A double-layer integral equation for the surface tractions on a body moving in a viscous fluid is derived, allowing for the incorporation of a background flow and/or the presence of a plane wall. The Lorentz reciprocal theorem is used to link the surface tractions on the body to integrals involving the background velocity and stress fields on an imaginary bounding sphere (or hemisphere for wall-bounded flows). The derivation requires the velocity and stress fields associated with numerous fundamental singularity solutions which we provide for free-space and wall-bounded domains. Two sample applications of the method are discussed: we study the tractions on an ellipsoid moving near a plane wall, which provides a more detailed understanding of the well-studied glancing and reversing trajectories, and we explore a new problem, erosion of bodies by a viscous flow, in which the surface is ablated at a rate proportional to the local viscous shear stress. Simulations and analytical estimates suggest that a spherical body in a uniform flow first reduces to nearly (but not exactly) the drag minimizing profile of a body in a Stokes flow and then vanishes in finite time. The shape dynamics of an eroding body in a shear flow and near a wall are also investigated. Stagnation points of the flow on the body lead generically to the formation of cusps, whose number depends on the flow configuration and/or the presence of a nearby wall.pdf J. Comput. Phys.
Recent experiments and numerical simulations have shown that certain types of microorganisms "reflect" off of a flat surface at a critical angle of departure, independent of the angle of incidence. The nature of the reflection may be active (cell and flagellar contact with the surface) or passive (hydrodynamic) interactions. We explore the billiard-like motion of such a body inside a regular polygon and show that the dynamics can settle on a stable periodic orbit, or can be chaotic, depending on the swimmer's departure angle and the domain geometry. The dynamics are often found to be robust to the introduction of weak random fluctuations. The Lyapunov exponent of swimmer trajectories can be positive or negative, can have extremal values, and can have discontinuities depending on the degree of the polygon. A passive sorting device is proposed that traps swimmers of different departure angles into separate bins. We also study the external problem of a microorganism swimming in a patterned environment of square obstacles, where the departure angle dictates the possibility of trapping or diffusive trajectories.pdf Physica D
The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the interface gradient. The gradient of the level set function represents the stretching of the interface, which is of critical importance in many physical applications. The proposed locally gradient-preserving reinitialization (LGPR) method involves the solution of three PDEs of Hamilton-Jacobi type in succession; first the signed distance function is found using a traditional reinitialization technique, then the interface gradient is extended into the domain by a transport equation, and finally the new level set function is achieved with the solution of a generalized reinitialization equation. We prove the well-posedness of the Hamilton-Jacobi equations, with possibly discontinuous Hamiltonians, and propose numerical schemes for their solutions. A subcell resolution technique is used in the numerical solution of the transport equation to extend data away from the interface directly with high accuracy. The reinitialization technique is computationally inexpensive if the PDEs are solved only in a small band surrounding the interface. As an important application, the LGPR method will enable the application of the local level set approach to the Eulerian Immersed boundary method.pdf J. Sci. Comput.
Liquid crystals (LCs) are anisotropic, viscoelastic fluids that can be used to direct colloids (e.g., metallic nanorods) into organized assemblies with unusual optical, mechanical, and electrical properties. In past studies, the colloids have been sufficiently rigid that their individual shapes and properties have not been strongly coupled to elastic stresses imposed by the LCs. Herein, we explore how soft colloids (micrometer-sized shells formed from phospholipids) behave in LCs. We reveal a sharing of strain between the LC and shells, resulting in formation of spindle-like shells and other complex shapes and also, tuning of properties of the shells (e.g., barrier properties). These results hint at previously unidentified designs of reconfigurable soft materials with applications in sensing and biology.pdf PNAS
We describe the controlled transport and delivery of polymer microparticles and non-motile eukaryotic cells by swimming bacteria suspended in nematic liquid crystals. The bacteria are shown to push reversibly attached cargo in a stable, unidirectional path (or along a complex patterned director field) over exceptionally long distances. Numerical simulations and an analytical prediction for swimming speeds provide a mechanistic insight into the hydrodynamics of the system. This study lays the foundation for using cargo-carrying bacteria in engineering applications and for understanding interspecies interactions in polymicrobial communities.pdf Soft Matter
