Applied Mathematics Colloquia Series
Archive

Fall 1999

September

09/03-11/30
Michael J. Shelley
Courant Institute
New York University

The hydrodynamics of elastic filaments in a stokes flow

In this talk, Dr. Shelley will discuss the interaction of a viscous fluid with elastic filament; a problem of fundamental importance in physics and biology (e.g., biological fibers, motility of microorganisms, phase-transition of liquid crystals). Motivated by the pattern formation during the growth of liquid crystals in isotropic smectic-A phase transition, Dr. Shelley will consider the nonlocal Stokesion (inertialess) dynamics of a growing elastic filament immersed in a fluid. The nonlocal interactions of the filament with itself are taken into account by a modification of Keller-Rubinow's slender body-theory. Given this new formulation, Dr. Shelley will show that there exists a buckling instability driven by the growth of the filament. Eventually, the coupling of the buckling instability with the non-local interaction of the filament with itself and the fluid leads to a space-filling labyrinthine pattern. Theoretical predictions and new numerical methods will be used to study the long-time dynamics of the system. (Abstract by A. Goriely).

09/10-11/30
Koen Visscher
Department of Physics
University of Arizona

Single kinesin molecule studied with a molecular force clamp

Kinesin is a two-headed motor protein that moves processively along microtubules in discrete steps of -8 nm, driven by the hydrolysis of ATP. Molecular details of the mechanochemistry of this motor remain obscure. How is chemical energy stored in ATP coupled to mechanical displacement? Does the number of ATP molecules required for movement increase when kinesin has to work against an opposing force? To shed light on these questions, a force clamp was constructed based on a feedback-driven optical tweezers capable of maintaining constant loads on single moving kinesin molecules. This novel instrument provides unprecedented resolution of molecular motion and permits mechanochemical studies under controlled external loads. Analysis of records of kinesin motion under variable ATP concentrations and loads revealed several new features. 1) Kinesin stepping is tightly coupled to ATP hydrolysis over a wide range of forces, requiring only a single hydrolysis per 8 nm step. 2) Kinesin stall forces depend upon the ATP concentration. 3) Increased loads reduce the maximum velocity as anticipated but also raise the apparent Michaelis-Menten constant. This last result indicates that the kinesin cycle contains at least one load-dependent transition affecting the rate at which ATP molecules bind and subsequently commit to hydrolysis. But, it is likely that at least one other load-dependent rate exists, affecting turnover number. Together, these findings necessitate revisions to current models and understanding of kinesin motion.

09/17-11/30
Monika Nitsche
Department of Mathematics and Statistics
University of New Mexico

Numerical simulations of vortex rings using the vortex sheet model

The vortex sheet is a mathematical model used to simulate shear flows. During the first half of this talk I will describe how vortex sheet motion is computed and what the inherent difficulties are. I then apply the model to simulate the formation of vortex rings. We first validate the model by comparison with experimental data validates and then use it to gain information about the flow. The focus will be on recent results concerning the fluid behaviour near the vortex core.

09/24-11/30
Lorenz Kramer
The University of Bayreuth, Germany

Simple theory of anisotropic convection

The speaker will introduce and discuss the modulational equations to describe these systems in an "extended" weakly nonlinear range that includes the relevant secondary instabilities. The equations are distinctively different from those for isotropic systems (e.g. Rayleigh Benard convection in simple fluids). The physical realizations are found in particular in electroconvection in liquid crystals, which has been, and still is being studied experimentally fairly intensively.

October

10/01-11/30
Paul Dubois
Lawrence Livermore National Laboratory

Object technology for scientific computing

Object technology is finally making inroads into scientific programming. In this talk we will cover four areas in which this technology is being brought to bear on the problems of creating large scientific simulations:

  • Expression template technology enables Fortran-speed code that is nevertheless highly abstract.
  • Object design can make mathematical libraries much more usable.
  • Object-based Fortran can bring some of the benefits of the object revolution to the Fortran programmer.
  • Steering using an object-oriented scripting language is creating "plug and play" architectures for scientific programs, responding to the growing pressures for omnibus models and rapid response to business opportunities.
10/08-11/30
Harrison Barrett
Department of Radiology
University of Arizona

Inverse problems with discrete data sets

10/15-11/30
Joseph Pedlosky
Woods Hole Oceanographic Institution

The tunneling of Rossby waves in the ocean

Oceanic flows of the very large scale associated with the general circulation often encounter planetary scale islands (e.g. Australia). The mid-ocean ridge system with segments separated by transform fault gaps present other examples of islands in the stream of planetary flow. A review will be given of the special dynamical features that arise when such geographical and topographic islands are considered. Both steady flows and time dependent, wave problems are examined.

In the steady flow case recirculations are generated by the presence of the island; the explanation for which follows directly from Kelvin's theorem and certain peculiar features of large scale oceanic flows as they are affected by the earth's rotation. Both the theory and confirming laboratory experiments will be described.

In the case of waves, interest is directed to the dynamics of large scale, westward propagating Rossby waves and the unexpected near-transparency of long meridional barriers to such waves as long as at least two small gaps are present in the barrier. Again, Kelvin's theorem is the essential ingredient for understanding the ability of the wave to pass through the barrier. The normal modes of ocean basins containing such barriers are described.

