Abstract: We investigate the "flapping flag" instability through a model for an inextensible flexible sheet in an inviscid 2D flow with a free vortex sheet. We solve the fully-nonlinear dynamics numerically and find a transition from a power spectrum dominated by discrete frequencies to an apparently continuous spectrum of frequencies. We compute the linear stability domain which agrees with previous approximate models in scaling but differs by large multiplicative factors. We also find hysteresis, in agreement with previous experiments.

Abstract: Motivated by the intriguing motility of spirochetes of helically shaped bacteria that screw through viscous fluids due to the action of internal periplasmic flagella, we examine the fundamental fluid dynamics of superhelices translating and rotating in a Stokes fluid. A superhelical structure may be thought of as a helix whose axial centerline is not straight, but also a helix. We examine the particular case in which these two superimposed helices have different handedness, and employ a combination of experimental, analytic, and computational methods to determine the rotational velocity of superhelical bodies being towed through a very viscous fluid. We find that the direction and rate of the rotation of the body is a result of competition between the two superimposed helices; for small axial helix amplitude, the body dynamics is controlled by the short-pitched helix, while there is a crossover at larger amplitude to control by the axial helix.We find far better, and excellent, agreement of our experimental results with numerical computations based upon the method of Regularized Stokeslets than upon the predictions of classical resistive force theory.

Abstract: Liquid crystal drops dispersed in a continuous phase of silicone oil are generated with a narrow distribution in droplet size in microfluidic devices both above and below the nematic-to-isotropic transition temperature. Our experiments show that the surface properties of the channels can be critical for droplet formation. We observe different dynamics in liquid crystal droplet generation and coalescence, and distinct droplet morphology on altering the microchannel surface energy. This is explained by the thermodynamic description of the wetting dynamics of the system. The effect of the nematic-to-isotropic transition on the formation of liquid crystal droplets is also observed and related to the capillary number. We also investigate how the nematic droplet size varies with the flow rate ratio and compare this behaviour with a Newtonian reference system. The effect of the defect structures of the nematic liquid crystal can lead to distinctly different scaling of droplet size in comparison with the Newtonian system. When the nematic liquid crystal phase is stretched into a thin filament before entering the orifice, different defect structures and numbers of defect lines can introduce scatter in the drop size. Capillary instabilities in thin nematic liquid crystal filament have an additional contribution from anisotropic effects such as surface gradients of bending stress, which can provide extra instability modes compared with that of isotropic fluids.

Abstract: The orientational order and dynamics in suspensions of self-locomoting slender rods are investigated numerically. In agreement with previous theoretical predictions, nematic suspensions of swimming particles are found to be unstable at long wavelengths as a result of hydrodynamic fluctuations. Nevertheless, a local nematic ordering is shown to persist over short length scales and to have a significant impact on the mean swimming speed. Consequences of the large-scale orientational disorder for particle dispersion are also discussed.

Abstract: Numerical simulations reveal the formation of singular structures in the polymer stress field of a viscoelastic fluid modeled by the Oldroyd-B equations driven by a simple body force. These singularities emerge exponentially in time at hyperbolic stagnation points in the flow and their algebraic structure depends critically on the Weissenberg number. Beyond a first critical Weissenberg number the stress field approaches a cusp singularity, and beyond a second critical Weissenberg number the stress becomes unbounded exponentially in time. A local approximation to the solution at the hyperbolic point is derived from a simple ansatz, and there is excellent agreement between the local solution and the simulations. Although the stress field becomes unbounded for a sufficiently large Weissenberg number, the resultant forces of stress grow subexponentially. Enforcing finite polymer chain lengths via a FENE-P penalization appears to keep the stress bounded, but a cusp singularity is still approached exponentially in time.

Abstract: It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability -- termed the "stretch-coil" instability -- when moving in the neighborhood of a hyperbolic stagnation point of the flow.  When the stagnation point is embedded in an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline flows.  It is also found that the flow is segregated into transport regions around hyperbolic stagnation points and their manifolds, and closed entrapment regions around elliptic points.

Abstract: We study the dynamics of surface waves on a semitoroidal ring of water that is excited by vertical vibration. We create this specific fluid volume by patterning a glass plate with a hydrophobic coating, which confines the fluid to a precise geometric region. To excite the system, the supporting plate is vibrated up and down, thus accelerating and decelerating the fluid ring along its toroidal axis. When the driving acceleration is sufficiently high, the surface develops a standing wave, and at yet larger accelerations, a traveling wave emerges. We also explore frequency dependencies and other geometric shapes of confinement.

Click on the image for a movie from the experiment

Abstract: We study the behavior of an elastic loop embedded in a flowing soap film. This deformable loop is wetted into the film and is held fixed at a single point against the oncoming flow. We interpret this system as a two-dimensional flexible body interacting in a two-dimensional flow. This coupled fluid-structure system shows bistability, with both stationary and oscillatory states. In its stationary state, the loop remains essentially motionless and its wake is a von Ka´rma´n vortex street. In its oscillatory state, the loop sheds two vortex dipoles, or more complicated vortical structures, within each oscillation period. We find that the oscillation frequency of the loop is linearly proportional to the flow velocity, and that the measured Strouhal numbers can be separated based on wake structure.

