  In this paper bulk viscosity is introduced to describe the effects of cosmic
non-perfect fluid on the cosmos evolution and to build the unified dark energy
(DE) with (dark) matter models. Also we derive a general relation between the
bulk viscosity form and Hubble parameter that can provide a procedure for the
viscosity DE model building. Especially, a redshift dependent viscosity
parameter $\zeta\propto\lambda_{0}+\lambda_{1}(1+z)^{n}$ proposed in the
previous work by X.H.Meng and X.Dou in 2009\cite{md} is investigated
extensively in this present work. Further more we use the recently released
supernova dataset (the Constitution dataset) to constrain the model parameters.
In order to differentiate the proposed concrete dark energy models from the
well known $\Lambda$CDM model, statefinder diagnostic method is applied to this
bulk viscosity model, as a complementary to the $Om$ parameter diagnostic and
the deceleration parameter analysis performed by us before. The DE model
evolution behavior and tendency are shown in the plane of the statefinder
diagnostic parameter pair \{$r,s$\} where the fixed point represents the
$\Lambda$CDM model. The possible singularity property in this bulk viscosity
cosmology is also discussed to which we can conclude that in the different
parameter regions chosen properly, this concrete viscosity DE model can have
various late evolution behaviors and the late time singularity could be
avoided. We also calculate the cosmic entropy in the bulk viscosity dark energy
frame, and find that the total entropy in the viscosity DE model increases
monotonously with respect to the scale factor evolution, thus this monotonous
increasing property can indicate an arrow of time in the universe evolution,
though the quantum version of the arrow of time is still puzzling.
