Calcium (Ca2+) dynamics in mammalian sperm are directly linked to motility. These dynamics depend on diffusion, nonlinear fluxes, Ca2+ channels specific to the sperm flagellum, and other signaling molecules. The goal of this work is to couple Ca2+ dynamics to a mechanical model of a motile sperm within a viscous, incompressible fluid. We will present recent progress on elements of this integrative model.

Keywords: fish swimming, vortices, krill, fins

Abstract: We consider two problems related to the propulsion of flexible surfaces in vortex wakes. First, we present a simple model of a trout swimming in a cylinder wake, which saves energy by slaloming through a vortex street. We find analytic solutions and compare with previous experiments and numerics. Second, we study ``inverted drafting'' in flags, in which the drag force on one flag is increased by excitation from the wake of another. The types of drafting and dynamics (synchronization and erratic flapping) depend on the separation distance between the flags. We may also discuss recent work on krill swimming and an optimal design for fish fins.

The traditional paradigm for the lab-based study of animal behavior has been to place an organism in a restricted environment and to understand it through the lens of a discrete set of allowed behaviors. While these limitations allow for more focused questions and higher data throughput, the particularities of these restrictions almost inevitably have implications for the resulting output. Moreover, the choice of a discretized classification by which the experiment describes the animal’s movements is often fraught with arbitrariness and anthropomorphism -- as well as the fact that a discrete set of behaviors may not even exist in the first place. What we aim to do here is to apply a data-driven approach to parameterize the behavior of a genetic model species, the fruit fly Drosophila melanogaster. Borrowing techniques from computer vision, machine learning, nonlinear dynamics, and statistical physics, our goal is to film long sequences of the insects moving within a relatively unrestricted environment and to translate these sequences into time-series data. From these time-series, we hope to build a descriptive language for fly behavior that can provide insight into how these creatures move, groom, communicate, mate, and otherwise live their lives.

The locomotion of biological microorganisms has been the object of much research over the last half of a century. Although significant progress has been made in the study of motion in Newtonian fluids, many biological cells such as bacteria often encounter viscous environments with suspended microstructures or macromolecules. The physics of micro-propulsion in such a non-Newtonian viscoelastic fluid has only recently started to be addressed. In our work, we present a numerical study of the motility of an axisymmetric spherical squirmer in a polymeric flow. The microswimmer that we consider here is driven by a purely tangential distortion on the outer surface reproduced as non-homogenous boundary condition on a rigid sphere. We solve the hydrodynamic Stokes equation (zero Reynolds number) with the extra stress term generated by advection and stretching of polymers. As transport equation for the polymeric stress, we use here the nonlinear Giesekus model. The swimming speed is lower in a visco-elastic fluid and is asymptotically recovering for large Weissenberg numbers approaching values only about 15% smaller than in a Newtonian fluid. Interestingly, the efficiency is seen to significantly increase as the viscosity of the polymeric fluid is increased.

Many biological systems use flexible filaments moving in a viscous fluid to achieve motility or fluid transport. Examples include bacteria that use flagella to swim, paramecium that use cilia for motion and cilia on the wall of the lungs used for particulate removal. In tall of these cases, the filaments are observed to synchronize their motion, and since there is no chemical or physiological coordination, this synchronization is thought to be mediated by hydrodynamic forces. We demonstrate the use of a model system that captures all of the essential features in hydrodynamic synchronization, including multiple filaments interacting through viscous stresses, as well as the presence of structural flexibility in each filament, which is known to be a necessary component for synchronization. We explore the dynamics of hydrodynamic synchronization in this modelsystem using experiments, numerical simluations that employ regularized Stokeslets, and a simple analtyical model. All three approaches give consistent results and point to the relative roles of viscous stresses, structural flexibility in the synchronization dynamics.

Work with Ross Hatton.

Keywords: locomotion, snake robots, gait design

Abstract: Animals often use gaits — cyclic changes in shape producing a net displacement — to move through their environments. In robotics, we are interested in planning motions for artificial systems that can match or exceed the locomotive capabilities of animals. A fundamental question of locomotion is "What are the characteristics of a useful gait?" The geometric mechanics community has made significant progress in answering this question, identifying functions of the system shape that capture the net displacements induced by broad classes of gaits without having to individually test each possible motion. In this talk, we first introduce these results with the aim of separating them from the specialized language of differential geometry and making them accessible to a broader audience. Following this introduction, we then examine how the choice of generalized coordinates quality of the locomotion functions, a question which has not previously been addressed.

Joint work with John O. Kessler (Physics Department, University of Arizona, Tucson, AZ 85721).

Keywords: bacteria, interacting self propelled particles, collective behavior, PIV, bio-fluid-dynamics

Abstract: Spatial order and fast collective coherent dynamics of populations of swimming bacteria emerges from local interactions and from flows generated by the organisms’ locomotion. The transition from dilute, to intermediate, to high concentrations of cells will be demonstrated by movie clips. Analyses of these data, presented as probability density functions for swimming velocity, show that the low concentration phase which exhibits swimming speeds characteristic of individual bacteria, arrives at the anomalously high speed phase, ZBN, the ZoomingBioNematic, via an intermediate phase that exhibits surprisingly low mean speeds. The origin of this phenomenon relates to scattering and the known dynamics of velocity flipping. Particle Imaging Velocimetry (PIV) was used for analysis of mixing, and of collective velocities, correlated with alignment within coherent patches.

Supported by the Department of Energy, grant DOE-W-31-109-ENG-38.

Keywords: fish/eddy interaction, vorticity, evolution of control surfaces and function, turbulence, fish responses.

Abstract: The natural habitats of fishes are characterized by water movements driven by a multitude of physical processes of either natural or human origin. The resultant unsteadiness is exacerbated when flow interacts with surfaces, such as the bottom and banks, and protruding objects, such as corals, boulders, and woody debris. There is growing interest in the impacts on performance and behavior of fishes swimming in "turbulent flows." The ability of fishes to stabilize body postures and their swimming trajectories is thought to be important in determining species distributions and densities, and hence resultant assemblages in various habitats. Furthermore, water flow, structure and vorticity are related to the shape of the body and fins of fishes swimming largely in relatively steady flows. Adaptations to minimize energy losses would be anticipated. We suggest such mechanisms may be found in varying the length of the propulsive wave, stiffening propulsive surfaces, and shifting to use of the median and paired fins when swimming at low speeds. The archetypal streamlined "fish" shape reduces destabilizing forces for fishes swimming into eddies. Understanding impacts of turbulence and vorticity on fishes is important as human practices modify water movements, and as turbulence-generating structures ranging from hardening shorelines to control erosion, through designing fish deterrents, to the design of fish passageways become common.

There has been much recent interest in understanding the dynamics of low-Reynolds-number swimmers near no-slip boundaries and free capillary surfaces. This talk will present theoretical ideas for studying the dynamics of such swimmers within the framework of simple two dimensional models. The 2D models are developed using the methods of complex analysis and afford significant analytical advantages while still capturing the essential mechanisms of (certain) fully three dimensional situations. In many cases the approach yields explicit nonlinear dynamical systems that can be directly studied. The models can rationalize behaviour observed in numerical and laboratory experiments and can provide predictions for the swimmer dynamics in more complicated situations. We will survey results on swimming near a no-slip wall [joint work with Y. Or], swimming in other conned geometries [joint work with O. Samson] and swimming near a free capillary surface [joint work with S. Lee, O. Samson, A. Hosoi and E. Lauga].

Using a nonlocal nematic potential, we present a kinetic theory for nonhomogeneous suspensions of micoswimmers. We then study the steady states to an imposed weak shear. Our results show that the activity parameter will result in permeative or oscillatory flow behaviors at the leading order depending the Leslie tumbling parameter. The linear viscoelasticity is also calculated, which is similar to chiral liquid crystals.

