)
)
)
)
)
)
)Previous Talks: 2001
We formulate a reaction-diffusion model for bacterial branching growth, and use this model to study possible scenarios during the life of a colony: The use of chemotactic signaling, the appearance of chirality, the emergence of mutations and the stress of antibiotics.
Elucidation of the electrostatic properties of biomolecules has become a standard practice in molecular biophysics. Foremost among the models used to evaluate the electrostatic potential is the Poisson-Boltzmann equation, however, existing solution methods have limited the scope of accurate calculations to relatively small biomolecular systems. Two new numerical techniques will be presented which enable the parallel solution of the Poisson-Boltzmann equation for supramolecular structures orders of magnitude larger in size than those accessible with traditional methods. As a demonstration of this methodology, electrostatic potentials have been calculated for large microtubule and ribosome structures. The results point to the likely role of electrostatics in a variety of activities of these structures.
We consider two-dimensional turbulence for the case in which a drag force linear in the velocity is present. Such a drag force occurs in all cases where two dimensionality is justified. We find modifications from the dragless power law exponent of the wavenumber-energy-spectrum associated with the enstrophy cascade. The case with drag is also predicted to desplay intermittency (in contrast to the case without drag). Comparisons with numerical simulations will be given.
Expression of any single gene in a living cell is dependent upon presence of the products of other genes, known as transcription factors. This cross-regulation is essential for development and function of an organism. Mathematically, it means that the master equations for the protein concentrations are coupled in a highly non--linear manner. Interestingly, analysis of this nonlinear system can be considerably simplified in the limit when the maximum concentrations of the transcription factors are much larger than the thresholds of their catalytic (inhibiting) action. In particular, the search of the fixed points of this system is reduced to a "semi-linear" problem, somewhat analogous to electric circuit with ideal diodes. We have developed an intuitive diagrammatic representation of the problem, which allows one to find the fixed points, and to study their stability. In this framework, I will discuss the criterion for multistability of a genetic network. It will be argued that the multistability is crucial for understanding many important aspects of the problem, such as robustness, cell fate differentiation, and pattern formation. As an example, I will apply some of these ideas to early development of Drosophila embryo.
SMALL IS DIFFERENT --FROM ELECTRONS TO NANOJETS
That the properties of materials depend on size is commonly expected and often observed. At sufficiently small sizes such dependencies may go beyond mere scaling with size, manifesting themselves in physical and chemical behavior that is new and different from that found at larger sizes. Such circumstances, when small is different in an essential way , may occur when one (or more) of the physical dimensions of the material aggregate approaches a length-scale characteristic to a physical phenomenon (with different phenomena being characterized by different length-scales), and similarly in the time domain. Associated with the above is the sensitivity of sufficiently small materials aggregates to shape. Not only can one affect the properties of a confined system by varying its shape, but, most importantly, is the spontaneous shape-selection occurring in such systems, originating from the ability of finite systems to adjust their shape (and structure) in order to minimize their (free) energy. Basic research of these and related issues underlies future technologies, from nano-scale machines, nanotribological systems, cellular injections, and nanocatalysis, to miniaturization of electronic circuitry and novel information storage and retrieval systems.In this talk we discuss and illustrate the above issues through large-scale classical and quantum mechanical simulations of several nano-scale systems. Topics include: (i) Formation mechanisms, mechanical, and quantized conductance properties of metal and semiconductor nanowires and their interconnections [1]; (ii) Atomic-scale friction, control of friction, and nanotribological processes in lubricated junctions [2]; (iii) Generation, stability and breakup of nanojets [3]; (iv) Catalysis by small gold and palladium clusters [4]; (v) Spontaneous symmetry breaking leading to formation of crystallized clusters (electron molecules) in individual two-dimensional quantum dots, and quantum-dot-molecules [5], (vi) Emergence of magnetism in free and surface-supported small palladium clusters [6], and (vii) Charge Transport in DNA (Science, August 19, 2001).
References
1. U. Landman et al, Microscopic Mechanisms and Dynamics of Adhesion, Microindentation and Fracture, Science 248, 454 (1990); U. Landman et al., Metal-Semiconductor Nanocontacts: Silicon Nanowires, Phys. Rev. Lett. 85, 1958 (2000).
