Hydromechanics of Swimming

Metachronal swimming of mantis shrimp: kinematics and interpleopod vortex interactions

Mantis shrimp swim via metachronal rowing, a pattern in which the pleopods (swimming limbs) stroke sequentially, starting with the last pair and followed by anterior neighbors. A similar swimming pattern is used at various sizes, Reynolds numbers, and advance ratios by diverse organisms including ciliates, ctenophores, copepods, krill, and lobsters. Understanding this type of locomotion is important because it is widespread and may inspire the design of underwater vehicles where efficiency, robustness, and maneuverability are desired. However, detailed measurements of the flow around free-swimming, metachronally rowing organisms are scarce, especially for organisms swimming in a high Reynolds number regime (Re ≥ 104). In this study, we present time-resolved, planar PIV measurements of a swimming peacock mantis shrimp (Odontodactylus scyllarus). Simultaneous kinematics measurements of the animal, which had body and pleopod lengths of 114 mm and 20 mm, respectively, reveal mean swimming speeds of 0.2-1.9 m s - 1 and pleopod beat frequencies of 3.6-13 Hz, corresponding to advance ratios of 0.75-1.84 and body-based Reynolds numbers of 23,000-217,000. Further, the animal's stroke is not purely metachronal, with a long phase lag between initiation of the first and fifth pleopod power strokes. Flow measurements in the sagittal plane show that each stroking pleopod pair creates a posteriorly moving tip vortex which evades destruction by the recovery strokes of other pleopod pairs. The vortex created by the anteriormost pleopod pair is the strongest and, owing to the animal's high advance ratio, is intercepted by the power stroke of the posteriormost pleopod pair. The vortex strength increases as a result of this interaction, which may increase swimming speed or efficiency. A relationship for vortex interception by the posterior pleopod is proposed that relates the phase lag between the interacting pleopods to the beat frequency, distance between those pleopods, and speed of the vortex relative to the animal. We describe this interaction with a novel parameter called the interpleopod vortex phase matching Strouhal number StIVPM which is equal to the phase lag between interacting pleopods. This new nondimensional parameter may be useful in predicting the conditions where a constructive interaction may occur in other species or in physical models. Finally, we relate the advance ratio to the Reynolds number ratio, the ratio between the body-based Reynolds number and the pleopod-based Reynolds number. The importance of these parameters in promoting the interpleopod vortex interactions identified here, in dynamically scaled experiments, and in wake signatures behind schooling metachronal swimmers is discussed.

Development of the Lunate-Shaped Caudal Fin in White Shark Embryos

The lunate-shaped caudal fin in lamnid sharks is a morphological specialization for their thunniform mode of locomotion, but its developmental process during gestation has been poorly investigated. Observations of 21 embryonic specimens of the white shark (Carcharodon carcharias) revealed that their caudal fin morphology drastically changes from strongly heterocercal to lunate-shaped through ontogeny. This morphological change involves (1) rapid elongation of the ventral lobe, (2) increased upward curvature of the vertebra within the caudal fin, and (3) formation of keels at both lateral sides of the caudal fin base. These morphological changes are probably shared among the members of the family Lamnidae and are in contrast with the developmental process of the heterocercal tail in the lamniform Carcharias taurus, in which the caudal fin morphology is almost unchanged through the late gestation period. Anat Rec, 301:1068-1073, 2018. © 2018 Wiley Periodicals, Inc.

