Publications 2010-2014

Collective Surfing of Chemically Active Particles
by H. Masoud and M. Shelley
in Physical Review Letters 112, 128304 (2014)
Abstract: We study theoretically the collective dynamics of immotile particles bound to a 2D surface atop a 3D  fluid layer. These particles are chemically active and produce a chemical concentration fi eld that creates surface-tension gradients along the surface. The resultant Marangoni stresses create flows that carry the particles, possibly concentrating them. For a 3D diff usion-dominated concentration fi eld and Stokesian fluid we show that the surface dynamics of active particle density can be determined using nonlocal 2D surface operators. Remarkably, we also show that for both deep or shallow fluid layers this surface dynamics reduces to the 2D Keller-Segel model for the collective chemotactic aggregation of slime mold colonies. Mathematical analysis has established that the Keller-Segel model can yield finite-time, fi nite-mass concentration singularities. We show that such singular behavior occurs in our finite-depth system, and study the associated 3D  flow structures.

Hydrodynamic capture of microswimmers into sphere-bound orbits
by D. Takagi, J. Palacci, A. Braunschweig, M. Shelley, and J. Zhang
in Soft Matter 10, 1784–1789 (2014)

Abstract: Self-propelled particles can exhibit surprising non-equilibrium behaviors, and how they interact  with obstacles remains an important open problem. We show experimentally that chemically propelled micro-rods can be captured, with little decrease in their speed, into close orbits around solid  spheres resting on a horizontal plane. This short-range interaction is consistent with a model, based  on lubrication theory, of a force- and torque-free swimmer driven by a surface slip and moving near  a solid surface. This study reveals the crucial aspects of interactions of self-propelled particles with  passive objects, and brings into question the use of colloidal tracers as probes of active matter.

Auto-chemotactic active suspensions: modeling, analysis and simulations
by E. Lushi, R. Goldstein, and M. Shelley
submitted (2014)

Abstract: Microorganisms can preferentially orient and move along gradients of a chemo-attractant (i.e., chemotax) while colonies of many microorganisms can collectively undergo complex dynamics in response to chemo-attractants that they themselves produce. For colonies or groups of micro-swimmers we investigate how an "auto-chemotactic" response that should lead to swimmer aggregation is a ffected by the non-trivial  fluid  flows that are generated by collective swimming. For this, we consider chemotaxis models based upon a hydrodynamic theory of motile suspensions that are fully coupled to chemo-attractant production, transport, and di usion. Linear analysis of isotropically ordered suspensions reveals both an aggregative instability due to chemotaxis that occurs independently of swimmer type, and a hydrodynamic instability when the swimmers are "pushers". Nonlinear simulations show nonetheless that hydrodynamic interactions can signi cantly modify the chemotactically-driven aggregation dynamics in suspensions of "pushers" or "pullers". Di fferent states of the dynamics resulting from these coupled interactions in the colony are discussed.

Self-similar Evolution of a Body eroding in a Fluid Flow
by M. Moore, L. Ristroph, S. Childress, J. Zhang, and M. Shelley
in Physics of Fluids 25, 116602 (2013)

Abstract: Erosion of solid material by flowing fluids plays an important role in shaping land-forms, and in this natural context is often dictated by processes of high complexity.  Here, we examine the coupled evolution of solid shape and fluid flow within the idealized setting of a cylindrical body held against a fast, unidirectional flow, and eroding under the action of fluid shear stress. Experiments and simulations both show self-similar evolution of the body, with an emerging quasi-triangular geometry that is an attractor of the shape dynamics. Our fluid erosion model, based on Prandtl boundary layer theory, yields a scaling law that accurately predicts the body's vanishing rate. Further, a class of exact solutions provides a partial prediction for the body's terminal form as one with a leading surface of uniform shear stress. Our simulations show this predicted geometry to emerge robustly from a range of di fferent initial conditions, and allow us to explore its local stability. The  sharp, faceted features of the terminal geometry defy the intuition of erosion as a globally smoothing process.

