  We study the imprints of AGN feedback and physical viscosity on the
properties of galaxy clusters using hydrodynamical simulation models carried
out with the TreeSPH code GADGET-2. Besides self-gravity of dark matter and
baryons, our approach includes radiative cooling and heating processes of the
gas component and a multiphase model for star formation and SNe feedback.
Additionally, we introduce a prescription for physical viscosity in GADGET-2,
based on a SPH discretization of the Navier-Stokes and general heat transfer
equations. Adopting the Braginskii parameterization for the shear viscosity
coefficient, we explore how gas viscosity influences the properties of
AGN-driven bubbles. We also introduce a novel, self-consistent AGN feedback
model where we simultaneously follow the growth and energy release of massive
black holes embedded in a cluster environment. We assume that black holes
accreting at low rates with respect to the Eddington limit are in a radiatively
inefficient regime, and that most of the feedback energy will appear in a
mechanical form. Thus, we introduce AGN-driven bubbles into the ICM with
properties, such as radius and energy content, that are directly linked to the
black hole physics. This model leads to a self-regulated mechanism for the
black hole growth and overcomes the cooling flow problem in host halos, ranging
from the scale of groups to that of massive clusters. (Abridged)
