organisms. The relative duration of the pauses between power strokes was observed to increase with organism size. We demonstrate that this is an inherent property of swimming by alternating power strokes and pauses. We finally show that the Strouhal number is in the range of peak propulsion efficiency, again suggesting that copepods are optimally designed for rapid escape jumps.