Swimming with stiff legs at low Reynolds number

Omnidirectional propulsion in a metachronal swimmer

Aquatic organisms often employ maneuverable and agile swimming behavior to escape from predators, find prey, or navigate through complex environments. Many of these organisms use metachronally coordinated appendages to execute complex maneuvers. However, though metachrony is used across body sizes ranging from microns to tens of centimeters, it is understudied compared to the swimming of fish, cetaceans, and other groups. In particular, metachronal coordination and control of multiple appendages for three-dimensional maneuvering is not fully understood. To explore the maneuvering capabilities of metachronal swimming, we combine 3D high-speed videography of freely swimming ctenophores (Bolinopsis vitrea) with reduced-order mathematical modeling. Experimental results show that ctenophores can quickly reorient, and perform tight turns while maintaining forward swimming speeds close to 70% of their observed maximum-performance comparable to or exceeding that of many vertebrates with more complex locomotor systems. We use a reduced-order model to investigate turning performance across a range of beat frequencies and appendage control strategies, and reveal that ctenophores are capable of near-omnidirectional turning. Based on both recorded and modeled swimming trajectories, we conclude that the ctenophore body plan enables a high degree of maneuverability and agility, and may be a useful starting point for future bioinspired aquatic vehicles.

Mantis Shrimp Locomotion: Coordination and Variation of Hybrid Metachronal Swimming

Across countless marine invertebrates, coordination of closely spaced swimming appendages is key to producing diverse locomotory behaviors. Using a widespread mechanism termed hybrid metachronal propulsion, mantis shrimp swim by moving five paddle-like pleopods along their abdomen in a posterior to anterior sequence during the power stroke and a near-synchronous motion during the recovery stroke. Despite the ubiquity of this mechanism, it is not clear how hybrid metachronal swimmers coordinate and modify individual appendage movements to achieve a range of swimming capabilities. Using high-speed imaging, we measured pleopod kinematics of mantis shrimp (Neogonodactylus bredini), while they performed two swimming behaviors: burst swimming and taking off from the substrate. By tracking each of the five pleopods, we tested how stroke kinematics vary across swimming speeds and the two swimming behaviors. We found that mantis shrimp achieve faster swimming speeds through a combination of higher beat frequencies, smaller stroke durations, and partially via larger stroke angles. The five pleopods exhibit non-uniform kinematics that contribute to the coordination and forward propulsion of the whole system. Micro-hook structures (retinacula) connect each of the five pleopod pairs and differ in their attachment across pleopods-possibly contributing to passive kinematic control. We compare our findings in N. bredini to previous studies to identify commonalities across hybrid metachronal swimmers at high Reynolds numbers and centimeter scales. Through our large experimental dataset and by tracking each pleopod's movements, our study reveals key parameters by which mantis shrimp adjust and control their swimming, yielding diverse locomotor abilities.

Muscular adaptations in swimming scale worms (Polynoidae, Annelida)

Annelids are predominantly found along with the seafloor, but over time have colonized a vast diversity of habitats, such as the water column, where different modes of locomotion are necessary. Yet, little is known about their potential muscular adaptation to the continuous swimming behaviour required in the water column. The musculature and motility were examined for five scale worm species of Polynoidae (Aphroditiformia, Annelida) found in shallow waters, deep sea or caves and which exhibit crawling, occasional swimming or continuous swimming, respectively. Their parapodial musculature was reconstructed using microCT and computational three-dimensional analyses, and the muscular functions were interpreted from video recordings of their locomotion. Since most benthic scale worms are able to swim for short distances using body and parapodial muscle movements, suitable musculature for swimming is already present. Our results indicate that rather than rearrangements or addition of muscles, a shift to a pelagic lifestyle is mainly accompanied by structural loss of muscle bundles and density, as well as elongation of extrinsic dorsal and ventral parapodial muscles. Our study documents clear differences in locomotion and musculature among closely related annelids with different lifestyles as well as points to myoanatomical adaptations for accessing the water column.

