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The shear viscosity$$\eta $$$\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 200AGeV and Pb + Pb collisions at 2760AGeV, 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$$\sigma $$$\sigma$for parton scatterings, the$$\eta /s$$$\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$$\sigma =3$$$\sigma =3$mb that enables the transport model to approximately reproduce the elliptic flow data of the bulk matter, the average$$\eta /s$$$\eta /s$of the parton matter in partial equilibrium is found to be very small, between one to two times$$1/(4\pi )$$$1/\left(4\pi \right)$.