Microorganisms often encounter anisotropy, for example in mucus and biofilms. We study how anisotropy and elasticity of the ambient fluid affects the speed of a swimming microorganism with a prescribed stroke. Motivated by recent experiments on swimming bacteria in anisotropic environments, we extend a classical model for swimming microorganisms, the Taylor swimming sheet, actuated either by transverse or longitudinal traveling waves in a three-dimensional nematic liquid crystal without twist. We calculate the swimming speed and entrained volumetric flux as a function of the swimmer's stroke properties as well as the elastic and rheological properties of the liquid crystal. The behavior is quantitatively and qualitatively well-approximated by a hexatic liquid crystal except in the cases of small Ericksen number and in a nematic fluid with tumbling parameter near the transition to a flow-aligning nematic, where anisotropic effects dominate. We also propose a novel method of swimming or pumping in a nematic fluid by passing a traveling wave of director oscillation along a flat rigid wall.pdf Soft Matter
The sedimentation of a rigid particle near a wall in a viscous fluid has been studied numerically by many authors, but analytical solutions have been derived only for special cases such as the motion of spherical particles. In this paper the method of images is used to derive simple ordinary differential equations describing the sedimentation of arbitrarily oriented prolate and oblate spheroids at zero Reynolds number near a vertical or inclined plane wall. The reduced order system may be solved analytically in many situations, and full trajectories are predicted which compare favorably with complete numerical simulations, performed using a novel double layer boundary integral formulation, a method of stresslet images. The conditions under which the glancing and reversing trajectories, first observed by Russel et al. (1977), occur are studied for bodies of arbitrary aspect ratio. Several additional trajectories are also described: a periodic tumbling trajectory for nearly spherical bodies, a linearly stable sliding trajectory which appears when the wall is slightly inclined, and three-dimensional glancing, reversing, and wobbling.pdf J. Fluid Mech.
Motivated by recent experiments, we consider the hydrodynamic capture of a microswimmer near a stationary spherical obstacle. Simulations of model equations show that a swimmer approaching a small spherical colloid is simply scattered. In contrast, when the colloid is larger than a critical size it acts as a passive trap: the swimmer is hydrodynamically captured along closed trajectories and endlessly orbits around the colloidal sphere. In order to gain physical insight into this hydrodynamic scattering problem, we address it analytically. We provide expressions for the critical trapping radius, the depth of the "basin of attraction," and the scattering angle, which show excellent agreement with our numerical findings. We also demonstrate and rationalize the strong impact of swimming-flow symmetries on the trapping efficiency. Finally, we give the swimmer an opportunity to escape the colloidal traps by considering the effects of Brownian, or active, diffusion. We show that in some cases the trapping time is governed by an Ornstein-Uhlenbeck process, which results in a trapping time distribution that is well-approximated as inverse-Gaussian. The predictions again compare very favorably with the numerical simulations. We envision applications of the theory to bioremediation, microorganism sorting techniques, and the study of bacterial populations in heterogeneous or porous environments.pdf Soft Matter
Recent experiments have shown that floating paramagnetic beads, under the influence of an oscillating background magnetic field, can move along a liquid-air interface in a sustained periodic locomotion. The locomotion arises from a periodically induced dipole-dipole repulsion between the beads acting in concert with a capillary attraction. We investigate analytically and numerically the stability and dynamics of this magnetocapillary swimming, and explore other related topics including the steady and periodic equilibrium configurations of two and three beads. The swimming speed and system stability depend on a dimensionless measure of the relative repulsive and attractive forces which we term the magnetocapillary number. An oscillatory magnetic field may stabilize an otherwise unstable collinear configuration, and striking behaviors are observed in fast transitions to and from locomotory states, offering insight into the behavior and self-assembly of interface-bound micro-particles.pdf Soft Matter
We study helical bodies of arbitrary cross-sectional profile as they swim or transport fluid by the passage of helical waves. Many cases are explored: the external flow problem of swimming in a cylindrical tube or an infinite domain, the internal fluid pumping problem, and confined/unconfined swimming and internal pumping in a viscoelastic (Oldroyd-B) fluid. A helical coordinate system allows for the analytical calculation of swimming and pumping speeds and fluid velocities in the asymptotic regime of nearly-cylindrical bodies. In a Newtonian flow, a matched asymptotic analysis results in corrections to the swimming speed accurate to fourth-order in the small wave amplitude, and the results compare favorably with full numerical simulations. We find that the torque-balancing rigid body rotation generally opposes the direction of wave passage, but not always. Confinement can result in local maxima and minima of the swimming speed in the helical pitch, and the effects of confinement decrease exponentially fast with the diameter of the tube. In a viscoelastic fluid, we find that the effects of fluid elasticity on swimming and internal pumping modify the Newtonian results through the mode-dependent complex viscosity, even in a confined domain.pdf Phys. Fluids