On the rotation of porous ellipsoids in  simple shear  flows
by H. Masoud, H. Stone, and M. Shelley
in Rapids, Journal of Fluid Mechanics 733, R6 (2013)

Abstract: We study theoretically the dynamics of porous ellipsoids rotating in simple shear  flows. We also examine the orientational di ffusion of permeable ellipses and spheroids in the  absence of a background fl ow. We use the Brinkman-Debye-Bueche (BDB) model to  simulate fl ow within and through particles and solve the coupled Stokes-BDB equations  to calculate the overall  flow fi eld and the rotation rate of porous ellipsoids. Our results  show that the permeability has little e ffect on the rotational behavior of particles, and  that the Jeff rey's prediction of the angular velocity of impermeable ellipsoids in simple  shear  flows (Proc. Roy. Soc. Lond. A, vol. 102, 1922, pp. 161-179) remains an excellent  approximation, if not an exact one, for porous ellipsoids. Employing an appropriate  scaling, we present approximate expressions for the orientational diff usion of ellipses and  spheroids. Our findings can serve as basis for developing a suspension theory for non- spherical porous particles.

Active Suspensions and Their Nonlinear Models
by D. Saintillan and M. Shelley
in Comptes Rendes Physique 14, 497-517(2013)

Abstract: Active suspensions, such as suspensions of self-propelled microorganisms and related synthetic microswimmers, are known to undergo complex dynamics and pattern formation as a result of hydrodynamic interactions. In this review, we summarize recent e fforts to model these systems using continuum  kinetic theories. We first derive a basic kinetic model for a suspension of self-propelled rodlike particles and discuss its stability and nonlinear dynamics. We then present extensions of this model to analyze the e ffective rheology of active suspensions in external flows, the eff ect of steric interactions in concentrated systems, and the dynamics of chemotactically responsive suspensions in chemical fields.

Instabilities and nonlinear dynamics of concentrated active suspensions
by B. Ezhilan, M. J. Shelley, D. Saintillan
in Physics of Fluids 25, 070607 (2013).

Abstract: Suspensions of active particles, such as motile microorganisms and artificial microswimmers, are known to undergo a transition to complex large-scale dynamics at high enough concentrations. While a number of models have demonstrated that hydrodynamic interactions can in some cases explain these dynamics, collective motion in experiments is typically observed at such high volume fractions that steric interactions between nearby swimmers are significant and cannot be neglected. This raises the question of the respective roles of steric vs hydrodynamic interactions in these dense systems, which we address in this paper using a continuum theory and numerical simulations...  

Optimization of chiral structures for micro-scale propulsion
by E. Keaveny, S. Walker, and M. Shelley
in Nanoletters 13, 531-537 (2013)

Abstract: Recent advances in micro- and nano-scale fabrication techniques allow for the construction of rigid, helically-shaped micro-swimmers that can be actuated using applied magnetic fields. These swimmers represent the first steps toward the development of micro-robots for targeted drug delivery and minimally invasive surgical procedures. To assess the performance of these devices and improve on their design, we perform shape optimization computations to determine swimmer geometries that maximize speed in the direction of a given applied magnetic torque. We directly assess aspects of swimmer shapes that have been developed in previous experimental studies, including helical propellers with elongated cross-sections, and attached payloads. From these optimizations, we identify key improvements to existing designs that result in swimming speeds that are 250 - 470% of their original values. 

Dispersion of Self-Propelled Rods Undergoing Fluctuation-Driven Flips
by D. Takagi, A. Braunschweig, J. Zhang and M. Shelley
in Physical Review Letters 110, 038301 (2013)

Abstract: Synthetic microswimmers may someday perform medical and technological tasks, but predicting their motion and dispersion is challenging. Here we show that chemically propelled rods tend to move on a surface along large circles but curiously show stochastic changes in the sign of the orbit curvature. By accounting for fluctuation-driven flipping of slightly curved rods, we obtain analytical predictions for the ensemble behavior in good agreement with our experiments. This shows that minor defects in swimmer shape can yield major long-term effects on macroscopic dispersion.

