For more information about this meeting, contact Mark Levi, Leonid Berlyand, Alexei Novikov.
|Title:||The onset of steady vortices in Taylor-Couette flow: the role of approximate symmetry|
|Seminar:||Applied Analysis Seminar|
|Speaker:||David Schaeffer, Duke University|
|The onset of steady cellular motion in the flow between rotating concentric cylinders is the pre-eminent example of a hydrodynamic instability in an internal fluid flow. In his 1923 paper, Taylor obtained remarkable agreement between experimental results and theoretical predictions for the onset of instability in this geometry, agreement that remains unsurpassed in fluid mechanics to this day. This problem is often considered as an example of a pitchfork bifurcation in the Navier-Stokes equations. This view would be valid if the flow could be modeled as periodic along the length of the cylinder. More accurately, one should expect ‘end effects’ to disconnect the pitchfork of the idealized problem. Taylor’s own results and many subsequent experiments and simulations have appeared to confirm this ‘imperfect bifurcation’ conceptual model. However, end effects in the finite-length case are not a small perturbation of the problem with periodic boundary conditions. Specifically, no matter how long the apparatus, finite-length effects greatly perturb the disconnected branch of the pitchfork of the periodic model (what Benjamin calls anomalous-mode flows). On the other hand, in long cylinders these effects appear to change the connected branch (normal-mode flows) only minimally. We propose a resolution of this seemingly contradictory behavior in terms of a symmetry breaking bifurcation. The relevant symmetry, which is only approximate, is between two normal-mode flows with large, and nearly equal, numbers of cells, quite different from the familiar translational symmetry of the periodic problem.|
Room Reservation Information
|Date:||09 / 06 / 2011|
|Time:||04:00pm - 05:00pm|