Keywords: vortices, locomotion, swimming, flying

Abstract: The formation and shedding of fluid vortices is an inevitable consequence of movement for all but the smallest of swimming and flying organisms. Can animals use these vortices to enhance locomotion? If so, can their methods of vortex-enhanced propulsion be translated to engineered systems? This talk will describe experimental studies of jellyfish and numerical simulations of eels that suggest candidate mechanisms to enhance the efficiency and speed of locomotion by using vortices. A bio-inspired underwater vehicle is created to test the ideas in an engineering context. It appears that swimming and flying animals have significant opportunities to optimize their locomotion by making use of vortices.

Keywords: squid, scallop, jet propulsion, hydrodynamic efficiency, Antarctica, ontogeny, scaling

Abstract: Among multicellular animals, jet propulsion is nature’s simplest (and arguably its oldest) form of aquatic locomotion. Any flexible, hollow body girdled by circumferential muscle fibers, can, by expelling fluid through an orifice, produce thrust and thereby swim. Despite the fundamental simplicity of this locomotory mechanism, aspects of its realization in nature continue to provide insight into the physiology, ecology, and evolution of a wide variety of animals. In this talk, I report on two mollusks that use jet propulsion. The Antarctic scallop is one of only a few bivalves that can swim. Like its temperate and tropical cousins, it claps its shells together to expel a jet of water sufficiently powerful to lift both its internal organs and its dense calcium-carbonate shell off the seafloor. But the Antarctic scallop must perform this feat in water at -1.86 degrees C, a temperature at which muscle power is reduced and water’s viscosity is 1.43 times that of tropical water. Shell mass in the Antarctic scallop is much reduced relative to tropical scallops, but muscle mass is reduced even more. The only net advantage evident in Antarctic scallops is the increased resilience of the “spring” that opens the shell, suggesting that even slight increases in hydrodynamic efficiency can be selected during evolution. Increases in hydrodynamic efficiency may also play an important role in squid locomotion. Unlike most jet propulsors (e.g., jellyfish, salps, clams), squids can actively control the size of the orifice through which water is expelled. Appropriate narrowing of the jet orifice during mantle contraction can boost the efficiency of both the hydrodynamics of propulsion and the contraction of muscle. This potential increase in efficiency may be most important in small juvenile squid, for whom jet propulsion is otherwise very inefficient.

In this new implementation of the Immersed Interface Method, we solve for the coupled dynamics of free moving objects in a fluid. We test the code by comparing our results with experimental data on falling plates and cylinders. We further present the dynamics of an array of falling cylinders and contrast the results of the even and odd configuration.

We present a formulation for coupled solutions of fluid and body dynamics in problems of biolocomotion. This formulation unifies the treatment at moderate to high Reynolds number with the corresponding inviscid problem. By a viscous splitting of the Navier–Stokes equations, inertial forces from the fluid are distinguished from the viscous forces, and the former are further decomposed into contributions from body motion in irrotational fluid and ambient fluid vorticity about an equivalent stationary body. In particular, the added mass of the fluid is combined with the intrinsic inertia of the body to allow for simulations of bodies of arbitrary mass, including massless or neutrally buoyant bodies. The resulting dynamical equations can potentially illuminate the role of vorticity in locomotion, and provide new pathways toward reduced-order modeling. Examples of articulated or continuously deforming bodies undergoing swimming or flying kinematics are presented and discussed.

The ulcer-causing gastric pathogen Helicobacter pylori is able to swim through the viscoelastic mucus gel that coats the stomach wall, but its mechanism of locomotion through an acidic gel environment is poorly understood. This experimental study indicates that the helicoidal-shaped H. pylori achieves motility by altering the rheological properties of its environment. H. pylori locally raises the pH of its environment in order to survive in the acidic conditions of the stomach. This local change of pH affects the rheology of the surrounding mucus material. We show that gastric mucus is pH dependent, changing from a gel at acidic conditions (low pH) into a viscous solution at more neutral conditions (higher pH). Microscopy studies of the motility of H. pylori in gastric mucin show that the bacteria swim freely at high pH, and are strongly constrained at low pH.

Joint work with J. P. Celli, B. S. Turner, N. H. Afdahl, S. Keates, I. Ghiran, C. Kelly, G. H. McKinley, P. So, S. Erramilli, and R. Bansil.

http://dx.doi.org/10.1073/pnas.0903438106

In an effort towards an understanding of the generation and control of vertebrate locomotion, including the role of the CPG and its interactions with reflexive feedback, muscle mechanics, and external fluid dynamics, we study a simple vertebrate, the lamprey. Lamprey body undulations are a result of a wave of neural activation that passes from head to tail, causing a wave of muscle activation. These active forces are mediated by passive structural forces. We present a model that includes the complete fluid-structure interaction problem, in which the body is elastic and deforms according to both internal muscular forces and external fluid forces. The model uses an immersed-boundary framework for solving the Navier-Stokes equations of fluid motion, and includes a nonlinear muscle model, an elastic body, and an adaptive solver that is accurate at length and velocity scales that are appropriate for swimming lampreys. The effects of various body and environmental properties, including tapered and uniform body shapes, different body stiffness, varying muscle parameters, and a range of viscosities are examined as they relate to swimming dynamics and energy requirements. (This is joint work with Chia-yu Hsu, Eric Tytell, Thelma Williams, Tyler McMillen and Avis Cohen.)

Keywords: hydrodynamics, lift-based propulsion, leading edge tubercles, dolphin, whale, manta, vorticity control

Abstract: Optimization of energy by large aquatic animals (e.g., dolphins, whales, manta) requires adaptations that control hydrodynamic flow to reduce drag, and improve thrust production and efficiency. Although streamlining of the body and appendages minimizes drag, highly derived aquatic animals utilize mechanisms of propulsion and control based on lift generation, which maximizes thrust production and minimizing drag. Oscillations of the wing-like fins and flukes, which are hydrofoils, generate thrust throughout the stroke cycle and maintain a propulsive efficiency over 80%. This high efficiency is dependent on spanwise and chordwise bending and management of swimming kinematics to control vorticity. Control of vorticity to improve locomotor performance for maneuverability is enhanced by modification to the leading edge of control surfaces. The humpback whale (Megaptera novaeangliae) flippers are unique because of the presence of large tubercles along the leading edge, which gives this surface a scalloped appearance. The tubercles function to produce vortical flows over the surface of the flipper and control lift characteristics at high angles of attack, where stall would occur. The potential benefits from mimicking these biological innovations as applied to engineered systems operating in fluids are high speeds, vorticity control, reduced detection, energy economy, and enhanced maneuverability.

Throughout the natural world, cells and organisms use flagella and cilia to propel fluid, achieve cell motility and a range of other functions. The mechanism regulating their waveform, however, is a long-standing biological problem. Indeed, the emergence of such flagellar undulatory waves is a consequence of an intricate balance of fluid dynamic viscous forces, flagellar elastic resistance, and the active bending generated by internal motor proteins. By using theoretical modelling and numerical computations accounting for high curvatures observed physiologically, we show that flagellar compression due to viscous friction and the internal forces may initiates an effective buckling behaviour that leads to an asymmetric flagellar beating and, consequently, switching sperm swimming from a straight migratory trajectory to a circling path. These results demonstrate that observed asymmetric flagellar beating may arise due to a dynamic instability, and not necessarily require some intrinsic asymmetric in the beating mechanism. This behaviour may be important in understanding mammalian sperm trapping, potentially a crucial step in natural fertilisation.