Experimental investigation and propulsion control for a bio-inspired robotic undulatory fin

Undulatory fin propulsion, inspired by the locomotion of aquatic species such as electric eels and cuttlefish, holds considerable potential for endowing underwater vehicles with enhanced propulsion and maneuvering abilities, to address the needs of a growing number of applications. However, there are still gaps in our understanding of the effect of the fin undulations' characteristics on the generated thrust, particularly within the context of developing propulsion control strategies for such robotic systems. Towards this end, we present the design and experimental evaluation of a robotic fin prototype, comprised of eight individually-actuated fin rays. An artificial central pattern generator (CPG) is used to produce the rays' undulatory motion pattern. Experiments are performed inside a water tank, with the robotic fin suspended from a carriage, whose motion is constrained via a linear guide. The results from a series of detailed parametric investigations reveal several important findings regarding the effect of the undulatory wave kinematics on the propulsion speed and efficiency. Based on these findings, two alternative strategies for propulsion control of the robotic fin are proposed. In the first one, the speed is varied through changes in the undulation amplitude, while the second one involves simultaneous adjustment of the undulation frequency and number of waves. These two strategies are evaluated via experiments demonstrating open-loop velocity control, as well as closed-loop position control of the prototype.

Measurement of hydrodynamic force generation by swimming dolphins using bubble DPIV

Attempts to measure the propulsive forces produced by swimming dolphins have been limited. Previous uses of computational hydrodynamic models and gliding experiments have provided estimates of thrust production by dolphins, but these were indirect tests that relied on various assumptions. The thrust produced by two actively swimming bottlenose dolphins (Tursiops truncatus) was directly measured using digital particle image velocimetry (DPIV). For dolphins swimming in a large outdoor pool, the DPIV method used illuminated microbubbles that were generated in a narrow sheet from a finely porous hose and a compressed air source. The movement of the bubbles was tracked with a high-speed video camera. Dolphins swam at speeds of 0.7 to 3.4 m s−1 within the bubble sheet oriented along the midsagittal plane of the animal. The wake of the dolphin was visualized as the microbubbles were displaced because of the action of the propulsive flukes and jet flow. The oscillations of the dolphin flukes were shown to generate strong vortices in the wake. Thrust production was measured from the vortex strength through the Kutta–Joukowski theorem of aerodynamics. The dolphins generated up to 700 N during small amplitude swimming and up to 1468 N during large amplitude starts. The results of this study demonstrated that bubble DPIV can be used effectively to measure the thrust produced by large-bodied dolphins.

Built for rowing: frog muscle is tuned to limb morphology to power swimming

Rowing is demanding, in part, because drag on the oars increases as the square of their speed. Hence, as muscles shorten faster, their force capacity falls, whereas drag rises. How do frogs resolve this dilemma to swim rapidly? We predicted that shortening velocity cannot exceed a terminal velocity where muscle and fluid torques balance. This terminal velocity, which is below Vmax, depends on gear ratio (GR = outlever/inlever) and webbed foot area. Perhaps such properties of swimmers are 'tuned', enabling shortening speeds of approximately 0.3Vmax for maximal power. Predictions were tested using a 'musculo-robotic' Xenopus laevis foot driven either by a living in vitro or computational in silico plantaris longus muscle. Experiments verified predictions. Our principle finding is that GR ranges from 11.5 to 20 near the predicted optimum for rowing (GR ≈ 11). However, gearing influences muscle power more strongly than foot area. No single morphology is optimal for producing muscle power. Rather, the 'optimal' GR decreases with foot size, implying that rowing ability need not compromise jumping (and vice versa). Thus, despite our neglect of additional forces (e.g. added mass), our model predicts pairings of physiological and morphological properties to confer effective rowing. Beyond frogs, the model may apply across a range of size and complexity from aquatic insects to human-powered rowing.

Changes in wingstroke kinematics associated with a change in swimming speed in a pteropod mollusk, Clione limacina

