  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.