Studying human spermotozoa motility is a subject of growing importance due to human male subfertility and the fact that the in-vitro fertilisation interventions that bypass normal sperm motility are invasive and entail significant risk for the healthy female partner, as well as being economically prohibitive for many. We present examples of how fluid and continuum dynamics can provide novel insights concerning the mechanics of human spermatozoon behaviour, focussing on the interpretation of recent high resolution imaging.

Legged locomotion on flowing ground ({em e.g.}~granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bio-inspired legged robots display speed relative to body size on hard ground comparable to high performing organisms like cockroaches but suffer significant performance loss on flowing materials like sand. In laboratory experiments we study the performance (speed) of a small (2.3~kg) six-legged robot, SandBot, as it runs on a bed of granular media (1~mm poppy seeds). For an alternating tripod gait on the granular bed, standard gait control parameters achieve speeds at best two orders of magnitude smaller than the 2~body lengths/s ($approx 60$~cm/s) for motion on hard ground. However, empirical adjustment of these control parameters away from the hard ground settings, restores good performance, yielding top speeds of 30~cm/s. Robot speed depends sensitively on the packing fraction $phi$ and the limb frequency $omega$, and a dramatic transition from rotary walking to slow swimming occurs when $phi$ becomes small enough and/or $omega$ large enough. We propose a kinematic model of the rotary walking mode based on generic features of penetration and slip of a curved limb in granular media. The model captures the dependence of robot speed on limb frequency and the transition between walking and swimming modes but highlights the need for a deeper understanding of the physics of granular media. Journal paper: Li et al, PNAS, 2009.

In this talk I will summarize our recent progress in experiments and models of the locomotion of a sand-swimming lizard, the sandfish ({em Scincus scincus}). We use high speed x-ray imaging to study how the 10 cm-long sandfish swims at 2 body-lengths/sec within sand, a granular material that displays solid and fluid-like behavior. Below the surface the lizard no longer uses limbs for propulsion but generates thrust to overcome drag by propagating an undulatory traveling wave down the body. While viscous hydrodynamics can predict swimming speed in fluids like water, an equivalent theory for granular drag is not available. To predict sandfish swimming speed, we develop an empirical resistive force model by measuring drag force on a small cylinder oriented at different angles relative to the displacement direction and summing these forces over the animal movement profile. The model correctly predicts the animal's wave efficiency (ratio of forward speed to wave speed) as approximately 0.5. The empirical model agrees with a more detailed (and more accurate) numerical simulation: a multi-segment model of the sandfish coupled to a multi-particle Molecular Dynamics simulation of the granular media. The agreement between models and the prediction of biological wave efficiency demonstrate that the non-inertial swimming occurs in a frictional fluid and the Molecular Dynamics simulation allows us to visualize the self-generated fluid surrounding the sandfish as it swims. We use the principles discovered to construct a sand-swimming physical model (a robot) which, like in our empirical and multi-particle numerical model, swims fastest using the preferred sandfish wave pattern.

Experiments and simulations indicate that suspensions of swimming microorganisms can exhibit complex phenomena, including pattern-forming instabilities, large scale fluid motions and enhanced passive scalar transport. This talk is an overview of theoretical and computational work describing some of these phenomena. Emphasis will be placed on analysis of various correlation functions associated with the dynamics. The talk concludes with a brief presentation of new directions in this area, including the coordinated motion of collections of bacterial flagella.

This is joint work with Patrick T. Underhill and Pieter J. A. Jansssen.

A nonlinear unsteady Darcy's equation to include inertial effects of a Hele-Shaw flow and the conditions under which it reduces to the classical Darcy's law are discussed. In the absence of surface tension, we derive a generalized Polubarinova-Galin equation for flows in a circular Hele-Shaw cell using the method of conformal mapping. The linear stability of the base-flow state is examined by perturbing the conformal mapping in form of polynomial modes. We find that small inertia always has the tendency to stabilize the interface.

Keywords: active suspensions, kinetic theory

Abstract: One of the challenges in modeling the transport properties of complex fluids (e.g. many biofluids, polymer solutions, particle suspensions) is describing the interaction between the suspended micro-structure with the fluid itself. Here I will focus on understanding the dynamics of active suspensions, like swimming bacteria or artificial micro-swimmers. Using a recently derived kinetic model, I have investigated the linearized structure of such an active system near a state of uniformity and isotropy. I will show that system instability can arise only from the dynamics of the first azimuthal mode in swimmer orientation, that the growth of fluctuations for a suspension of anterior actuated swimmers is associated with a proliferation of oscillations in swimmer orientation, that diffusion acts as a smoothing parameter, and that at small-scales the system is controlled independently of the nature of the suspension. Finally a prediction about the onset of the instability as a function of the volume concentration of anterior actuated swimmers and a comparison with numerical simulations is made.

Joint work with Andrew K. Dickerson, Zachary G. Mills, Paul C. Foster (School of Mechanical Engineering, Georgia Institute of Technology).

While much attention has been devoted to the ability of animals to propel themselves through fluids, less work has been done on how they exploit fluids in their grooming habits. The problem of how animals clean and dry themselves involves complex flexible surfaces (hair, skin), unsteady speeds, and wetting/de-wetting of drops and fluid ligaments. In this experimental investigation, we investigate the ability of dogs, rats, mice and other hirsute mammals to rapidly oscillate their bodies in order to shed water droplets, nature's analogy to the spin cycle on a washing machine. High-speed videography and fur-particle tracking is employed to determine the angular position of the animal's shoulder skin as a function of time. We formulate the conditions for de-wetting and propulsion of water drops based on the balance of the forces of surface tension, centripetal forces (which tend to pull drops normally from the skin) and angular-acceleration forces (which tend to slide drops). We find that smaller animals shake fastest: specifically, shaking frequency scales as the shoulder radius to the -1/2 power, as is required for centripetal forces on drops to remain constant as animals grow. An important consideration in this process is the looseness of the skin with respect to the body, whose presence increases the peak speed and acceleration of their fur. The energy expenditure and remaining water moisture content of self-drying mammals is estimated.

Plasma membrane blebs are dynamic cytoskeleton-regulated cell protrusions that have been implicated in apoptosis, cytokinesis, and cell movement. A variety of theoretical and experimental results support the hypothesis that nonapoptotic membrane blebbing plays a central role for cell migration and cancer cell invasion. For cancer cells crawling through a 3D matrix, there are two morphologically distinguished modes of invasion: one that appears as a mesenchymal cell movement that relies on proteolytic degradation of the surrounding matrix and another that adopts a rounded, more amoeboid mode of motility that frequently is accompanied by cell blebbing. Here we will focus on the latter mode--cell migration through a 3D matrix by cell blebbing.

To date, because of the limitation of experimental systems, most mechanisms governing plasma membrane blebbing are derived from 2D standard cell cultures that display blebbing under certain conditions. For example, much has been done for blebbing cells crawling on a substrate by adhesion. When lifted to 3D environment, the mechanisms may be totally different, and in the absence of interaction between the cell and the extracellular matrix, we now come to the classic physical problem: swimming in low Reynolds number fluid. One possible propulsive mechanism in a low Reynolds number environment is cyclic deformation of the cell membrane. Two of the most important biological phenomena related to this cyclic cell shape deformation are the observed shape oscillations and cell blebbing. The former one may be well generalized to a single swimming sphere at low Reynolds number that exerts self-propulsion by means of small amplitude, high-frequency waves over the surface. However, for the cell blebbing problem, one sphere model may not be adequate. Instead, a minimal model may comprise several connecting spheres that can exchange mass among them.

In the poster, we will first present a brief review of fundamental theories of swimming at low Reynolds number, together with some previous models. Next we will advance several of my models and discuss their properties.