On the role of phase lag in multi-appendage metachronal swimming of euphausiids

Metachronal paddling is a common method of drag-based aquatic propulsion, in which a series of swimming appendages are oscillated, with the motion of each appendage phase-shifted relative to the neighboring appendages. Ecologically and economically important euphausiid species such as Antarctic krill (Euphausia superba) swim constantly by stroking their paddling appendages (pleopods), with locomotion accounting for the bulk of their metabolic expenditure. They tailor their swimming gaits for behavioral and energetic needs by changing pleopod kinematics. The functional importance of inter-pleopod phase lag (ϕ) to metachronal swimming performance and wake structure is unknown. To examine this relation, we developed a geometrically and dynamically scaled robot ('krillbot') capable of self-propulsion. Krillbot pleopods were prescribed to mimic published kinematics of fast-forward swimming (FFW) and hovering (HOV) gaits ofE. superba, and the Reynolds number and Strouhal number of the krillbot matched well with those calculated for freely-swimmingE. superba. In addition to examining published kinematics with unevenϕbetween pleopod pairs, we modifiedE. superbakinematics to uniformly varyϕfrom 0% to 50% of the cycle. Swimming speed and thrust were largest for FFW withϕbetween 15%-25%, coincident withϕrange observed in FFW gait ofE. superba. In contrast to synchronous rowing (ϕ= 0%) where distances between hinged joints of adjacent pleopods were nearly constant throughout the cycle, metachronal rowing (ϕ> 0%) brought adjacent pleopods closer together and moved them farther apart. This factor minimized body position fluctuation and augmented metachronal swimming speed. Though swimming speed was lowest for HOV, a ventrally angled downward jet was generated that can assist with weight support during feeding. In summary, our findings show that inter-appendage phase lag can drastically alter both metachronal swimming speed and the large-scale wake structure.

Metachronal Coordination of Multiple Appendages for Swimming and Pumping

As a strategy for creating fluid flow, metachronal motion is widespread across sizes and species, including a broad array of morphologies, length scales, and coordination patterns. Because of this great diversity, it has not generally been viewed holistically: the study of metachrony for swimming and pumping has historically been taxonomically siloed, in spite of many commonalities between seemingly disparate organisms. The goal of the present symposium was to bring together individuals from different backgrounds, all of whom have made substantial individual contributions to our understanding of the fluid dynamics of metachronal motion. Because these problems share a common physical-mathematical basis, intentionally connecting this community is likely to yield future collaborations and significant scientific discovery. Here, we briefly introduce the concept of metachronal motion, present the benefits of creating a research network based on the common aspects of metachrony across biological systems, and outline the contributions to the symposium.

Spatiotemporal Asymmetry in Metachronal Rowing at Intermediate Reynolds Numbers

In drag-based swimming, individual propulsors operating at low Reynolds numbers (where viscous forces dominate over inertial forces) must execute a spatially asymmetric stroke to produce net fluid displacement. Temporal asymmetry (that is, differing duration between the power vs. recovery stroke) does not affect the overall generated thrust in this time-reversible regime. Metachronal rowing, in which multiple appendages beat sequentially, is used by a wide variety of organisms from low to intermediate Reynolds numbers. At the upper end of this range, inertia becomes important, and increasing temporal asymmetry can be an effective way to increase thrust. However, the combined effects of spatial and temporal asymmetry are not fully understood in the context of metachronal rowing. To explore the role of spatiotemporal asymmetry in metachronal rowing, we combine laboratory experiments and reduced-order analytical modeling. We measure beat kinematics and generated flows in two species of lobate ctenophores across a range of body sizes, from 7 mm to 40 mm in length. We observe characteristically different flows in ctenophores of differing body size and Reynolds number, and a general decrease in spatial asymmetry and increase in temporal asymmetry with increasing Reynolds number. We also construct a one-dimensional mathematical model consisting of a row of oscillating flat plates whose flow-normal areas change with time, and use it to explore the propulsive forces generated across a range of Reynolds numbers and kinematic parameters. The model results show that while both types of asymmetry increase force production, they have different effects in different regions of the parameter space. These results may have strong biological implications, as temporal asymmetry can be actively controlled while spatial asymmetry is likely to be partially or entirely driven by passive fluid-structure interaction.