This book serves as an introduction to the continuum mechanics and mathematical modeling of complex fluids in living systems. The form and function of living systems are intimately tied to the nature of surrounding fluid environments, which commonly exhibit nonlinear and history-dependent responses to forces and displacements. With ever-increasing capabilities in the visualization and manipulation of biological systems, research on the fundamental phenomena, models, measurements, and analysis of complex fluids has taken a number of exciting directions. In this book, many of the world's foremost experts explore key topics such as:
In this chapter we introduce the fundamental concepts in Newtonian and complex fluid mechanics, beginning with the basic underlying assumptions in continuum mechanical modeling. The equations of mass and momentum conservation are derived, and the Cauchy stress tensor makes its first of many appearances. The Navier-Stokes equations are derived, along with their inertialess limit, the Stokes equations. Models used to describe complex fluid phenomena such as shear-dependent viscosity and viscoelasticity are then discussed, beginning with generalized Newtonian fluids. The Carreau-Yasuda and power-law fluid models receive special attention, and a mechanical instability is shown to exist for highly shear-thinning fluids. Differential constitutive models of viscoelastic flows are then described, beginning with the Maxwell fluid and Kelvin-Voigt solid models. After providing the foundations for objective (frame-independent) derivatives, the linear models are extended to mathematically sound nonlinear models including the upper-convected Maxwell and Oldroyd-B models, and others. A derivation of the upper-convected Maxwell model from the kinetic theory perspective is also provided. Finally, normal stress differences are discussed, and the reader is warned about common pitfalls in the mathematical modeling of complex fluids.Chapter Springer
The swimming behavior of bacteria and other microorganisms is sensitive to the physical properties of the fluid in which they swim. Mucus, biofilms, and artificial liquid-crystalline solutions are all examples of fluids with some degree of anisotropy that are also commonly encountered by bacteria. In this paper, we study how liquid-crystalline order affects the swimming behavior of a model swimmer. The swimmer is a one-dimensional version of G. I. Taylor's swimming sheet: an infinite line undulating with small-amplitude transverse or longitudinal traveling waves. The fluid is a two-dimensional hexatic liquid-crystalline film. We calculate the power dissipated, swimming speed, and flux of fluid entrained as a function of the swimmer's waveform as well as properties of the hexatic film, such as the rotational and shear viscosity, the Frank elastic constant, and the anchoring strength. The departure from isotropic behavior is greatest for large rotational viscosity and weak anchoring boundary conditions on the orientational order at the swimmer surface. We even find that if the rotational viscosity is large enough, the transverse-wave swimmer moves in the opposite direction relative to a swimmer in an isotropic fluid.pdf Phys. Rev. E
Suspensions of sedimenting slender fibres in a viscous fluid are known to be unstable to fluctuations of concentration. In this paper we develop a theory for the role of fibre flexibility in sedimenting suspensions in the asymptotic regime of weakly-flexible bodies (large elasto-gravitation number). Unlike the behaviour of straight fibres, individual flexible filaments rotate as they sediment, leading to an anisotropic base-state of fibre orientations in an otherwise homogeneous suspension. A mean-field theory is derived to describe the evolution of fibre concentration and orientation fields, and we explore the stability of the base-state to perturbations of fibre concentration. We show that fibre flexibility affects suspension stability in two distinct and competing ways: the anisotropy of the base-state renders the suspension more unstable to perturbations, while individual particle self-rotation acts to prevent clustering and stabilizes the suspension. In the presence of thermal noise, the dominant effect depends critically upon the relative scales of flexible fibre self-rotation compared to rotational Brownian motion.pdf J. Fluid Mech.
Rotating helical bodies of arbitrary cross-sectional profile and infinite length are explored as they swim through or transport a viscous fluid. The Stokes equations are studied in a helical coordinate system, and closed form analytical expressions for the force-free swimming speed and torque are derived in the asymptotic regime of nearly cylindrical bodies. High-order accurate expressions for the velocity field and swimming speed are derived for helical bodies of finite pitch angle through a double series expansion. The analytical predictions match well with the results of full numerical simulations, and accurately predict the optimal pitch angle for a given cross-sectional profile. This work may improve the modeling and design of helical structures used in microfluidic manipulation, synthetic microswimmer engineering, and the transport and mixing of viscous fluidspdf Phys. Fluids
The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectional thickness are considered. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes. These are shown to match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. The instability travels as a wave along the filament opposite the direction of gravity as it grows and the predicted growth rates are shown to compare favourably with numerical simulations. The linear eigenmodes of the governing equation are also studied, which agree well with the finite-amplitude buckled shapes arising in simulationspdf J. Fluid Mech.
