Michael Shearer - US grants

This research program is concerned with the theoretical and numerical study of systems of nonlinear partial differntial equations which change type by losing strict hyperbolicity. Specific efforts include the adaptation and implementation of the emerging shock capturing numerical methods of high accuracy. This involves the approximate solution of generalized Riemann problems for which recent theoretical insights obtained by the PI will be used. Constitutive laws leading to equations of mixed type will also be investigated. Results from this research program will have applications in the areas of multiphase flows, elastic solids and magneto- hydrodynamics.

This grant is under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences. It is for the purchase of special purpose equipment dedicated to the support of research in the mathematical sciences. In general, this equipment is required by several research projects, and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers, and with substantial cost-sharing from the institution submitting the proposal. This is an instance of university, industrial, and government cooperation in the support of basic research in the mathematical sciences. The equipment in this project will be utilized in the following projects in the mathematics department: connections between finite simple groups and nonassociative algebras; hyperbolic partial differential equations describing plastic flow; closed form solutions of differential equations; and some problems in optimization, with various applications.

The proposed research involves the study of instability and illposedness for systems of equations that govern plastic flow in two and three space dimensions and the study of nonstrictly hyperbolic systems in one space dimension that model the evolution of undercompressive shock waves. Solution techniques will be drawn from the theory of dynamical systems and bifurcation theory, and there are plans to attack some of the problems numerically. Many important physical phenomena are modelled by systems of equations in which dissapative effects such as viscosity can be safely neglected. The proposer will study such "hyperbolic" systems using a panoply of analytical and numerical techniques in order to investigate problems in the flows of plastics and in the dynamics of certain types of shock waves.

The principal investigators plan to develop three courses during the period of the award. The courses are designed to interest students in aspects of Scientific Computation through computer algebra, numerical bifurcation, and partial differential equations. The students holding fellowships will have various educational tasks, such as writing up clear lecture notes and assignment solutions, software testing, maintenance and consultation. They will also run a weekly tutorial for lectures and demonstrations concerning the use of software for applications in the courses. The three courses will be loosely coordinated through a weekly seminar, in which the principal investigators, and later the students, will give presentations on topics of common interest, and will lead discussion. The award recommendation is for a one-time contribution to equipment and student support for this laudable activity, consistent with the gold of High Performance Computing and Communications.

ABSTRACT: MULTIDIMENSIONAL PROBLEMS IN DYNAMIC PLASTICITY This interdisciplinary proposal addresses three broad scientific issues regarding the dynamics of granular flow: (1) instability, including pattern formation in the post-instability regime, (2) fluctuations, and (3) computation of granular flows with multiple scales. Current and planned experiments include: (a) constitutive tests using a biaxial apparatus with the capability of measuring the speed of sound and of continuously monitoring the deformation with x-rays, (b) further study of porosity waves, (c) experiments directed toward isolating the causes of the instabilities of shaken granular material, and (d) experiments probing various aspects of fluctuations in granular flow, including stress chains and 1/f noise. Based on the fact that the governing PDE of nonassociative plasticity become ill-posed at moderate strains, current analytical work seeks (a) to generalize previous one-dimensional work, driven by both mathematical considerations and the need to establish a sound theoretical framework for numerical simulations of two-and-three-dimensional phenomena and (b) to relate ill-posedness to various experimental phenomena such as porosity waves and shear banding. Related problems in metal plasticity are also being studied analytically. Numerical work includes both (a) continuum and (b) molecular-dynamics (MD) computations. The key effort in the former is to complete a code for simulating shear-band formation and propagation, especially in the biaxial test; this code includes front tracking and mesh refinement at the shear band. MD computations have the immediate goal of gathering quantitative information about fluctuations in granular flow, particularly the variation of such fluctuations with length scale. In the long range, it is planned to develop a hybrid code that solves continuum equations in regions where the solution is smooth and invokes MD in regions of rapid change. Many areas of applied engineer ing stand to benefit from progress on the fundamental questions addressed in this project, including (1) particle handling and transport, (2) soil mechanics, (3) materials forming, and (4) geotechnical engineering. The following elaborates on area (1). An estimated 40%, or $61 billion, of the value added by the chemical industry is linked to particle technology. A study by the Rand Corporation found that, because of inability to accurately predict powder behavior, solids-producing manufacturing plants performed on average at 63% of design capacity, compared to 84% for liquids-producing plants. In economic terms, this difference is staggering. (Regarding future competitiveness, the U.S. should note that Germany and Japan lead the world in particle-technology research.) Fundamental understanding of the flow of granular materials would help (a) in finding ways to control industrial problems and (b) in developing new, more efficient industrial processes. To illustrate (a): one of the difficulties of granular flow is that quite different behavior may result from apparently identical circumstances, especially when scale-up is involved (e.g., from laboratory-scale experiments to an industrial silo). Most existing theories attempt to describe the flow only in term of average quantities, ignoring deviations from these averages. The focus in this project on fluctuations and length scales offers the possibility of being able to predict, and design for, the full range of behavior of real materials-handling systems. To illustrate (b): the experiments in this project with shaken granular material have led to ideas for two planned applications for patents, one in obtaining uniform mixing of multi-sized particles and the other in exposing a large surface area to the surrounding gas (as in a fluidized bed but not requiring fluid flow). These applications are being explored in concert with researchers in industry.