Universal Features in Panarthropod Inter-Limb Coordination during Forward Walking

Terrestrial animals must often negotiate heterogeneous, varying environments. Accordingly, their locomotive strategies must adapt to a wide range of terrain, as well as to a range of speeds to accomplish different behavioral goals. Studies in Drosophila have found that inter-leg coordination patterns (ICPs) vary smoothly with walking speed, rather than switching between distinct gaits as in vertebrates (e.g., horses transitioning between trotting and galloping). Such a continuum of stepping patterns implies that separate neural controllers are not necessary for each observed ICP. Furthermore, the spectrum of Drosophila stepping patterns includes all canonical coordination patterns observed during forward walking in insects. This raises the exciting possibility that the controller in Drosophila is common to all insects, and perhaps more generally to panarthropod walkers. Here, we survey and collate data on leg kinematics and inter-leg coordination relationships during forward walking in a range of arthropod species, as well as include data from a recent behavioral investigation into the tardigrade Hypsibius exemplaris. Using this comparative dataset, we point to several functional and morphological features that are shared among panarthropods. The goal of the framework presented in this review is to emphasize the importance of comparative functional and morphological analyses in understanding the origins and diversification of walking in Panarthropoda. Introduction.

Hybrid metachronal rowing augments swimming speed and acceleration via increased stroke amplitude

Numerous aquatic invertebrates use drag-based metachronal rowing for swimming, in which closely spaced appendages are oscillated starting from the posterior, with each appendage phase-shifted in time relative to its neighbor. Continuously swimming species such as Antarctic krill generally use "pure metachronal rowing" consisting of a metachronal power stroke and a metachronal recovery stroke, while burst swimming species such as many copepods and mantis shrimp typically use "hybrid metachronal rowing" consisting of a metachronal power stroke followed by a synchronous or nearly synchronous recovery stroke. Burst swimming organisms need to rapidly accelerate in order to capture prey and/or escape predation, and it is unknown whether hybrid metachronal rowing can augment acceleration and swimming speed compared to pure metachronal rowing. Simulations of rigid paddles undergoing simple harmonic motion showed that collisions between adjacent paddles restrict the maximum stroke amplitude for pure metachronal rowing. Hybrid metachronal rowing similar to that observed in mantis shrimp (Neogonodactylus bredini) permits oscillation at larger stroke amplitude while avoiding these collisions. We comparatively examined swimming speed, acceleration, and wake structure of pure and hybrid metachronal rowing strategies by using a self-propelling robot. Both swimming speed and peak acceleration of the robot increased with increasing stroke amplitude. Hybrid metachronal rowing permitted operation at larger stroke amplitude without collision of adjacent paddles on the robot, augmenting swimming speed and peak acceleration. Hybrid metachronal rowing generated a dispersed wake unlike narrower, downward-angled jets generated by pure metachronal rowing. Our findings suggest that burst swimming animals with small appendage spacing, such as copepods and mantis shrimp, can use hybrid metachronal rowing to generate large accelerations via increasing stroke amplitude without concern of appendage collision.

Geometric Methods for Efficient Planar Swimming of Copepod Nauplii

Copepod nauplii are larval crustaceans with important ecological functions. Due to their small size, they experience an environment of low Reynolds number within their aquatic habitat. Here we provide a mathematical model of a swimming copepod nauplius with two legs moving in a plane. This model allows for both rotation and two-dimensional displacement by the periodic deformation of the swimmer’s body. The system is studied from the framework of optimal control theory, with a simple cost function designed to approximate the mechanical energy expended by the copepod. We find that this model is sufficiently realistic to recreate behavior similar to those of observed copepod nauplii, yet much of the mathematical analysis is tractable. In particular, we show that the system is controllable, but there exist singular configurations where the degree of non-holonomy is non-generic. We also partially characterize the abnormal extremals and provide explicit examples of families of abnormal curves. Finally, we numerically simulate normal extremals and observe some interesting and surprising phenomena.