An actuated elastic sheet interacting with passive and active structures in a viscoelastic  fluid
by J. Chrispell, M. Shelley, L. Fauci
in Physics of Fluids 25, 013103

Abstract: We adapt the classic Taylor swimming sheet set-up to investigate both the transient and long-time dynamics of an actuated elastic sheet immersed in a viscoelastic  fluid as it interacts with neighboring structures. While the preferred kinematics of the sheet are speci ed, the  flexible sheet interacts with the surrounding fluid and other structures, and its realized kinematics emerges from this coupling. We use an immersed boundary framework to evolve the Oldroyd-B/Navier-Stokes equations and capture the spatial and temporal development of viscoelastic stresses and sheet shape. We compare the dynamics when the actuated sheet swims next to a free elastic membrane, with and without bending rigidity, and next to a fixed wall. We demonstrate that the sheets can exploit the neighboring structures to enhance their swimming speed and efficiency, and also examine how this depends upon fluid viscoelasticity. When the neighboring structure is likewise an actuated elastic sheet, we investigate the viscoelastic dynamics of phase-locking.

Sculpting of an erodible body by flowing water
by L. Ristroph, M. Moore, S. Childress, M. Shelley, and J. Zhang
in Proceedings of the National Academy of Science, USA 109, 19606-19609
Abstract: Erosion by flowing fluids carves striking landforms on Earth and also provides important clues to the past and present environments of other worlds. In these processes, solid boundaries both influence and are shaped by the surrounding fluid, but the emergence of morphology as a result of this interaction is not well understood. Here, we study the coevolution of shape and flow in the context of erodible bodies molded from clay and immersed in fast flowing water. Though commonly viewed as a smoothing process, we find that erosion sculpts pointed and corner-like features that persist as the solid shrinks. We explain these observations using flow visualization and a fluid mechanical model in which the local shear stress dictates the rate of material removal. Experiments and simulations show that this interaction ultimately leads to self-similarly receding boundaries and a unique front surface characterized by nearly uniform shear stress. This tendency toward conformity of stress offers a principle for understanding erosion in more complex geometries and flows, such as those present in nature.

Collective Chemotactic Dynamics in the Presence of Self-Generated Fluid Flows
by E. Lushi, R. Goldstein, and M. Shelley
in Rapid Communications, Physical Review E 86, 040902 (2012)

Abstract: In microswimmer suspensions locomotion necessarily generates fluid motion, and it is known that such flows can lead to collective behavior from unbiased swimming. We examine the complementary problem of how chemotaxis is affected by self-generated flows. A kinetic theory coupling run-and-tumble chemotaxis to the flows of collective swimming shows separate branches of chemotactic and hydrodynamic instabilities for isotropic suspensions, the first driving aggregation, the second producing increased orientational order in suspensions of “pushers” and maximal disorder in suspensions of “pullers.” Nonlinear simulations show that hydrodynamic interactions can limit and modify chemotactically driven aggregation dynamics. In puller suspensions the dynamics form aggregates that are mutually repelling due to the nontrivial flows. In pusher suspensions chemotactic aggregation can lead to destabilizing flows that fragment the regions of aggregation. 

A weak-coupling expansion for viscoelastic fluids applied to dynamic settling of a body
by M. Moore and M. Shelley
in Journal of Non-Newtonian Fluid Mechanics 183-184, 25-36 (2012)

Abstract: The  flow of viscoelastic fluids is an area in which analytical results are difficult to attain, yet can provide invaluable information. We present a weak-coupling expansion that allows for semi-analytical computations of viscoelastic fluid flows coupled to immersed structures. We apply the expansion to the transient benchmark problem of a rigid sphere settling from rest through a viscoelastic fluid using the Oldroyd-B model, and we recover the previously observed transient behavior. The theory presented here is in contrast to the retarded motion, or low Weissenberg number, expansions that have received much attention, and one advantage is that the weak-coupling expansion off ers information for order-one Weissenberg number. The expansion's limit of validity is closely related to the diluteness criterion for a Boger fluid. We extend the classical settling problem to include a time-dependent body-force, and show how the introduction of the forcing time-scale modi fies the body-dynamics.