Fundamental and Applied Problems in Granular Flow Michael Shearer/David Schaeffer This project focuses on three aspects of the flow of granular materials: an investigation of fluctuations, flows and stresses in industrial silos, and liquefaction of soils. The various issues will be addressed using an interdisciplinary approach involving analysis, computation and experiment. The latter two parts will also involve input from industrial and geotechnical experts. Details of each subproject follow. Subproject 1, fluctuations in sheared granular materials: Recent work has shown that fluctuations of forces and to some extent velocities can be very large for moderate scale systems. The experimental part of this project will provide additional quantitative characterizations of these fluctuations for modest scale slowly sheared systems. In addition, new experiments will be constructed of a Couette type that will probe force fluctuations on larger length and time scales. These experiments will be integrated into ongoing work to model force fluctuations by lattice type models, and computations using novel hybrid molecular dynamics and finite element codes. Subproject 2, flow in industrial silos: In collaboration with engineers at the firm Jenike and Johanson, Inc. the co-PI's of this project will analyze flows in a spatial region that corresponds to the shape of a typical hopper. This analysis will be based both on Coulomb materials and on critical state soil mechanics (CSSM). Some of the aspects under study will include an investigation of the relationship between CSSM and Coulomb models, shock and rarefaction wave solutions, boundary value problems for hopper flow, and stability of such solutions. An important application is the design of flow corrective devices. Subproject 3, liquefaction of soils: This phenomenon corresponds to the abrupt loss of load-bearing capacity of a loose, water-saturated soil, possibly leading to a massive landslide. Real world s oil failure/liquefaction will be investigated in collaboration with G. Gudehus and his associates. This project will combine experiments, mathematical analysis, computer simulation and industrial/geotechnical expertise to better understand the flow of granular materials. The area of study is of considerable importance to technical processes involving all types of granular materials, including but by no means limited to chemical process industries, and to the handling of coal, ores, food grains, and pharmaceuticals. Many aspects of the above processes are not fully understood, leading in some cases to enormous financial losses. Also under consideration are geotechnical issues such as the stability of embankments, as well as the stability of soils under earthquake conditions. The project will involve the application of existing theories for granular materials to such fundamental problems as flows in hoppers and stability of soils in landfills. New models will be developed in order to take into account some important aspects of granular flows such as fluctuations of forces. Recent experiments in this lab have shown that fluctuations, which are not accounted for in existing models, can be very strong and may well be necessary to provide safe and reliable design criteria for industrial devices involving granular flows. The models will be tightly linked to the experimental data, on the one hand, and, on the other hand, will be the basis of computer solutions for relevant technical problems.

0073841
Shearer

A combined experimental, analytical, and computational study of
fundamental problems in the dynamics of thin viscous films and fluid
interfaces is proposed. The recent discovery of stable undercompressive
waves in driven films has created the opportunity for a unique
collaboration between experiments and mathematical theory. This research
program will include related studies of solid-liquid-vapor interfaces,
moving contact lines, and surface tension effects. Analytical and
computational studies will be integrated with a series of experiments that
includes a search for undercompressive waves in a spin coating geometry,
motion of contact lines near room- temperature critical points, and high-
speed video imaging of the dynamics of singularity formation in finite-
time rupture of fluid interfaces. Mathematical analysis will include models
for film rupture, stability of driven contact lines, and numerical analysis of
schemes for computing these problems.