2. B. Bhushan, J.N. Israelachvili and U. Landman, Nanotribology: Friction, Wear and Lubrication at the Atomic Scale, Nature 374, 607 (1995); J. Gao, W.D. Luedtke, and U. Landman, Friction Control in Thin-Film Lubrication, J. Phys. Chem. Chem. B 102, 5033 (1998).
3. M. Moseler and U. Landman, Formation, Stability and Breakup of Nanojets, Science 289, 1165 (2000).
4. A. Sanchez et al., When Gold is not Noble: Nanoscale Gold Catalysts, J. Phys. Chem. A 103, 9573 (1999); S. Abbet, U. Heiz, H. Hakkinen, and U. Landman, CO Oxdidation on a Single Pd Atom Model Catalyst, Phys. Rev. Lett, 86, 5950 (2001).
5. C. Yannouleas and U. Landman, Spontaneous Symmetry Breaking in Quantum Dots and Dot-Molecules, Phys. Rev. Lett. 82, 5325 (1999); ibid., Collective and Independent-Particle Motion in Two-Electron Artificial Atoms, Phys. Rev. Lett. 85, 1726 (2000); Coupling and Dissociation in Artificial Molecules, Euor.. Phys. J D 16, 373 (2001).
6. M. Moseler, H. Hakkinen, R.N. Barnett, and U. Landman, Structural and Spin Isomers of Neutral and Anionic Palladium clusters, Phys. Rev. Lett. 86, 2545 (2001).
Tightening of Knots and Dynamics of Topological Constraints in Granular Chains
For polymer systems, it has been conjectured that entropic effects lead to a spontaneous tightening of knots. However, this cannot be directly observed, only indirectly through radius of gyration. I will instead be discussing a system for which such an observation is possible. This system is a vibrated granular chain, and combines aspect of polymer and granular systems.I will discuss the entropic reasons for tightening in equilibrium. Experimental results will be given, which show a much sharper behavior than expected for an equilibrium system, as well as a clear breakdown of detailed balance. To interpret these results, I will give a dynamical explanation for the tightening, qualitatively valid for both equilibrium and nonequilibrium processes. This approach is hoped to elucidate the role of entropy in nonequilibrium systems.
Secrets of Alien Technology Revealed!-- or Chirality Transformations Propagating on Bacterial Flagella
Chemotaxis in many bacterial species is made possible by the remarkable and bizarre dynamics of their multiple, rotating, helical flagella. They bundle and de-bundle as their rotary motors episodically change rotational direction. When the flagella are bundled, the bacterium moves linearly, but the dissolution of the bundle leads to a tumbling event that effectively randomizes the cell's orientation. The motor reversal that initiates the tumbling not only torques the flagella oppositely, but also reverse the chirality of the filament, turning a left-handed helix into a right-handed helix. Hotani has performed careful experiments on helical flagella in external flows and he observed that regions within the filament periodically flip to the opposite chirality, and that those domains propagate stably downstream. I'll present a dynamical model for this phenomenon based on the existence of two competing locally stable states of opposite chirality whose interaction with the flow is through the torque they produce. The model displays a number of the key features seen in the experiments.
Shedding and interaction of solitons in imperfect medium
The propagation of a soliton pattern through one-dimensional medium with weakly disordered dispersion is considered. Solitons, perturbed by this disorder, radiate. The emergence of a long-range interaction between the solitons, mediated by the radiation, is reported. Basic soliton patterns are analyzed. The interaction is triple and is extremely sensitive to the phase mismatch and relative spatial separations within the pattern. The phenomenon is a generic feature of any problem explaining adiabatic evolution of solitons through a medium with frozen disorder. This is a joint work with I. Gabitov, I. Kolokolov, and V. Lebedev to appear in October issue of JETP Lett.