Slithering Locomotion
by David Hu and Michael Shelley
in Natural Locomotion in Fluids and on Surfaces: Swimming, Flying, and Sliding,
edited by S. Childress, A. Hosoi,  W. Schultz, and J. Wang, Springer, New York (2012)

Abstract: Limbless terrestrial animals propel themselves by sliding their bellies along the ground. Although the study of dry solid-solid friction is a classical subject, the mechanisms underlying friction-based limbless propulsion have received little attention. We review and expand upon our previous work on the locomotion of snakes, who are expert sliders. We show that snakes use two principal mechanisms to slither on surfaces. First, their bellies are covered with scales that catch upon ground asperities, providing frictional anisotropy. Second, they are able to lift parts of their body slightly off the ground when moving. This reduces undesired frictional drag and applies greater pressure to the parts of the belly that are pushing the snake forwards. We review a theoretical framework that may be adapted by future investigators to understand other kinds of limbless locomotion.

Oscillations of a Layer of Viscoelastic Fluid under Steady Forcing
by B. Liu, M. Shelley, and J. Zhang
in Journal of Non-Newtonian Fluid Mechanics 175-176, 38-43 (2012)

Visco PIV
Abstract: We study the dynamics of a layer of viscoelastic fluid, in the Stokesian regime, that is driven from below by a 4 x 4 checkerboard pattern of rotating and counter-rotating disks. At low disk rotation rate (low Weissenberg number) the fluid flow response is slaved to the geometry of this forcing and divides into many steadily rotating cells, each contained within invariant manifolds issuing from hyperbolic stagnation points. As the rotation rate increases these fluid cells begin to oscillate periodically in a synchronized fashion. At a yet higher rotation rate, this temporally periodic flow disappears and is replaced by a richer, "turbulent" dynamics where the flow is delocalized from the forcing and has fluid cells that are continuously destroyed and reformed.

Fluid-Structure Interactions: Research in the Courant Institute's Applied Mathematics Laboratory
by S. Childress, M. Shelley, and J. Zhang
in Communications in Pure and Applied Mathematics 65, 1697-1721 (2012)

Abstract: The Applied Mathematics Laboratory is a research laboratory within the Mathematics Department of the Courant Institute. It was established to carry out physical experiments, modeling, and associated numerical studies in a variety of problems of interest to Courant faculty, postdocs, and graduate and undergraduate students. Most of the research to date has involved fluid mechanics, and we focus in this paper on the work that relates to the interaction of fluids with rigid, moveable, or flexible bodies.

Emergence of Coherent Structures and Large-Scale Flows in Motile Suspensions
by D. Saintillan and M. Shelley
in Journal of the Royal Society Interface 9, 571-585 (2012)

Abstract: The emergence of coherent structures, large-scale flows, and correlated dynamics in suspensions of motile particles such as swimming micro-organisms or artificial microswimmers is studied using direct particle simulations. Simulations are performed with periodic boundary conditions for various system sizes and suspension volume fractions, and clearly demonstrate a transition to large-scale correlated motions in suspensions of rear-actuated swimmers, or pushers, above a critical volume fraction and system size. This transition, which is not observed in suspensions of head-actuated swimmers, or pullers, is characterized by a sudden and sharp increase in fluid velocity correlation lengths, number density fluctuations, particle velocities, mixing efficiency, and passive tracer diffusivities. These observations are all consistent with and confirm for the first time a prediction from our previous mean-field kinetic theory, which states that instabilities will arise in uniform isotropic suspensions of pushers when the product of the linear system size with the suspension volume fraction exceeds a given threshold. Good quantitative agreement is found between the theoretically predicted threshold and its measured value in our simulations.

Experiments and Theory of Undulatory Locomotion in a Simple Structured Medium
by T. Majmudar, E. Keaveny, J. Zhang and M. Shelley
in Journal of the Royal Society Interface 9, 1809-1823 (2012)

Abstract: Undulatory locomotion of microorganisms through geometrically complex, fluidic environments is ubiquitous in nature and requires the organism to negotiate both hydrodynamic effects and geometric constraints. To understand locomotion through such media, we experimentally investigate swimming of the nematode Caenorhabditis elegans through fluid-filled arrays of micro-pillars and conduct numerical simulations based on a mechanical model of the worm that incorporates hydrodynamic and contact interactions with the lattice. We show that the nematode’s path, speed, and gait are significantly altered by the presence of the obstacles and depend strongly on lattice spacing. These changes and their dependence on lattice spacing are captured, both qualitatively and quantitatively, by our purely mechanical model. Using the model, we demonstrate that purely mechanical interactions between the swimmer and obstacles can produce complex trajectories, gait changes, and velocity fluctuations, yielding some of the life-like dynamics exhibited by the real nematode. Our results show that mechanics, rather than biological sensing and behavior, can explain some of the observed changes in the worm’s locomotory dynamics.

