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)
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