Space, maps and the evolution of navigation
I am interested in the evolution of spatial representation. I present here a radical revision of how the mammalian brain sees, understands and maps its external world. This process, mediated by the hippocampus, has implications not only for the evolution of spatial navigation but for the evolution of a complex cognitive trait from simple antecedents. Based on a new model of hippocampal evolution, I propose that the hippocampus maps space with two independent representations, that are mediated by the 'old' (dentate gyrus) and the 'new' (Ammon's horn) hippocampal structures. The two maps are based on different stimuli: distributed and discrete stimuli, respectively. The coactivation of these parallel maps leads to the cognitive map, which I argue is a property that can only emerge from the collaboration of these two maps. The parallel map theory of hippocampal function has important implications: it explains paradoxes of spatial learning in rats with hippocampal lesions, patterns of sex differences in learning in rats, and why the hippocampus continues to generate new cells, even in the adult mammal. It also predicts sex differences in humans, not only in how men and women navigate the world, but also in abstract thought processes that are spatially encoded. Thus the understanding of a primitive universal, the need to navigate in space, may have important consequences for the development of higher cognitive processes.
Comparing simulations with the real world
Faraday Wave Patterns
Standing waves, parametrically excited on the free surface of a fluid byperiodic acceleration, are found in a wide variety of patterns: squares,hexagons, quasipatterns, superlattices, etc. The more exotic of these patterns are readily obtained in laboratory experiments by employing a periodic forcing function that has (at least) two frequency components. This introduces a number of control parameters to the problem: the amplitudes, frequencies and relative phases of the forcing components. Our aim is to understand the role of each in the pattern formation problem in a systematic fashion. We use methods of equivariant bifurcation theory to make some headway with this, focusing on the weakly inviscid situation. Our analysis identifies a subtle stabilization mechanism for superlattice patterns that involves resonant interactions of waves driven by the two frequency components. We also use general symmetry arguments to derive the form of the resonant coefficients in the relevant bifurcation problem in several cases, showing how these depend strongly on the forcing frequency ratio and on the relative phase of the two driving terms. Additional scaling laws follow from the weakly broken time reversal symmetry and an underlying Hamiltonian structure. The predicted scaling laws, etc. are confirmed by numerically calculating coefficients for the resonant triad amplitude equations from the quasipotential formulation of the gravity-capillary wave problem due to Zhang and Vinals. This talk highlights collaborative work with Jeff Porter and Chad Topaz.
From synthetic towards real fully developed turbulence -- learning from phenomenology
Turbulence is one of those notorious problems in classical and nonlinear physics, which so far has not surrendered to the cohorts of physicists. Over almost a century many tactics have been invented to derive the statistical properties of fully developed turbulence from the fundamental hydrodynamical equations, but with little success. Contrary to these theoretical top-down approaches, a bottom-up approach is presented. The emphasis is put on understanding the data and the relationship between various, apparently different data-motivated models first. From there, more sophisticated phenomenological models are developed, which are expected to bridge the gap between data and fundamental hydrodynamical equations.
Clustering and anomalous diffusion in a granular gas
Granular gases spontaneously separate into dense and dilute regions. Here we experimentally and theoretically demonstrate that the cluster formation and its breakage are fundamentally different due to the lack of time reversability: For a vibro-fluidized granular gas in N connected compartments the cluster formation process is gradual, via several metastable states, whereas the collapse of the cluster is very abrupt. The observed cluster lifetime (as a function of the driving intensity) is analytically calculated within a flux model, making use of the self-similarity of the process. After collapse, the cluster diffuses out into the uniform distribution in a self-similar way, with an anomalous diffusion exponent 1/3.
Uses of Optimal Control in Comparisons of Experiments and Simulations
It is often the case that successful simulations of experiments result from a collaboration of the experimentalists and the modelers, because the relation between the two groups allows transfer of details about the weaknesses of the tools that each group must use. The aim of the work discussed in this talk is move part of this relationship to the experiment and the simulation by allowing the simulation to use partial and flawed experimental information, such as isolated measurements, shadow-like projections and/or qualitative information to confine simulation results. We will give TOY examples in which we have only qualitative information about initial conditions and a few isolated measurements which illustrate how we can guess initial conditions and experimental parameters that give agreement between the experiments and the simulations. Techniques that incorporate uncertainty in the measured values will be examined.
BE WARNED, this is a report on work that is quite preliminary. If you are expecting an encyclopedic view, you will be disappointed. On the other hand, since we don't know what we are doing, your insights will be much appreciated.
