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Abstracts and Talk Materials
Geometrical Singularities and Singular Geometries
July 14 - 25, 2008

Thin sheets have a tendency to bend rather than stretch or compress. A non-uniform lateral growth or shrinking of a plate prescribes a 2D non Euclidean "target" metric on it. In order to reduce stretching and compression the plate bends into a 3D form. In this work we experimentally study the variations in sheet configurations with decreasing thickness. We observe two types of behaviours: sheets with imposed positive Gaussian curvature have a weak thickness dependence, their bending content is bounded and their total bending energy scales like thickness to the third power. On the other hand, sheets with imposed negative Gaussian curvature, undergo a set of bifurcations, as the sheets obtain configurations with increasing number of nodes as thickness decreases. As a result their bending content increases with decreasing thickness, causing the bending energy to scale like thickness squared.

Joint with Efi Efrati and Efran Sharon.

Crumpling of thin sheets is an everyday problem for the frustrated scientist. A familiar phenomenon in crumpled 2D sheets is the localization of deformations in the sheet to a singular network of 0D (developable cone) and 1D (ridge) structures. In this work we directly measure the crumpled configuration of a thin elastic sheet confined inside a symmetric sphere, free of gravitation and any other external forces or constrains. We observe the dynamic evolution of structures in the sheet as confinement ratio increases, and analyze the statistical nature of the elastic energy localization around singularities.

Capillary winding
July 16, 2008

When a liquid droplet is deposited on a flexible sheet, the sheet may deform and spontaneously wrap the droplet. We propose to address a problem in connection with this "capillary origami" experiment: does a flexible rod put in contact with a liquid droplet spontaneously winds itself around the droplet? In the positive situation, what is the maximum length that can be packed inside the droplet? We will finally try to connect this problem to damping issues in spider webs.

Starting with a swirling flow we prove the existence of unique turbulent solutions of the stochastically driven Navier-Stokes equation in three dimensions. These solutions are not smooth but Hölder continuous with index 1/3. The turbulent solutions give the existence of an invariant measure that determines the statistical theory of turbulence including Kolmogorov´s scaling laws. We will discuss how the invariant measure can be approximated leading to a implicit formula that can be used to compare with simulations and experiments.

Low dimensional elastic manifolds (such as rods 1D or sheets 2D) have been drawing a lot of attention lately. When confined into environments smaller than their size at rest, elastic objects sustain large deformations involving many fascinating mechanisms such as energy condensation from large length-scales to small singular structures, topological self-avoidance, complex energetical landscapes... One only needs to crumple a piece of paper to observe the extreme complexity of fold patterns generated. This begs the question: Is there an underlying statistical mechanics foundation?

By studying the isotropic compaction of elastic rods in a 2D space via experiments and numerical simulations, we have been able to gain some encouraging insight into this question. It turns out that the rod can be decomposed into more basic elements whose energy distribution display a Boltzmann-law like behavior containing an "effective" temperature. The comparison between experiments and numerical simulations ensure of the robustness of this result. Moreover thermal equilibration does occur between geometrically different subsystems. These results put on firm ground the underlying statistical nature of crumpling phenomena and their implications will be discussed in light of some recent theoretical work.

Bubble pinch-off at high pressures
December 31, 1969

See 7/14 abstract.

Bubble pinch-off at high pressures
December 31, 1969

We present high-speed videos and numerical simulations of the pinch-off of high-pressure gas bubbles (Xenon) in an exterior inviscid fluid (water). Previously we have studied the pinch-off of conventional air bubbles in water [1]. The density ratio between the interior gas and exterior fluid is D. In the simple case of large D~1000, the pinch-off is similar to that of a water drop pinching-off in air described by a power-law in time with an exponent of 2/3, while at small D~0.001, the pinch-off is that of an air bubble in water with an exponent close to 1/2. By using Xenon as a working gas, we are able to span a wide ranging of density ratios simply by increasing the pressure of the gas. A high-pressure (~100atm) chamber with optical access through sapphire windows was constructed in order to view the pinch-off. The numerical simulations are performed assuming perfectly inviscid fluids using boundary-integral techniques. In the simulations, Shear instabilities in the interface are seen for intermediate density ratios [2]. Comparisons between experimental and numerical results will be discussed.

[1] J.C. Burton, R. Waldrep, and P. Taborek. Phys. Rev. Lett. 94, 184502, (2005). [2] D. Leppinen and J.R. Lister. Phys. Fluids 15, 568, (2003).

We will consider boundary value problems of elastic systems presenting singular behavior in the bulk and on the boundary. Two types of constitutive equations of elastomers will be addressed: isotropic and nematic liquid crystals. In the former, we will study slender domains subject to large strain, that cause loss of hyperelasticity of the material. In the second case, the appearence of isotropic liquid crystal defects follows as a result of the nonconvexity of the free energy of the elastomer.

Motivated by some estimates of vorticity growth in axisymmetric flows without swirl, we re-examine the paired vortex model of singularity formation proposed by Pumir and Siggia for Euler flows in three dimensions. The problem is reformulated as a generalized system of differential equations. No supporting solutions of the system are known, and it is suggested that core deformation remains the most likely mechanism preventing the formation of a singularity.

Ablative Rayleigh-Taylor instability
July 15, 2008

Ablative Rayleigh-Taylor (R-T) instability is a special feature of the acceleration phase in inertial confinement fusion (ICF). Ablation stabilizes the disturbances with small wavelength, introducing a marginal wavelength. Due to a large temperature ratio, the conduction length-scale varies strongly across the wave, and the attention is limited to the intermediate acceleration regimes for which the length-scale of the marginal wavelength is in-between the smallest and the largest conduction length-scale. The analysis is performed for a strong temperature dependence of thermal conductivity. At the leading order, the ablation front appears as a vortex sheet separating two potential flows 1, 2, and the free boundary problem takes the form of an extension of the pure R-T instability with unity Atwood number and zero surface tension. It shows also some analogies with the Kelvin-Helmholtz instability described by the Birkhoff-Rott equation. However, the hot flow of ablated matter introduces a damping at small wavelength which has a form different from the usual damping (as the surface tension for example). The nonlinear patterns are obtained by the same boundary integral method as used for revisiting the R-T instability 3. Unfortunately, a curvature singularity develops within a finite time, even though the short wavelengths are stabilised. Scaling laws are derived from numerical fitting and a self-similarity solution of the problem is exhibited close to the critical time 4. The occurrence of a curvature singularity indicates that the modifications to the inner structure of the vortex sheet can no longer be neglected. A non-local curvature effect is obtained by pushing the asymptotic analysis to the next order 5. The corresponding small pressure correction is showed to prevent the occurrence of the curvature singularity within a finite time.