Closer appendage spacing augments metachronal swimming speed by promoting tip vortex interactions

Numerous species of aquatic invertebrates, including crustaceans, swim by oscillating multiple closely spaced appendages. The coordinated, out-of-phase motion of these appendages, known as "metachronal paddling", has been well-established to improve swimming performance relative to synchronous paddling. Invertebrates employing this propulsion strategy cover a wide range of body sizes and shapes, but the ratio of appendage spacing (G) to the appendage length (L) has been reported to lie in a comparatively narrow range of 0.2 < G/L ≤ 0.65. The functional role of G/L on metachronal swimming performance is unknown. We hypothesized that for a given Reynolds number and stroke amplitude, hydrodynamic interactions promoted by metachronal stroke kinematics with small G/L can increase forward swimming speed. We used a dynamically scaled self-propelling robot to comparatively examine swimming performance and wake development of metachronal and synchronous paddling under varying G/L, phase lag, and stroke amplitude. G/L was varied from 0.4 to 1.5, with the expectation that when G/L is large, there should be no performance difference between metachronal and synchronous paddling due to a lack of interaction between vortices that form on the appendages. Metachronal stroking at non-zero phase lag with G/L in the biological range produced faster swimming speeds than synchronous stroking. As G/L increased and as stroke amplitude decreased, the influence of phase lag on the swimming speed of the robot was reduced. For smaller G/L, vortex interactions between adjacent appendages generated a horizontally-oriented wake and increased momentum fluxes relative to larger G/L, which contributed to increasing swimming speed. We find that while metachronal motion augments swimming performance for closely spaced appendages (G/L < 1), moderately spaced appendages (1.0 ≤ G/L ≤ 1.5) can benefit from metachronal motion only when the stroke amplitude is large.

Metachronal motion across scales: current challenges and future directions

Metachronal motion is used across a wide range of organisms for a diverse set of functions. However, despite its ubiquity, analysis of this behavior has been difficult to generalize across systems. Here we provide an overview of known commonalities and differences between systems that use metachrony to generate fluid flow. We also discuss strategies for standardizing terminology and defining future investigative directions that are analogous to other established subfields of biomechanics. Lastly, we outline key challenges that are common to many metachronal systems, opportunities that have arisen due to the advent of new technology (both experimental and computational), and next steps for community development and collaboration across the nascent network of metachronal researchers.

Emergent metachronal waves using tension-driven, fluid-structure interaction models of tomopterid parapodia

Metachronal waves are ubiquitous in propulsive and fluid transport systems across many different scales and morphologies in the biological world. Tomopterids are a soft-bodied, holopelagic polychaete that use metachrony with their flexible, gelatinous parapodia to deftly navigate the midwater ocean column that they inhabit. In the following study, we develop a three-dimensional, fluid-structure interaction model of a tomopterid parapodium to explore the emergent metachronal waves formed from the interplay of passive body elasticity, active muscular tension, and hydrodynamic forces. After introducing our model, we examine the effects that varying material properties have on the stroke of an individual parapodium. We then explore the temporal dynamics when multiple parapodia are placed sequentially and how differences in the phase can alter the collective kinematics and resulting flow field.

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.

Metachronal Swimming with Flexible Legs: A Kinematics Analysis of the Midwater Polychaete Tomopteris