The motion of a rotating helical body in a viscoelastic fluid is considered. In the case of force-free swimming, the introduction of viscoelasticity can either enhance or retard the swimming speed and locomotive efficiency, depending on the body geometry, fluid properties, and the body rotation rate. Numerical solutions of the Oldroyd-B equations show how previous theoretical predictions break down with increasing helical radius or with decreasing filament thickness. Helices of large pitch angle show an increase in swimming speed to a local maximum at a Deborah number of order unity. The numerical results show how the small-amplitude theoretical calculations connect smoothly to the large-amplitude experimental measurements.pdf Phys. Rev. Lett.
When a flexible filament is confined to a fluid interface, the balance between capillary attraction, bending resistance, and tension from an external source can lead to a self-buckling instability. We perform an analysis of this instability and provide analytical formulae that compare favorably with the results of detailed numerical computations. The stability and long-time dynamics of the filament are governed by a single dimensionless elastocapillary number quantifying the ratio between capillary to bending stresses. Complex, folded filament configurations such as loops, needles, and racquet shapes may be reached at longer times, and long filaments can undergo a cascade of self-folding events.pdf Soft Matter
The swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.pdf J. Fluid Mech.
In addition to conventional planar and helical flagellar waves, insect sperm flagella have also been observed to display a double-wave structure characterized by the presence of two super- imposed helical waves. In this paper, we present a hydrodynamic investigation of the locomotion of insect spermatozoa exhibiting the double-wave structure, idealized here as superhelical waves. Resolving the hydrodynamic interactions with a non-local slender body theory, we predict the swimming kinematics of these superhelical swimmers based on experimentally collected geometric and kinematic data. Our consideration provides insight into the relative contributions of the major and minor helical waves to swimming; namely, propulsion is owing primarily to the minor wave, with negligible contribution from the major wave. We also explore the dependence of the propulsion speed on geometric and kinematic parameters, revealing counterintuitive results, particularly for the case when the minor and major helical structures are of opposite chirality.pdf J. R. Soc. Interface
Most bacteria swim through fluids by rotating helical flagella which can take one of 12 distinct polymorphic shapes, the most common of which is the normal form used during forward swimming runs. To shed light on the prevalence of the normal form in locomotion, we gather all available experimental measurements of the various polymorphic forms and compute their intrinsic hydrodynamic efficiencies. The normal helical form is found to be the most efficient of the 12 polymorphic forms by a significant margin - a conclusion valid for both the peritrichous and polar flagellar families, and robust to a change in the effective flagellum diameter or length. Hence, although energetic costs of locomotion are small for bacteria, fluid mechanical forces may have played a significant role in the evolution of the flagellum.pdf Phys. Rev. Lett.
The locomotion of a body through an inviscid incompressible fluid, such that the flow remains irrotational everywhere, is known to depend on inertial forces and on both the shape and the mass distribution of the body. In this paper we consider the influence of fluid viscosity on such inertial modes of locomotion. In particular we consider a free body of variable shape and study the centre-of-mass and centre-of-volume variations caused by a shifting mass distribution. We call this recoil locomotion. Numerical solutions of a finite body indicate that the mechanism is ineffective in Stokes flow but that viscosity can significantly increase the swimming speed above the inviscid value once Reynolds numbers are in the intermediate range 50-300. To study the problem analytically, a model which is an analogue of Taylor's swimming sheet is introduced. The model admits analysis at fixed, arbitrarily large Reynolds number for deformations of sufficiently small amplitude. The analysis confirms the significant increase of swimming velocity above the inviscid value at intermediate Reynolds numbers.pdf J. Fluid Mech.
Phys. Fluids "Research Highlights."
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, which corresponds to the potential flow created by a source dipole at the sphere center. 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.pdf Phys. Fluids
Selected to appear in the Virtual Journal of Biological Physics Research, April 15, 2010.
Motivated by recent advances in vesicle engineering, we consider theoretically the locomotion of shape-changing bilayer vesicles at low Reynolds number. By modulating their volume and membrane composition, the vesicles can be made to change shape quasi-statically in thermal equilibrium. When the control parameters are tuned appropriately to yield periodic shape changes, which are not time- reversible, the result is a net swimming motion over one cycle of shape deformation. For two classical vesicle models (spontaneous curvature and bilayer coupling), we numerically determine the sequence of vesicle shapes through an enthalpy minimization, as well as the fluid-body interactions by solving a boundary integral formulation of the Stokes equations. For both models, net locomotion can be obtained either by continuously modulating fore-aft asymmetric vesicle shapes or by crossing a continuous shape-transition region and alternating between fore-aft asymmetric and fore-aft symmetric shapes. The obtained hydrodynamic efficiencies are similar to those of other low Reynolds number biological swimmers and suggest that shape-changing vesicles might provide an alternative to flagella-based synthetic microswimmers.pdf Soft Matter
Selected to appear in the Virtual Journal of Biological Physics Research, March 15, 2010.
