  Many galaxy clusters pose a "cooling-flow problem", where the observed X-ray
emission from their cores is not accompanied by enough cold gas or star
formation. A continuous energy source is required to balance the cooling rate
over the whole core volume. We address the feasibility of a gravitational
heating mechanism, utilizing the gravitational energy released by the gas that
streams into the potential well of the cluster dark-matter halo. We focus here
on a specific form of gravitational heating in which the energy is transferred
to the medium thorough the drag exerted on inflowing gas clumps. Using
spheri-symmetric hydro simulations with a subgrid representation of these
clumps, we confirm our earlier estimates that in haloes >=10^13 solar masses
the gravitational heating is more efficient than the cooling everywhere. The
worry was that this could overheat the core and generate an instability that
might push it away from equilibrium. However, we find that the overheating does
not change the global halo properties, and that convection can stabilize the
cluster by carrying energy away from the overheated core. In a typical rich
cluster of 10^{14-15}solar masses, with ~5% of the accreted baryons in gas
clumps of ~10^8 solar masses, we derive upper and lower limits for the
temperature and entropy profiles and show that they are consistent with those
observed in cool-core clusters. We predict the density and mass of cold gas and
the level of turbulence driven by the clump accretion. We conclude that
gravitational heating is a feasible mechanism for preventing cooling flows in
clusters.