Aquatic animals have developed a wide array of adaptations specific to life underwater, many of which are related to moving in the water column. Different swimming methods have emerged, such as lift-based flapping, drag-based body undulations, and paddling. Patterns occur across scales and taxa, where animals with analogous body features use similar locomotory methods. Metachronal paddling is one such wide-spread propulsion mechanism, occurring in taxa as diverse as ctenophores, crustaceans, and polychaetes. Sequential movement of multiple, near identical appendages allows for steady swimming through phase-offsets between adjacent propulsors. The soft-bodied, holopelagic polychaete Tomopteris has two rows of segmental appendages (parapodia) positioned on opposite sides along its flexible body that move in a metachronal pattern. The outer one-third of their elongate parapodia consist of two paddle-like pinnules that can be spread or, when contracted, fold together to change the effective width of the appendage. Along with metachronal paddling, tomopterid bodies undulate laterally, and by using high speed video and numerical modeling, we seek to understand how these two behaviors combine to generate effective swimming. We collected animals using deep-diving remotely operated vehicles, and recorded video data in shore- and ship-based imaging laboratories. Kinematics were analyzed using landmark tracking of features in the video data. We determined that parapodia are actively moved to generate thrust and pinnules are actively spread and contracted to create differences in drag between power and recovery strokes. At the same time, the body wave increases the parapodium stroke angle and extends the parapodia into undisturbed water adjacent to the body, enhancing thrust. Based on kinematics measurements used as input to a 1 D numerical model of drag-based swimming, we found that spreading of the pinnules during the power stroke provides a significant contribution to propulsion, similar to the contribution provided by the body wave. We conclude that tomopterids combine two different propulsive modes, which are enabled by their flexible body plan. This makes their anatomy and kinematics of interest not only for biologists, but also for soft materials and robotics engineers.

A tail's tale: Biomechanical roles of dorsal thoracic spine of barnacle nauplii

Many marine invertebrates have complex life histories that begin with a planktonic larval stage. Similar to other plankton, these larval invertebrates often possess protruding body extensions, but their function beyond predator deterrence is not well-documented. For example, the planktonic nauplii of crustaceans have spines. Using the epibiotic pedunculate barnacle Octolasmis spp., we investigated how the dorsal thoracic spine affects swimming and fluid disturbance by comparing nauplii with their spines partially removed against those with intact spines. Our motion analysis showed that amputated Octolasmis spp. swam slower, in jerkier trajectories, and were less efficient per stroke cycle than those with intact spines. Amputees showed alterations in limb beat pattern: larger beat amplitude, increased phase lag, and reduced contralateral symmetry. These changes might partially help increase propulsive force generation and streamline the flow, but were insufficient to restore full function. Particle image velocimetry further showed that amputees had a larger relative area of influence, implying elevated risk by rheotactic predator. Body extensions and their interactions with limb motion play important biomechanical roles in shaping larval performance, which likely influences the evolution of form.

Swimming kinematics of cyprids of the barnacle Balanus glandula

Larvae of barnacles typically pass through naupliar and cyprid planktonic stages before settlement and metamorphosis. As the final larval stage, cyprids swim much faster than nauplii and in turbulent fluid environments with high shears as they seek habitat. Cyprids swim with six pairs of reciprocating thoracic appendages and use two anterior antennules during settlement. Our understanding of how thoracic appendages generate movement is limited due to short stroke intervals (∼5 ms) that impede observations of the shape and trajectory of appendages. Here, we used high speed videography to observe both free-swimming and tethered cyprids of the intertidal acorn barnacle Balanus glandula to produce a comprehensive description of thoracic appendage swimming kinematics. Cyprids used a drag-based method of swimming: their six pairs of thoracic appendages moved through metachronal power strokes and synchronous recovery strokes similar to the thoracopod motions in calanoid copepods during escape swimming. During the power stroke, plumose setae on each appendage pair spread laterally into a high surface area and high drag paddle composed of a meshwork of fused setules. This interconnected setal array collapsed into a low surface area and low drag shape during the recovery stroke. These effective swimming appendages allowed cyprids to move upwards at an average speed of 1.4 cm s-1 (about 25 body lengths s-1) with an average beat frequency of 16 beats s-1, and reach an instantaneous velocity of up to 6 cm s-1. Beat frequency of the thoracic appendages was significantly associated with speed, with higher beat frequencies indicating faster swimming speed. At their average speed, cyprids moved at the intermediate Reynolds number of ∼10, in which both viscous and inertial forces affected movement. Cyprids could alter swimming direction by sweeping the posterior-most appendage pair to one side and beating the remaining thoracic appendages synchronously through the power stroke with greater motion on the outside of their turn. These results greatly enhance our understanding both of cyprid motility and how small planktonic organisms can use swimming appendages with fused setule arrays to reach high swimming speeds and affect directional changes.

