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.
CHAOS: CLASSICAL AND QUANTUM webbook