  The shear viscosity $\eta$ of a quark-gluon plasma in equilibrium can be
calculated analytically using multiple methods or numerically using the
Green-Kubo relation. It has been realized, which we confirm here, that the
Chapman-Enskog method agrees well with the Green-Kubo result for both isotropic
and anisotropic two-body scatterings. We then apply the Chapman-Enskog method
to study the shear viscosity of the parton matter from a multi-phase transport
model. In particular, we study the parton matter in the center cell of central
and midcentral Au+Au collisions at $200A$ GeV and Pb+Pb collisions at $2760A$
GeV, which is assumed to be a plasma in thermal equilibrium but partial
chemical equilibrium. As a result of using a constant Debye mass or cross
section $\sigma$ for parton scatterings, the $\eta/s$ ratio increases with time
(as the effective temperature decreases), contrary to the trend preferred by
Bayesian analysis of the experimental data or pQCD results that use
temperature-dependent Debye masses. At $\sigma=3$ mb that enables the transport
model to approximately reproduce the elliptic flow data of the bulk matter, the
average $\eta/s$ of the parton matter in partial equilibrium is found to be
very small, between one to two times $1/(4\pi)$.
