Campuses:

<span class=strong>Reception and Poster Session</span><br><br/><br/><b>Poster submissions welcome from all participants</b><br><br/><br/><a href=http://www.ima.umn.edu/visitor-folder/contents/workshop.html#poster><b>Instructions</b></a>

Monday, March 23, 2009 - 5:00pm - 6:30pm
Lind 400
  • Local existence of solutions to a PDE model of criminal

    behavior

    Nancy Rodriguez (University of California, Los Angeles)
  • Tear film dynamics on an eye-shaped domain: Pressure

    boundary conditions

    Richard Braun (University of Delaware)
    We model the evolution of human tear film during relaxation (after a blink) using lubrication
    theory and explore the effects of viscosity, surface tension, gravity and boundary conditions
    that specify the pressure. The governing nonlinear partial differential equation is solved on
    an overset grid by a method of lines using finite differences in space and an adaptive second order
    backward difference formula solver in time. Our two-dimensional simulations, calculated
    in the Overture framework, display sensitivity in the flow around the boundary to both our
    choice between two different pressure boundary conditions and to the presence of gravity. The
    simulations recover features seen in one-dimensional simulations and capture some experimental
    observations including hydraulic connectivity around the lid margins.
  • A multigrid method for the dual formulation of total

    variation-based image restoration

    Jamylle Carter (San Francisco State University)
    We present a multigrid method for solving the dual formulation of the Total Variation-based problem in image restoration. Flat regions of the desired image contribute to the slow convergence of the widely-used Chambolle method. Numerical results confirm that the multigrid method with a modified Chambolle smoother is many orders of magnitude faster than the original Chambolle method.
  • Head-on impact of liquid drops
    Wendy Zhang (University of Chicago)
    When two point particles collide, the outcome is governed entirely by energy and momentum conservation, with no dependence on the detailed interaction potential. Here we use a Volume of Fluid (VOF) simulation to examine what happens in the analogous case when two liquid drops collide. At low speeds, the liquid drops rebounce elastically, just as seen for point particles. At high speeds, however, a liquid sheet is ejected along the impact plane. When ambient gas pressure is low, both simple estimates and simulation show that the ejection is dominated by inertial effects. This idea enables us to collapse the pressure variation within the liquid drop at early times. In addition we find that surface tension effects are confined to the rim of the expanding sheet and acts primarily to slow the radial expansion.
  • Dewetting of thin liquid films
    Andreas Münch (University of Nottingham)
    We present results on various aspects of thin film models for dewetting films involving high order equations and systems of equations. These include results on the rim instability and the shape of the rim where the liquid dewets, as well as the occurence of non-classical shocks for fast dewetting where inertia becomes important.
  • On instabilities of finite-size films and rivulets
    Lou Kondic (New Jersey Institute of Technology)
    Joint work with J. Diez, A. Gonzalez, and R. Rack.

    We discuss the influence of finite size effects
    on the breakup process involving finite-size films
    and rivulets. For films, we show that the breakup process
    due to finite size effects can be related to the so-called
    nucleation mode of instability of infinite films.
    We also consider coupling of different modes of instabilities,
    and the competition between them.
    Next, we revisit the classical problem of rivulet instability and
    discuss whether finite size effects may be important in determining
    relevant breakup mechanisms. We apply our results to rupture of
    nano-scale metal lines irradiated by repeated laser pulses and
    discuss relevance of the considered process to self-assembly on
    nanoscale.

