Recent X-ray observations reveal growing evidence for heating by active
galactic nuclei (AGN) in clusters and groups of galaxies. AGN outflows play a
crucial role in explaining the riddle of cooling flows and the entropy problem
in clusters. Here we study the effect of AGN on the intra-cluster medium in a
cosmological simulation using the adaptive mesh refinement FLASH code. We pay
particular attention to the effects of conductivity and viscosity on the
dissipation of weak shocks generated by the AGN activity in a realistic galaxy
cluster. Our 3D simulations demonstrate that both viscous and conductive
dissipation play an important role in distributing the mechanical energy
injected by the AGN, offsetting radiative cooling and injecting entropy to the
gas. These processes are important even when the transport coefficients are at
a level of 10% of the Spitzer value. Provided that both conductivity and
viscosity are suppressed by a comparable amount, conductive dissipation is
likely to dominate over viscous dissipation. Nevertheless, viscous effects may
still affect the dynamics of the gas and contribute a significant amount of
dissipation compared to radiative cooling. We also present synthetic Chandra
observations. We show that the simulated buoyant bubbles inflated by the AGN,
and weak shocks associated with them, are detectable with the Chandra
observatory.Comment: accepted to ApJ, minor change