  Over the past several years, numerous examples of X-ray cavities coincident
with radio sources have been observed in so-called "cool core" clusters of
galaxies. Motivated by these observations, we explore the evolution and the
effect of cavities on a cooling intracluster medium (ICM) numerically, adding
relevant physics step by step. In this paper we present a first set of
hydrodynamical, high resolution (1024^3 effective grid elements),
three-dimensional simulations, together with two-dimensional test cases. The
simulations follow the evolution of radio cavities, modeled as bubbles filled
by relativistic plasma, in the cluster atmosphere while the ICM is subject to
cooling. We find that the bubble rise retards the development of a cooling flow
by inducing motions in the ICM which repeatedly displace the material in the
core. Even bubbles initially set significantly far from the cluster center
affect the cooling flow, although much later than the beginning of the
simulation. The effect is, however, modest: the cooling time is increased by at
most only 25%. As expected, the overall evolution of pure hydrodynamic bubbles
is at odds with observations, showing that some additional physics has to be
considered in order to match the data.
