(Abridged) Ideal hydrodynamic models of the intracluster medium (ICM) in the core regions of galaxy clusters fail to explain both the observed temperature structure of this gas, and the observed morphology of radio-galaxy/ICM interactions. It has recently been suggested that, even in the presence of reasonable magnetic fields, thermal conduction in the ICM may be crucial for reproducing the temperature floor seen in many systems. If this is indeed correct, it raises the possibility that other transport processes may be important. With this motivation, we present a numerical investigation of the buoyant evolution of AGN-blown cavities in ICM that has a non-negligible shear viscosity. We use the ZEUS-MP code to follow the 3-d evolution of an initially static, hot bubble in a beta-model ICM atmosphere with varying degrees of shear viscosity. With no explicit viscosity, it is found that the combined action of Rayleigh-Taylor and Kelvin-Helmholtz instabilities shred the ICM cavity and one does not reproduce the intact and detached ``ghost cavities observed in systems such as Perseus-A. On the other hand, even a modest level of shear viscosity can be important in quenching the fluid instabilities and maintaining the integrity of the bubble. In particular, we show that the morphology of the NW ghost cavity found in Perseus-A can be reproduced, as can the flow pattern inferred from the morphology of H-alpha filaments. Finally, we discuss the possible relevance of ICM viscosity to the fact that many of the active ICM cavities are not bounded by strong shocks.
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