Liquid films and moving contact lines arise in problems ranging from
industrial design of paints and microchip fabrication to medical
applications including contact lenses and the lining of the lung. All of
these problems involve interactions across widely different length-scales
in which the physical laws are not clearly understood. This is a
collaboration between researchers from the Mathematics (Bertozzi,
Witelski) and Physics (Behringer) Departments at Duke University and
the Mathematics Department at North Carolina State University (Shearer).
This effort combines mathematical modeling, analysis, and numerical
simulation with new laboratory experiments to study fundamental
problems in driven films and moving contact lines. Computational and
mathematical models will direct the design of experiments investigating
new phenomena in spin coating processes and dewetting films. The
program will involve undergraduates, graduate students, postdoctoral
associates, and visiting scientists from other institutions. This research will
foster curriculum developments in the Departments of Mathematics,
Physics and the Center for Nonlinear and Complex Systems at Duke
University.

This research effort addresses a spectrum of fundamental and applied problems in the slow flow of granular materials. It is organized into four projects, chosen partly because of their importance in the field of granular materials, but also because they raise intriguing mathematical and scientific issues of broader significance. The first project attacks a fundamental physical problem: How to include micromechanical effects in a continuum description of granular flow, especially the effect of velocity fluctuations. The second project concerns the mathematics surrounding multidimensional continuum models for granular flow, specifically the issue of extracting mathematically rigorous information from ill-posed partial differential equations. The third project proposes to extend Jenike's radial solution for flows in axisymmetric hoppers to conical hoppers with a general cross section. The fourth project deals with flows of fine granular materials, where the interstitial gas significantly affects the flow. The research program involves coordinated efforts in modeling, analysis, numerical simulations, and experiment.

At the heart of this research project is a basic question concerning the flow of granular materials: "What behaviors of slowly flowing granular material can be understood in terms of a continuum formulation?" This question may be viewed as an attempt to reconcile continuum models, used in industrial design and engineering problem solving, with discrete models, introduced to understand the results of small- to medium-scale physical experiments. Continuum models are highly desirable since they are much more tractable analytically and computationally than particle dynamics simulations, which treat the discreteness of the flow directly. An important issue that arises from this basic question, and which is addressed in this project, is the form that a continuum description should take. This issue has been the subject of debate in the engineering literature ever since Janssen in 1895 demonstrated that stresses in a column of granular material do not increase indefinitely with depth, but approach asymptotically to a constant. The research program has significance well beyond the context of granular materials in mathematics as well as physics. The project is supported by a long-standing industrial collaboration.

NSF Award Abstract - DMS-0138923
Mathematical Sciences: Nonlinear Differential Equations, Mechanics and Bifurcation

Abstract

0138923 Shearer

This award supports U.S. participants in the conference on Nonlinear Differential Equations, Mechanics, and Bifurcation held at Duke University on May 20-22, 2002. The purpose of the conference is to explore connections between the fields of ordinary and partial differential equations, mechanics and bifurcation, and industrial mathematics, and to encourage the involvement of young mathematical scientists in these areas.
The speakers include established leaders in these fields, as well as promising young mathematicians and scientists. The meeting is a collaboration between the Research Triangle universities of Duke University, North Carolina State University, and the University of North Carolina at Chapel Hill.

The central theme of the conference is the role of nonlinearity in physical systems, especially elasticity, granular materials, and fluid flow. The interplay between experiments, modeling, theoretical mechanics, mathematical analysis, numerical simulation, and industrial design has proved particularly fruitful in recent years. This conference brings together scientists and mathematicians from all of these areas. The conference is designed to include lively discussion of the interplay between discoveries of new experimental phenomena, proposed mathematical models, theoretical developments and challenges, and corresponding innovations in numerical methods. Bifurcation and pattern formation are observed in experiments with solids, fluids and granular materials, and the mathematical techniques to analyze phenomena have grown quite sophisticated. This connection between experiment and theory, coupled to numerical simulation, occurs throughout the conference.

This collaborative project among Duke University, the University of California at Los Angeles, and North Carolina State University involves research into an array of interrelated problems in the fluid dynamics of thin viscous films. It combines analytical, computational, and experimental approaches to fundamental issues concerning surfactant spreading, patterned surfaces, vibrational forcing, Marangoni flows, flows on curved surfaces, and spin-coating flows. Methods include the theory of nonlinear partial differential equations, scientific computing, mathematical modeling, asymptotics, and experiments. Connections will be made with applications such as the role of surfactants in lung physiology, and the emerging field of microfluidics. The weekly Focused Research Group meeting, a central organizing feature of the project, combines research and education by promoting discussion and interaction across disciplines, and among faculty, students, and post-docs.