We numerically investigate the bundling between flagella. For this, we consider two flexible helices next to each other. Each helix is modeled as several prolate spheroids connected at the tips by springs. On the first spheroid, a constant torque is applied. Torsion springs at the connections provide bending and twisting resistance. Hydrodynamic interactions are incorporated via a modified non-singular Stokeslet. Additionally, there is a repulsive force and torque, based on the Gay-Berne potential to prevent crossing of the flagella.

Our results provide some insights in the details of the bundling process. In the initial stage, rotlet interactions between the rotating helices ensures that both deflect each other. Due to the end point fixation, this deflection combined with the rotlet interaction leads to the flagella rotating around each other. Longer simulations show that the tips of a flagella pair only rotate about once around each other, in contrast with a more complicated entwinement suggested before. Flagella closer together bundle faster.

Most micro-organisms often swim in a variety of complex environments as their natural habitat. For instance, Paramecium tend to congregate and swim near the boundaries. We investigate the locomotion of Paramecium in confined geometries while comparing its motion in the un-bounded fluid. A modified theoretical model based on Taylor’s sheet is developed to study the boundary effect on its motion. During experiments, we introduce Paramecium in capillary tubes of different sizes and measure the influence of the tube diameter on the swimming velocity. The data from the experiments is compared with the theoretical model to test its validity and understand the ciliary locomotion of organisms in a confined channel.

The passive locomotion of a body placed in the flow of periodically-generated vortices is studied. This work is motivated by recent experimental evidence that live and dead trout exploit the vortices in the wake of an oscillating cylinder to swim upstream. We consider a simple model of a rigid body interacting dynamically with point vortices introduced periodically into the flow to emulate the shedding of vortices from an external source. We show the existence of periodic solutions where the body `swims' passively against the flow by extracting energy from the ambient vortices. We also find solutions where the body holds station in the incoming wake. However, for bodies of elongated geometries, rotational instabilities may hinder their motion. We propose active feedback control strategies to overcome these instabilities. (This is joint work with my graduate student Babak G. Oskouei)

Models for aquatic locomotion generally seek to balance fidelity and scope with analytical or computational tractability. When the goal in model development is a platform for model-based feedback control design, analytical structure is essential to provide a point of access for most current design techniques, but some fidelity may be sacrificed as long as the scope of the model encompasses the range of situations under which control will be applied. This talk will describe a model for simplified fishlike swimming based on the Hamiltonian equations governing the interaction of a free deformable body with a system of point vortices in a planar ideal fluid. The use of this model in designing motion-control strategies for a biologically inspired robotic vehicle will be discussed, with a particular focus on the realization of energy-efficient gaits for solitary swimming and energy-harvesting methods for controlled schooling.

Keyword: waves, turbulence, walking, swimming, larvae, crabs

Abstract: Turbulent ambient currents and waves in marine habitats impose forces on organisms. The locomotory performance of organisms swimming in the water column and moving across the substratum is affected by environmental fluid dynamic forces. Therefore, the functional significance of morphological features and kinematics of locomoting organisms can best be understood if studied under the range of flow conditions they experience in their natural habitats. Three examples will be discussed: 1) Many bottom-dwelling marine animals produce microscopic larvae that are dispersed to new sites by ambient water currents. How do these weakly-swimming larvae carried in wavy, turbulent water flow manage to land on the sea floor in suitable habitats? 2) Horseshoe "crabs", Limulus polyphemus, gather in the surf zone to mate. How can they crawl in the waves without being pushed in the wrong direction or overturned by the back-and-forth flow of the waves? 3) Shore crabs, Grapsus tenuicrustatus, also live on wave-swept shores, where they spend part of their time in air and part underwater. How do they run in air (where gravity predominates) versus underwater (where hydrodynamic forces are important), and what happens when a wave hits?

We perform physical experiments with self-propelled rods which undergo directed random motion on a substrate motivated by collective behavior in various active living systems such as bacterial colonies and hoofed animal herds. In particular, we examine the persistent random motion of rods as a function of the area fraction ϕ and study the effect of steric interactions on their diffusion properties. Self-propelled rods of length l and width w are fabricated with a spherocylindrical head attached to a beaded chain tail, and show directed motion on a vibrated substrate. The mean square displacement on the substrate grows linearly with time t for ϕ<w/l, before displaying caging as ϕ is increased, and stops well below the close packing limit. Direction autocorrelations decay progressively slower with ϕ. We describe the observed decrease of SPR propagation speed c(ϕ) with a tube model [1]. Further, we discuss the observed collective behavior such as aggregation at the boundaries and swirling motion which arise because of physical interactions between individuals [2].

[1]: "Concentration Dependent Diffusion of Self-Propelled Rods," A. Kudrolli, Phys. Rev. Lett. 104, 088001 (2010).

[2]: "Swarming and swirling in self-propelled granular rods," A. Kudrolli, G. Lumay, D. Volfson, and L. Tsimring, Phys. Rev. Lett. 100, 058001 (2008).

Same abstract as the contributed talk.

This experimental work is investigating a new and unique passive boundary-layer separation control methodology derived from shark skin, functioning at the micro-scale level. The skin and denticles (scales) of sharks represent over 400 million years of natural selection for swimming efficiency. Evolutionary adaptations in the morphological structure of the shark skin, to develop unique boundary layer control (BLC) mechanisms, stem from the ensuing decrease in drag, probable increase in fin performance (e.g. thrust production) and enhanced turning agility for fast-swimming sharks. Shark denticles have been documented to be capable of bristling. A bristled microgeometry mimicking shark skin results in the formation of a system of interlocking embedded cavity vortices. Three mechanisms have been hypothesized which lead to boundary layer control via deterrence of separation over the shark skin. The first mechanism is the formation of embedded micro-vortices that increase momentum in the very near-wall region due to the resulting partial slip condition. The second mechanism is that the preferential flow direction inherent in the surface geometry inhibits global flow reversal and leads to passive actuation via denticle bristling. The third mechanism involves turbulence augmentation, or an additional energizing of flow, in the near-wall region and cavities, leading to higher partial slip velocities. The study involves engineers, working together with biologists Dr. Phil Motta (University of South Florida) and Dr. Robert Hueter (Mote Marine Laboratory), to fully comprehend the morphological bristling mechanism of shark denticles. Initial results for scale angles and morphology on hammerhead and shortfin mako sharks along with flow measurements over shark skin models embedded in a turbulent boundary layer will be presented.

To study the hydrodynamics of swimming of multi-flagellated bacteria, such as Escherichia coli, we develop a simulation method using a bead-spring model to account for the hydrodynamic and the mechanical interactions between multiple flagella and the cell body, the reversal of the rotation of a flagellum in a tumble and associated polymorphic transformations of the flagellum. This simulation reproduces the experimentally observed behaviors of E. coli, namely, a three-dimensional random-walk trajectory in run-and-tumble motion and steady clockwise swimming near a wall. Here we show using a modeled cell that the polymorphic transformation of flagellum in a tumble facilitates the reorientation of the cell, and that the time-averaged flow field near a cell in a run has double-layered helical streamlines. Moreover, the instantaneous flow field, which is strongly time-dependent, is more than 10-fold larger in magnitude than the time-averaged flow, large enough to affect the migration behavior of surrounding chemoattractants, with the Péclet number for these molecules being larger than one near a swimming cell.