High dimensional crumpling: A mathematician's apology (withapologies to G. H. Hardy).
There has been much recent activity on the problem of crumpling of elastic membranes, both here at the Univ. of Chicago, and at many other places. I will give an (elementary) overview of some of the mathematical questions that come up through the study of crumpled sheets. I will then discuss some of the tools and techniques that are used to study this problem. Finally, I will give my (very biased) perspective on some of the recent results that have been obtained, and the many questions that spring from these results.
Continuum description of avalanches in granular media
A continuum theory of partially fluidized granular flows is developed. The theory is based on a combination of the equations for the flow velocity and shear stresses coupled with the order parameter equation which describes the transition between flowing and static components of the granular system. We apply this theory to several important granular problems: avalanche flow in deep and shallow inclined layers, and shear granular flows between two plates. We carry out quantitative comparisons between the theory and experiment.
Nonlinear evolution of unstable fluid interface
Every day, whenever water flows out from an overturned cap, we observe the Rayleigh-Taylor instability. Turbulent mixing caused by this instability is a long-standing problem in many physical and technological applications (inertial confinement fusion, supernova, flames, etc.). The cascades of energy and the dynamics of large-scale coherent structure are fundamental issues. This large-scale structure is a periodic array of regular bubbles and singular spikes. To study its dynamics we propose new approach based on symmetry theory. First we analyze the local properties, and derive from the conservation laws a dynamical system governing the bubble dynamics. Due to formation of the singularities, the regular asymptotic solution to the dynamical system for a family, and we choose the fastest stable solution in the family as the physically dominant one. The dependence of the bubble motion on the acceleration history as well as on the flow symmetry is analyzed. It is shown that 3D bubbles in RTI conserve near-circular contour, and the 3D-2D dimensional crossover is discontinuous. Then we consider global properties, and study the structural stability and the transitions associated with the growth of the flow length scale. Both local and global analysis lead us to a conclusion that in RT turbulent mixing a balance between the inverse and direct cascades is required to keep isotropy of the flow. The theory eliminates discrepancies between previous approaches, explains existing experiments, predicts new ways of the bubble front evolution, and establishes control parameters to be monitored in experiments.
Computational issues in object detection in images
A central problem in computer vision is the detectionof objects of a particular class in complex images containing multiple objects. Detection refers not only to computing the location, but other parameters describing the particular instantiation of the object in the image. This can have the form of a non-linear deformation defined on a `prototypical' example of the object. I will describe two continuum based variational problems for finding the deformation of a prototype to a simple image containing only one object, at more or less the scale of the prototype. These are solved using a coarse to fine gradient descent procedure. In complex images location and scale need to be identified prior to the computation of the deformation. This is impossible to do with the original continuum based cost functions. Instead a coarse and discrete approximation to these functions is formulated, allowing for very efficient identification of candidate locations and scales.
JFI Symposium
Polymer brushes under shear
Motivated by experiments performed both with a surface forces apparatus and by neutron scattering, we discuss theoretically the behavior of grafted polymer layers under a hydrodynamic shear.
In the existing models, chain configurations are stationnary; we take explicitly into account the diffusion of the chain end points in the direction perpendicular to the grafting surface; this diffusive motion allows an exchange of the chain configurations. Each chain is subjected to the shear only when its end point stands in the thin region at the edge of the grafted layer where the flow penetrates.
We calculate both the extension of a chain in the direction of the flow and the tension on the grafting surface. As long as the relevant Deborah number (the dimensionless shear rate) remains small, these quantities remain small of the order of hte values that they reach by thermal fluctuations. This is in agreement with the recent neutron scattering experiments but in strong disagreement with the old surface force experiment where a strong deformation of the chains is observed. Our mean field approach also predicts a thinning of the grafted layer due to the shear.
Some free surface problems solved by the boundary integral method
The following free surface problems will be presented, two forpotential flow and two for Stokes flow. 1. The design of the pressure pulse for a drop-on-demand ink-jet-printer, and the associated pinch-off. 2. How cusps are rounded by surface tension in the Hele-Shaw flow of an initially circular blob of fluid withdrawn through a non-concentric sink. 3. The collision of two deformable drops in a viscous suspension undergoing shear. 4. The rheology of an emulsion at moderate concentration.
