Applied Mathematics Colloquia Series
Archive

Spring 2000

January

01/21
Charles R. Doering
Department of Mathematics
University of Michigan

Laminar and turbulent energy dissipation in a shear layer

The rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. For sufficiently large values of the suction rate, a steady laminar flow is absolutely stable at all shear rates. For sufficiently small but nonzero suction, however, the laminar flow is linearly unstable at high Reynolds numbers. We find that the rigorous upper bound on the energy dissipation rate---valid even for turbulent (and weak) solutions of the Navier-Stokes equations---scale precisely the same as the dissipation in the laminar solution in the zero viscosity limit. Both the laminar and any possible turbulent flows display a finite nonvanishing residual dissipation as the viscosity is decreased. This is a manifestation of so-called Kolmogorov-type scaling in which the energy dissipation rate becomes independent of the viscosity at high Reynolds numbers. This result establishes the sharpness of the upper bound's scaling in the vanishing viscosity limit---for these boundary conditions. It also provides a mathematical illustration of the delicacy of corrections to high Reynolds number scaling (such as the logarithmic terms as appearing in the Prandtl-von Karman "law of the wall") to perturbation in the boundary conditions. This is joint work with Edward A. Spiegel (Columbia University) and Rodney A. Worthing (University of Michigan).

February

02/04
John A. Strain
Department of Mathematics
University of California, Berkeley

Modular methods for moving interfaces

Professor Strain will present a fast modular numerical method for solving general moving interface problems. It simplifies code development by providing a black-box solver which moves a given interface one step with given normal velocity. The method combines an efficiently redistanced level set approach, a problem-independent velocity extension, and a second-order semi-Lagrangian time stepping scheme. Adaptive quadtree meshes concentrate computational effort on the interface, so an N-element interface costs only O(N log N) work per time step. Numerical results show that the method computes accurate viscosity solutions to a wide variety of difficult geometric moving interface problems involving merging, anisotropy, faceting, nonlocality and curvature.

02/11
Byron Goldstein
Los Alamos National Laboratory

Biosensors: from detectors to analytic instruments

Biosensors detect biomolecules by using a biological recognition mechanism coupled with a physical transduction technique. Some biosensors are used purely as detection devices to indicate when a specific biomolecule is present above some threshold concentration, the limit of the detector's sensitivity. Other biosensors are used as analytical instruments in basic research to obtain detailed quantitative information about a biological interaction. The most popular such instrument is an optical biosensor called a BIACORE that is now in wide use as a tool for extracting reaction rates from the observed time course of a biomolecular reaction. In the BIACORE, one reactant in solution, the analyte, flows past a second reactant attached to a sensor surface. Reliable interpretation of the resulting binding data depends on understanding the roles of flow, diffusion, surface attachment, and reaction. We have derived conditions under which simple models, consisting of ordinary differential equations for binding at the sensor surface, can be used to separate information about the chemical reaction from confounding factors inherent in the biosensor design. To test these models we have developed a computer model that can simulate BIACORE experiments. This model accounts for the flow and diffusion of the analyte in solution, the chemical reaction at the sensor surface and the geometry of the BIACORE. Our approach is to use the computer model to simulate BIACORE experiments and then see if the simple models can fit the simulated data and yield the correct chemical rate constants.

02/18
Shripad Tuljapurkar
Mountain View Research

Transition dynamics in human populations

Population momentum is the ratio of a population's ultimate size after a demographic transition to its initial size before the transition. For an instantaneous drop to replacement fertility, Nathan Keyfitz found a simple expression $M_{K}$ for the momentum. However, as Keyfitz pointed out, `no one thinks that any country will drop immediately to stationary reproduction patterns'. I present exact analytical results developed jointly with Nan Li concerning the momentum of a population whose demographic transition completes within a finite time. I show that relatively simple formulae are accurate approximations to population momentum for transitions that take as long as 100 years. I show also that the speed of fertility decline makes a substantial difference to population momentum.

March

03/03
C. David Levermore
Department of Mathematics
University of Arizona

The mathematics of scales

Science is often faced with describing phenomena that can be modeled mathematically on a variety of scales. For example, gases can be modeled as atoms or as a continuum, epidemics by individuals or populations. The mathematics of scales strives to understand the connections between such models. A number of examples will be presented of successes (few) and failures (many). More questions will be raised than answered.