  • High order geometric and potential driving PDEs for image and

    surface analysis

    Guowei Wei (Michigan State University)
    A family of high-order geometric and potential driving evolution equations was
    introduced and applied to image analysis and biomolecular surface formation.
    Coupled geometric PDEs were introduced for image edge detection.
  • Step evolution for crystals of finite size: The ADL case
    Hala Al Hajj Shehadeh (New York University)
    We study the step evolution of crystal structures relaxing toward flat
    surface when the number of steps is finite. We assume that the mass
    transport process on the structure's surface is attachment-detachment
    limited (ADL).
    We propose a fourth order PDE for the slope of the profile as a function
    of its height. This PDE is derived from the step equations of motion. The
    solution is asymptotically self-similar. We prove existence and
    uniqueness of the self-similar solution in the discrete setting.
  • Microfluidics enhanced novel materials synthesis
    Amy Shen (University of Washington)
    The flow of complex fluids in confined geometries produces rich and new phenomena due to the interaction between the intrinsic length-scales of the fluid and the geometric length-scales of the device. In this poster, we will show three examples to illustrate how self-assembly, confinement, and flow can be used to control fluid microstructure and enhance the controlled synthesis of bio-compatible nanomaterials and supramolecular hydrogels.
  • Asymptotic dynamics of attractive-repulsive swarms
    Andrew Bernoff (Harvey Mudd College)Chad Topaz (Macalester College)
    We classify and predict the asymptotic dynamics of a class of swarming models. The model consists of a conservation equation in one dimension describing the movement of a population density field. The velocity is found by convolving the density with a kernel describing attractive-repulsive social interactions. The kernel's first moment and its limiting behavior at the origin determine whether the population asymptotically spreads, contracts, or reaches steady-state. For the spreading case, the dynamics approach those of the porous medium equation. The widening, compactly-supported population has edges that behave like traveling waves whose speed, density and slope we calculate. For the contracting case, the dynamics of the cumulative density approach those of Burgers' equation. We derive an analytical upper bound for the finite blow-up time after which the solution forms one or more δ-functions.
  • The viscous N-vortex problem: A generalized Helmholtz-Kirchhoff approach
    David Uminsky (Boston University)
    We give a convergent expansion of solutions of the two-dimensional,
    incompressible Navier-Stokes equations which generalizes the
    Helmholtz-Kirchhoff point vortex model to systematically include the
    effects of both viscosity and finite core size. The evolution of each
    vortex is represented by a system of coupled ordinary differential
    equations
    for the location of its center, and for the coefficients in the expansion
    of the vortex with respect to a basis of Hermite functions. The
    differential equations for the evolution of the moments contain only
    quadratic nonlinearities and we give explicit combinatorial formulas for
    the coefficients of these terms. We also show that in the limit of
    vanishing viscosity and core size we recover the classical
    Helmholtz-Kirchhoff
    point vortex model.
  • A gradient flow approach to a free boundary droplet model
    Natalie Grunewald (Rheinische Friedrich-Wilhelms-Universität Bonn)
    We consider a quasi–stationary free boundary droplet model.
    This model does not satisfy a comparison principle and can have
    non unique solutions. Nevertheless it can be seen as a gradient flow
    on the space of possible supports of the drop.
    The gradient flow formulation leads to a natural time discretization,
    which we employ to show the existence of
    a weak form of viscosity solutions for the model.
  • Statistical models of criminal behavior: The effects of

    law enforcement actions

    Paul Jones (University of California, Los Angeles)
    We continue the study, initiated in Short et al., of criminal
    activities as described by an agent based model with dynamical target
    affinities. Here we incorporate effect of law enforcement agents on
    the spatial distribution and overall level of crime in simulated urban
    settings. Our focus is on a two–dimensional lattice model of
    residential burglaries, where each site (target) is characterized by a
    dynamic attractiveness to burglary and where criminal and law
    enforcement agents are represented by random walkers. The dynamics of
    the criminal agents and the attractiveness field are, with certain
    modifications to be detailed, as described in Short et al. Here the
    dynamics of enforcement agents are affected by the attractiveness
    field via a biasing of the walk the detailed rules of which define a
    deployment strategy. We observe that law enforcement agents, if
    properly deployed, will in fact reduce the total amount of crime, but
    their relative effectiveness depends on their numbers, the deployment
    strategy used, and spatial distribution of criminal activity.
  • Thin fluid films with surfactant
    Ellen Peterson (North Carolina State University)Michael Shearer (North Carolina State University)
    Thin liquid films driven by surface tension are considered, both when
    gravity plays a significant role, as on an inclined plane, and when it is
    less significant, on a horizontal substrate. Motion of the film is
    modeled in the lubrication approximation by a fourth order system of PDE.
    In the case of a horizontal substrate, we examine the influence of
    insoluble surfactant both experimentally and numerically. In the
    experiments, we visualize surfactant using fluorescence, and its effect on
    the thin film using a laser. The numerical code tracks the edge of the
    surfactant as it propagates. We also analyze the stability of a thin film
    wave traveling down an inclined plane driven by both surfactant and
    gravity. Numerical results show the propagation of small disturbances,
    thereby substantiating the analysis. This is joint work with Karen
    Daniels, Dave Fallest, Rachel Levy and Tom Witelski.
  • Eulerian indicators for predicting mixing efficiency
    Rob Sturman (University of Leeds)
    Mixing is inherently a Lagrangian phenomenon, a property of the movement of fluid particles. Many different methods exist for measuring, quantifying and predicting the quality of a mixing process, all involving evolution of individual trajectories. We propose indicative tools which are formulated using only Eulerian information, and illustrate their use briefly on a variety of different model mixers.
  • Effect of boundary conditions on mixing efficiency
    James Springham (University of Leeds)Rob Sturman (University of Leeds)
    We consider the mixing of fluid by chaotic advection. Many well-studied examples may be modeled by a class of dynamical systems known as linked-twist maps. The mathematical discipline of ergodic theory studies concepts such as mixing which will be familiar to experimentalists. New analytical results for linked-twist maps suggest mixing rates similar to those observed experimentally and numerically.
  • On the planar extensional motion of an inertially-driven

