  Using Grad's method, we calculate the entropy production and derive a formula
for the second-order shear viscosity coefficient in a one-dimensionally
expanding particle system, which can also be considered out of chemical
equilibrium. For a one-dimensional expansion of gluon matter with Bjorken boost
invariance, the shear tensor and the shear viscosity to entropy density ratio
$\eta/s$ are numerically calculated by an iterative and self-consistent
prescription within the second-order Israel-Stewart hydrodynamics and by a
microscopic parton cascade transport theory. Compared with $\eta/s$ obtained
using the Navier-Stokes approximation, the present result is about 20% larger
at a QCD coupling $\alpha_s \sim 0.3$(with $\eta/s\approx 0.18$) and is a
factor of 2-3 larger at a small coupling $\alpha_s \sim 0.01$. We demonstrate
an agreement between the viscous hydrodynamic calculations and the microscopic
transport results on $\eta/s$, except when employing a small $\alpha_s$. On the
other hand, we demonstrate that for such small $\alpha_s$, the gluon system is
far from kinetic and chemical equilibrium, which indicates the break down of
second-order hydrodynamics because of the strong noneqilibrium evolution. In
addition, for large $\alpha_s$ ($0.3-0.6$), the Israel-Stewart hydrodynamics
formally breaks down at large momentum $p_T\gtrsim 3$ GeV but is still a
reasonably good approximation.