The presentation will be partly about mechanics, including some unresolved problems, and partly about numerical methods, including some recent ideas such as the use of B-splines to represent the unknown surface distributions in 2D calculations and the use of radial basis functions to calculate the curvature of a 3D surface.
Two Problems in Internet Architecture
Most talks in this series describe how one can use computation to help answer scientific questions. This talk will instead be about using science to improve the computational infrastructure. I will discuss two design questions relating to the Internet architecture. These questions involve extending the Internet architecture to improve Quality-of-Service (QoS); one extension is to allow bandwidth reservation and the other is to use multiple priority levels for streaming media. The emphasis will be on using extremely simple models to explore qualitative aspects of these design issues. This talk will be self-contained, and no knowledge of networking or of Internet architecture is required.
Studying protein folding with simplified and atomically detailed simulations
My research aims to elucidate how proteins attain their folded conformation within biologically relevant time scales. Proteins sample a vast number of conformations on their way to the folded state and the study of folding is best approached from a statistical standpoint. Concepts borrowed from the well-established field of statistical mechanics have provided considerable insight into the folding problem. The energy landscape of the protein may be described as a minimally rough surface, in which a strong energy bias towards the native state "funnels" the protein towards its biologically active conformation. Roughness of the surface, which hinders this funneling, can be due to both energetic factors (associated with the formation of incorrect but stabilizing interactions) and to topological factors (geometric constraints leading to the premature formation of native interactions). I will discuss the origins of topological roughness (or "frustration") and how the native state "shape" of the protein affects the folding process. My approach involves a combination of off-lattice minimalist models in which the protein is described in a coarse-grained manner and fully atomic models, which provide a detailed representation of both the protein and the solvent.
Things That Think
The digital revolution has given us a clear distinction between hardware and software, between channels and the content they carry, between physical science and computer science, but it is right at these boundaries between the bits of the digital world and the atoms of our physical world that the most compelling opportunities and problems in information technology lay. I will discuss the science underlying the integration of information with its physical properties over length scales from atomic nuclei to planetary networks, and discuss its implications for the life of people, and their machines. Examples will be drawn from projects addressing global development, creative expression, and appropriate interfaces.
Knowledge about What?
Quantum Mechanics and Computation
I will try to describe quantum computation to computer scientists unfamiliar with quantum mechanics, without boring quantum physicists unfamiliar with its recent application to computation. People familiar with both subjects will have to settle for the pleasure of understanding everything they hear, though they may find my point of view amusing. Or irritating.
Avoidance and detachment in bailout embeddings
Many interesting problems involve a combination of Hamiltonian mechanics with some dissipative dynamics. In this work, a large space contains a friction-free lower dimensional region embedded within it. A particle located in that embedded region will undergo a Hamiltonian dynamics. Particles with trajectories in some nearby regions are drawn into the space. Other nearby orbits make the particle escape, 'bail out', of this neighborbood of the Hamiltonian region and go someplace far away. Escape and reinjection into the neighborhhood permit all kinds of interesting orbital behaviors.
We can design this space so that orbits of our choosing bail out, and others stay. Specifically, we build our system so the KAM trajectories stay near the embedded region. (KAM trajectories are particularly interesting motion of the Hamiltonian system, which show marginal stability and an integrable behavior.) A particle initially on the embedded region is subjected to small amounts of noise, which then serves to move it slightly out of the embedding region. Such a particle typically bails out of the neighborhood of the embedded region after some time, except if it lies on a KAM orbit. This procedure thus allows us to "surgically" separate KAM orbits from the other Hamiltonian orbits.
We analyze this bailout in terms of the fluctuation amplitude around the stable embedding for infinitesimally small noise. We show that the bailout process consists of two distinct phases, an avoidance phase, where the fluctuations are small but whose amplitude acquire arbitrarily large prefactors, and a detachment phase, where the fluctuations become finite in size even for infinitesimally small noise.
