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-dimensional evolution of an initially static, hot bubble in a

-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 rapidly 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 (corresponding to approximately 25% of the
Spitzer value) 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

filaments. Finally, we discuss the possible relevance of
ICM viscosity to the fact that many of the active ICM cavities
(i.e., those currently associated with active radio-lobes) are not
bounded by strong shocks, the so-called ``shock problem''.