Click on the image for a movie from the experiment

Abstract: We study the sedimentation of two identical but nonspherical 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 shapes.

Click on the image for movies referenced in this publication

Abstract: A common strategy for locomotion through a fluid uses appendages, such as wings or fins, flapping perpendicularly to the direction of travel. This is in marked difference to strategies using propellers or screws, ciliary waves, or rowing with limbs or oars which explicitly move fluid in the direction opposite to travel. Flapping locomotion is also never observed for microorganisms moving at low Reynolds number. To understand the nature of flapping locomotion we study numerically the dynamics of a simple body, flapped up and down within a viscous fluid and free to move horizontally. We show here that, at sufficiently large  frequency Reynolds number,  unidirectional locomotion emerges as an attracting state for an initially nonlocomoting body. Locomotion is generated in two stages: first, the fluid field loses symmetry by the classical von Karman instability; and second, precipitous interactions with vortical structures shed in previous flapping cycles  push  the body into locomotion. Body mass and slenderness play central and unexpected roles in each stage. Conceptually, this work demonstrates how locomotion can be transduced from the simple oscillations of a body through an interaction with its fluid environment.

Abstract: By immersing a compliant yet self-supporting sheet into flowing water, we study a heavy, stream-lined and elastic body interacting with a fluid.  We find that above a critical flow velocity a sheet aligned with the flow begins to flap with a Strouhal frequency consistent with animal locomotion.  This transition is subcritical.  Our results agree qualitatively with a simple fluid dynamical model that predicts linear instability at a critical flow speed.  Both experiment and theory emphasize the importance of body inertia in overcoming the stabilizing effects of finite rigidity and fluid drag.

Click on the image for a movie from the experiment

Abstract: Ten years ago Hou, Lowengrub and Shelley published a state-of-the-art boundary integral simulation of a classical viscous fingering problem, the Saffman-Taylor instability. In terms of complexity and level of detail, those computations are still among the most ramified and accurately computed interfacial instability patterns that have appeared in the literature. Since 1994, the computational power of a standard workstation has increased a hundredfold as predicted by Moore's law. The purpose of this Note is to consider Moore's law and its consequences in computational science, and in particular, its impact on studying the Saffman-Taylor instability. We illustrate Moore's law and fast algorithms in action by presenting the worlds largest viscous fingering simulation to date.

Abstract: We study the stretch flow of a thin layer of Newtonian liquid constrained between two circular plates. The evolution of the interface of the originally circular bubble is studied when lifting one of the plates at a constant velocity and the observed pattern is related to the measured lifting force. By comparing experimental results to numerical simulations using a Darcy's law model we can account for the fully non-linear evolution of the observed fingering pattern. One observes an initial destabilization of the interface by growth of air fingers due to a Saffman Taylor like instability and then a coarsening of the pattern towards a circular interface until complete debonding of the two plates occurs. Numerical simulations reveal that when relating the observed patterns to the lifting force not only the number of fingers but also the amplitude of the fingering growth has to be taken into account. This is in agreement with the experimental observations.

See also: Hele-Shaw Flow and Pattern Formation in a Time-Dependent Gap by M.J. Shelley, F.-R. Tian, and K. Wlodarski, Nonlinearity , 10, 1471, (1997).

Abstract: In this study we consider the unsteady separated flow of an inviscid fluid around a falling flat plate of small thickness and high aspect ratio. The motion of the plate, which is initially released from rest, is unknown in advance and is determined as part of the solution. The flow solution is assumed two-dimensional and to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. Throughout its motion, the plate continually sheds vorticity from each of its two sharp edges and the unsteady Kutta condition, which states the fluid velocity must be bounded everywhere, is applied at each edge. The coupled equations of motion for the plate and its trailing vortex wake are derived and are shown to depend only on a modified Froude number.

Abstract: We present a novel moving overset grid scheme for the accurate and efficient long-time simulation of an air bubble displacing a non-Newtonian fluid in the prototypical thin film device, the Hele-Shaw cell. We use a two-dimensional generalization of Darcy's law that accounts for shear thinning of a non-Newtonian fluid. In the limit of weak shear thinning, the pressure is found from a ladder of two linear elliptic boundary value problems, each to be solved in the whole fluid domain. A moving body fitted grid is used to resolve the flow near the interface, while most of the fluid domain is covered with a fixed Cartesian grid. Our use of body-conforming grids reduces grid anisotropy effects and allows the accurate modeling of boundary conditions.