Wax Tectonics
The floor of the earth's ocean has been created over the past 150 million years by plate tectonics. Continental plates are moving apart and new ocean floor is perpetually solidified at the mid-ocean ridges. From satellite data and ship soundings we have an excellent knowledge of the bathymetry of the ocean floors and find three distinct morphological features: transform faults and fracture zones, a spreading velocity dependent mid ocean ridge profile, and microplates. Midocean ridge dynamics combines the full complexity of fluid flow, phase transformation, and fracture. I will show in my talk that a simple table top experiment can capture the complex dynamics of earth like processes. Our experiments pose a challenge to theorists: Is it possible to develop a consistent theoretical model that captures the complexity of our table top experiments and possibly that of the earth.
Asymptotic flamelets and large scale simulations of turbulent premixed flames.
[Joint work with Boualem Khouider (UdM) and Andy Majda (Courant).]
Turbulence enhances the speed of propagation of premixed flames via mixing in the preheat zone. Capturing this phenomenon is a huge computational challenge because, to do so accurately, one would need to resolve the wide range of length scales induced by turbulence, chemical reactions, and their interaction.
On the other hand, a rigorous homogenization approach to describe the turbulent flame propagation in the asymptotic limit of a very thin reaction zone has been developed by Majda and Souganidis (1994) in an idealized context. In this talk, I will describe our attempts at bridging the gap between the mathematical understanding in the asymptotic limit and practical simulations requirements.
This involves purely numerical issues (designing a novel, robust numerical solver for the effective Hamiltonian of the flame for a variety of flows); fundamental modeling issues (using the numerical database and formal asymptotics to parameterize the burning speed turbulent enhancement as a function of a flame residence time - this lead us to a new quantitative explanation for the so-called "bending" effect); large scale computational issues (possible interpretation of the homogenized limit not just as a mathematical concept by also in the practical context of the validation of large scale simulations by comparison with resolved computations at finite values of the flame thickness.)
Mean field approximation and a small parameter inturbulence theory
Numerical and physical experiments on two-dimensional (2d) turbulenceshow that the differences of transverse components of velocity field are well described by Gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume the existence of a critical space-dimensionality d=dc, at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At dThe expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d=dc. The resulting equation describes experimental data on two-dimensional turbulence and demonstrate onset of intermittency as d-dc>0 and r/L -> 0 in three-dimensional flows in close agreement with experimental data. In addition, some new exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point pdf of transverse velocity components in developing as well as in the large-scale stabilized two-dimensional turbulence is a gaussian.
The possible physics (mechanics) of walking
Robots have motors and people have muscles. What for? To guide motions and to make up for lost mechanical energy. How much guidance is fundamentally needed for repetitive tasks? How much energy needs to be supplied for what losses? One approach to understanding the need for motors and controllers is what can be done without them.
Tad McGeer demonstrated (1988-1993) with simple computational models and with physical devices that uncontrolled human-like walking motions can be achieved with, to put it simply, sticks and hinges that walk downhill. The motions of these toy-like devices are energetically efficient (low specific transport cost) and stable (limit-cycles with linearized stability). We have found that, in principle, some of these devices can walk on arbitrarily small slopes and thus approach perfect efficiency and that Robot configurations that have this efficiency are reminiscent of the human design. These models can also limp (period 2), waltz (period 3) , and stumble (chaos). One of our devices has the unintuitive feature that it has no stable standing posture, yet can walk stably.
The basic theory is not novel: numerical search for limit cycles and numerical evaluation of their stability. However, two morals seem to be exposed: locomotion efficiency is based on avoidance of impacts, and stability comes from utilizing non-holonomic constraints.
ActiveSpaces: The Access Grid, Active Mural and Advanced Visualization Systems
At Argonne, Chicago and elsewhere work has begun to explore the concept of integrated whole room scale visual environments. These environments consist of group work rooms that have been augmented with multiple displays including: large-format whole wall displays (e.g. ActiveMural our high-resolution rear projected tiled display), driven by PC clusters, or multi-processor visualization engines, semi-immersive or immersive displays (Workbenches, ImmersaDesks, CAVEs), multiple desktop devices, and multiple front projection systems. These rooms may also have active or passive tracking systems, multiple channels of audio support, and support for multiple wireless hand-held controllers and navigation devices.
