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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. |
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Abstract: Synthetic micro- and nanoscale
motors can move autonomously, like motile
microorganisms, when they are powered by chemical fuels,
electric fields, and magnetic fields... |
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Abstract: In suspensions of flagellated
microorganisms cellular 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
a ffected 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 instability for isotropic
suspensions, the first driving aggregation, the second
producing increased orientational order in suspensions
of "pushers". Simulations of the long-time nonlinear
dynamics show that hydrodynamic interactions can limit
and modify chemotactically-driven aggregation dynamics.
In pusher suspensions chemotactic aggregation can lead
to destabilizing time-dependent flows with fragmented
regions of accumulation. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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). |
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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. |
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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. |
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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. |