03/24
Philip Holmes
Program in Applied & Computational Mathematics
Princeton University

Models for insect locomotion, or why cockroaches get away

03/31
Katja Lindenberg
Department of Chemistry and Biochemistry
University of California, San Diego

Nonclassical diffusion-limited kinetics in constrained geometries

Simple chemical reactions in low dimensions under a great variety of conditions lead to the spontaneous formation of spatial patterns and to associated "anomalous" (usually slower) rate laws for the global densities of the reacting species. The kinetic anomalies arise as a consequence of the spatial distribution of the reactants. The standard exponents reflect a random spatially homogeneous distribution; the law of mass action presumes that such a distribution is maintained at all times, even as the reaction proceeds. This, in turn, presumes an efficient mixing mechanism. What might such a mixing mechanism be? Examples include physical stirring, convection, and diffusion. Deviations from a homogeneous distribution arise when there is no mixing or when there is insufficient mixing. Diffusion is not a particularly effective mixing mechanism, especially in low dimensions. As a result, initial spatial density fluctuations lead to spatial ordering at later times, and this spatial ordering in turn is manifested through nonclassical rate laws. In this lecture we discuss the underlying ideas that lead to this behavior, as well as the different physical manifestations of spatial ordering. We present analytical and numerical methods that have been used to address these problems. We also briefly mention experimental evidence (not plentiful) of these effects, and some completely different contexts (unrelated to chemical reactions) where these same descriptions and effects are applicable.

April

04/07
Chris Jordan
VCAPP
Washington State University

The mechanics of undulatory swimming: lessons from virtual and robotic leeches

Swimming with whole body undulations involves a mechanical interaction between the organism's tissues and its fluid surroundings. Unfortunately, we do not fully understand the form of this interaction, nor do we understand how variation in an organism's morphology and physiology may affect this interaction. It is readily apparent that the internal and external components of the swimming system are tightly coupled, and that the coupling plays a major role in determining the swimming behavior exhibited by whole body undulators. However, it is the coupled nature of the internal and external mechanics that makes the problem so challenging. Studied in isolation, the internal and external mechanical components of the swimming system may not be representative of their in situ behavior. To address this issue, as well as the form of the coupling and its sensitivity to the organism's morphology and physiology, I am taking a number of approaches. One, the 'virtual leech', is a mathematical model of a flexible body constructed from elements that have the mechanical properties of both active and passive biological materials. The model is coupled to a simplistic representation of a fluid, however the approach explicitly accounts for the internal and external mechanics, as well as their interactions. Other approaches are essentially refinements on the above model: a finite-element representation of a soft-bodied organism coupled to a Navier-Stokes fluid solver, and a mechanical undulator with prescribed morphology, kinematics and swimming speed with which I can measure the fluid-body interactions as a function of body form and swimming kinematics.

04/14
Gil Strang
Department of Mathematics
Massachusetts Institute of Technology

Partly random graphs and small world networks

It is almost true that any two people in the US are connected by less than six steps from one friend to another. What are models for large graphs with such small diameters ?

Watts and Strogatz observed (in Nature, June 1998) that a few random edges in a graph could quickly reduce its diameter (longest distance between two nodes). We report on an analysis by Newman and Watts (using mathematics of physicists) to estimate the diameter with an N-cycle and M random shortcuts, 1 << M << N.

We also study a related model, which adds N edges around a second (but now random) cycle. The average distance between pairs becomes nearly A log n + B. The eigenvalues of the adjacency matrix are surprisingly close to an arithmetic progression; for each cycle they would be cosines, the sum changes everything.

We will discuss some of the analysis (with Alan Edelman and Henrik Eriksson at MIT) and also some applications. We also report on the surprising eigenvalue distribution for trees (large and growing ) found by Li He and Xiangwei Liu. And a nice work by Jon Kleinberg discusses when the short paths can actually be located efficiently.

04/21
Martin Bazant
Department of Mathematics
Massachusetts Institute of Technology

Renormalization groups and central limit theorems in percolation

Percolation is used in almost every area of science as the simplest model for spatial disorder. The model amounts to randomly coloring each of N sites in a graph (e.g. the hypercubic lattice) either black or white with probability p and then identifying "clusters" of adjacent black sites. This very simple coin-flipping game has a rather nontrivial "phase transition" in the limit N -> oo which is apparent in the expected size of the largest cluster S: For p less than some critical value p_c, the largest cluster is negligably small, E[S] = O(log N), while for p > p_c it occupies a nonzero fraction of the system E[S] = O(N). (At p = p_c it is a fractal.) Although this scaling of the mean is well-known, however, the scaling of higher moments and the limiting shapes of the probability distribution of S are not. Here, these quantities are derived analytically and checked numerically on square lattices of up to 30 million sites. A "renormalization-group" picture of percolation is proposed that draws on classical ideas from probability theory as well as the modern theory of critical phenomena.

04/28
Sidney A. Coon
Department of Physics
New Mexico State University

Three-nucleon interactions and light nuclei

Few-body systems are the simplest systems where predictions of realistic models can be accurately computed and the results compared to complete experimental measurements. As an example, I will discuss the construction of a three-nucleon interaction, called the Tucson-Melbourne interaction in the literature, which is based on the interchange of virtual pions. This interaction is subjected to the constraints of the chiral symmetry of the pion-nucleon (strong) interaction and of pion-nucleon scattering data. This three-body force is meant to be used in the non-relativistic many-nucleon Schroedinger equation. I will describe contemporary mathematical techniques to solve the three and four-nucleon bound states (nuclei) with realistic two- and three-body forces and conclude with recent bound state (and some continuum) results.