In pteropod mollusks, the gastropod foot has evolved into two broad, wing-like structures that are rhythmically waved through the water for propulsion. The flexibility of the wings lends a tremendous range of motion, an advantage that could be exploited when changing locomotory speed. Here, we investigated the kinematic changes that take place during an increase in swimming speed in the pteropod mollusk Clione limacina. Clione demonstrates two distinct swim speeds: a nearly constant slow swimming behavior and a fast swimming behavior used for escape and hunting. The neural control of Clione's swimming is well documented, as are the neuromuscular changes that bring about Clione's fast swimming. This study examined the kinematics of this swimming behavior at the two speeds. High speed filming was used to obtain 3D data from individuals during both slow and fast swimming. Clione's swimming operates at a low Reynolds number, typically under 200. Within a given swimming speed, we found that wing kinematics are highly consistent from wingbeat to wingbeat, but differ between speeds. The transition to fast swimming sees a significant increase in wing velocity and angle of attack, and range of motion increases as the wings bend more during fast swimming. Clione likely uses a combination of drag-based and unsteady mechanisms for force production at both speeds. The neuromuscular control of Clione's speed change points to a two-gaited swimming behavior, and we consider the kinematic evidence for Clione's swim speeds being discrete gaits.

Hydrodynamics of pulsed jetting in juvenile and adult brief squid Lolliguncula brevis: evidence of multiple jet `modes' and their implications for propulsive efficiency

The dynamics of pulsed jetting in squids throughout ontogeny is not well understood, especially with regard to the development of vortex rings, which are common features of mechanically generated jet pulses (also known as starting jets). Studies of mechanically generated starting jets have revealed a limiting principle for vortex ring formation characterized in terms of a`formation number' (F), which delineates the transition between the formation of isolated vortex rings and vortex rings that have `pinched off'from the generating jet. Near F, there exists an optimum in pulse-averaged thrust with (potentially) low energetic cost, raising the question: do squids produce vortex rings and if so, do they fall near F, where propulsive benefits presumably occur? To better understand vortex ring dynamics and propulsive jet efficiency throughout ontogeny, brief squid Lolliguncula brevis ranging from 3.3 to 9.1 cm dorsal mantle length (DML) and swimming at speeds of 2.43–22.2 cms–1 (0.54–3.50 DMLs–1) were studied using digital particle image velocimetry (DPIV). A range of jet structures were observed but most structures could be classified as variations of two principal jet modes: (1) jet mode I, where the ejected fluid rolled up into an isolated vortex ring; and (2) jet mode II, where the ejected fluid developed into a leading vortex ring that separated or `pinched off' from a long trailing jet. The ratio of jet length [based on the vorticity extent(Lω)] to jet diameter [based on peak vorticity locations (Dω)] was <3.0 for jet mode I and>3.0 for jet mode II, placing the transition between modes in rough agreement with F determined in mechanical jet studies. Jet mode II produced greater time-averaged thrust and lift forces and was the jet mode most heavily used whereas jet mode I had higher propulsive efficiency, lower slip, shorter jet periods and a higher frequency of fin activity associated with it. No relationship between Lω/Dω and speed was detected and there was no apparent speed preference for the jet modes within the speed range considered in this study; however, propulsive efficiency did increase with speed partly because of a reduction in slip and jet angle with speed. Trends in higher slip, lower propulsive efficiency and higher relative lift production were observed for squid <5.0 cm DML compared with squid ≥5.0 cm DML. While these trends were observed when jet mode I and II were equally represented among the size classes, there was also greater relative dependence on jet mode I than jet mode II for squid <5.0 cm DML when all of the available jet sequences were examined. Collectively, these results indicate that ∼5.0 cm DML is an important ontogenetic transition for the hydrodynamics of pulsed jetting in squids. The significance of our findings is that from early juvenile through to adult life stages, L. brevis is capable of producing a diversity of vortex ring-based jet structures, ranging from efficient short pulses to high-force longer duration pulses. Given that some of these structures had Lω/Dωs near F,and F represented the delineation between the two primary jet modes observed, fluid dynamics probably played an integral role in the evolution of squid locomotive systems. When this flexibility in jet dynamics is coupled with the highly versatile fins, which are capable of producing multiple hydrodynamic modes as well, it is clear that squid have a locomotive repertoire far more complex than originally thought.

