I. 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.
II. 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.
III. Interface dynamics in incompressible inviscid flow.
Supplemental Reserve Room Reading:
Audience: 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, esp in part III on interface dynamics.
Prerequisites: Complex variables, elementary real analysis, working knowledge of ODEs, Fourier transform and Fourier series. Basic knowledge of PDEs is useful but not required.
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.
Time and Place: Physics Building room 120, MW, 2:20PM-3:35
Homework policy: All homework must be turned in IN CLASS on the day it is due. No partial credit will be given for late homework assignments. The final grade will be based on homework grades and on the final project or exam grade that will weigh as several homework assignments.