Motile eukaryotic cells propel themselves in viscous fluids by passing waves of bending deformation down their flagella. An infinitely long flagellum achieves a hydrodynamically optimal low-Reynolds number locomotion when the angle between its local tangent and the swimming direction remains constant along its length. Optimal flagella therefore adopt the shape of a helix in three dimensions smooth and that of a sawtooth in two dimensions nonsmooth. Physically, biological organisms or engineered microswimmers must expend internal energy in order to produce the waves of deformation responsible for the motion. Here we propose a physically motivated derivation of the optimal flagellum shape. We determine analytically and numerically the shape of the flagellar wave which leads to the fastest swimming for a given appropriately defined energetic expenditure. Our novel approach is to define an energy which includes not only the work against the surrounding fluid, but also (1) the energy stored elastically in the bending of the flagellum, (2) the energy stored elastically in the internal sliding of the polymeric filaments which are responsible for the generation of the bending waves microtubules, and (3) the viscous dissipation due to the presence of an internal fluid. This approach regularizes the optimal sawtooth shape for two-dimensional deformation at the expense of a small loss in hydrodynamic efficiency. The optimal waveforms of finite-size flagella are shown to depend on a competition between rotational motions and bending costs, and we observe a surprising bias toward half-integer wave numbers. Their final hydrodynamic efficiencies are above 6%, significantly larger than those of swimming cells, therefore indicating available room for further biological tuning.pdf Phys. Fluids
To better understand the role of wing and fin flexibility in flapping locomotion, we study through experiment and numerical simulation a freely moving wing that can pitch passively as it is actively heaved in a fluid. We observe a range of flapping frequencies corresponding to large horizontal velocities, a regime of underperformance relative to a clamped (non-pitching) flapping wing, and a surprising, hysteretic regime in which the flapping wing can move horizontally in either direction !despite left/right symmetry being broken by the specific mode of pitching". The horizontal velocity is shown to peak when the flapping frequency is near the immersed system's resonant frequency. Unlike for the clamped wing, we find that locomotion is achieved by vertically flapped symmetric wings with even the slightest pitching flexibility, and the system exhibits a continuous departure from the Stokesian regime. The phase difference between the vertical heaving motion and consequent pitching changes continuously with the flapping frequency, and the direction reversal is found to correspond to a critical phase relationship. Finally, we show a transition from coherent to chaotic motion by increasing the wing's aspect ratio, and then a return to coherence for flapping bodies with circular cross section.pdf Phys. Fluids
A means of swimming in a viscous fluid is presented, in which a swimmer with only two links rotates around a joint and then rehinges in a periodic fashion in what is here termed rehinging locomotion. This two-link rigid swimmer is shown to locomote with an efficiency similar to that of Purcell's well-studied three-link swimmer, but with a simpler morphology. The hydrodynamically optimal stroke of an analogous flexible biflagellated swimmer is also considered. The introduction of flexibility is found to increase the swimming efficiency by up to 520% as the body begins to exhibit wavelike dynamics, with an upper bound on the efficiency determined by a degeneracy in the limit of infinite flexibilitypdf Phys. Rev. E
Featured in "Outside JEB", J. Exp. Biol., 2009.
Motivated by recent experiments on the hovering of passive bodies, we demonstrate how a simple shape-changing body can hover or ascend in an oscillating background flow. We study this ratcheting effect through numerical simulations of the two-dimensional Navier-Stokes equations at intermediate Reynolds number. This effect could describe a viable means of locomotion or transport in such environments as a tidal pool with wave-driven sloshing. We also consider the velocity burst achieved by a body through a rapid increase in its aspect ratio, which may contribute to the escape dynamics of such organisms as jellyfish.pdf Phys. Fluids
We study the sedimentation of two identical but non-spherical particles sedimenting in a Stokesian fluid. Experiments and numerical simulations reveal periodic orbits wherein the bodies mutually induce an in-phase rotational motion accompanied by periodic modulations of sedimentation speed and separation distance. We term these "tumbling orbits" and find that they appear over a broad range of body shapespdf Phys. Rev. E