Keywords: Low Reynolds number; locomotion; symmetry-breaking; synchronization; cilia; optimization

Abstract: Fluid mechanics plays a crucial role in many cellular processes. One example is the external fluid mechanics of motile cells such as bacteria, spermatozoa, algae, and essentially half of the microorganisms on earth. The most commonly-studied organisms exploit the bending or rotation of a small number of flagella (short whip-like organelles, length scale from a few to tens of microns) to create fluid-based propulsion. As a difference, Ciliated microorganisms swim by exploiting the coordinated surface beating of many cilia (which are short flagella) distributed along their surface. In this talk, we consider two instances of symmetry-breaking arising in small-scale locomotion. First, we address the observed flagellar synchronization between eukaryotic cells swimming in close proximity. By using a two-dimensional model, we show analytically and computationally that synchronization between co-swimming cells can be driven by hydrodynamic interactions alone if there is a geometrical symmetry-breaking displayed by the their flagellar waveforms. In a second part, we pose the problem of ciliary propulsion as an optimization problem. Specifically, for a spherical body, we compute numerically and theoretically the time-periodic tangential deformations of the body surface which leads to swimming of the body with optimal hydrodynamic efficiency. We show that this calculation leads to symmetry-breaking in the surface actuation, and the emergence of waves, reminiscent of the metachronal waves displayed by real biological cilia.

Joint work with W. B. Dickson², J. L. van Leeuwen¹, and M. H. Dickinson².

As they descend, the autorotating seeds of maples and some other trees generate unexpectedly high lift, but how they attain this elevated performance is unknown. To elucidate the mechanisms responsible, we measured the three-dimensional flow around dynamically scaled models of maple and hornbeam seeds. Our results indicate that these seeds attain high lift by generating a stable leading-edge vortex (LEV) as they descend. The compact LEV, which we verified on real specimens, allows maple seeds to remain in the air more effectively than do a variety of nonautorotating seeds. LEVs also explain the high lift generated by hovering insects, bats, and possibly birds, suggesting that the use of LEVs represents a convergent aerodynamic solution in the evolution of flight performance in both animals and plants.

¹ Experimental Zoology Group, Wageningen University, 6709 PG Wageningen, Netherlands.

² Bioengineering and Biology, California Institute of Technology, Pasadena, CA 91125, USA.

Science 12 June 2009: Vol. 324. no. 5933, pp. 1438 - 1440 DOI: 10.1126/science.1174196 http://www.sciencemag.org/cgi/content/abstract/324/5933/1438

We consider the stirring of an inviscid or Stoksian fluid caused by the locomotion of bodies through it. The swimmers are approximated by non-interacting cylinders or spheres moving steadily along straight lines. We find the displacement of fluid particles caused by the nearby passage of a swimmer as a function of an impact parameter. We use this to compute the effective diffusion coefficient from the random walk of a fluid particle under the influence of a distribution of swimming bodies. We find good agreement between theoretical results with simulations and identifies regions of dominant contribution to mixing. Also It is shown that Stokesian squirmers yields a great boost in effective diffusivity.

Joint work with Jean-Luc Thiffeault and Stephen Childress.

Suspensions of micro-swimmers display complex dynamics in response to chemical substances. They preferentially move and orient toward gradients of such a chemo-attractant in a process called chemotaxis. We present a new chemotaxis model based on the kinetic theory of swimmer suspensions in low Reynolds number fluid that is coupled to the chemo-attractant dynamics. The chemotactic response is included as a phenomenon due to fluxes in the individual swimmer. Entropy and linear stability analysis indicates a chemotaxis-induced instability at finite wavelengths for pusher and puller bacteria alike and regardless of their shape ratio. Nonlinear dynamics are investigated using numerical simulations of the full system in two dimensions. We observe aggregation in suspensions of pullers and mixing in suspensions of pushers.

Using three-dimensional computer simulations, we examine hovering aerodynamics of flexible planar wings oscillating at resonance. We model flexible wings as tilted elastic plates whose sinusoidal plunging motion is imposed at the plate root. Our simulations reveal that large-amplitude, resonance oscillations of elastic wings drastically enhance aerodynamic lift and efficiency of low-Reynolds-number plunging. Driven by a simple sinusoidal stroke, flexible wings at resonance generate a hovering force comparable to that of small insects that employ a very efficient, but much more complicated stroke kinematics. Our results indicate the feasibility of using flexible wings driven by a simple harmonic stroke for designing efficient microscale flying machines.

When studying the mechanics of swimming and flying, engineers and scientists often pose questions in the form of optimization strategies. This approach has been quite useful when trying to understand the kinematics of insect flight or the frequency that fish beat their tails. Understanding the kinematics and the morphology of animals that multitask is not as straightforward. For example, the elaborate tails of male guppy fish are likely not optimized for swimming efficiency or speed, but they do increase the likelihood of attracting a mate. In this presentation, the fluid dynamics of the currents generated by the upside down jellyfish *Cassiopea sp. *will be presented in the context of swimming and feeding. Medusae of this genus are unusual in that they typically rest upside down on the ocean floor and pulse their bells to generate feeding currents, only swimming when significantly disturbed. The pulsing kinematics and fluid flow around these upside-down jellyfish is investigated using a combination of videography, digital particle image velocimetry, and numerical simulation. There is no evidence of the formation of a train of vortex rings as observed in oblate medusae exhibiting rowing propulsion. Instead, significant mixing occurs around and directly above the oral arms and secondary mouths. Numerical simulations using the immersed boundary method agree with experimental measurements and suggest that the presence of porous oral arms induce net horizontal flow towards the bell and the absence of coherent vortex structures. The implications of these results on feeding and swimming efficiency will be discussed.

Spines and other thin projections from cell surfaces literally expand the volume of fluid with which a cell interacts and may provide effective levers on which the flow can act. We use an immersed boundary formulation to solve the coupled phytoplankton-fluid system to predict the 3D trajectories of the cells within a background flow. We examine the effect of spines on the period of rotation of phytoplankton in linear shear flow.

Joint work with John O. Dabiri.

We study the dynamics and stability of the vortex wakes of swimming fish, with the aim of quantifying the role of vortex dynamics in determining the performance and limitations of fish-like swimmers (cf the work on vortex rings and their relation to the performance of jetting swimmers by Gharib et al 1998, Krueger and Gharib 2003, Linden and Turner 2004, Dabiri et al 2010). In order to enable studies of the dynamics and stability of these vortex flows, we analyze wake kinematics using Lagrangian coherent structures (LCS) (Haller 2000, 2001). We computed FTLE field in both two and three dimensions to extract the 2D and 3D LCS in the wake of a numerically-simulated, self-propelled anguilliform swimmer (Kern and Koumoutsakos 2006). The attracting and repelling LCS in the flow were found to clearly bound the vortices shed by the swimmer (Green et al 2010, Shadden et al 2006), and the shedding of two vortex ring per cycle and formation of a double row of vortex rings in the wake was observed. Fluid transport in the wake was studied using passive drifters seeded into the flow, and we observed the formation of slender lobes along the length of the swimmer, which "pull" fluid into the wake such that fluid particles inside each of these lobes are entrained into separate vortices in the swimmer's wake. Changes in the LCS in a flow are known to correspond to changes in the structure and dynamics of the underlying vortices (Green et al 2010, O'Farrell and Dabiri 2010), thus future work will focus on analysis of changes in LCS structure as indicators of changes in the dynamics and stability of the underlying vortex flow.

References: 1) M. Gharib. E. Rambod, and K. Shariff. A universal time scale for vortex ring formation. J. Fluid Mech., 360:121-140, 1998. 2) P. S. Krueger and M. Gharib. The significance of vortex ring formation to the impulse and thrust of a starting jet. Phys. Fluids. 15:1271-81, 2003. 3) P. F. Linden and J. S. Turner. 'Optimal' vortex rings and aquatic propulsion mechanisms. Proc. R. Soc. B, 271:647-53, 2004. 4) J. O. Dabiri, S. P. Colin, K. Katija, and J. H. Costello. A wake-based correlate of swimming performance and foraging behavior in seven co-occurring jelly fish species. J. Exp. Biol., 13 (8): 1217-25, 2010. 5) G. Haller. Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos, 10:99, 2000. 6) G. Haller. Distinguished material surfaces and coherent structures in three-dimensional flows. Physica D, 149:1851–61, 2001. 7) S. Kern and P. Komoutsakos. Simulations of optimized anguilliform swimming. J. Exp. Biol., 209:4841–57, 2006. 8) M. A Green, C. W. Rowley, and A. J. Smits. Using hyperbolic Lagrangian Coherent Structures to investigate vortices in bio-inspired fluid flows. Chaos, 20:017509, 2010. 9) S. .C. Shadden, J. O. Dabiri, and J. E. Marsden. Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys. Fluids., 18:047105, 2006. 10) C. O’Farrell and J. O. Dabiri. A Lagrangian approach to identifying vortex pinch-off. Chaos, 20:017513, 2010.

