  Cooling flows in galaxy clusters and isolated elliptical galaxies are a
source of mass for fueling accretion onto a central supermassive black hole. We
calculate the dynamics of accreting matter in the combined gravitational
potential of a host galaxy and a central black hole assuming a steady state,
spherically symmetric flow (i.e., no angular momentum). The global dynamics
depends primarily on the accretion rate. For large accretion rates, no simple,
smooth transition between a cooling flow and an accretion flow is possible; the
gas cools towards zero temperature just inside its sonic radius, which lies
well outside the region where the gravitational influence of the central black
hole is important. For accretion rates below a critical value, however, the
accreting gas evolves smoothly from a radiatively driven cooling flow at large
radii to a nearly adiabatic (Bondi) flow at small radii. We argue that this is
the relevant parameter regime for most observed cooling flows. The transition
from the cooling flow to the accretion flow should be observable in M87 with
the {\it Chandra X-ray Observatory}.
