  We consider QCD at temperatures T near Tc, where the theory deconfines. We
distinguish between a "complete" Quark Gluon Plasma (QGP), where the vacuum
expectation value of the renormalized Polyakov loop is near unity, essentially
constant with T, and the "semi"-QGP, where the loop changes strongly with T.
Lattice simulations indicate that in QCD, there is a semi-QGP from below Tc to
a few times that. Using a semi-classical model, we compute the shear viscosity,
eta, to leading order in perturbation theory. We find that near Tc, where the
expectation value of the Polyakov loop is small, that eta/T^3 is suppressed by
two powers of the loop. For heavy ions, this suggests that during the initial
stages of the collision, hydrodynamic behavior at the LHC will be characterized
by a shear viscosity which is significantly larger than that at RHIC.
