Summer 2001

Fluid Dynamics


This is a preliminary version of the course announcement. For an up-to-date version visit the class page in German language.

Summary:
This course is an introduction to the theory of fluids. The main focus are the Euler equations for incompressible inviscid fluids (``water'') as the most the fundamental model for fluids. As time permits, we will discuss the effect of viscosity, and of other physical effects that are often of crucial importance in applications.

Although coming from a mathematical perspective, we will emphasize the interplay of Modeling, Analysis, Simulation, and Experiment, which is central to current research in the field.


Language:
German or English, depending on the participants.

Hours:
Two hours lecture and one hour tutorial per week. Participartion in the tutorial is strongly encouraged.

Main Textbook:
  • C. Marchioro and M. Pulvirenti: ``Mathematical Theory of Incompressible Nonviscous Fluids,'' Springer-Verlag, 1994.

Supplementary Texts:
  • V. I. Arnold and B. A. Khesin, ``Topological Methods in Hydrodynamics,'' Springer-Verlag, 1998.
  • A. J. Chorin and J. E. Marsden: ``A Mathematical Introduction to Fluid Mechanics,'' Springer-Verlag, 1989.
  • G. K. Batchelor, ``An Introduction to Fluid Dynamics,'' Cambridge University Press, 1967.
  • P. G. Saffman: ``Vortex Dynamics,'' Cambridge University Press, 1992.

Prerequisites:
Working knowledge of vector analysis (Analysis II/IV or Electrodynamics). The course will draw on a variety of ideas from other subject areas, in particular Classical Mechanis, Electrodynamics, Differential equations, Functional Analysis, Numerical Analysis and Differential Geometry. However, I will present the necessary tools as part of the course and, while seeking mathematical rigor whenever possible, try to avoid unnecessary abstraction.


List of Topics

Derivation of the Euler equations:
From particle dynamics to diffeomorphism groups, Hamilton's principle of stationary action, Helmholtz decomposition, Eulerian versus Lagrangian viewpoint, vorticity, symmetries and conservation laws, boundary conditions.
Existence and uniqueness of solutions:
Contraction mapping theorem, Picard iterations on a Banach space, quasi-Lipschitz vector fields, regularity of solutions, global existence in 3D: the Beale-Kato-Majda condition.
Vortex dynamics:
Point vortex dynamics, vortex motion in the presence of boundaries, numerical vortex methods, vortex sheets and the Kelvin-Helmholtz instability; vortex motion in 3D: radial vorticity distribution, ``2 1/2D'' flows, vortex filaments.
Viscous Fluids:
Navier-Stokes equations, Leray's weak solutions, the Stokes-operator and boundary conditions, inviscid limits, visco-elasticity and other non-newtonian effects.
Issues in geophysical fluid dynamics:
Coriolis acceleration, inhomogeneous fluids, thermodynamics, Boussinesq- and hydrostatic approximations, shallow water theory.
Optional topics:
Introduction to hydrodynamic stability theory, Lagrangian averaging, numerics of the Navier-Stokes equations, LES (large eddy simulation) versus DNS (direct numerical simulalation, gas dynamics.


Last modified: 2000/10/22
Marcel Oliver (oliver@uni-tuebingen.de)