Colloidal suspensions consist of micron sized solid particles suspended in a solvent. The particles are Brownian so that the suspension as a whole behaves as a thermal system governed by the laws of statistical mechanics. The thermodynamic nature of these systems allows scientists to use colloidal suspensions as models for investigating numerous processes that typically take place on the atomic scale but are often very difficult to investigate. In this talk I will describe how we use confocal microscopy techniques to investigate the structure and dynamics of these systems and gain an understanding of dislocation nucleation and transport in colloidal crystals. Such dislocations are examples of singular point defects in 2D crystals and line defects in 3D crystals.

We consider examples of melts of corpora, that is collections of compacts each having finitely many degrees of freedom, such as articulated particles or n-gons. We associate to the melt the moduli spaces of the corpora, compact metric or pseudometric spaces equipped with a Borel probability measure representing the phase space measure. We consider probability distributions on the moduli spaces of such corpora, we associate a free energy to them, and show that under general conditions, the zero temperature limit of free energy minimizers are delta functions concentrated on a single corpus, the ur-corpus. We give a selection principle for the ur-corpus. This is a generalization of the isotropic to nematic transition but we suggest that this language is appropriate for a larger class of n-body interactions. This is work in progress with Andrej Zlatos.

Topological singularities in optical waves
July 24, 2008

Understanding of complicated spatial patterns emerging from wave interference, scattering and diffraction is frequently aided by insight from topology: the isolated places where some fundamental physical quantity -- such as optical phase in a complicated light field -- is undefined (or singular) organize the rest of the field. In scalar wave patterns, the optical phase is undefined at nodes at points in 2D, and lines in 3D, in general whenever 3 or more waves interfere. Similar singularities occur in optical polarization fields, and these quantized defects bear some morphological similarity to defects in other systems, such as crystal dislocations, diclinations and quantum vortices in condensed matter physics, etc.

I will describe the features of these optical singularities, concentrating on three cases. The first will be three-dimensional optical speckle, familiar as the mottled pattern in reflected laser light. Natural speckle volume is filled with a dense tangle of nodal phase singularity lines. We have found in computer simulations that these lines have several fractal scaling properties. Secondly, by controlling the interference using diffractive holograms in propagating laser light, I will show how these nodal lines can be topologically shaped to give a range of loops, links and knots. Finally, I will describe the natural polarization pattern that occurs in skylight (due to Rayleigh scattering in the atmosphere), originally discovered in the 1800s by Arago, Babinet and Brewster. This pattern contains polarization singularities, whose global geometry has several physical interpretations and analogs.

Elastic theory of unconstrained non-Euclidean plates
December 31, 1969

Joint work with Raz Kupferman and Eran Sharon.

Non-Euclidean plates are thin elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints and may assume complicated equilibrium shapes even in the absence of external forces. We present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. The elastic theory of non-Euclidean plates offers new theoretical insights to the spontaneous buckling phenomena. The buckling transition shows both sub-critical and super-critical bifurcations for different geometries. A new length-scale, unique to the Non-Euclidean plate theory is shown to be important to the post buckled behavior.

Elastic theory of non-Euclidean plates
July 21, 2008

Thin elastic sheets are very common in both natural and man-made structures. The configurations these structures assume in space are often very complex and may contain many length scales, even in the case of unconstrained thin sheets. We will show that in some cases, a simple intrinsic geometry leads to complex three-dimensional configurations, and discuss the mechanism shaping thin elastic sheets through the prescription of an intrinsic metric.

Current reduced (two-dimensional) elastic theories devised to describe thin structures treat either plates (flat bodies having no structure along their thin dimension) or shells (non-flat bodies having a non-trivial structure along their thin dimension). We propose the concept of non-Euclidean plates, which are neither plates nor shells, to approximate many naturally formed thin elastic structures. We derive a thin plate theory which is a generalization of existing linear plate theories for large displacements but small strains, and arbitrary intrinsic geometry. We conclude by surveying some experimental results for laboratory-engineered non-Euclidean plates.

A catalogue of singularities
July 14, 2008

We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of an infinite dimensional dynamical system. We point out that analysing the dynamics close to the fixed point is a useful way of classifying the structure of the singularity. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. We will point out unsolved problems, present perspectives, and try to look at the role of geometry in singularity formation.

A circular disc impacting on a water surface creates a remarkably vigorous jet. Upon impact an axisymmetric air cavity forms and eventually pinches off in a single point halfway down the cavity. Immediately after closure two fast sharp-pointed jets are observed shooting up- and downwards from the closure location, which by then has turned into a stagnation point surrounded by a locally hyperbolic flow pattern. Counter-intuitively, however, this flow is not the mechanism feeding the two jets. Using boundary-integral simulations we show that only the inertial focussing of the liquid colliding along the entire surface of the cavity provides enough energy to eject the high-speed jets. With this in mind we show how the natural description of a collapsing void (using a line of sinks along the axis of symmetry) can be continued after pinch-off to obtain a quantitative analytical model of jet formation.

Hydraulic jump in a flowing soap film
July 15, 2008

Joint work with S. Steers, J. Larkin, A. Prescott (University of Pittsburgh), T. Tran, G. Gioia, P. Chakraborty, G. Gioia, and N. Goldenfeld (University of Illinois, Urbana).