These room-sized environments can be linked via the national "Grid" to form compelling collaborative visualization environments (e.g. "The Access Grid"). We believe these systems represent a new type of visual application development target and delivery mechanism. We call these ensembles ActiveSpaces. In this talk I will explore with the audience some of the ideas we are working on to facilitate the delivery of high-end scientific visualization to groups of users and to create new types of electronically augmented spaces explicitly designed to support rapid collaborative exploration and visual analysis of complex data.
Investigating The Topological Transition In The Selective Withdrawal Problem
In the selective withdrawal experiment we lower a straw so that its tip hangs above a water-oil interface. We then withdraw the oil through the straw. When the withdrawal rate is low the interface below the straw deforms into a hump with a flow stagnation point located at the hump peak. As the withdrawal rate is increased, this hump grows in height and the curvature at the hump peak becomes very large. At the transition flow rate the interface makes a topological transition from being bounded to being unbounded in the vertical direction. The water is then entrained in a thin spout along with the oil and the stagnation point moves from the interfacial boundary to the interior of the lower fluid. For two-fluid systems with different straw diameters, viscosity ratios, density differences and surface tensions, I will present data showing interesting scaling and hysteretic behavior in the transition-flow-rate dependence on the straw heights. I will then discuss the scaling behavior in the steady state hump shapes as the system approaches the transition point. Finally, I will point out some experimental problems related to selective withdrawal and will try to draw the audience into a discussion of these experimental issues.
Search for jamming signatures and force chains in the simplest granular system
A very simple system that supports stress like a granular material is a pack of frictionless spheres of random sizes, deposited one at a time in a periodic box. David Head (University of Edinburgh) Alexei Tkachenko (Bell Labs) and I have been studying jamming and force chains in simulations of this system in two dimensions. First I'll describe how the simulation achieves a mechanically stable state by a simplified relaxation process that requires no translational motion. Next I'll show how forces propagate in this granular medium, corroborating previous theoretical postulates. The support for an applied small point force is concentrated along diagonal lines leading to the bottom: so-called light cones. The distribution of contact forces resembles those reported in physical jammed systems of simulated emulsions or real glass beads. We report how the force distribution evolves under various conditions of loading. We also report our search for organization of the contact forces into force chains, in which strong forces tend to occur in opposing pairs on a given particle.
Experiments and Theory for Foam Drainage and Coarsening
The evolution of a foam is determined by the drainage flow of the continuous (liquid) phase and the coarsening (aging) of the gas bubbles. Free drainage experiments with slow and fast-coarsening gases show markedly different dynamics and elucidate the importance of the coupling of these two effects. Strong coarsening leads to accelerated drainage; however, the liquid flow also becomes self-limiting and cannot exceed a maximum drainage rate. A simple physical model incorporating foam drainage and diffusive coarsening shows quantitative agreement with experiment.
Defect Statistics in the Two Dimensional Complex Ginsburg-Landau Model
The statistical correlations between defects in the two dimensional complex Ginsburg-Landau model are discussed in the defect coarsening regime. In particular the defect-velocity probability distribution is determined. The spiral arms of the defects lead to a very different behavior for the order parameter correlation function in the scaling regime compared to the results for the related dissipative model.
Patterns in extended, periodically forced systems : A Continuum coupled map approach.
This is joint work with Ed Ott. We propose that an useful approach to the modeling of periodically forced extended systems is through continuum coupled maps (CCMs). CCMs are discrete time, continuous space models, mapping a continuous spatially varying field Xi_n(x) from time n to time n+1.
The efficacy of CCM models is illustrated by application to experiments of Umbanhowar et al. on vertically vibrated granular layers. We first derive an appropriate CCM model for this system, using simple physical considerations (essentially dimensional analysis). We then present a framework for the analysis of pattern selection in CCM models using a truncated modal expansion. Through the analysis, we show how the model reproduces the observed experimental behavior. We also obtain some other results (scaling laws) that are experimentally testable predictions from our model. We conclude with a discussion of the limitations of our model for the vibrated granular layer systems, and extensions of this approach to other (non-granular) periodically forced, strongly dissipative systems.