The Craik-Criminale class of solutions to the incompressible
Navier-Stokes and similar equations
Bruce Fabijonas
Department of Mathematics, Southern Methodist University,
Dallas, TX.
Tuesday, June 25, 2002 15:00-16:00,
Room 145, NIST North (820)
Gaithersburg
Tuesday, June 25, 2002 13:00-14:00,
Room 4511
Boulder
Abstract:
Navier-Stokes equations expressed as a sum of a `base' flow which is linear
in the spatial coordinates in an unbounded domain and a `disturbane' in the
form of a standing wave. A classic example of such a solution is Bayly's
elliptic instability. Bayly (1986) perturbed a rotating column of inviscid,
incompressible fluid with elliptic streamlines by a modulated standing wave
and found that for certain orientations of the wavevector, the amplitude
grew exponentially in time in the linearized equations. The mechanism for
this instability is parametric resonance. What Bayly failed to realize is
that his solutions, including those which grow unbounded in time, are exact
solutions to the nonlinear equations. The speaker's 1997 PhD thesis
analyzed the stability of Bayly's solutions, and is the first known
stability analysis of a Craik-Criminale solution.
This talk will focus on the speaker's recent work in this field, including a
stability analysis of a Craik-Criminale solution for the magnetohydrodynamic
equations. This talk will also address Craik-Criminale solutions which
appear in the Lagrangian averaged Navier-Stokes-alpha model of turbulence,
also known as the viscous Camassa-Holm equations, and discuss how the model
alters Bayly's elliptic instability.
Contact: D. W. LozierNote: Visitors from outside NIST must contact
Robin Bickel; (301) 975-3668;
at least 24 hours in advance.
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