A soap film flows vertically downward under gravity and in a steady state. At all lengths of the film, its thickness h(x) decreases as the distance x from the top reservoir increases. But then h(x) abruptly starts to increase and its downward flow velocity u(x) correspondingly decreases to very small value. To explain this nonmonotonic behavior in h(x) and u(x), it is necessary to invoke the film's elasticity; one has a type of Marangoni effect. The transition from subcritical flow speed to a supercritical one at the thickening point, is akin to the classical hydraulic jump. This transition will be explained, but other findings, also to be described, are not yet understood.

For general non-linear elliptic PDEs, e.g. non-linear rubber elasticity, linear stability analysis is false. This is because of the possibility of point-instabilities. A point-instability is a non-linear instability with zero amplitude threshold that occurs while linear stability still holds. Examples include cavitation, fracture, and the formation of a crease, a self-contacting fold in an otherwise free surface. Each of which represents a kind of topological change. For any such PDE, a point-instability occurs whenever a certain auxiliary scale-invariant problem has a non-trivial solution. E.g. when sufficient strain is applied at infinity in a rubber (half-)space to support a single, isolated crease, crack, cavity, etc. Owing to scale-invariance, when one such solution exists, an infinite number or geometrically similar solutions also exist, so the appearance of one particular solution is the spontaneous breaking of scale-invariance. We then identify this (half-)space with a point in a general domain. The condition that no such solutions exist is called point-coercivity, and can be formulated as non-linear eigenvalue problem that predicts the critical stress for fracture, etc. And when point-coercivity fails for a system, the system is susceptible to the nucleation and self-similar growth of some kind of topological defect. Viewing fracture, etc. as symmetry breaking processes explains their macroscopic robustness.

Point-coercivity is similar to, but more general than, quasi-convexity, as it can be formulated for any elliptic PDE, not just Euler-Lagrange systems (i.e. for out-of-equilibrium systems, and so defining meta-stability in a general sense). Indeed, these are just two examples of a host of point-conditions, the study of which might be called point-calculus. Time allowing, I will show that for almost any elliptic PDE, linear- and point-instabilities exhaust the possible kinds of instabilities. The lessons learned from elliptic systems will be just as valid for parabolic and hyperbolic systems since the underlying reason linear analysis breaks down – taking certain limits in the wrong order holds for these systems as well.

Calabi-Yau manifolds are currently being studied in theoretical physics to unify Einstein's general relativity and quantum mechanics. Vibrating strings in string theory live in 10-dimensional spacetime, with four of these dimensions being 3-dimensional observable space plus time and six additional dimensions being a Calabi-Yau manifold. In this talk, I will discuss orbifold singularity on a Calabi-Yau variety and the topology of crepant resolutions using the McKay Correspondence.

We present the continuum theory on smectic C liquid crystals and apply it to see the switching dynamics in a simple geometry. We prove the existence and uniqueness of traveling wave solutions and show that the dynamic model exhibits a slightly faster switching time than the static model. We also investigate the instability in smectic A liquid crystals when the magnetic field is applied in the direction parallel to the layers. When liquid crystals are confined in a slab with the thickness $d$, we derive analytic estimates for the magnetic field strength, at which the undeformed state loses its stability.

Joint work with H. Kim and T. Funada.

The problem of radial fingering in two phase gas/liquid flow in a Hele-Shaw cell under injection or withdrawal is studied here. The problem is analyzed as a viscous potential flow VPF in which the potential flow analysis of Paterson 1981 and others is augmented to account for the effects of viscosity on the normal stress at the gas/liquid interface. The unstable cases in which gas is injected into liquid or liquid is withdrawnfrom gas lead to fingers. This stability problem was previously considered by other authors with the viscous normal stress neglected. Here we show that the viscous normal stress should not be neglected; the normal stress changes the speed of propagation of the undisturbed interface, it changes the growth rate, and the numbers of fingers that grow the fastest and the cut-off number above which fingers can not grow.

I will discuss different singular behaviors that arise when one consider the impact of drop on thin liquid films or solid surface. For instance, singularities can be observed for low velocity impacts on super-hydrophobic surface, related to classical surface singularities. I will then discuss in more details the condition of prompt splash when an impact is made on a thin liquid film. Self-similar behaviors are then exhibited which allow a simplified understanding of empirical scaling laws.

Dear Participants: We've arranged a video conference with the parallel conference in Cargese on The Geometry and Mechanics of Growth in Biological Systems

We can ask them about the news from their meeting and tell them what is happening here. Several people there were interested in coming to our meeting, but the conflicting times prevented this.

The teleconference is set for 12:30 noon our time in the meeting room EE/CS 3-180. It will be 19:30 their time. The moderator at our end is Christophe Josserand.

To see their conference program go to http://www.lps.ens.fr/~adda/Cargese2008/home.html and click on "lecturers" "scientific committee" and "program." "program" page doesn't exist yet, but we hope it will be added in the meantime.

Please come today at 12:30 if you'd like and we'll see what happens.

Best, Tom Witten

Scaling properties of singular matrix eigenvectors
July 16, 2008

Joint work with Hui Dai and Zachary Geary.

Singular Toeplitz matrices have been extensively used to calculate singularities in classical phase transition problems. We calculate the eigenfunctions of these matrices in the limiting case in which the matrices are large. (This limit applies to long-distance correlations in the phase transition problems.) The large N calculation is done by putting together two N= infinity calculations. Both algebraic and logarithmic singularities are found in the resulting eigenfunctions. (Supported by the NSF via the University of Chicago MRSEC.)

The theory of smectic liquid crystals is notoriously difficult to study. Thermal fluctuations render them disordered through the Landau-Peierls instability, lead to anomalous momentum dependent elasticity, and make the nematic to smectic-A transition enigmatic, at best. I will discuss recent progress in studying large deformations of smectics which necessitate the use of nonlinear elasticity in order to preserve the underlying rotational symmetry. By recasting the problem of smectic configurations geometrically it is often possible to exploit toplogical information or, equivalently, boundary conditions, to confront these highly nonlinear problems. Specifically, I will discuss edge dislocations, disclination networks in three-dimensionally modulated smectics, and large angle twist grain boundary phases. Fortuitously, it is possible to make intimate comparison with experimental systems!