A model of cytoplasmically-driven microtubule-based motion in the single-celled C. elegans embryo
by T. Shinar, M. Mano, F. Piano, and M. Shelley
in Proceedings of the National Academy of Science, USA 108, 10508-10513 (2011)

Abstract: We present a model of cytoplasmically-driven microtubule-based pronuclear motion in the single-celled C. elegans embryo. In this model, a centrosome pair at the male pronucleus initiates stochastic microtubule (MT) growth. These MTs encounter motor proteins, distributed throughout the cytoplasm, that attach and exert a pulling force. The consequent MT-length dependent pulling forces drag the pronucleus through the cytoplasm. On physical grounds, we assume that the motor proteins also exert equal and opposite forces on the surrounding viscous cytoplasm, here modeled as an incompressible Newtonian fluid constrained within an ellipsoidal eggshell. This naturally leads to streaming flows along the MTs. Our computational method is based on an immersed boundary formulation which allows for the simultaneous treatment of fluid flow and the dynamics of structures immersed within. Our simulations demonstrate that the balance of MT pulling forces and viscous nuclear drag is sufficient to move the pronucleus, while simultaneously generating minus-end directed flows along MTs that are similar to the observed movement of yolk granules toward the center of asters. Our simulations show pronuclear migration, and moreover, a robust pronuclear centration and rotation very similar to that observed in vivo. We find also that the confinement provided by the eggshell significantly affects the internal dynamics of the cytoplasm, increasing by an order of magnitude the forces necessary to translocate and center the pronucleus.

Modeling and Simulation of Liquid Crystal Elastomers
by W. Zhu, Michael Shelley, and Peter Palffy-Muhoray
in Physical Review E 83, 051703 (2011)

Abstract: We consider a continuum model describing the dynamic behavior of nematic liquid crystal elastomers (LCEs) and implement a numerical scheme to solve the governing equations. In the model, the Helmholtz free energy and Rayleigh dissipation are used, within a Lagrangian framework, to obtain the equations of motion. The free energy consists of both elastic and liquid crystalline contributions, each of which is a function of the material displacement and the orientational order parameter. The model gives dynamics for the material displacement, the scalar order parameter and the nematic director, the latter two of which correspond to the orientational order parameter tensor. Our simulations are carried out by solving the governing equations using an implicit-explicit scheme and the Chebyshev polynomial method. The simulations show that the model can successfully capture the shape changing dynamics of LCEs that have been observed in experiments, and also track the evolution of the order parameter tensor.

A Stokesian Viscoelastic Flow: Transition to Oscillations and Mixing
by B. Thomases, M. Shelley, and J.-L. Thiffeault
in Physica D 240, 1602-1614 (2011)

Abstract: To understand observations of low Reynolds number mixing and flow transitions in viscoelastic fluids, we study numerically the dynamics of the Oldroyd-B viscoelastic fluid model. The fluid is driven by a simple time-independent forcing that, in the absence of viscoelastic stresses, creates a cellular flow with extensional stagnation points. We find that at O(1) Weissenberg number these flows lose their slaving to the forcing geometry of the background force, become oscillatory with multiple frequencies, and show continual formation and destruction of small-scale vortices. This drives flow mixing, the details of which we closely examine. These new flow states are dominated by a single-quadrant vortex, which may be stationary or cycle persistently from cell to cell.