Swimming kinematics and hydrodynamics of barnacle larvae throughout development

Changes in size strongly influence organisms' ecological performances. For aquatic organisms, they can transition from viscosity- to inertia-dominated fluid regimes as they grow. Such transitions are often associated with changes in morphology, swimming speed and kinematics. Barnacles do not fit into this norm as they have two morphologically distinct planktonic larval phases that swim differently but are of comparable sizes and operate in the same fluid regime (Reynolds number 100-101). We quantified the hydrodynamics of the rocky intertidal stalked barnacle Capitulum mitella from the nauplius II to cyprid stage and examined how kinematics and size increases affect its swimming performance. Cyprids beat their appendages in a metachronal wave to swim faster, more smoothly, and with less backwards slip per beat cycle than did all naupliar stages. Micro-particle image velocimetry showed that cyprids generated trailing viscous vortex rings that pushed water backwards for propulsion, contrary to the nauplii's forward suction current for particle capture. Our observations highlight that zooplankton swimming performance can shift via morphological and kinematic modifications without a significant size increase. The divergence in ecological functions through ontogeny in barnacles and the removal of feeding requirement likely contributed to the evolution of the specialized, taxonomically unique cyprid phase.

A Numerical Study of Metachronal Propulsion at Low to Intermediate Reynolds Numbers

Inspired by the forward swimming of long-tailed crustaceans, we study an underwater propulsion mechanism for a swimming body with multiple rigid paddles attached underneath undergoing cycles of power and return strokes with a constant phase-difference between neighboring paddles, a phenomenon known as metachronal propulsion. To study how inter-paddle phase-difference affects flux production, we develop a computational fluid dynamics model and a numerical algorithm based on the immersed boundary method, which allows us to simulate metachronal propulsion at Reynolds numbers (RE) ranging from close to 0 to about 100. Our main finding is that the highest average flux is generated when nearest-neighbor paddles maintain an approximate 20%–25% phase-difference with the more posterior paddle leading the cycle; this result is independent of stroke frequency across the full range of RE considered here. We also find that the optimal paddle spacing and the number of paddles depend on RE; we see a qualitative transition in the dynamics of flow generated by metachronal propulsion as RE rises above 80. Roughly speaking, in terms of average flux generation, a tight paddle spacing is preferred when RE is less than 10, but a wider spacing becomes clearly favored when RE is close to or above 100. In terms of efficiency of flux generation, at RE 0.1 the maximum efficiency occurs at two paddles, and the efficiency decreases as the number of paddles increases. At RE 100 the efficiency increases as the number of paddles increases, and it appears to saturate by eight paddles, whereas using four paddles is a good tradeoff for both low and intermediate RE.

Rotational Maneuvers of Copepod Nauplii at Low Reynolds Number

Copepods are agile microcrustaceans that are capable of maneuvering freely in water. However, the physical mechanisms driving their rotational motion are not entirely clear in small larvae (nauplii). Here we report high-speed video observations of copepod nauplii performing acrobatic feats with three pairs of appendages. Our results show rotations about three principal axes of the body: yaw, roll, and pitch. The yaw rotation turns the body to one side and results in a circular swimming path. The roll rotation consists of the body spiraling around a nearly linear path, similar to an aileron roll of an airplane. We interpret the yaw and roll rotations to be facilitated by appendage pronation or supination. The pitch rotation consists of flipping on the spot in a maneuver that resembles a backflip somersault. The pitch rotation involved tail bending and was not observed in the earliest stages of nauplii. The maneuvering strategies adopted by plankton may inspire the design of microscopic robots, equipped with suitable controls for reorienting autonomously in three dimensions.