What's new for microstructure
July 21, 2008

Cellular structures coarsen according to a local evolution law, a gradient flow or curvature driven growth, for example, limited by space filling constraints, which give rise to random changes in configuration. Composed of volumes, facets, their boundaries, and so forth, they are ensembles of singlular structures. Among the most challenging and ancient of such systems are polycrystalline granular networks, especially those which are anisotropic, ubiquitous among engineered materials. It is the problem of microstructure. These are large scale metastable, active across many scales. We discuss recent work in this area, especially the discovery and the theory of the GBCD, the grain boundary character distribution, which offers promise as a predictive measure of texture related material properties. There are many mathematical challenges and the hint of universality.

The convective Cahn-Hilliard equation has gathered a lot of attention in recent years. While its solutions have been analyzed thoroughly, we show that within the context of quantum dot dynamics, a higher order convective Cahn-Hilliard equation brings along strong similarities in the solution structure. The parameter plane for stationary solutions highlights the familiar kinks and further multi-hump solutions. The coarsening rates from numerical simulations, where small pyramids vanish while bigger ones continue growing, are comparable, but show a large initial logarithmic regime for small deposition rates.

Modeling of the self-assembly of quantum dots raised much attention in recent years and leads to complicated PDEs describing the growth of these nano-structures. A simple model results in a higher order driven Cahn-Hilliard equation that brings along strong similarities in the solution structure to the known convective Cahn-Hilliard equation. The parameter plane for stationary solutions highlights the familiar kinks and further multi-hump solutions. For the one-hump heteroclinic connections we summarize an exponential asymptotics approach that gives an analytical expression for the width of the ''slope-bubble'' and far-field parameter expressions. The coarsening rates from numerical simulations, where small pyramids vanish while bigger ones continue growing, show two logarithmic regimes with a fast transition regime in between.

We will discuss the packing and folding of a confined beaded chain vibrated in a flat circular container as a function of chain length, and compare with random walk models from polymer physics. Time permitting, we will briefly discuss crumpling and folding structures obtained with paper and elastic sheets obtained with a laser-aided topography technique. We have shown that the ridge length distribution is consistent with a hierarchical model for ridge breaking during crumpling.

About a decade ago, biophysicists observed an approximately linear relationship between the combinatorial complexity of knotted DNA and the distance traveled in gel electrophoresis experiments [1]. Modeling the DNA as tightly knotted rope of uniform thickness, it was suggested that lengths of such tight knots (rescaled to have unit thickness) would grow linearly with crossing numbers, a simple measure of knot complexity. It turned out that this relationship is more subtle: some families of knots have lengths growing as the the 3/4 power of crossing numbers, others grow linearly, all powers between 3/4 and 1 can be realized as growth rates, and it could be proven that that the power cannot exceed 2 [2-5]. It is still unknown whether there are families of tight knots whose lengths grow faster than linearly with crossing numbers, but the largest power has been reduced to 3/2 [6]. We will survey these and more recent developments in the geometry of tightly packed or knotted ropes, as well as some other physical models of knots as flattened ropes or bands which exhibit similar length versus complexity power laws, some of which we can now prove are sharp [7].

References:

[1] Stasiak A, Katritch V, Bednar J, Michoud D, Dubochet J "Electrophoretic mobility of DNA knots" Nature 384 (1996) 122

[2] Cantarella J, Kusner R, Sullivan J "Tight knot values deviate from linear relation" Nature 392 (1998) 237

[3] Buck G "Four-thirds power law for knots and links" Nature 392 (1998) 238

[4] Buck G, Jon Simon "Thickness and crossing number of knots" Topol. Appl. 91 (1999) 245

[5] Cantarella, J, Kusner R, Sullivan J "On the minimum ropelength of knots and links" Invent. Math. 150 (2002) 257

[6] Diao Y, Ernst C, Yu X "Hamiltonian knot projections and lengths of thick knots" Topol. Appl. 136 (2004) 7

[7] Diao Y, Kusner R [work in progress]

The idea that a single, rotating, self-gravitating mass — like a star — can evolve into a pair of masses orbiting one another — like a double-star — was suggested over a century ago. The elaboration of the mathematical details led to negative results and most astronomers abandoned this idea in the 1920's. The negative results are not decisive, however, and we discuss alternative mathematical formulations of this problem and their prospects for positive outcomes.

Untangling elastic knots by the energy decreasing flow
December 31, 1969

We introduce a method for untangling given smooth knots by the geometric flow. Namely, we set up an energy decreasing flow for the total energy of knots, which consists of elastic energy and the Moebius energy. We show that, as the initial knots are smooth, the evolving of knots would remain smooth for all time during the flow, and would asymptotically approach an equilibrium configuration of knots in the same knot type as the initial ones.

Much of the work on capillary pinch-off, and on other fluid-mechanical problems with changes in topology, has focused on situations that lead to finite-time singularities in the neighbourhood of which there is some kind of similarity solution. Capillary instability in the absence of gravity of an axisymmetric layer of fluid coating a circular cylinder is, by contrast, an example of an infinite-time singularity. Even more unusually, film rupture proceeds through an episodic series of oscillations that form a diverging geometrical progression in time, each of which reduces the remaining film thickness by a factor of about 10.

Crofton measure in Minkowski geometry and singularities
December 31, 1969

The classical Crofton formula is a fundamental formula with the idea of measuring length by integrating intersection numbers. From the perspective of integral geometry, we are interested in Crofton measures for Minkowski geometry. Singularity on Crofton measures arises as the strict convexity of unit ball in Minkowski space fails. In this post, We'll focus on the Minkowski p-space, give explicit Crofton formulas, and discuss about the singularity case. This is some of the work under advisories of Dr. Joseph H. G. Fu.