Aquatic wing flapping at low Reynolds numbers: swimming kinematics of the Antarctic pteropod, Clione antarctica

We studied swimming kinematics of the Antarctic pteropod, Clione antarctica, to investigate how propulsive forces are generated by flexible oscillating appendages operating at low Reynolds numbers (10<Re<100). We filmed ten ascending individuals at 125 frames s(-1) from two orthogonal views, and reconstructed three-dimensional coordinates of the wing tip and body. Each half-stroke of flapping consisted of distinct power and recovery phases, which were of approximately equal duration in both the upstroke and the downstroke. As pteropods ascended, the body traced a sawtooth path when viewed laterally. The magnitude of these oscillations decreased with body mass, and larger animals (operating at Re>25) exhibited gliding during the recovery phase of each half-stroke. Maximum translational and rotational accelerations of the body occurred at the initiation of each power phase, suggesting that rotational circulation, the acceleration reaction, and wake recapture may all potentially contribute to vertical force production. Individual contributions of these mechanisms cannot, however, be assessed from these kinematic data alone. During recovery phases of each half-stroke, C. antarctica minimized adverse drag forces by orienting the wings parallel to flow and by moving them along the body surface, possibly taking advantage of boundary layer effects. Vertical force production was altered through changes in the hydrodynamic angle of attack of the wing that augmented drag during the power phase of each half-stroke. At higher translational velocities of the body, the inclination of the power phase also became more nearly vertical. These results indicate that, in addition to serotonin-mediated modulation of wingbeat frequency reported previously in Clione, geometric alteration of wingbeat kinematics offers a precise means of controlling swimming forces.

Economy and efficiency of swimming at the surface with fins of different size and stiffness

The aim of this study was to investigate how fins with varying physical characteristics affect the energy cost and the efficiency of aquatic locomotion. Experiments were performed on ten college swimmers who were asked to swim the dolphin kick while using a monofin (MF) and to swim the front crawl kick with a small-flexible fin (SF), a large-stiff fin (LS) and without fins (BF, barefoot). The energy expended to cover one unit distance (C) was highest for BF (C=10.6+/-1.8 kJ m(-1) kg(-1) at 0.8 m s(-1)) and decreased by about 50% with LS, 55% with SF and 60% with MF, allowing for an increase in speed (for a given metabolic power) of about 0.4 m s(-1) for MF and of about 0.2 m s(-1) for SF and LS (compared with BF). At any given speed, the fins for which C was lower were those with the lowest kick frequency (KF): KF=1.6+/-0.22 Hz at 0.8 m s(-1) (for BF) and decreased by about 40% for SF, 50% for LS and 60% for MF. The decrease in KF from BF to SF-LS and MF was essentially due to the increasing surface area of the fin which, in turn, was associated with a higher Froude efficiency (eta(F)). eta(F) was calculated by computing the speed of the bending waves moving along the body in a caudal direction (as proposed for the undulating movements of slender fish): it increased from 0.62+/-0.01 in BF to 0.66+/-0.03 in SF and 0.67+/-0.04 in LS reaching the highest values (0.76+/-0.05) with MF. No single fin characteristic can predict a swimmer's performance, rather the better fin (i.e. MF) is the one that is able to reduce most KF at any given speed and hence to produce the greatest distance per kick (d=v/KF). The latter indeed increased from 0.50+/-0.01 m in BF to about 0.90+/-0.05 m in SF and LS and reached values of 1.22+/-0.01 m in MF.

The mechanical scaling of coasting in zebrafish (Danio rerio)

Many fish species span two or three orders of magnitude in length during the growth from larvae to adults, and this change may have dramatic consequences for locomotor performance. We measured how the performance of coasting changes over the life history of zebrafish (Danio rerio) and examined the scaling of mechanics underlying this change. Adult zebrafish coast disproportionately further and faster and maintain their speed for a longer duration than do larvae and juveniles. Measurements of drag on tethered dead fish suggest that adult fish operate in an inertial regime by coasting at relatively high Reynolds numbers (Re > 1000), and in vivo drag measurements showed adults to operate with a drag coefficient (Cinert approximately 0.024) that was consistent with previously published estimates. However, drag scaled differently at lower Re values than those assumed in previous studies. We found a viscous regime at Re < 300, which corresponds to the routine coasting of larvae and juveniles. Despite these changes in hydrodynamics over growth, a mathematical model of coasting mechanics suggests that the disproportionately longer coasting of adults is caused primarily by their large body mass and high speed at the beginning of coasting. We therefore propose that changes in coasting performance with growth are dictated primarily by the scaling of momentum rather than resulting from hydrodynamic changes. These results provide an opportunity for new interpretations of function in the growth and evolution of fish.

