Mathematics 228: Mathematical Fluid
Dynamics (Spring 2003)
Instructor
Andrea Bertozzi
Description
This course is designed
to give an overview of fluid dynamics from a mathematical viewpoint, and to
introduce the student to areas of active research in fluid dynamics. This
course is aimed at first year mathematics graduate students, although
students in related fields are encouraged to participate. The course will
include some elementary discussion of experiments and numerical methods,
especially in part III on interface dynamics.
Fluid dynamics is one of the most important areas of applied mathematics in
which ODE's, PDE's, and scientific computing are applied. Although this
course is not a prerequisite for any later courses (except special Topics
courses), it serves as important background for all students interested in
applied mathematics.
Syllabus
- The Euler and Navier-Stokes equations.
-
Symmetry groups, particle trajectories, The role of vorticity.
Shear, deformation, and rotation. Conserved quantities.
Leray's formulation of incompressible flows and Hodge's decomposition of
vector fields.
- Vorticity.
The vorticity stream form for 2D flows.
Radial eddies for the Euler and Navier-Stokes equations.
Examples of 3D flows with nontrivial vortex dynamics.
``2 1/2'' D flows. 3D Axisymmetric flows and the role of swirl.
3D Beltrami flows. The role of vorticity in 3D.
Introduction to singular integral operators in hydrodynamics.
Reformulation of the Euler equation as an
introgro-differential equation for the particle trajectories.
- Interface dynamics in incompressible inviscid flow.
- Discontinuous vorticity
Vortex patches. Some elementary examples. Contour dynamics.
- Discontinuous velocity
Potential flow. Introduction to vortex sheets.
The Birkhoff-Rott equation and the Kelvin-Helmholtz instability.
Requirements
Homework assignments will be due roughly every 1.5
weeks. Depending on the size of the class, there will be either a final
exam or a final project.
Prerequisites
Complex variables, elementary real analysis, working
knowledge of ODEs, Fourier transform and Fourier series.
Basic knowledge of PDEs is useful but not required.
Text
Supplemental Reserve Room Reading
- A Mathematical Introduction to Fluid Mechanics
by A. J. Chorin and J. E. Marsden
- Acheson, Elementary Fluid Dynamics
- Landau and Lifshitz, Fluid Mechanics
Course Website
For more information see http://www.math.duke.edu/~bertozzi/hydrodynamics.html
Return to:
Course List *
Math Graduate Program *
Department of Mathematics *
Duke University
Last modified: 24 October 2002