Plasticity in metals and alloys is a mature discipline in the mechanics of materials. However, it appears that current theoretical modeling lacks predictive power. If a new form of steel, say, is fabricated, there appears to be no way of predicting its deformation and fracture behavior as a function of temperature, and/or cyclic loading. The root of this problem appears to be with the paucity of controlled experimental measurements, as opposed to visualizations, of the properties of dislocations, the defects that are responsible for plastic deformation of crystals. Indeed, the tool of choice in this area is transmission electron microscopy, which involves an intrusive measurement of specially prepared samples. Is it possible to develop non intrusive tools for the measurement of dislocation properties? Could ultrasound be used to this end? This talk will highlight recent developments in this line of thought.

Specific results include a theory of the interaction of elastic, both longitudinal and transverse, bulk as well as surface, waves with dislocations, both in isolation and in arrays of large numbers, in two and three dimensions. Results for the isolated case can be checked with experimental results obtained using stroboscopic X-ray imaging. The theory for the many-dislocations case constitutes a generalization of the standard Granato-Lücke theory of ultrasound attenuation in metals, and it provides an explanation of otherwise puzzling results obtained with Resonant Ultrasound Spectroscopy (RUS). Application of the theoretical framework to low-angle grain boundaries, that can be modeled as arrays of dislocations, provides an understanding of recently obtained results concerning the power law behavior of acoustic attenuation in polycrystals. Current developments of instrumentation that may lead to a practical, non-intrusive probe of plastic behavior will be described.

Introductory talk
July 14, 2008

See 7/14 abstract.

We examine a simple hard disc fluid with no long range interactions on the two dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is frustrated, allowing us to construct a tractable model of disordered monodisperse hard discs. We extend free area theory and the virial expansion to this regime, deriving the equation of state for the system, and compare its predictions with simulation near an isostatic packing in the curved space.

We recently developed a thin film model that describes the rupture and dewetting of very thin liquid polymer films where slip at the liquid/solid interface is very large. In this talk, we investigate the singularity formation at the moment of rupture for this model, where we identify different similarity regimes.

Impact!
July 16, 2008

The difficulty of constructing ordered states on spheres was recognized by J. J. Thomson, who discovered the electron and then attempted regular tilings of the sphere in an ill-fated attempt to explain the periodic table. We first discuss how protein packings in buckled virus shells solve a related “Thomson problem”. We then describe the grain boundary scars that appear on colloidosomes, drug delivery vehicles that represent another class of solution to this problem. The remarkable modifications in the theory necessary to account for thermal fluctuations in crumpled amorphous shells of spider silk proteins will be described as well. We then apply related ideas to the folding strategies and shapes of pollen grains during dehydration when they are released from the anther after maturity. The grain can be modeled as a pressurized high-Young-modulus sphere with a weak sector and a nonzero spontaneous curvature. In the absence of such a weak sector, these shells crumple irreversibly under pressure via a strong first order phase transition. The weak sectors (both one and three-sector pollen grains are found in nature) eliminate the hystersis and allow easy rehydration at the pollination site, somewhat like the collapse and subsequent reassembly of a folding chair.

In this talk, we first discuss mathematical modeling of ferroelectric liquid crystals with existence results. With a special geometry, we consider a one dimensional problem with an applied field. We present existence of finitely many equilibrium branches and nested hysteresis loops between the polarization and applied fields, which show finer structures of ferroelectric materials.

In this poster, we present mathematical modeling of ferroelectric liquid crystals which have attracted my scientists due to their potential applications. In systems of such materials, the spontaneous or permanent polarization comes into play so that one should account for the effect of the polarization in the model. We discuss some of mathematical results together with various effects of the governing energy functional.

This talk will discuss certain singularities that arise in the solution to boundary value problems involving the swelling of otherwise hyperelastic solids. In this setting, both non-uniform swelling and constrained swelling give rise to nonhomogeneous deformation in the absence of externally applied load. The standard singularities that are encountered in nonlinear elasticity may occur, such as cavitation. Additional singularities also arise, such as loss of smoothness associated with the concentration of deformation on singular surfaces.

When a physical object, which is perceived as a singularity on a certain level of mathematical description, is set into motion, a paradox may arise rendering dynamic description impossible unless the singularity is resolved by introducing new physics in the singular core. This situation, appearing in diverse physical contexts, necessitates application of multiscale matching methods, employing a simpler long-scale model in the far field and a short-scale model with more detailed physical contents in the core of the singularity. The law of motion can be derived within this approach by applying a modified Fredholm alternative in a region large compared to the inner and small compared with the outer scale, and evaluating the boundary terms which determine both the driving force and dissipation. I give examples of applying this technique to both topological (vortices) and non-topological (contact lines) singularities.

A free material surface which supports surface diffusion becomes unstable when put under external non-hydrostatic stress. Since the chemical potential on a stressed surface is larger inside an indentation, small shape fluctuations develop because material preferentially diffuses out of indentations. When the bulk of the material is purely elastic one expects this instability to run into a finite-time cusp singularity. It is shown here that this singularity is cured by plastic effects in the material, turning the singular solution to a regular crack.

A few viscous jets
July 17, 2008

We present a few situations involving a viscous jet or thread, and its deformation under a constraint. We first discuss the case of a thread initially horizontal that deforms in the field of gravity (the so-called viscous catenary first described by Mahadevan). Then, we consider a jet hitting a bath of the same liquid: here again, we show that understanding of the jet deformations is essential to capture dynamic phenomena such as air entrainment.

We present a simple one-dimensional equation modeling slender jets of a Newtonian fluid in Stokes flow. It would be desirable to have a proof linking the asymptotics of surface tension driven breakup to the behavior of the initial condition near the thinnest point of the jet. Despite the apparent simplicity of the equations, the problem is open. I shall discuss some partial results.

I will talk about a work in collaboration with G. During and C. Josserand on the long-time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence (successfully applied in the past to plasma, optics, and hydrodynamic waves), this nonlinear wave system evolves at long times with a slow transfer of energy from one mode to another. We derived a kinetic equation for the spectral transfer in terms of the second order moment. We show that such a theory describes the approach to an equilibrium wave spectrum and represents also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations that confirm this scenario. Finally, I will discuss recent experiments by A. Boudaoud and collaborators and N. Mordant.

