# <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.