The central theme of this project is the study of how surface tension forces govern the motion of micro-scale fluids in many different settings. Progress in this basic research on thin liquid films will lead to new theoretical understanding of the physics of fluid flow on a microscopic scale, which is crucial to the emerging technology of microfluidic devices. These "labs on a chip" are transforming biotechnology in much the same way that microelectronics has transformed the computing industry. Contributions from this project will help to identify fluid flows and surface properties relevant to specific devices, as well as establishing the theoretical underpinnings for future research in this area. In addition to microfluidics, research will also be undertaken in the modeling of surfactant transport in the liquid lining of the lung. In developing new theory for this important physiological application, the research group will focus on how the surfactant changes surface tension, giving rise to a force that mobilizes the fluid coating of the lung walls. Development of theory, experiments, and numerical simulations in this project will involve ongoing interactions with scientists and engineers directly connected with the applications.

Proposal: DMS-0244488
PI: Michael Shearer
Institution: North Carolina State University
Title: FRG: Collaborative Research: Physical, Mathematical and Engineering Problems in Slow Granular Flows

ABSTRACT

This proposal addresses a spectrum of fundamental and applied problems in the slow flow of granular materials. It includes coordinated efforts in experiment, mathematical analysis, modeling, and numerical simulation, supported by a long-standing industrial collaboration. Continuum models of granular materials are desirable from several points of view, but the associated partial differential equations are typically ill-posed, a mathematical difficulty that has severely hampered progress, especially for multidimensional flow. This ill-posedness reflects a real instability, the tendency of the deformation in granular flow to localize into shear bands. Fundamental issues surrounding the continuum description will be addressed through inter-related projects that investigate: (i) the role of discreteness and nonlinearity in regularizing the ill-posed continuum description, (ii) the formulation from micromechanical considerations, based on experiments and MD simulations, of a continuum model that accounts for microscopic velocity fluctuations, (iii) mathematical predictions and experimental tests of multidimensional steady-state hopper flows, and (iv) the settling of powders, an industrially significant prototype two-phase flow problem.

The research program attacks different aspects of a basic question: What behavior of slowly flowing granular material can be understood by modeling the material as a continuum? This is particularly an issue for industrial settings such as the flow of agricultural grains in a converging hopper: When a silo is discharged, the granular material flows somewhat like a continuous fluid, but the forces on the hopper walls are unlike those exerted by a fluid due to the solid-like properties of the material and its discreteness. Scientifically, we wish to understand these properties, especially the role that the discreteness of the material plays in the models. Among other contributions, this research is expected to improve predictive capabilities in the materials-handling industry and thereby increase efficiency in manufacturing. The group will create a research environment that integrates research with the training of students and post-docs, through weekly group meetings, participation in conferences and a new course on granular materials. Collaboration across fields will be facilitated by the proximity of the mathematics and physics departments at Duke University, and by the existing collaborations between PIs at NC State and Duke Universities.

Shearer
DMS-0604047

Recent results suggest that there are strong connections
between the science and mathematics of thin liquid films and the
flow of thin layers of granular materials. The investigators
explores these connections through modeling, analysis, numerical
simulation and experiments. They study multidimensional
nonlinear systems of partial differential equations of avalanche
models, and the connection to experiments, to gain new insights
into fundamental aspects of granular and avalanche mechanics. In
related studies, the investigators formulate and analyze systems
of partial differential equations for thin liquid films driven by
surfactants or controlled by temperature. These equations give
rise to mathematical theory addressing new wave structures
observed in experiments and simulations.

The investigators explore strong connections between the
science and mathematics of thin liquid films and the flow of thin
layers of granular materials. Thin films are important in
medical applications such as surfactant replacement therapy, and
in microscopic coating flows that arise in industry and
manufacturing. Granular materials occur over a large range of
length scales, from pharmaceuticals and agricultural grains to
rocks and gravel in landslides. The results of this project
further the understanding of issues of both specialized
scientific interest and of broad significance to society, such as
the identification of mechanisms of wave propagation and patterns
in surfactant transport in human lungs, and the separation by
size of rocks and other particles in avalanches. This project
incorporates training of mathematics and physics graduate and
undergraduate students in mathematics, simulation, and
experimental physics in a multidisciplinary research program.
The participation of under-represented groups builds on recent
successes. Results from this project are disseminated through
journal publications, special sessions and lectures at
international conferences, articles in popular science
publications, and through a project web page.