Swimming and flying animals locomote by coupling changes in body shape to complex fluid flows. For example, insects achieve flight by rapidly flapping and flipping their wings back and forth. In executing aerial maneuvers, insects break symmetries in wing motion, leading to fluid force imbalances and thus body motion. Here, we present a new automated method that efficiently extracts the body and wing motion of insects from flight videos taken from several views. We apply this method of Hull Reconstruction Motion Tracking (HRMT) to maneuvering insects and find that, unlike helicopters and airplanes, tiny fruit flies often generate lateral forces during flight. We propose that lateral forces can be induced by asymmetries in wing angle of attack and that flies achieve this by imposing a phase difference between the flipping over of its left and right wings.

Welcome to the IMA
July 14, 2008

The dynamics of droplet breakup in Newtonian fluids are described by the Navier-Stokes equation. Previous experiments have shown that in many cases the breakup dynamics follow a self-similar behavior where successive drop profiles can be scaled onto one another. In visco-elastic systems however, the Navier-Stokes equation is not sufficient to describe breakup. In this talk we will describe droplet breakup in a visco-elastic surfactant system which forms micellar, lamellar, and reverse-micellar phases at various concentrations. We present results of the dynamics of breakup in this system and compare these to previously studied Newtonian systems.

Under rapid periodic pacing, cardiac cells typically undergo a period-doubling bifurcation in which action potentials of short and long duration alternate with one another. If these action potentials propagate in a fiber, the short-long alternation may suffer abrupt reversals of phase at various points along the fiber, a phenomenon called (spatially) discordant alternans. Either stationary or moving patterns are possible. Echebarria and Karma proposed an approximate equation to describe the spatiotemporal dynamics of small-amplitude alternans in a class of simple cardiac models, and they showed that an instability in this equation predicts the spontaneous formation of discordant alternans. We show that for certain parameter values a degenerate steady-state/Hopf bifurcation occurs at a multiple eigenvalue. Generically, such a bifurcation leads one to expect chaotic solutions nearby, and we perform simulations that find such behavior. Chaotic solutions in a one-dimensional cardiac model are rather surprising--typically chaos in the cardiac system has occurred from the breakup of spiral waves in two dimensions.

When a flow converges above the interface of two miscible, viscous liquids, the stresses on the surface must negotiate to determine the dynamics. If the converging flow is stronger than the density stratification, a thin tendril of the lower layer is entrained within the upper. The width of the tendril is much smaller than any imposed length scales and is controlled by a balance of the viscous stress and the stratification. We derive a simple scaling law for the volume flux through the tendril. A more precise, long-wavelength model of the tendril dynamics reveals that information about the geometry of the converging flow must be included to ensure the problem has a unique solution.

We identify a new mechanism by which dynamics near a singularity can preserve detailed information about its early history. As an air bubble breaks apart, small deviations from cylindrical symmetry in its initial shape are encoded by vibrational modes whose amplitudes become fixed as the disconnection approaches. In contrast, the phase of each mode “chirps,” changing more and more rapidly in time. Thus information about the initial phase relations between the different modes is scrambled. Additionally, we investigate the effect of an up/down asymmetry in the neck shape about the minimum. Close to disconnection, the neck resembles two cones joined by a short cylindrical segment. A slight difference in the cone angles causes a vertical motion of the minimum which is coupled to the radial collapse.

Drop impact dynamics
December 31, 1969

Joint work with Christophe Josserand, Stephane Zaleski, and Wendy W. Zhang.

We examine the impact of a liquid drop onto a smooth and dry solid surface in the regime of negligible atmospheric effects. This study is motivated by recent experiments showing that, at sufficiently low ambient air pressures, drop impact at a speed of several m/s does not produce a splash [1,2]. Instead liquid is ejected from the drop in a thin sheet that expands outwards along the solid surface without lifting upwards off the surface. This non-splashing ejection dynamics is inaccessible in previous atmospheric-air-pressure experiments. Here we simulate the impact of a 20 cP liquid drop at 1/10th of ambient pressure. As in the experiments, no splash is formed. We find that liquid is ejected outwards from the contact region in an axisymmetric stagnation-point flow. Viscous effects at the solid surface modify this flow and create a boundary layer which is spatially uniform and thickens over time. The sheet evolution slows as the boundary layer thickens, eventually saturating at a constant value, one consistent with the boundary layer thickness over the impact time-scale. Finally we show that, as a result of this saturation, the rim of the sheet grows at the same rate over time regardless of how rapidly the sheet expands.

[1] L. Xu, W. W. Zhang & S. R. Nagel, Phys. Rev. Lett. 94 184505 (2005). [2] C. Stevens, N. Keim, W. W. Zhang & S. R. Nagel, APS-DFD.FC003S (2007).

Liquid flow driven by light scattering
December 31, 1969

Joint work with Alexis Casner, Regis Wunenburger, Wendy W. Zhang, and Jean-Pierre Delville.

We demonstrate that light scattering can drive a bulk flow. A near-critical phase-separated liquid experiences large fluctuations in its index of refraction. These fluctuations scatter light and transfer momentum from the light to the fluid. The resultant flow deforms the soft interface between the two fluid phases. We demonstrate agreement between the observed deformations and the predicted deformations from a model flow driven by this mechanism.

We describe an approach for the construction of singular solutions to the 3D Euler equations for complex initial data. The approach is based on a numerical simulation of complex traveling wave solutions with imaginary wave speed, originally developed by Caflisch for axisymmetric flow with swirl. Here, we simplify and generalize this construction to calculate traveling wave solutions in a fully 3D (nonaxisymmetric) geometry. Our new formulation avoids a numerical instability that required the use of ultra-high precision arithemetic in the axisymmetric flow calculations. This is joint work with Russ Caflisch.

We present an overview of a variational approach to modelling fracture initiation in the framework of nonlinear elasticity. The underlying principle is that energy minimizing deformations of an elastic body may develop singularities when the body is subjected to large boundary displacements or loads. These singularities often bear a striking resemblance to fracture mechanisms observed in polymers.