Caudal fin allometry in the white shark Carcharodon carcharias: implications for locomotory performance and ecology

Allometric scaling analysis was employed to investigate the consequences of size evolution on hydrodynamic performance and ecology in the white shark Carcharodon carcharias. Discriminant analysis using the power equation y=ax(b) was negative for caudal fin span (S) versus fork length (FL) in C. carcharias. In contrast in two delphinid species, Delphinus capensis and Tursiops aduncus, the span of the flukes versus fork length rises in positive allometric fashion, and strong positive allometry of S versus [corrected] radicalA (area) was also recorded. The latter reflects a high lift/drag ratio. S versus radicalA in C. carcharias displays negative allometry and consequently a lower lift/drag ratio. A lower aspect ratio (AR) caudal fin in C. carcharias compared to that of the delphinids (mean 3.33 and 4.1, respectively) and other thunniform swimmers provides the potential for better maneuverability and acceleration. The liver in sharks is frequently associated with a buoyancy function and was found to be positively allometric in C. carcharias. The overall findings suggest that the negatively allometric caudal fin morphometrics in C. carcharias are unlikely to have deleterious evolutionary fitness consequences for predation. On the contrary, when considered in the context of positive liver allometry in C. carcharias it is hereby suggested that buoyancy may play a dominant role in larger white sharks in permitting slow swimming while minimizing energy demands needed to prevent sinking. In contrast hydrodynamic lift is considered more important in smaller white sharks. Larger caudal fin spans and higher lift/drag ratio in smaller C. carcharias indicate greater potential for prolonged, intermediate swimming speeds and for feeding predominantly on fast-moving fish, in contrast to slow-swimming search patterns of larger individuals for predominantly large mammalian prey. Such data may provide some answers to the lifestyle and widespread habitat capabilities of this still largely mysterious animal.

An energy balance of front crawl

With the aim of computing a complete energy balance of front crawl, the energy cost per unit distance (C = Ev(-1), where E is the metabolic power and v is the speed) and the overall efficiency (eta(o) = W(tot)/C, where W(tot) is the mechanical work per unit distance) were calculated for subjects swimming with and without fins. In aquatic locomotion W(tot) is given by the sum of: (1) W(int), the internal work, which was calculated from video analysis, (2) W(d), the work to overcome hydrodynamic resistance, which was calculated from measures of active drag, and (3) W(k), calculated from measures of Froude efficiency (eta(F)). In turn, eta(F) = W(d)/(W(d) + W(k)) and was calculated by modelling the arm movement as that of a paddle wheel. When swimming at speeds from 1.0 to 1.4 m s(-1), eta(F) is about 0.5, power to overcome water resistance (active body drag x v) and power to give water kinetic energy increase from 50 to 100 W, and internal mechanical power from 10 to 30 W. In the same range of speeds E increases from 600 to 1,200 W and C from 600 to 800 J m(-1). The use of fins decreases total mechanical power and C by the same amount (10-15%) so that eta(o) (overall efficiency) is the same when swimming with or without fins [0.20 (0.03)]. The values of eta(o) are higher than previously reported for the front crawl, essentially because of the larger values of W(tot) calculated in this study. This is so because the contribution of W(int) to W(tot )was taken into account, and because eta(F) was computed by also taking into account the contribution of the legs to forward propulsion.