Applying a Second-Kind Boundary Integral Equation for Surface Tractions in Stokes Flow
by E. Keaveny and M. Shelley
in Journal of Computational Physics 230, 2141-2159 (2011)

Abstract: A second-kind integral equation for the tractions on a rigid body moving in a Stokesian fluid is established using the Lorentz reciprocal theorem and a completed second-kind integral equation for a double-layer density. A second-order collocation method based on the trapezoidal rule is applied to the integral equation after appropriate singularity reduction. For translating prolate spheriods with various aspect ratios, the scheme is used to explore the effects of the choice of completion flow on the error in the numerical solution, as well as the condition number of the discretized integral operator. The approach is applied to obtain the velocity and viscous dissipation of rotating helices of circular cross section. These results are compared with both local and non-local slender body theories. Motivated by the design of artificial micro-swimmers, similar simulations are performed on previously unstudied helices of non-circular cross-section to determine the dependence of the velocity and propulsion efficiency on the cross-section aspect ratio and orientation.

Dynamics of Complex Bio-Fluids
by C. Hohenegger and M. Shelley
in New Trends in the Physics and Mechanics of Biological Systems. Edited by M. Ben-Amar, A. Goriely, M. Muller, and L. Cugliandolo, Oxford University Press (2011)

Introduction: Complex fluids, such as polymer solutions, particulate suspensions, and many biological fluids, form a broad class of liquids whose mechanical and dynamical properties must be described on multiple length scales. In recent years, biological flow phenomena involving complex fluids, such as peristaltic pumping and sperm motility in the reproductive tracts, have received much attention. It has also been fruitful to consider systems such as bacterial baths as complex fluids themselves when describing them at the macroscopic scale. One of the most challenging issues when characterizing the transport properties of these systems is capturing the interactions between the fluid and the suspended microstructures (e.g. polymer coils, colloidal particles, flexible and rigid fibers, and “active” particles such as bacteria). This is important, as it is these couplings that lead to very complicated dynamical structures and large-scale flow associated with mixing or enhanced swimming efficiency. In many cases, numerical simulations are as challenging as model development, since complex fluid systems can have many degrees of freedom.

Flapping and Bending Bodies Interacting with Fluid Flows
by M. Shelley and J. Zhang
in Annual Review of Fluid Mechanics
43, 449-465 (2011)
              Buddhist monks in argument.
Abstract: The flapping or bending of a flexible planar structure in a surrounding fluid flow, which includes the flapping of flags and the self-streamlining of flexible bodies, constitutes a central problem in the field of fluid-body interactions. Here we review recent, highly detailed experiments that reveal new nonlinear phenomena in these systems, as well advances in theoretical understanding, resulting in large part from the rapid development of new simulational methods that fully capture the mutual coupling of fluids and flexible solids.

Focused Force Transmission through an Aqueous Suspension of Granules
by B. Liu, M. Shelley, and J. Zhang
in Physical Review Letters 105, 188301
                      cover of PRL, vol 105
Abstract: We investigate force transmission through a layer of shear-thickening fluid, here a concentrated aqueous cornstarch suspension. When a solid body is pushed through this complex fluid and approaches its containing wall, a hardened volume of the suspension is observed that adds to the leading side of the body. This volume leads to an imprint on the wall which is made of molding clay. By studying the geometry of the hardened volume, inferred by the imprint shapes, we find that its geometry is determined by the size and speed of the body. By characterizing the response of the clay to deformation we show that the force transmitted through the suspension to the wall is localized. We also study other aspects of this dynamical hardening of the suspension, such as the effect of the substrate and body shape, and its relaxation as the imposed straining is stopped.

Viscoelastic fluid response can increase the speed and efficiency of a free swimmer
by J. Teran, L. Fauci, and M. Shelley
in Physical Review Letters 104, 038101 (2010)

Abstract: Microorganisms navigate through complex environments such as biofilms and mucosal tissues and tracts. To understand the effect of a complex media upon their locomotion, we investigate numerically the effect of fluid viscoelasticity on the dynamics of an undulating swimming sheet. First, we recover recent small amplitude results for infinite sheets that suggest that viscoelasticity impedes locomotion. We find the opposite result when simulating free swimmers with large tail undulations, with both velocity and mechanical efficiency peaking for Deborah numbers near one. We associate this with regions of highly stressed fluid aft of the undulating tail.

