  The acceleration and energy dissipation terms due to the shear viscosity have
been implemented and tested in GodunovSPH. The double summation method has been
employed to avoid the well known numerical noise of the second derivative in
particle based codes. The plane Couette flow with various initial and boundary
conditions have been used as tests, and the numerical and analytical results
show a good agreement. Not only the viscosity--only calculation, but the full
hydrodynamics simulations have been performed, and they show expected results
as well. The very low kinematic viscosity simulations show a turbulent pattern
when the Reynolds number exceeds $\sim$$10^2$. The critical value of the
Reynolds number at the transition point of the laminar and turbulent flows
coincides with the previous works approximately. A smoothed dynamic viscosity
has been suggested to describe the individual kinematic viscosity of particles.
The infinitely extended Couette flow which has two layers of different
viscosities has been simulated to check the smoothed dynamic viscosity, and the
result agrees well with the analytic solution. In order to compare the standard
SPH and GodunovSPH, the two layers test has been performed again with a density
contrast. GodunovSPH shows less dispersion than the standard SPH, but there is
no significant difference in the results. The results of the viscous ring
evolution has also been presented as well, and the numerical results agrees
with the analytic solution.