10/22-11/30
Guenter Ahlers
Center for Nonlinear Science and Department of Physics
University of California, Santa Barbara

Pattern formation near onset of Rayleigh-Bénard convection: Some simple unexplained results

Our goals as experimentalists in the field of pattern formation fall into three categories:

  1. Test already existing theoretical predictions;
  2. Find qualitatively new phenomena which are simple enough to be understood in the not too distant future;
  3. Design simple "idealized" experiments illustrating important general phenomena.

This talk will illustrate how these goals are pursued, by discussing a number of studies of pattern formation under carefully controlled conditions near the onset of convection in a shallow horizontal layer of a fluid heated from below.

The first part of the talk will be devoted to experimental results for convection-roll selected in a cylindrical sample which had an interior section of uniform spacing. For Rayleigh numbers above the critical value in the interior, straight or slightly curved rolls were selected. It is observed that in some regimes, the pattern repeatedly formed defects. The defects moved in the direction of the perturbation associated with the instability responsible for their formation.

In the second half of the talk, the effect of a Coriolis force due to rotation of a sample with rigid sidewalls (no ramp) about a vertical axis will be considered. At small dimensionless rotation rates appropriate Prandtl numbers domain chaos was found immediately above a supercritical bifurcation. The dependence of the time and length scales of the chaotic state on the Rayleigh number differed from the theoretically expected dependence. For other values, the patterns differed from the theoretically expected chaotic state. Instead, rotating square patterns were found. Finally, some opportunities for future work will be mentioned.

10/29-11/30
Alan Newell
Department of Mathematics
University of Arizona

Wave turbulence in optics and oceans, in sound, semiconductor lasers and the solar wind

November

11/05-30
Peter E. Crouch
College of Engineering and Applied Sciences
Arizona State University

Ridid body equations, double bracket equations and optimal control

In this talk we introduce optimal control problems, or variational problems on adjoint orbits of certain Lie Groups, and demonstrate a form of extremal equations, with a particular Lie Bracket structure. These equations may be exhibited as Hamiltonian equations or Euler Lagrange equations, for a special form of metric on the adjoint orbits. A special subproblem may be stated as an optimal control problem on certain symmetric spaces, which yields additional structure in the extremal equations. These equations are then further related to the Left and Right generalized rigid body equations, through a nontraditional form of the equations, which we call the symmetric rigid body equations. These equations are locally equivalent to the classical equations, which are geodesic equations on the Lie group. A corresponding discrete time optimal control problem is introduced and the extremal trajectories exhibited. Their relation to the continuous time optimal control problems above is discussed, as are their relation to the Moser-Veselov symplectic integrator for the generalized rigid body equations. Our formulation of the corresponding discrete flow displays the same symmetry as does our nontraditional, symmetric form for the generalized rigid body equations, shedding new light on the Moser-Veselov equations.

11/12-30
Michael B. Brenner
Department of Mathematics
Massachusetts Institute of Technology

Towards an effective theory of sedimentation

Consider a large group of particles falling very slowly together in a viscous fluid. The interesting and important question is to understand how to formulate an effective theory for the sediment. The difficulty is that even though the mixture might be very dilute the interactions between particles are long ranged, with velocities far from a single particle falling off as $r^{-1}$. Recently it has been realized that the assumptions used in presently accepted effective theories imply that the velocity fluctuations of the particles (relative to the mean flow) diverge in an infinite system. On the other hand, several types of experiments (e.g. (Segre, Herboltzheimer and Chaikin, Phys. Rev. Lett. 79, 2574(1997)) have measured finite fluctuations, independent of system size. Through theory, scaling arguments and computer simulations, this talk will explain what is known and not known about this discrepency, and in general about what is the correct effective theory for a sediment.

11/19-30
Giovanni Gallavotti
Department of Physics
La Sapienza, Roma

Nonequilibrium statistical mechanics and turbulence: The chaotic hypothesis

The chaotic hypothesis is an extension of the ergodic hypothesis to mechanical systems in stationary states but out of equilibrium (e.g. an ionized gas in an electric field which generates a stationary current). The hypothesis is that such systems behave "essentially" as uniformly hyperbolic systems. This is an idea that goes back to the early '70's (Ruelle) and which recently has produced some concrete "predictions" and is therefore worth pushing to the extreme. The talk will illustrate the few successes and many problems and difficulties that have been generated by the attempts to understand better the chaotic hypothesis.

December

12/03
J. Nathan Kulz
Department of Applied Mathematics
University of Washington

Dynamics, stability, and bifurcation of topological solitons in the optical parametric oscillator

We consider the dynamics associated with topological solitons (localized structures) of the optical parametric oscillator which models the parametric exchange of energy between optical fields at a fundamental and second harmonic frequency. Simulations show that this nonlinear interaction can support stable front structures as well as localized, bistable solitary wave solutions. We perform a systematic study of the bifurcation structure and stability analysis of both solitary wave and front solutions which arise. The stability analysis is carried out for the onset of instability which arises from a Ginzburg-Landau description as well as a modified Swift-Hohenberg description at resonance. The analysis, which is carried out in 1-D, can be utilized in predicting the dynamical behavior in 2-D systems. Further, the theoretical conclusions provide important practical predictions which are verified via extensive numerical simulations.