    liquid sheet

    Linda Smolka (Bucknell University)
    We derive a time-dependent exact solution of the free surface problem
    for the Navier-Stokes equations that describes the planar extensional
    motion of a viscous sheet driven by inertia. The linear stability of
    the exact solution to one- and two-dimensional symmetric perturbations
    is examined in the inviscid and viscous limits within the framework of
    the long-wave or slender body approximation. Both transient growth
    and long-time asymptotic stability are considered. For one-dimensional
    perturbations in the axial direction, viscous and inviscid sheets are
    asymptotically marginally stable, though depending on the Reynolds and
    Weber numbers transient growth can have an important effect. For
    one-dimensional perturbations in the transverse direction, inviscid
    sheets are asymptotically unstable to perturbations of all wavelengths.
    For two-dimensional perturbations, inviscid sheets are unstable to
    perturbations of all wavelengths with the transient dynamics
    controlled by axial perturbations and the long-time dynamics
    controlled by transverse perturbations. The asymptotic stability of
    viscous sheets to one-dimensional transverse perturbations and to
    two-dimensional perturbations depends on the capillary number (Ca);
    in both cases, the sheet is unstable to longwave transverse perturbations
    for any finite Ca. This work is in collaboration with Thomas P. Witelski.
  • Memory as vibration in a disconnecting air bubble
    Wendy Zhang (University of Chicago)
    Focusing a finite amount of energy dynamically into a vanishingly
    small amount of material requires that the initial condition be
    perfectly symmetric. In reality, imperfections are always present and
    cut-off the approach towards the focusing singularity. The
    disconnection of an underwater bubble provides a simple example of
    this competition between asymmetry and focusing. We use a combination
    of theory, simulation and experiments to show that the dynamics near
    disconnection contradicts the prevailing view that the disconnection
    dynamics converges towards a universal, cylindrically-symmetric
    singularity. Instead an initial asymmetry in the shape of the bubble
    neck excites vibrations that persist until disconnection. We argue
    that such memory-encoding vibrations may arise whenever initial
    asymmetries perturb the approach towards a singularity whose dynamics
    has an integrable form.
  • Shape optimizer needed
    Andreas Savin (Université de Paris VI (Pierre et Marie Curie))
    Motivation:

    It is possible to relate the concept of chemical bond to the
    region of three-dimensional space where the probability to find
    exactly one pair of electrons is maximal.

    Characteristics:

    - The computation of the probability for a given volume chosen
    can be time-consuming. It requires the eigenvalues of a matrix having
    elements computed from integrals over the volume.


    - The shape derivatives can vary strongly from one part of the
    delimiting surface to another.


    - Multiple solutions exist by the nature of the problem.
    However, the user might have a good intuition of what they are and
    choose a good starting volume.
  • Phase-field model of self-assembled copolymer monolayer
    Hsiang-Wei Lu (Harvey Mudd College)
    We develop a phase field model that incorporates the polymer vitrification and diffusion in the self-assembly of polymer blends. Simulation shows the different polymers in the blend cooperate to self-assemble into nanoscale features with varying dimension. The feature dimensions can be tuned by adjusting the blend composition and the surface concentration.
  • Instabilities and Taylor dispersion in isothermal binary thin

    fluid films

    Burt Tilley (Franklin W. Olin College of Engineering)
    Joint work with Z. Borden, H. Grandjean, L.
    Kondic, and A.E. Hosoi.

    Experiments with glycerol-water thin films flowing down an inclined
    plane reveal a localized instability that is primarily
    three-dimensional. These transient structures, referred to as
    dimples, appear initially as nearly isotropic depressions on the
    interface. A linear stability analysis of a binary mixture model in
    which barodiffusive effects dominate over thermophoresis (i.e. the Soret
    effect) reveals unstable modes when the components of the mixture have
    different bulk densities and surface tensions. This instability occurs
    when Fickian diffusion and Taylor dispersion effects are small, and is
    driven by solutalcapillary stresses arising from gradients in
    concentration of one component, across the depth of the film.
    Qualitative comparison between the experiments and the linear stability
    results over a wide range of parameters is presented.




  • Numerical study of the parameters α and β

    in the Navier–Stokes-αβ equations for turbulence

    Eliot Fried (McGill University)
    We perform numerical studies of the
    Navier–Stokes-αβ equations, which are based on a
    general framework for fluid-dynamical theories with gradient
    dependencies. Specifically, we examine the effect of the length
    scales α and β on the energy spectrum in
    three-dimensional statistically homogeneous and isotropic
    turbulent flows in a periodic cubic domain, including the
    limiting cases of the Navier–Stokes-α and Navier–Stokes
    equations. A significant increase in the accuracy arises for
    β the grid resolution.