The dynamics of a simple micro-swimmer model near a no-slip wall is formulated and analyzed. The model consists of an assemblage of spheres where propulsion is generated by rotation of the spheres. The geometric structure of the dynamics is analyzed, and stability properties of translation parallel to the wall are derived. The results are demonstrated through simulations and motion experiments on a macro-scale robotic swimmer in viscous fluid.



I will also present results of a recent joint work with Darren Crowdy on utilizing complex analysis to formulate an explicit two-dimensional dynamic model of a treadmilling swimmer near a wall, and discuss ongoing work on extension to shape-changing controlled swimmers such as Purcell’s three-link swimmer model near a wall.

A set of linearized equations of motion, using a spring-link model, is derived for undulatory swimming. The transverse translational and rotational equations of motion give natural deformation modes of the body which feeds energy to the axial translational motion. It is found, consistent with prior work, that anisotropy in drag is required to enable swimming. The first three deformation modes are excited the most and consequently contribute most to the forward swimming velocity. Typical imposed frequencies, for the case of eel considered here, are found to be lower than the lowest natural frequency of the deformation modes of the body. Thus, lower modes are found to be more easily triggered by the muscle forcing.

A novel constraint-based formulation to simulate self-propulsion has been developed. The numerical approach is used to obtain the following results. A drag-thrust decomposition is found in the propulsion by undulatory ribbon fins of gymnotiform and balistiform swimmers. The height of the ribbon fin of a gymnotiform swimmer seems optimized with respect to the mechanical Cost of Transport (COT). This can be explained based on the drag-thrust decomposition. Optimization based on COT is also found to work in case of pectoral fin movements of larval zebrafish.

The underlying basis of how swimming organisms propel themselves forward against resistance from the surrounding fluid has been studied for almost a century. Many traditional analyses have centered on decomposing the total force on a swimming body into drag and thrust. The validity of this decomposition has been controversial since it is not expected to hold for finite Reynolds number swimming. Yet, we report an approximate drag-thrust decomposition for one class of undulatory propulsors - the ribbon fins of gymnotiform and balistiform swimmers. The conclusion is based on high-resolution numerical simulations to calculate the force acting on an undulatory ribbon fin of the black ghost knifefish (Apteronotus albifrons). We show that drag-thrust decomposition is possible because there is very little spatial overlap between the drag-associated flow field and the thrust-associated flow field. This decomposition is different from the decomposition due to Lighthill that has been widely discussed in literature over the past four decades.

The results above are used to interrogate balistiform and gymnotiform swimmers that move by undulating elongated ribbon fins attached to a body that is held nearly rigid. The question of whether this evolutionary adaptation may have a hydrodynamic basis was considered by Lighthill and Blake. They proposed, based on Lighthill's elongated body theory, that the ability of the ribbon fin to generate thrust is enhanced by the presence of a rigid body. This mechanism, commonly referred to as “momentum enhancement”, has been widely discussed in literature over the past two decades. Our results show that there is no momentum enhancement. This is explained by noting that the dominant mechanism of thrust generation by ribbon fins is different from that assumed in the theoretical approach of Lighthill. Nevertheless, many features of the morphology of gymnotiform and balistiform swimmers do appear to have a hydrodynamic basis. Specifically, it is found that the observed height of the ribbon fin, for a given body size, is such that the mechanical energy spent per unit distance, i.e., the mechanical cost of transport (COT) is optimized.

Many open issues remain. First, it remains to be explored whether the drag-thrust decomposition can be extended to anguilliform and carangiform swimming. Second, while we have found optimal fin height for gymnotiform and balistiform swimmers for a given body size, it is still unclear whether keeping part of the body rigid is hydrodynamically better compared to a mode of swimming where the entire body is undulated (like in anguilliform swimming). Preliminary results interrogating these aspects will be discussed.

Jellyfish represents a group of animals that have an axisymmetric body and swim by periodically contracting the body and generating axisymmetric jets and vortices. In this study, jellyfish is modeled as an axisymmetric swimmer with a thin, flexible body. The wake vortex generated by the swimmer is approximated by a circular vortex sheet. Using this approach, the fluid dynamics and characteristics of the fluid wake are investigated. Swimming performance is also evaluated to quantify the effects of body shape and swimming modes. The study provides insights on fluid dynamical basis of jellyfish swimming and how certain body kinematics of jellyfish enhance the swimming performance.

Flying insects execute aerial maneuvers through subtle manipulations of their wing motions. Here, we measure the free-flight kinematics of fruit flies and determine how they modulate their wing pitching to induce sharp turns. By analyzing the torques these insects exert to pitch their wings, we infer that the wing hinge acts as a torsional spring that passively resists the wing’s tendency to flip in response to aerodynamic and inertial forces. To turn, the insects asymmetrically change the spring rest angles to generate asymmetric rowing motions of their wings. Thus, insects can generate these maneuvers using only a slight active actuation that biases their wing motion.

For animals and machines alike, maintaining balance during flight is a crucial and demanding task. The need for airplane flight stability led to a schism between aviators who sought built-in, or passive, stability and those who emphasized the need for active controls. How has this tension played out for the first flyers, the insects? Our group combines table-top experiments on fruit flies and lap-top physically-based simulations to study insect flight stability and control. First, we show how directly perturbing the flight of insects unlocks the physics of flapping-wing flight and also reveals some remarkable properties of these critters’ sensory-neural systems. Second, we argue that these sophisticated fight control systems are largely sculpted by the physical requirement of stability. This idea leads to a general theory that links the body plans of insects with the controllers that must suppress the growth of instabilities, and we apply this theory to a variety of modern insects, flapping-wing robots, and even the prehistoric insects that were the first to take to the air.

Joint work with Amir Alizadeh Pahlavan.

The effects of an external shear flow on the dynamics and pattern formation in a dilute suspension of swimming micro-organisms are investigated using a linear stability analysis and three-dimensional numerical simulations, based on the kinetic model previously developed by Saintillan and Shelley [``Instabilities, pattern formation, and mixing in active suspensions,'' Phys.~Fluids textbf{20}, 123304 (2008)]. The external shear flow is found to damp the instabilities that occur in these suspensions by controlling the orientation of the particles. We demonstrate that the rate of damping is direction-dependent: it is fastest in the flow direction, but slowest the direction perpendicular to the shear plane. As a result, transitions from three- to two- to one-dimensional instabilities are observed to occur as shear rate increases, and above a certain shear rate the instabilities altogether disappear. The density patterns and flow structures that arise at long times in the suspensions are also analyzed from the numerical simulations using standard techniques from the literature on turbulent flows. The imposed shear flow is found to have an effect on both density patterns and flow structures, which typically align with the extensional axis of the external flow. The disturbance flows in the simulations are shown to exhibit similarities with turbulent flows. However, the flows described herein are also significantly different owing to the strong predominance of large scales, as exemplified by the very rapid decay of the kinetic energy spectrum, an effect further enhanced after the transitions to two- and one-dimensional instabilities.