The program "Mathematics of Materials" provides a wide range of interdisciplinary research and training opportunities for undergraduates, graduate students, and postdocs in the mathematical sciences. The research activities are organized around five topics that play a fundamental role in emerging technologies: multifunctional materials, polymers and composites including carbon nanotubes, orthopaedic biomaterials, dynamics of thin material layers, and material behavior of laser welding. Within each topic, investigations focus on fundamental model development, mathematical solution, numerical simulation, and control design. Each project involves substantial collaboration with experimental colleagues that provides interdisciplinary training opportunities to mathematics students and postdocs. The training component is designed to prepare students and postdocs for the varied roles of interdisciplinary research mathematicians. This includes training modules that introduce participants to research and career topics not typically covered in coursework, targeted courses on topics pertaining to research areas and national research agendas, and participation in summer internships and national conferences.

"Mathematics of Materials" is an interdisciplinary research training group program designed for undergraduates, graduate students and postdocs in the mathematical sciences. The research activities are organized around five topics that play a fundamental role in emerging
technologies: multifunctional materials, polymers and composites including carbon nanotubes, orthopaedic biomaterials, dynamics of thin material layers, and material behavior of laser welding. The training component involves a coordinated set of activities designed to prepare students and postdocs for the varied roles of interdisciplinary research mathematicians.
This includes training modules that introduce participants to research and career topics not covered in traditional coursework, new courses pertaining to the five research topics and areas of national need, and participation in summer internships and national conferences spanning multiple disciplines. The objective of the program is to attract and train highly qualified students and postdocs for academic and nonacademic careers at the interface between applied mathematics, materials science, engineering, physics, and advanced technology.

The science of thin liquid films has developed rapidly in recent years, with applications to coating flows, biofluids, microfluidic engineering, and medicine. In this project, we will focus on three groups of free-surface flow problems using a combination of mathematical modeling, analysis and numerical simulation, coupled to carefully chosen quantitative experiments. The three areas are: (i) flows driven by surface tension gradients, (ii) the role of inertia in regimes where lubrication is inadequate to model thin film flow, and (iii) vibrating thin films. Advances in our understanding of theoretical issues surrounding modeling these flows, and the exploration of new features of the flows through experiments will have direct applications to improving control and reliability of manufacturing processes in high-tech industries and to biomedical systems.

The flow of thin liquid layers or films occurs in contexts including industrial coating flows, microfluidic engineering, and medicine. In this project, we focus on properties of thin liquid films such as the role of surface tension in driving flow, and the behavior of thin layers under rapid vibration. Surface tension is an important mechanism in the dynamics of mucus layers in the lung, and is central to the treatment of premature infants by surfactant replacement therapy. The control of thin layer flow, through the temperature dependence of surface tension or using rapid vibrations, has the potential to eliminate damaging nonuniformities in industrial coating processes. Each of these areas and applications pose substantial scientific challenges that will be met with a collaborative effort between mathematicians and physicists, exploring theoretical and experimental methods in tandem. The project includes graduate and undergraduate student training in an interdisciplinary research program that will develop coordinated advanced skills in physics, engineering, and applied mathematics.

This research into the mathematics of deformation and flow of materials is motivated by applications to subsurface flows of fluids such as groundwater, oil and gas, avalanche dynamics, including the processing of granular materials in tumblers and chutes, the motion of microscopic fluid droplets, and the role of shock waves in the propagation of signals, for example in optical fibers. The mathematics linking these applications concerns the modeling of continuous processes with nonlinear equations, and the study of wave-like solutions of the equations. Recent experiments and new models have opened up opportunities for fresh insights into these complex systems. The research project will include graduate student training in mathematics and physics, with an emphasis on clear presentation of results at scientific conferences.

This research project on nonlinear models of continuum mechanics focuses on four areas where recent developments provide opportunities for progress. The investigation of dispersive and dissipative shock waves will take place in the context of models of magnetization and porous media flow. New models of two phase flow possess singular nonlinearities that stimulate innovations in analysis and numerical simulation. The study of contact lines of droplets on soft elastic substrates involves the formulation of models for asymmetric drops that can be solved by the method of transforms and dual integral equations. A conjectured mechanism for the motion and control of microscopic droplets is to induce spatial variations in the elastic modulus of the substrate, thereby providing a driving force on the droplet. Granular flow models, specifically with poly-disperse grain sizes, will be analyzed using the theory of nonlinear partial differential equations. The equations will incorporate recent descriptions of stress distribution in granular materials, coupled to an established dynamic mechanism that quantifies how grains of different sizes accumulate in distinct layers.