12.207/18.354  Nonlinear Dynamics II: Continuum Systems

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12.207/18.354  Nonlinear Dynamics II: Continuum Systems

Spring 2011

Instructor: Tristan Gilet

Lecture:  TR 9.30-11 AM  (2-135)
Office Hours:  TR 5-6 PM  (2-336)      

Information: 

This course introduces the basic ideas for understanding the dynamics of continuum systems, in the context of specific examples taken from a range of different fields. Our goal will be to explain the general principles, and also to illustrate them via important physical effects. A parallel goal of this course is to give you an introduction to mathematical modeling.

The first part of the course will study diffusion, to demonstrate how continuum descriptions arise from averaging microscopic degrees of freedom. The equations of motion that we derive for continuum systems are typically nonlinear partial differential equations, for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion.

We shall then study the `calculus of variations', which is a minimization approach to finding solutions of continuous systems. We are familiar with these techniques for discrete systems and now adapt the ideas to help us with continuous systems. First we examine the classical brachiostrome problem posed by Bernoulli, and then consider an array of problems in many different physical systems (e.g. orientation of domains in a ferromagnet, shapes of soap films, bending of elastic beams, etc.)

Having appreciated the subtleties of constructing continuum descriptions of systems with many degrees of freedom, we will proceed to study the stability of solutions. Time permitting, we shall first study neutral stability to develop an understanding of wave motion (water waves, sound waves, gravity waves) before proceeding to study a range of instabilities (Rayleigh-Taylor,Turing, Taylor-Couette etc.).

In the second half of the course we will examine singular perturbations. We will see how Prandtl's boundary layer resolved nineteenth century mysteries about dissipation in low viscosity fluids, and led to an understanding of airplane flight. We will also study the Ekman layer and its role in controlling rotating flows, leading to an understanding of the dynamics of both the atmosphere and the ocean.

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