Experiments indicate that voids may form in polymer samples (that appear macroscopically perfect) when the samples are subjected to large tensile stresses. This phenomenon of cavitation can be viewed as the growth of infinitesimal pre-existing holes in the material or as the spontaneous creation of new holes in an initially perfect body. In this talk we adopt both viewpoints simultaneously. Mathematically, this is achieved by the use of deformations whose point singularities are constrained to be at certain fixed points (the "flaws" in the material). We show that, under suitable hypotheses, the energetically optimal location for a single flaw can be computed from a singular solution to a related problem from linear elasticity.

One intriguing consequence of the above approach is that cavitation may occur at a point which is not energetically optimal. We show that such a disparity will produce configurational forces (of a type previously identified in the context of defects in crystals) and conjecture that this may provide a mathematical explanation for crack initiation.

Much of the above work is joint with S.J. Spector (S. Illinois University).

Long-wave unstable thin-film equations exhibit rich dynamical behavior: Solutions can spread indefinitely, converge to a steady droplet configuration or blow up in finite time. We will discuss the properties of scaling solutions that govern the blowup dynamics. In particular, we will present how energy based methods can be used to study the stability of selfsimilar blowup solutions as well as other dynamical properties of the blowup solutions. Strong connections to studies of blowup behavior in other equations will be indicated.

Experiments on elastomers have shown that triaxial tensions can induce a material to exhibit holes that were not previously evident. Analytic work in nonlinear elasticity has established that such cavity formation may indeed be an elastic phenomenon: sufficiently large prescribed boundary deformations yield a hole-creating deformation as the energy minimizer whenever the elastic energy is of slow growth.

In this lecture the speaker will discuss the use of isoperimetric arguments to establish that a radial deformation, producing a spherical cavity, is the energy minimizer in a general class of isochoric deformations that are discontinuous at the center of a ball and produce a (possibly non-symmetric) cavity in the deformed body. The key ingredient is a new radial-symmetrization procedure that is appropriate for problems where the symmetrized mapping must be one-to-one in order to prevent interpenetration of matter.

See 7/14 abstract.

Recently , we have presented a numerical and analytical investigation of 2D inviscid pinch-off [1]. The asymptotic collapse of the pinching region is characterized by an anomalous, non-rational similarity exponent, indicating the existence of self-similarity of the second kind. Numerical solutions of the boundary integral equations show that the height of the pinch region shrinks faster than the width, so that the singularity can be described by a slender approximation. The partial differential equations obtained from this approximation lead to a nonlinear eigenvalue problem for the value of the similarity exponent = 0.6869±0.0003. We have also found a simple experimental system consistent with our 2D theory: thin liquid alkane lenses on the surface of water. For sufficiently small negative values of the spreading coefficient S, the dynamics of pinch-off is accurately described by our theory for an ideal 2D sheet. Successive profiles of the pinching region obtained from high speed video can be collapsed onto a single curve using non-isotropic scaling and irrational exponents characteristic of self similarity of the second kind. For larger negative values of S, the scaling exponents approach the value 2/3 expected for 3D inviscid pinch-off.

[1] "Two Dimensional Inviscid Pinch-off: An Example of Self-similarity of the Second Kind", J.C. Burton and P. Taborek, Phys. Fluids 19, 102109 (2007).

Joint work with Ovidiu Costin, G. Luo.

We consider a new approach to a class of evolutionary PDEs where question of global existence or lack of it is tied to the asymptotics of solution to a non-linear integral equation in a dual variable whose solution has been shown to exist a priori. This integral equation approach is inspired by Borel summation of a formally divergent series for small time, but has general applicability and is not limited to analytic initial data.

In this approach, there is no blow-up in the variable p, which is dual to 1/t or some power 1/tn; solutions are known to be smooth in p and exist globally for p in R+. Exponential growth in p, for different choice of n, signifies finite time singularity. On the other hand, sub-exponential growth implies global existence.

Further, unlike PDE problems where global existence is uncertain, a discretized Galerkin approximation to the associated integral equation has controlled errors. Further, known integral solution for p in [0, p0], numerically or otherwise, gives sharper analytic bounds on the exponents in p and hence better estimate on the existence time for the associated PDE.

We will also discuss particular results for 3-D Navier-Stokes and discuss ways in which this method may be relevant to numerical studies of finite time blow-up problems.

Singular jets in free-surface flows
July 17, 2008

Free-surface 'singular jetting' occurs in geometries where flow focusing accelerates the free surface symmetrically towards a line or a point. This is known to occur in a number of configurations, such as during the collapse of free-surface craters and of granular cavities as well as for capillary waves converging at the apex of oscillating drops. Drops impacting onto super-hydrophobic surfaces also generate such jets. We will show recent work on characterizing such jetting, in well-known and new jetting configurations. High-speed video imaging, with frame-rates up to 1,000,000 fps, will be presented and used for precise measurement of jet size and velocity. The focus will be on three well-controlled flow-configurations: During the crater collapse following the impact of a drop onto a liquid pool and after the pinch-off of a drop from a vertical nozzle. Finally, we will show a new apex jet which is generated by the impact of a viscous drop onto a lower-viscosity pool.

Recent experiments by Kantsler et al. [Phys. Rev. Lett. 99, 178102 (2007)] have shown that the relaxational dynamics of a vesicle in external elongation flow is accompanied by the formation of wrinkles on a membrane. Motivated by these experiments we present a theory describing the dynamics of a wrinkled membrane. The formation of wrinkles is related to the dynamical instability induced by negative surface tension of the membrane. For quasispherical vesicles we perform analytical study of the wrinkle structure dynamics. We derive the expression for the instability threshold and identify three stages of the dynamics. The scaling laws for the temporal evolution of wrinkling wavelength and surface tension are established, confirmed numerically, and compared to experimental results.

Singularities in 2D void collapse
December 31, 1969

Joint work with Lipeng Lai and Wendy W. Zhang.

