PHY458 Geometric Methods in Fluid Mechanics
S. G. Rajeev
Mon/Wed at 10:25-11:40am in Meliora 219
First Class on Mon, Sep 8 2014
Office Hours: Mondays /Wednesdays 11:45-12:45 pm
I will be using Blackboard to post Notes and Problems.
Synopsis
This course describes several ways in which geometrical methods can be applied to fluid mechanics. At an elementary level,
it involves visualizing advection in terms of integral curves of vector fields. Or using curvilinear co-ordinates to solve
Euler and Navier-Stokes equations in various geometries. Much of the material is old, although not usually presented geometrically.
Some new ideas,speculations and directions of research will be presented on each topic.
The deepest contribution to the subject to date is by V.I. Arnold: his realization of Euler equations as geodesic equations of the
diffeomorphism group and the computation of its sectional curvature.That this is negative in most directions explains the instability common in fluid mechanics.
Also fundamental is the work of Holm, Marsden and Ratiu on the hamiltonian formulation of Euler-Poincare equations.
Along the way, we will touch on Smale's work on chaos; Aref's remarkably simple model of chaotic advection; how the topology of braids can be used to design mixing machines
(e.g., for pizza dough) in mechanical engineering; and delve deep into the geometry of geodesics on a Lie group.
Pre-requisites
A basic knowledge of Riemannian geometry (e.g., a course on General Relativity AST231 or PHY413) and of fluid mechanics (e.g., PHY457) will be helpful. However, I will review what is needed and build the course from the ground up.
References
There is no one textbook that covers this ground. A mix of books and review articles will be used instead, along with my own notes.
Books
• V. I. Arnold and B. A. Khesin Topological Methods in Hydrodynamics Springer (1999)
• S. G. Rajeev Advanced Mechanics From Euler's Determinism to Arnold's Chaos Oxford(2013)
Journal Articles
• H. Aref Stirring by Chaotic Advection J. Fluid Mech. 143, 1-21 (1984)
• S. Smale Differentiable Dynamical Systems Bull Amer Math Soc 73, 747-817 (1967)
• V. I. Arnold Ann. Inst. Poly. Grenoble 16, 319-361 (1966)
• J. Milnor Curvatures of the Left Invariant Metrics on Lie Groups Adv. Math. 29, 293-329 (1976)
Online Articles
Syllabus
Syllabus