Workshop on Dynamical Systems and Continuum Physics

November 14-16, 2007
Organizer: L Tuckerman (PMMH-ESPCI, France)

Dynamical systems theory provides astonishingly faithful representations of systems from continuum physics. This workshop will focus on recent research in laminar and turbulent hydrodynamics and in some more discrete systems, notably granular media and foams.

In laminar hydrodynamics, transitions in rotating, sheared, and heated fluids provided motivation for the development of bifurcation theory during much of the 20th century. The successive transitions in Taylor-Couette flow and Rayleigh-Benard convection have long been classified as pitchfork and Hopf bifurcations. Attention has shifted to more complicated scenarios involving tori and heteroclinic orbits, and to open flows such as wakes and shear layers.

The Navier-Stokes equations continue to govern turbulent hydrodynamics; the challenge is to explain the transition to chaotic behavior, i.e. turbulence, displayed by these well-known deterministic equations, especially in the absence of linear instability.  Phenomenological laws developed for engineering purposes cannot be derived from first principles and, in addition, do not predict or describe transition. Patterns involving multiple turbulent structures or the coexistence of turbulent with laminar regions provide interesting new puzzles.

Granular media and foams, on the cutting edge of current continuum physics, are not as well understood as hydrodynamics.  Granular media display patterns such as segregation, lattices and waves, as well as various seeming paradoxes. Efforts have been made to formulate equations governing granular media at appropriate scales intermediate between molecular dynamics and continuum fields.  In foams, the elementary constituents are bubbles, governed by the competition between surface tension and pressure. Similar themes are present in highly viscous fluids as well.