  (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.