We simulate how a void immersed in an inviscid fluid collapses when its shape is perturbed from a perfect circle. We find that a weak distortion grows into a strong distortion and ends in a singular shape. Numerics show two types of singular shapes: a contact singularity characterized by two portions of the void surface touching in a finite amount of time and a cusp singularity. Which of the two singular shapes is attained varies non-monotonically with the size of the initial perturbation.

A practical consequence of the breakup of a liquid jet by the pinch-off singularity is the redistribution of volume. To the extent that volume concentrates into drops in the streamwise direction, pinch-off can lead to coarsening. The fundamental redistribution of volume by surface tension can be understood in the absence of pinch-off, however. We pose a simple model for the coarsening of connected spherical-cap drops in the absence of pinch-off. Our study shows that many properties of this simple model hold true for a general class of coupled elements.

A system of N drops with pinned contact lines is coupled through a network of conduits. The system coarsens in the sense that, as time progresses, the volume becomes increasingly localized and ends up primarily in a single 'winner' drop. Numerical simulations show that the identity of the winner can depend discontinuously on the initial condition and conduit network. This motivates a study of the corresponding N-dimensional dynamical system. An analysis of the system yields analytic expressions for the fixed points and their energy stability, which depend only on the characteristic pressure-volume response of each element. The dynamic stability is shown to be identical to the energy stability, thus characterizing the number of stable and unstable manifolds at each fixed point. To determine which of the stable fixed points will be the winner, separatrix manifolds of the attracting regions are found using a method which combines local information from the eigenvectors at fixed points with global information from invariant manifolds obtained from symmetry. This method is used to explain phenomena observed in the numerical simulations.

Fragmentation under impact
July 17, 2008

Fragmentation phenomena will be reviewed with a particular emphasis on processes occurring with liquids, those giving rise to drops (the case of solid fragmentation can discussed also, depending on the audience requests). Examples including impacts of different kinds, and raindrops will specifically illustrate the construction mechanism of the drop size distributions in the resulting spray.

Geometric theory of columnar phases on curved substrates
December 31, 1969

We study thin self-assembled columns constrained to lie on a curved, rigid substrate. The curvature presents no local obstruction to equally spaced columns in contrast with curved crystals for which the crystalline bonds are frustrated. Instead, the vanishing compressional strain of the columns implies that their normals lie on geodesics which converge (diverge) in regions of positive (negative) Gaussian curvature, in analogy to the focusing of light rays by a lens. We show that the out of plane bending of the cylinders acts as an effective ordering field.

We investigate the dynamics of a post-rupture thin liquid film dewetting on a hydrophobised substrate driven by Van-der-Waals forces. The stability of the three-phase contact line is discussed numerically and asymptotically in the framework of lubrication models by taking account of various degrees of slippage. The results are used to explain some experimentally observed patterns. Finally, we present some recent studies of the impact of slippage on the late stages of the dynamics. Here, we present some novel coarsening behaviour of arrays of interacting droplets.

Shape optimization plays a central role in engineering and biological design. However, numerical optimization of complex systems that involve coupling of fluid mechanics to rigid or flexible bodies can be prohibitively expensive (to implement and/or run). A great deal of insight can often be gained by optimizing a reduced model such as Reynolds' lubrication approximation, but optimization within such a model can sometimes lead to geometric singularities that drive the solution out of its realm of validity. We present new rigorous error estimates for Reynolds' approximation and its higher order corrections that reveal how the validity of these reduced models depend on the geometry. We use this insight to study the problem of shape optimization of a sheet swimming over a thin layer of viscous fluid.

Finite-time topological rupture occurs in many models in fluid and solid mechanics. We review and discuss some properties of the self-similar solutions for such problems. Unresolved issues regarding analytical forms of the solutions (stability and symmetry vs. asymmetry) and numerical calculation methods (shooting vs. global relaxation) will be highlighted. Further questions of interest arise in post-rupture coarsening dynamics of dewetting thin films.

Coarsening of dewetting thin films subject to gravity
December 31, 1969

Thin films of viscous fluid coating a hydrophobic substrate are unstable to pattern-forming instabilities. Rupture and dewetting forms an array of near-equilibrium droplets connected by ultra-thin fluid layers. In the absence of gravity, previous use of lubrication theory has shown that coarsening dynamics will ensue – the system will evolve by successively eliminating small drops to yield fewer larger drops. While gravity has only a weak influence on the initial thin film, we show that it has a significant influence on the later stages of the coarsening dynamics, dramatically slowing the rate of coarsening for large drops. Small drops are relatively unaffected, but as coarsening progresses, these aggregate into larger drops whose shape and dynamics are dominated by gravity. The change in the mean drop shape causes a corresponding gradual transition from power-law coarsening to a logarithmic behavior.

Fadeout of deposition in an evaporating drop
December 31, 1969

Remarkably, the evaporation of a droplet containing a nonvolatile solute creates a singular deposition profile: an arbitrarily large fraction of the solute becomes concentrated at the the perimeter. As the volume fraction of solute increases from zero, the width of the deposition band increases. The density profile has a distinctive form, with a smooth fadeout at the trailing edge. We show that this fadeout is of powerlaw form. For the simplest and most common case, the predicted power is -7. The power law is governed by the stagnation region of the lateral flow as the drop evaporates. It suffices to know two quantities: a) the evaporating flux J(0) at this stagnation point relative to its average over the drop and b) the height at the stagnation point relative to its average. We describe conditions for achieving a range of power laws.

Studies on the break-up of a liquid drop or an air bubble reveal that the dynamics prior to a singularity can have several forms, ranging from universal, with no memory of the initial state, or integrable, which has a complete memory. We find that how an air bubble disconnects from an underwater nozzle is associated with an unusually rich class of dynamics, one reflecting the integrable singularity of the cylindrically symmetric break-up. Dynamics that are weakly distorted from cylindrical symmetry support vibrations whose amplitudes freeze as disconnection approaches, thus encoding details about the initial distortion. As a result, even a slight asymmetry entirely changes the nature of the singularity. Instead of collapsing down to a point, the bubble neck evolves towards a double column shape.

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