Correlation between spatial-frequency and orientation selectivity in V1 cortex: implications of a network model
by W. Zhu, D. Xing, M. Shelley, and R. Shapley
Vision Research 50, 2261-2273 (2010)

Abstract: We addressed how spatial frequency and orientation selectivity coexist and co-vary in Macaque primary visual cortex (V1) by simulating cortical layer 4C-alpha of V1 with a large-scale network model and then comparing the model's behavior with a population of cells we recorded in layer 4C-alpha. We compared the distributions of orientation and spatial frequency selectivity, as well as the correlation between the two, in the model with what we observed in the 4C-alpha population. We found that 1) in the model, both spatial frequency and orientation selectivity of neuronal firing are greater and more diverse than the LGN inputs to model neurons; 2) orientation and spatial frequency selectivity co-vary in the model in a way very similar to what we observed in layer 4C-alpha neurons; 3) in the model, orientation and spatial frequency selectivity co-vary because of intra-cortical inhibition. The results suggest that cortical inhibition provides a common mechanism for selectivity in multiple dimensions.

Surprising Behaviors in Flapping Locomotion with Passive Pitching
by S. Spagnolie, L. Moret, J. Zhang, and M. Shelley
in Physics of Fluids
22, 041903 (2010)

Abstract: 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 heaved in a fluid. We observe a range of flapping frequencies corresponding to very efficient locomotion, a regime of under-performance relative to a rigid (non-pitching) 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). Unlike for the rigid 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.

Searching for autocoherence in the cortical network with a time-frequency analysis of the local field potential
by Samuel P. Burns, Dajun Xing, Michael J. Shelley, and Robert M. Shapley
in Journal of Neuroscience 30, 4033-4047 (2010)

Abstract: Gamma-band peaks in the power spectrum of local field potentials (LFP) are found in multiple brain regions. It has been theorized that gamma oscillations may serve as a ‘clock’ signal for the purposes of precise temporal encoding of information and ‘binding’ of stimulus features across regions of the brain. Neurons in model networks may exhibit periodic spike firing or synchronized membrane potentials that give rise to a gamma-band oscillation that could operate as a ‘clock’. The phase of the oscillation in such models is conserved over the length of the stimulus. We define these types of oscillations to be ‘autocoherent’. We investigated the hypothesis that autocoherent oscillations are the basis of the experimentally observed gamma-band peaks: the autocoherent oscillator (ACO) hypothesis. To test the ACO hypothesis, we developed a new technique to analyze the autocoherence of a time-varying signal. This analysis used the continuous Gabor transform to examine the time evolution of the phase of each frequency component in the power spectrum. Using this analysis method, we formulated a statistical test to compare the ACO hypothesis with measurements of the LFP in macaque primary visual cortex, V1. The experimental data were not consistent with the ACO hypothesis. Gamma-band activity recorded in V1 did not have the properties of a ‘clock’ signal during visual stimulation. We propose instead that the source of the gamma-band spectral peak is the resonant V1 network driven by random inputs.

Stability of Active Suspensions
by C. Hohenegger and M. Shelley
in Physical Review E
81, 046311(2010)

Abstract: We study theoretically the stability of "active suspensions", modeled here as a Stokesian fluid in which are suspended motile particles. The basis of our study is a kinetic model recently posed by Saintillan & Shelley (2008) where the motile particles are either ``Pushers'' or ``Pullers''. General considerations suggest that, in the absence of diffusional processes, perturbations fromuniform isotropy will decay for Pullers, but grow unboundedly for Pushers, suggesting a possible ill-posedness. Hence, we investigate the structure of this system linearized near a state of uniform isotropy. The linearized system is non-normal and variable coefficient, and not wholly described by an eigenvalue problem, in particular at small length-scales. Using a high wave-number asymptotic analysis, we show that while long-wave stability depends upon the particular swimming mechanism, short-wave stability does not, and that the growth of perturbations for Pusher suspensions is associated not with concentration fluctuations, as we show these generally decay, but with a proliferation of oscillations in swimmer orientation. These results are also confirmed through numerical simulation, and suggest that the basic model is well-posed, even in the absence of translational and rotational diffusion effects. We also consider the influence of diffusional effects in the case where the rotational and translational diffusion coefficients are proportional and inversely proportional respectively to the volume fraction and predict the existence of a critical volume fraction or system size for the onset of the long-wave instability in a Pusher suspension. We find reasonable agreement between the predictions of our theory and numerical simulations of rod-like swimmers by Saintillan & Shelley (2007).


