Mathematics 228: Mathematical Fluid Dynamics (Spring 2002)

Instructor

Andy Bernoff

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.

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.

Here is a brief outline of possible topics:

I. Mixing
    A. Advection-Diffusion Equation
    B. Exact Solutions
    C. Chaotic Advection
    D. Shear/Diffusion Mechanisms
    E. Taylor Diffusion
II. Inviscid Irrotational Fluids
    A. Potential Flow
    B. Free Surface Waves
    C. Rayleigh-Taylor instability
    D. Helmholtz instability of a liquid cylinder
    E. Parametric instability of a free surface -- the Faraday problem
    F. Amplitude Equations (Nonlinear Schroedinger and Korteweg-deVries)
III.  Dynamics of Vorticity (2D)
    A. The vorticity equation
    B. Circulation
    C. Dynamics of Point Vortices
    D. Motion of a vortex filament
III. Viscous Flows
    A. Navier-Stokes equations
    B. Viscous shear flows
    C. Stokes Flow
    D. Thin film equations
    E. Viscous effects in vortices

Prerequisites

A working knowledge of basic complex variables, Fourier series, Fourier transforms and some basic PDE's ( i.e. Laplace's equation, the heat equation, and the wave equation). Since Prof. Bernoff will not be here till the beginning of the Spring semester, students having questions about the prerequisites may e-mail bertozzi@math.duke.edu in the meantime.

Text(s)

The course textbook will be  Elementary Fluid Dynamics by  D. J. Acheson.

Supplementary texts will be suggested during the term.

Course Website

For more information see http://www.math.hmc.edu/~ajb/fluids/

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Last modified: 17 October 2001