Active particle suspensions, of which a bath of swimming bacteria is a paradigmatic example, are characterized by complex dynamics involving strong fluctuations and large-scale correlated motions. These motions, which result from the many-body interactions between particles, are biologically relevant as they impact mean particle transport, mixing and diffusion, with possible consequences for nutrient uptake and the spreading of bacterial infections. To analyze these effects, a kinetic theory is presented and applied to elucidate the dynamics and pattern formation arising from mean-field interactions. Based on this model, the stability of both aligned and isotropic suspensions is investigated. In aligned suspensions, an instability is shown to always occur at finite wavelengths, in agreement with previous predictions and simulations. In isotropic suspensions, a new instability for the active particle stress is also found to exist, in which shear stresses are eigenmodes and grow exponentially at low wavenumbers, resulting in large-scale fluctuations in suspensions of pusher particles above a threshold concentration. Numerical simulations of the kinetic equations are also performed, and applied to study the long-time nonlinear dynamics, which are characterized by transient particles clusters that form and break up in time, as well as complex chaotic flows correlated on the system size. The predictions from the kinetic model are then tested using direct particle simulations accounting for multi-body hydrodynamic interactions between model microswimmers: these simulations confirm the existence of a transition to correlated motions and large-scale flows above a certain volume fraction, as demonstrated by a sharp increase in density fluctuations, velocity correlation lengths, and mean particle velocities. The effect of this transition on fluid mixing is also investigated, and the emergence of large-scale flows is shown to significantly enhance convective mixing. To conclude, consequences of particle activity on the effective rheology of the suspensions are briefly discussed. We demonstrate that the rheology is characterized by much stronger normal stress differences than in passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.

Many creatures navigate their world through undulation – the unidirectional propagation of bending waves along the body. Undulatory locomotion in a fluid is well studied, at least at low Reynolds number. There, undulation breaks time-reversal symmetry and an organism can locomote by using the anisotropy of fluid drag with respect to body shape. On land, limbless creatures such as snakes also use undulation to traverse "featureless" surfaces with relative ease. I will discuss theoretical models and experimental observations that illustrate how snakes accomplish this by using the frictional anisotropy provided their scales, as well as selective body lifting. To provide another example of an undulator in action, I will discuss some recent modeling and experiments that show how swimming nematodes interact with microfluidic environments filled with immovable obstacles.

Joint work with Ahammed Anwar Chengala.

The effects of flow shear on the swimming behavior of halophilic microalga Dunaliella primolecta is examined by an in-house developed digital holographic microscopy and microfluidic channel. To investigate the shear-induced response, the algal culture is injected into a channel with a cross section of 3.5 x 0.4 mm at several fluid flow rates, generating shear rates that are consistent with the energy dissipation levels in estuaries, coastal waters, and lakes. We quantified the kinematics of D. primolecta by the estimates of 3D swimming velocities, auto-correlation swimming velocities, kinetic spectral densities and swimming-induced dispersion. Preliminary analysis indicate that swimming velocities and dispersion were strongly mediated by local fluid shear rates. On-going analysis is aimed to reveal scaling parameters and functional relationships among small-scale fluid motion and microorganism motility characteristics.

Joint work with Harsh Agarwal.

There is growing interest in understanding microscale biophysical processes such as the kinematics and dynamics of swimming microorganisms, and their interactions with surrounding fluids. Statistically robust experimental observations on swimming characteristics of bacteria in a wall bounded shear flow are crucial for understanding cell attachment and detachment during the initial formation of a biofilm. In this paper, we integrate microfluidics and holography to measure 3-D trajectories of a model bacteria, Escherichia coli (AW405), subjecting to a carefully controlled shear flow. Experiments are conducted in a straight mchannel of 40x3x0.2 mm with shear rates up to 200 (1/s). Holographic microscopic movies recorded at 40X magnification and 15 fps are streamed real-time to a data acquisition computer for an extended period of time (>5 min) that allows us to examine long term responses of bacteria in the presence of flow shear. Three-dimensional locations and orientations of bacteria are extracted with a resolution of 0.185 μm in lateral directions and 0.5 μm in the wall normal direction. The 3-D trajectories are tracked by an in-house developed particle tracking algorithm. Over three thousand of 3D trajectories over a sample volume of 380×380×200 μm have been obtained for our control (quiescent flow). Swimming characteristics, i.e. swimming velocities, Lagrangian spectra, dispersion coefficients, is extracted to quantify the cell-flow and cell-wall interactions. Preliminary results have revealed that near wall hydrodynamic interactions, i.e. swimming in circles and reducing lateral migration, cause the reduction in wall-normal dispersion, subsequently are responsible for wall trapping and prompting attachment. On-going analysis is to understand the effects of shear flow on such a mechanism.

Interest is growing rapidly in understanding the swimming behaviour of micro-organisms near solid surfaces. This is an important aspect of biofilm initiation and has significant implications for the shipping, water and medical industries. Through boundary element methods, we can accurately simulate hydrodynamic interactions between a single bacterium and solid surfaces even when the separation distance is small. Past experiments and analytical arguments have shown that bacteria propelled by a flagellum tend to follow curved trajectories rather than straight when swimming near a surface. Our simulations verify this and we find that there can also be a stable separation from the wall, leading to circular orbits. We show that parameters controlling the shape of the swimmer can significantly influence this equilibrium distance. Hence, this model suggests that certain "designs" of bacteria accumulate at boundaries while others do not.

A number of findings from recent works are presented. At intermediate Reynolds number, where both inertia and viscous dissipation are important, we have observed through experiment and numerical simulation a number of counter-intuitive behaviors in a flapping wing system with passive pitching [joint work with L. Moret, M. Shelley, and J. Zhang]. The behavior of shape-changing bodies are also considered, along with consequences on vortex shedding and vortex-interaction dynamics, driven either by a recoil force from internal oscillations (recoil) or by an external background flow (hovering) [joint work with M. Shelley, S. Childress, and T. Tokieda]. Other problems are investigated at low Reynolds number, where viscous dissipation dominates inertial effects. These include the effects of elastic bending costs on the optimal swimming shape of slender bodies, the locomotion of bilayer vesicles, and the swimming behavior and efficiency of a fluid-extruding body ("jet propulsion") at zero Reynolds number [joint work with E. Lauga and A. Evans].

A body immersed in a highly viscous fluid can locomote by drawing in and expelling fluid through pores at its surface. We consider this mechanism of jet propulsion without inertia in the case of spheroidal bodies, and derive both the swimming velocity and the hydrodynamic efficiency. Elementary examples are presented, and exact axisymmetric solutions for spherical, prolate spheroidal, and oblate spheroidal body shapes are provided. In each case, entirely and partially porous (i.e. jetting) surfaces are considered, and the optimal jetting flow profiles at the surface for maximizing the hydrodynamic efficiency are determined computationally. The maximal efficiency which may be achieved by a sphere using such jet propulsion is 12.5%, a significant improvement upon traditional flagella-based means of locomotion at zero Reynolds number. Unlike other swimming mechanisms which rely on the presentation of a small cross section in the direction of motion, the efficiency of a jetting body at low Reynolds number increases as the body becomes more oblate, and limits to approximately 162% in the case of a flat plate swimming along its axis of symmetry. Our results are discussed in the light of slime extrusion mechanisms occurring in many cyanobacteria. (Joint work with E. Lauga)

Blebbing occurs when the cytoskeleton detaches from the cell membrane, resulting in the pressure-driven flow of cytosol towards the area of detachment and the local expansion of the cell membrane. Recent interest has focused metastatic cancer cells that use blebs for cell motility. We present a dynamic computational model of the cell that includes mechanics of and the interactions between the intracellular fluid, the actin cortex, and the cell membrane. The cortex is an active, elastic, permeable material, which moves with a velocity distinct from that of the background fluid. The Immersed Boundary Method is modified to account for the relative motion between the cortex and the fluid. The computational model is used to explore several hypotheses for bleb formation, and these simulations are compared to experimental results. Additionally, a pressure threshold for bleb initiation, which depends crucially on the constitutive law for the membrane, is predicted based on reduced analytic models. These predictions are further explored in the full computational model and identified with underlying cellular processes.

