Slow Burgers Vortices in Hot Conformal Fluids

Abstract

The quintessential vortex solution in (3+1)-dimensional nonrelativistic, incompressible fluid mechanics is the Burgers vortex. We show that, in a finite domain, conformal fluids also admit hot vortex solutions with everywhere nonrelativistic speeds. These are identical to Burgers' solution, except that their radius is reduced by a factor of 2/sqrt(3). A rough calculation indicates that at RHIC these vortices are indeed smaller than the fireball itself during thermalization. Similarly to the Burgers vortex, these solutions manifest vortex stretching which avoids short distance singularities and so suggests that conformal fluid flows share the same nonsingularity as solutions of the Navier-Stokes equations. Naively generalizing this calculation to an arbitrary equation of state w, we observe that the Burgers vortex radius diverges as w crosses -1. While it has been argued that such a crossover leads to an instability in certain perfect fluids, the absence of Burgers vortices and therefore vortex stretching suggests that, in addition to the well-studied big rip singularities, viscous phantom fluids generically develop vorticity singularities.