Shape Optimization of Peristaltic Pumping
by S. Walker and M. Shelley
Journal of Computational Physics 229, 1260-1291 (2010)

Abstract: Transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this; it is also important to many industrial processes. We present a variational method for optimizing the wave shape of a peristaltic pump. Specifically, we optimize the wave profile of a two dimensional channel containing a Navier-Stokes fluid with no assumption on the wave profile other than it is a traveling wave (e.g. we do not assume it is the graph of a function). Hence, this is an infinite-dimensional optimization problem. The optimization criteria consists of minimizing the input fluid power (due to the peristaltic wave) subject to constraints on the average flux of fluid and area of the channel. Sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming (SQP) method to find a solution of the first order KKT conditions.  We also use a merit-function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape. Our numerical implementation uses a finite element method for computing a solution of the Navier-Stokes equations, adjoint equations, as well as for the SQP method when computing perturbations of the channel shape. The walls of the channel are deformed by an explicit front-tracking approach. In computing funct,ional sensitivities with respect to shape, we use L2-type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature; we show error estimates for the boundary stress and tangent field approximations. As a result, we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature. Specifically, we see highly asymmetric wave shapes that are far from being sine waves. Many examples are shown for a range of fluxes and Reynolds numbers up to Re = 500 which illustrate the capabilities of our method.

LFP spectral peaks in V1 cortex: network resonance and cortico-cortical feedback
by Kukjin Kang, Michael Shelley, James Andrew Henrie and Robert Shapley
in Journal of Computational Neuroscience 29, 495-507 (2010).

Abstract: This paper is about how cortical recurrent interactions in primary visual cortex (V1) together with feedback from extrastriate cortex can account for spectral peaks in the V1 local field potential (LFP). Recent studies showed that visual stimulation enhances the γ-band (25–90 Hz) of the LFP power spectrum in macaque V1. The height and location of the γ-band peak in the LFP spectrum were correlated with visual stimulus size. Extensive spatial summation, possibly mediated by feedback connections from extrastriate cortex and long-range horizontal connections in V1, must play a crucial role in the size dependence of the LFP. To analyze stimulus-effects on the LFP of V1 cortex, we propose a network model for the visual cortex that includes two populations of V1 neurons, excitatory and inhibitory, and also includes feedback to V1 from extrastriate cortex. The neural network model for V1 was a resonant system. The model’s resonance frequency (ResF) was in the γ-band and varied up or down in frequency depending on cortical feedback. The model’s ResF shifted downward with stimulus size, as in the real cortex, because increased size recruited more activity in extrastriate cortex and V1 thereby causing stronger feedback. The model needed to have strong local recurrent inhibition within V1 to obtain ResFs that agree with cortical data. Network resonance as a consequence of recurrent excitation and inhibition appears to be a likely explanation for γ-band peaks in the LFP power spectrum of the primary visual cortex.

Modeling simple locomotors in Stokes flow
by A. Kanevsky, M. Shelley and A.-K. Tornberg
in Journal of Computational Physics 229, 958-977 (2010).

Abstract: Motivated by the locomotion of flagellated micro-organisms and by recent experiments of chemically driven nanomachines, we study the dynamics of bodies of simple geometric shape that are propelled by specified tangential surface stresses. We develop a mathematical description of the body dynamics based on a mixed-type boundary integral formulation. We also derive analytic axisymmetric solutions for the case of a single locomoting sphere and ellipsoid based on spherical and ellipsoidal harmonics, and compare our numerical results to these. The hydrodynamic interactions between two spherical and ellipsoidal swimmers in an infinite fluid are then simulated using second-order accurate spatial and temporal discretizations. We find that the near-field interactions result in complex and interesting changes in the locomotors’ orientations and trajectories. Stable as well as unstable pairwise swimming motions are observed, similar to the recent findings of Pooley et al.