In order to develop better methods for diagnosis and treatment of infertility, as well as safer contraceptives, more must be learned about how mammalian sperm move through the female reproductive tract. Crucial phases of mammalian sperm transport include passage through the cervix and uterotubal junction, storage of sperm in the oviductal storage reservoir, release from the reservoir, and location of the egg. There is some evidence for the existence of special passageways for sperm in the cervix, but this needs to be demonstrated and the mechanism of guiding sperm through the cervix needs to be elucidated. Passage of sperm through the uterotubal junction requires sperm to have certain proteins, but how these proteins function is not known. There is evidence that sperm must undergo motility hyperactivation in order to be released from the oviductal storage reservoir; however, the process is not understood. Finally, it is not clear whether there are chemotactic agents that emanate from the vicinity of the egg to modulate sperm flagellar beating patterns in order to guide them toward the egg. There are three main areas in which bioengineers can provide crucial help for elucidating these mysteries: (1) by developing a method for measuring and comparing sperm flagellar bending patterns, (2) by improving optical equipment for viewing the movement of sperm within the female reproductive tract, and (3) by developing chambers that mimic the physical environment of the tract so that molecular mechanisms that regulate sperm movement can be elucidated. USDA CSREES NRICGP 2008-35203-19031 and NIH 1RO3HD062471-01.

Joint work with Haixin Chang (Cornell University) .

Hyperactivation, a swimming pattern used by mammalian sperm in the oviduct, is essential for fertilization. It is characterized by highly asymmetrical flagellar beating and an increase of cytoplasmic Ca²⁺. We observed that some mouse sperm swimming in the oviduct produce high-amplitude pro-hook bends (bends in the direction of the hook on the head) while others produce high-amplitude anti-hook bends. Switching direction of the high-amplitude bends could serve to re-direct sperm toward oocytes. Our objective was to test the hypothesis that different Ca²⁺ cell signaling pathways produce pro-hook and anti-hook patterns. In vitro, sperm that hyperactivated during capacitation (a process that prepares sperm for fertilization) swam using large pro-hook bends, which resulted from influx of Ca²⁺ through plasma membrane CatSper channels. The anesthetic procaine and the K⁺-channel blocker 4-Aminopyridine (4-AP) also each induced large pro-hook bends. In contrast, thimerosal, which triggers Ca²⁺ release from an intracellular Ca²⁺ storage site, induced large anti-hook bends. When capacitated sperm were treated with thimerosal, 90% switched from pro-hook to anti-hook bending. Sperm loaded with the fluorescent Ca²⁺ indicator Fluo-4 AM revealed that thimerosal initiated a Ca²⁺ increase at the base of the flagellum, while 4-AP initiated an increase in the principal piece of the flagellum. Proteins were extracted from sperm for examination of phosphorylation patterns induced by Ca²⁺ signaling. Procaine and 4-AP treatments phosphorylated threonine and serine residues of some proteins, whereas thimerosal treatment dephosphorylated some proteins. Tyrosine phosphorylation was unaffected. We concluded that pro-hook hyperactivation, associated with sperm capacitation, can be modulated by a distinct Ca²⁺ signaling system to re-direct sperm toward oocytes. NIH 1R03HD062471-01.

Seed dispersal is the means by which plants expand and colonize new areas. To maximize their range, some plants have developed elaborate gliding, spinning or tumbling winged seedpods, whose aerodynamics enable them to extend their flight time and range. Such winged seedpods are often light and thin, which generally decreases their surface loading and hence their rate of descent. As a consequence, they can be flexible. We are broadly interested in elucidating the role of flexibility in passive flight. The influence of flexibility on the flight of autorotating winged seedpods is examined through an experimental investigation of tumbling rectangular paper strips freely falling in air. Our results suggest the existence of a critical length above which the wing bends. We develop a theoretical model that demonstrates that this buckling is prompted by inertial forces associated with the tumbling motion, and yields a buckling criterion consistent with that observed. We further develop a reduced model for the flight dynamics of flexible tumbling wings that illustrates the effect of aeroelastic coupling on flight characteristics and rationalizes experimentally observed variations in the wing's falling speed and range.

Keywords: locomotion, motion planning, verification, control, robotic birds, perching

Abstract: Locomotion in fluids (and on terrain) often involves complex nonlinear dynamics and non-trivial notions of stability including limit cycles and dynamically stable maneuvers. In this talk I will describe some new algorithms for automatically verifying stability (via a Lyapunov function) and estimating regions of attraction for dynamic nonlinear locomotion. These tools have important implications for motion planning and feedback design, which I will demonstrate by describing our attempts to build robots that fly like a bird and execute post- stall maneuvers to land on a perch.

I will first give an brief overview of the immersed interface method for fluid-solid interaction. I will then focus on the application of the method to numerical simulation of insect flight. In particular, I will present (1) a boundary condition capturing approach for the prescribed kinematics of an insect wing, (2) a matrix formulation of the Newton dynamics for insect flight, and (3) a coupling approach to couple the aerodynamics and Newton dynamics of insect flight.

Joint work with D. W. Murphy¹, D.R. Webster¹, S. Kawaguchi², R. King², and F. Sotiropoulos³.

The locomotion of Antarctic krill (Euphausia superba) is known to depend on the metachronal paddling of the animal’s five pairs of pleopods. A wave passing along these swimming appendages from posterior to anterior transfers momentum to the surrounding fluid, thus producing thrust. The kinematics of these pleopods, however, have not been fully characterized. Determining the kinematics of krill in various swimming modes will shed light on the fluid mechanics of krill locomotion and thereby deepen our understanding of krill sensing and schooling. High speed footage (250 fps) of freely swimming juvenile and adult Antarctic krill was acquired at the Australian Antarctic Division in Hobart, Tasmania. Various swimming modes were identified based on swimming angle and behavior, and two-dimensional kinematic parameters such as pleopod stroke frequency, amplitude, stroke overlap, and animal velocity were investigated as a function of these swimming modes. The variability of these parameters over time provides insight into the high sensitivity and responsiveness of krill to their hydrodynamic environment. Useful comparisons can also be made to previously gathered kinematics data for pacific krill (Euphausia pacifica), which live in a much lower viscosity environment. These parameters will prove necessary in future computational fluid dynamics (CFD) simulations of krill locomotion.

¹ School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0355 USA ² Australian Antarctic Division, Kingston, Tasmania, Australia 7050 ³ Saint Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55414

Joint work with Kourosh Shoele, Dept of Structural Engr, UCSD.

Fins of bony fishes are characterized by a skeleton-reinforced membrane structure consisting of a soft collagen membrane strengthened by embedded flexible rays. Morphologically, each ray is connected to a group of muscles so that the fish can control the rotational motion of each ray individually, enabling multi-degree of freedom control over the fin motion and deformation. We have developed a fluid-structure interaction model to simulate the kinematics and dynamic performance of a structurally idealized fin. This method includes a boundary-element model of the fluid motion and a fully-nonlinear Euler-Bernoulli beam model of the embedded rays. Using this model we studied thrust generation and propulsion efficiency of the fin at different combinations of parameters. Effects of kinematic as well as structural properties are examined. It has been illustrated that the fish’s capacity to control the motion of each individual ray, as well as the anisotropic deformability of the fin determined by distribution of the rays (especially the detailed distribution of ray stiffness), are essential to high propulsion performance. Finally, we note that this structural design is a recurring motif in nature. Several similar biostructures will be discussed.