Evolution of Feeding Shapes Swimming Kinematics of Barnacle Naupliar Larvae: A Comparison between Trophic Modes

A central goal in evolutionary biology is connecting morphological features with ecological functions. For marine invertebrate larvae, appendage movement determines locomotion, feeding, and predator avoidance ability. Barnacle larvae are morphologically diverse, and the morphology of non-feeding lecithotrophic nauplii are distinct from those that are planktotrophic. Lecithotrophic larvae have a more globular body shape and simplified appendages when compared with planktotrophs. However, little is known about whether and how such morphological changes affect kinematics, hydrodynamics, and ecological functions. Here, we compared the nauplii kinematics and hydrodynamics of a lecithotrophic Rhizocephalan species, Polyascus planus, against that of the planktotrophic nauplii of an intertidal barnacle, Tetraclita japonica. High-speed, micro-particle image velocimetry analysis showed that the Polyascus nauplii swam faster and had higher amplitude and more synchronous appendage beating than the Tetraclita nauplii. This fast swimming was accompanied by a faster attenuation of induced flow with distance, suggesting reduced predation risk. Tetraclita nauplii had more efficient per beat cycles with less backward displacement during the recovery stroke. This "anchoring effect" resulted from the anti-phase beating of appendages. This movement, together with a high-drag body form, likely helps direct the suction flow toward the ventral food capturing area. In sum, the tradeoff between swimming speed and predation risks may have been an important factor in the evolution of the observed larval forms.

Metachronal Swimming with Rigid Arms near Boundaries

Various organisms such as crustaceans use their appendages for locomotion. If they are close to a confining boundary then viscous as opposed to inertial effects can play a central role in governing the dynamics. To study the minimal ingredients needed for swimming without inertia, we built an experimental system featuring a robot equipped with a pair of rigid slender arms with negligible inertia. Our results show that directing the arms to oscillate about the same time-averaged orientation produces no net displacement of the robot each cycle, regardless of any phase delay between the oscillating arms. The robot is able to swim if the arms oscillate asynchronously around distinct orientations. The measured displacement over time matches well with a mathematical model based on slender-body theory for Stokes flow. Near a confining boundary, the robot with no net displacement every cycle showed similar behavior, while the swimming robot increased in speed closer to the boundary.

Hydrodynamics of metachronal paddling: effects of varying Reynolds number and phase lag

Negatively buoyant freely swimming crustaceans such as krill must generate downward momentum in order to maintain their position in the water column. These animals use a drag-based propulsion strategy, where pairs of closely spaced swimming limbs are oscillated rhythmically from the tail to head. Each pair is oscillated with a phase delay relative to the neighbouring pair, resulting in a metachronal wave travelling in the direction of animal motion. It remains unclear how oscillations of limbs in the horizontal plane can generate vertical momentum. Using particle image velocimetry measurements on a robotic model, we observed that metachronal paddling with non-zero phase lag created geometries of adjacent paddles that promote the formation of counter-rotating vortices. The interaction of these vortices resulted in generating large-scale angled downward jets. Increasing phase lag resulted in more vertical orientation of the jet, and phase lags in the range used by Antarctic krill produced the most total momentum. Synchronous paddling produced lower total momentum when compared with metachronal paddling. Lowering Reynolds number by an order of magnitude below the range of adult krill (250-1000) showed diminished downward propagation of the jet and lower vertical momentum. Our findings show that metachronal paddling is capable of producing flows that can generate both lift (vertical) and thrust (horizontal) forces needed for fast forward swimming and hovering.

Choreographed swimming of copepod nauplii

Small metazoan paddlers, such as crustacean larvae (nauplii), are abundant, ecologically important and active swimmers, which depend on exploiting viscous forces for locomotion. The physics of micropaddling at low Reynolds number was investigated using a model of swimming based on slender-body theory for Stokes flow. Locomotion of nauplii of the copepod Bestiolina similis was quantified from high-speed video images to obtain precise measurements of appendage movements and the resulting displacement of the body. The kinematic and morphological data served as inputs to the model, which predicted the displacement in good agreement with observations. The results of interest did not depend sensitively on the parameters within the error of measurement. Model tests revealed that the commonly attributed mechanism of 'feathering' appendages during return strokes accounts for only part of the displacement. As important for effective paddling at low Reynolds number is the ability to generate a metachronal sequence of power strokes in combination with synchronous return strokes of appendages. The effect of feathering together with a synchronous return stroke is greater than the sum of each factor individually. The model serves as a foundation for future exploration of micropaddlers swimming at intermediate Reynolds number where both viscous and inertial forces are important.