The evolution of hot interstellar gas in cluster-centered cD galaxies and the inflow of gas from the surrounding galaxy clusters are strongly coupled. Cooling flows arise inside the cD galaxy because the deep stellar potential and stellar mass loss increase the gas density and decrease the radiative cooling time within the galaxy. Recent X-ray observations of M87 in the Virgo Cluster and NGC 4874 in Coma reveal that the gas temperature beyond about 50 kpc from these cD galaxies is comparable to the virial temperature of the cluster, 3 or 9 keV, respectively, but within the optical galaxy the temperature drops to the galactic virial temperature similar to1 keV. We show that these steep thermal gradients on galactic scales follow naturally from the usual cooling inflow assumptions without recourse to thermal conductivity. However, most of the gas must radiatively cool ("dropout") before it flows to the galactic core; i.e., the gas must be multiphase. The temperature and density profiles observed in M87 and NGC 4874 can be matched with approximate gasdynamical models calculated over several gigayears with either globally uniform or centrally concentrated multiphase mass dropout. Recent XMM observations of M87 indicate single-phase flow at every radius with no apparent radiative cooling to low temperatures. Gasdynamical models can be made consistent with single-phase flow for r greater than or similar to 10 kpc, but to avoid huge central masses of cooled gas, we assume that some distributed cooling dropout occurs near the center of the flow, where the gas temperature is T similar to 1 keV. The evidence in X-ray spectra for multiphase cooling beginning at lower temperatures similar to1 keV may be less apparent than for higher temperatures and may have escaped detection. However, even if the mass of cooled gas is distributed within r less than or similar to 10 kpc, it is necessary that the mass of cooled gas not conflict with dynamical mass-to-light determinations. Because of small deviations from true steady-state flow, we find that the standard decomposition methods used by X-ray observers to determine the mass flow (M) over dot (r) may fail rather badly, particularly when the mass dropout decreases with radius in the flow. For this case the decomposition procedure gives the usual cooling flow result, (M) over dot proportional to r, which is quite unlike the true variation of (M) over dot (r) in our computed models.
