Nov 25 2002

Stability of hydrodynamical flows

Drazin book excerpts

Two sections are included here: the first gives a general overview, the second is a sample normal form, for one of the most common onsets of instability, bifurcation of a stationary solution into oscillating Hopf attracting cycle. It becomes quickly clear that there is no realistic hydrodynamics stability problem where one does not get swamped by a sea of details (perhaps Faraday waves are the simplest example) so I have chosen to illustrate linear stability analysis by a simple problem, a 1-d "flame flutter" problem that gives you a flavor of how you would proceed to investigate linear stability in a spatially extended system which is sort of 1-d Navier-Stokes (but not really):

Linear stability of spatiotemporal dynamical systems

Read section 2.4 of Chapter 3: Flows

Read section 4.3.1 of Chapter 4: Local stability

You might want to read other sections of these chapters to make sense of the material covered in the lecture.

From the Kuramoto-Sivasinsky example you learn that "viscous fluids" near the first onset of linear instability can be described by low-dimensional ODEs, and hopefully get some feeling why the 2-d stationary flow bifurcating into a Hopf limit cycle offers also a normal-form description of an "infinite-dimensional" PDE instability onset. It's a long (and so far unexplored) way from this first instability to an honest theory of turbulence, sorry.

Further links

CHAOS: CLASSICAL AND QUANTUM webbook