Abstract: Liquid crystal elastomers (LCEs) are rubbers whose constituent molecules are orientationally ordered.  Their salient feature is strong coupling between the orientational order and mechanical strain. For example, stretching or otherwise deforming an LCE sample changes the orientational order, which in turn changes bulk properties such as birefringence and dielectric susceptibility. Conversely, changing the orientational order gives rise to internal stress, which leads to strains that can change the shape of a sample. While orientational order can be affected by changes in temperature and other externally applied fields, light can also change the orientational order via a number of distinct processes. We demonstrate here that by dissolving azo dyes into an LCE sample, its mechanical deformation in response to visible light becomes large and very fast. Light induced bending of more than 60o has been observed on the timescale of tens of milliseconds; this is more than two orders of magnitude faster than previous results. Rapid light induced deformations allow LCE materials to interact with their environment in new and unexpected ways. We report here also the astonishing observation that when light from above is shined on a dye-doped LCE sample floating on water, the LCE "swims" away from the light.

Experiment and theory from the Courant Applied Mathematics Lab:

Nature abounds with organisms utilizing body flexibility in order to survive in flowing fluids.  We study aspects of this using a length of fiber optic glass -- a flexible body -- immersed in the the quasi- two-dimensional flow of a running soap film.  As the flow speed increases the shape of the flexible body bends and becomes more and more streamlined -- the two left panels -- and consequently the fluid drag on the body grows much more slowly than if it were rigid.  The rightmost figure shows the numerical solution of a mathematical model of a flexible body deformed by an surrounding  flow and wake.  This theory suggests an emerging self-similarity in shape arising from a balance of fluid and elastic forces at the tip.  This self-similarity yields a new, reduced drag law where drags grows as the 4/3's power, rather than the square, of the flow velocity.

Download Drag Reduction through Self-Similar Bending of a Flexible Body, by S. Alben, M. Shelley, and J. Zhang, in Nature 420, 479-481 (2002).

Nature News and Views:  Bend and Survive by Victor Steinberg

Download How Flexibility induces Streamlining in a Two-Dimensional Flow, by S. Alben, M. Shelley, and J. Zhang, in Physics of Fluids 16, 1694-1713 (2004).

Flapping Flags:
 

An experiment from the Courant Institute Applied Mathematics Lab:
 
 

A flapping filament in a quasi-two-dimensional soap film flow tunnel: The resulting motion is similar to a flapping flag. We study the dynamics of a single filament as well as multiple interacting filaments under the action of an incoming laminar flow. Two stable states are observed in this system.  For more details click on the picture.

Download Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind, by J. Zhang, S. Childress, A. Libchaber and M. Shelley, in Nature 408, 835-838 (2000).
 

Swinging Chains:

Time-exposure image of a swinging chain whose support is being vertically oscillated (experiments performed at the Pritchard Laboratory, Department of Mathematics, Penn State).

The numerical simulation of a swinging chain, using the mathematical model of an inextensible many-linked chain.

Abstract: When shaken vertically, a hanging chain displays a startling variety of distinct behaviors.  We find experimentally that a chain becomes unstable in tongue-like bands of parameter space, either to swinging or rotating pendular states, or to chaotic states.  Mathematically, the chain dynamics are described by a nonlinear wave equation.  A linear stability analysis predicts parametric instabilities within the well-known resonance tongues; their boundaries agree very well with experiment.  Fully nonlinear simulations of the 3D dynamics reproduce and elucidate many aspects of the experimental observations.  Experimentally, the chain is also observed to tie knots in itself, drastically modifying its dynamics.

Dynamic Patterns and Self-Knotting of a Driven Hanging Chain,  by Andrew Belmonte, Shaden Eldakar, Michael Shelley, and Chris Wiggins, in Physical Review Letters 87, 114301 (2001).  Download PDF

The pendular state:

Click here to see an animation of the experiment
Click here to see an animation of the simulation.

Some entertaining publicity

Abstract: We present a brief review of the application of boundary integral methods in two dimensions to multicomponent fluid flows and multiphase problems in materials science.We focus on the recent development and outcomes of methods which accurately and efficiently include surface tension. In fluid flows, we examine the effects of surface tension on the Kelvin¿Helmholtz and Rayleigh¿Taylor instabilities in inviscid fluids, the generation of capillary waves on the free surface, and problems in Hele-Shaw flows involving pattern formation through the Saffman¿Taylor instability, pattern selection, and singularity formation. In materials science, we discuss microstructure evolution in diffusional phase transformations, and the effects of the competition between surface and elastic energies on microstructure morphology. A common link between these different physical phenomena is the utility of an analysis of the appropriate equations of motion at small spatial scales to develop accurate and efficient time-stepping methods.

Other Research Descriptions and Downloads on:

• Surface Tension and the Kelvin-Helmholtz Instability

• Pattern formation in Hele-Shaw cells

• The Saffman-Taylor Instability and Non-Newtonian Hele-Shaw Flow

• Singularity Formation in Surface Tension Driven Flows