The ontogenetic scaling of hydrodynamics and swimming performance in jellyfish (Aurelia aurita)

It is not well understood how ontogenetic changes in the motion and morphology of aquatic animals influence the performance of swimming. The goals of the present study were to understand how changes in size, shape and behavior affect the hydrodynamics of jet propulsion in the jellyfish Aurelia aurita and to explore how such changes affect the ontogenetic scaling of swimming speed and cost of transport. We measured the kinematics of jellyfish swimming from video recordings and simulated the hydrodynamics of swimming with two computational models that calculated thrust generation by paddle and jet mechanisms. Our results suggest that thrust is generated primarily by jetting and that there is negligible thrust generation by paddling. We examined how fluid forces scaled with body mass using the jet model. Despite an ontogenetic increase in the range of motion by the bell diameter and a decrease in the height-to-diameter ratio, we found that thrust and acceleration reaction scaled with body mass as predicted by kinematic similarity. However, jellyfish decreased their pulse frequency with growth,and speed consequently scaled at a lower exponential rate than predicted by kinematic similarity. Model simulations suggest that the allometric growth in Aurelia results in swimming that is slower, but more energetically economical, than isometric growth with a prolate bell shape. The decrease in pulse frequency over ontogeny allows large Aurelia medusae to avoid a high cost of transport but generates slower swimming than if they maintained a high pulse frequency. Our findings suggest that ontogenetic change in the height-to-diameter ratio and pulse frequency of Aurelia results in swimming that is relatively moderate in speed but is energetically economical.

The hydrodynamics of locomotion at intermediate Reynolds numbers: undulatory swimming in ascidian larvae (Botrylloides sp.)

Understanding how the shape and motion of an aquatic animal affects the performance of swimming requires knowledge of the fluid forces that generate thrust and drag. These forces are poorly understood for the large diversity of animals that swim at Reynolds numbers (Re) between 10(0) and 10(2). We experimentally tested quasi-steady and unsteady blade-element models of the hydrodynamics of undulatory swimming in the larvae of the ascidian Botrylloides sp. by comparing the forces predicted by these models with measured forces generated by tethered larvae and by comparing the swimming speeds predicted with measurements of the speed of freely swimming larvae. Although both models predicted mean forces that were statistically indistinguishable from measurements, the quasi-steady model predicted the timing of force production and mean swimming speed more accurately than the unsteady model. This suggests that unsteady force (i.e. the acceleration reaction) does not play a role in the dynamics of steady undulatory swimming at Re approximately 10(2). We explored the relative contribution of viscous and inertial force to the generation of thrust and drag at 10(0)<Re<10(2) by running a series of mathematical simulations with the quasi-steady model. These simulations predicted that thrust and drag are dominated by viscous force (i.e. skin friction) at Re approximately 10(0) and that inertial force (i.e. form force) generates a greater proportion of thrust and drag at higher Re than at lower Re. However, thrust was predicted to be generated primarily by inertial force, while drag was predicted to be generated more by viscous than inertial force at Re approximately 10(2). Unlike swimming at high (>10(2)) and low (<10(0)) Re, the fluid forces that generate thrust cannot be assumed to be the same as those that generate drag at intermediate Re.

How fins affect the economy and efficiency of human swimming

The aim of the present study was to quantify the improvements in the economy and efficiency of surface swimming brought about by the use of fins over a range of speeds (v) that could be sustained aerobically. At comparable speeds, the energy cost (C) when swimming with fins was about 40 %lower than when swimming without them; when compared at the same metabolic power, the decrease in C allowed an increase in v of about 0.2 ms-1. Fins only slightly decrease the amplitude of the kick (by about 10 %) but cause a large reduction (about 40 %) in the kick frequency. The decrease in kick frequency leads to a parallel decrease of the internal work rate (Ẇint, about 75 %at comparable speeds) and of the power wasted to impart kinetic energy to the water (Ẇk, about 40 %). These two components of total power expenditure were calculated from video analysis (Ẇint) and from measurements of Froude efficiency(Ẇk). Froude efficiency(ηF) was calculated by computing the speed of the bending waves moving along the body in a caudal direction (as proposed for the undulating movements of slender fish); ηF was found to be 0.70 when swimming with fins and 0.61 when swimming without them. No difference in the power to overcome frictional forces(Ẇd) was observed between the two conditions at comparable speeds. Mechanical efficiency[Ẇtot/(Cv), where Ẇtot=Ẇk+Ẇint+Ẇd]was found to be about 10 % larger when swimming with fins, i.e. 0.13±0.02 with and 0.11±0.02 without fins (average for all subjects at comparable speeds).

