Please Join the Mathematics
Department for a Colloquium for General Audiences
A
Bird’s Eye View of the Vortex Problem
By
Jorge
Viveros Rogel
Georgia Tech
School of Mathematics
Thursday, November 10, 2005
3 pm D235
Cookies at 2:45 pm
Imagine a large portion of
three-dimensional Euclidean space filled with a fluid such that the fluid’s flow is
horizontal and the same at different depths: we can think of a fluid on a plane. The vorticity field of such a fluid is the curl of
its velocity field and in our scenario, it looks like a vector perpendicular to the plane. We are interested in the case in which the
vorticity field is zero except at certain points called ‘point vortices.’ For
ideal, incompressible fluids, point vortices move as particles while the flow very close
to each one of them is nearly circular. The
n-vortex problem is the study of the interaction between n point vortices. We give a brief derivation of the equations
of motion and also describe the cases of other geometries such as the sphere and the
cylinder. The main focus of the talk is a selection of interesting results such as the
existence of highly symmetric solutions and their stability (regular vortex n-gons, vortex
polyhedra and vortex crystals), “choreographic solutions” (and other
similarities between the n-vortex problem and the n-body problem of celestial mechanics),
and some unsuspected connections with other areas in mathematics such as potential and
frame theories.
