Abstract:

Vorticity tends to concentrate in a high-Reynolds-number Navier–Stokes fluid, reducing the system down to the dynamics of vortex filaments.  In the first part of the talk, I will give a novel dynamical model for variable thickness, viscous vortex filaments. The resulting equation can describe such varied phenomena as underwater bubble rings or the intricate “chandeliers” formed by ink dropping into a fluid. The underlying theory is an instance of the dissipative geodesic equation on a Kaluza–Klein Riemannian manifold, which extends the classical work in the dynamics of vortex filaments through the inclusion of viscous drag forces.  In the final part of the talk, I will return to the classical nondissipative vortex dynamics in the Da Rios limit and present new explicit formulas for its infinitely many conserved quantities.