Swimming mechanics and behavior of the shallow-water brief squidLolliguncula brevis

Although squid are among the most versatile swimmers and rely on a unique locomotor system, little is known about the swimming mechanics and behavior of most squid, especially those that swim at low speeds in inshore waters. Shallow-water brief squid Lolliguncula brevis, ranging in size from 1.8 to 8.9 cm in dorsal mantle length (DML), were placed in flumes and videotaped, and the data were analyzed using motion-analysis equipment. Flow visualization and force measurement experiments were also performed in water tunnels. Mean critical swimming speeds (Ucrit) ranged from 15.3 to 22.8 cm s–1, and mean transition speeds (Ut; the speed above which squid swim exclusively in a tail-first orientation) varied from 9.0 to 15.3 cm s–1. At low speeds, negatively buoyant brief squid generated lift and/or improved stability by positioning the mantle and arms at high angles of attack, directing high-speed jets downwards (angles &gt;50°) and using fin activity. To reduce drag at high speeds, the squid decreased angles of attack and swam tail-first. Fin motion, which could not be characterized exclusively as drag- or lift-based propulsion, was used over 50–95 % of the sustained speed range and provided as much as 83.8 % of the vertical and 55.1 % of the horizontal thrust. Small squid (&lt;3.0 cm DML) used different swimming strategies from those of larger squid, possibly to maximize thrust benefits from vortex ring formation. Furthermore, brief squid employed various unsteady behaviors, such as manipulating funnel diameter during jetting, altering arm position and swimming in different orientations, to boost swimming performance. These results demonstrate that locomotion in slow-swimming squid is complex, involving intricate spatial and temporal interactions between the mantle, fins, arms and funnel.

A continuous dynamic beam model for swimming fish

When a fish swims in water, muscle contraction, controlled by the nervous system, interacts with the body tissues and the surrounding fluid to yield the observed movement pattern of the body. A continuous dynamic beam model describing the bending moment balance on the body for such an interaction during swimming has been established. In the model a linear visco–elastic assumption is made for the passive behaviour of internal tissues, skin and backbone, and the unsteady fluid force acting on the swimming body is calculated by the 3D waving plate theory. The body bending moment distribution due to the various components, in isolation and acting together, is analysed. The analysis is based on the saithe (Pollachius virens), a carangiform swimmer. The fluid reaction needs a bending moment of increasing amplitude towards the tail and near–standing wave behaviour on the rear–half of the body. The inertial movement of the fish results from a wave of bending moment with increasing amplitude along the body and a higher propagation speed than that of body bending. In particular, the fluid reaction, mainly designed for propulsion, can provide a considerable force to balance the local momentum change of the body and thereby reduce the power required from the muscle. The wave of passive visco–elastic bending moment, with an amplitude distribution peaking a little before the mid–point of the fish, travels with a speed close to that of body bending. The calculated muscle bending moment from the whole dynamic system has a wave speed almost the same as that observed for EMG–onset and a starting instant close to that of muscle activation, suggesting a consistent matching between the muscle activation pattern and the dynamic response of the system in steady swimming. A faster wave of muscle activation, with a variable phase relation between the strain and activation cycle, appears to be designed to fit the fluid reaction and, to a lesser extent, the body inertia, and is limited by the passive internal tissues. Higher active stress is required from caudal muscle, as predicted from experimental studies on fish muscle. In general, the active force development by muscle does not coincide with the propulsive force generation on the tail. The stiffer backbone may play a role in transmitting force and deformation to maintain and adjust the movement of the body and tail in water.