  The failure of the XMM-Newton and Chandra X-ray telescopes to detect cooling
gas in elliptical galaxies and clusters of galaxies has led many to adopt the
position that the gas is not cooling at all and that heating by an active
nucleus in the central E or cD galaxy is sufficient to offset radiative
cooling. In this paper we explore an idealized limiting example of this point
of view in which hot, buoyant bubbles formed near the center return the
inflowing, radiatively cooling gas to distant regions in the flow. We show that
idealized steady state, centrally heated non-cooling flows can indeed be
constructed. In addition, the emission-weighted temperature profiles in these
circulating flows resemble those of normal cooling flows. However, these
solutions are valid only (1) for a range of bubble parameters for which there
is no independent justification, (2) for a limited spatial region in the
cooling flow and (3) for a limited period of time after which cooling seems
inevitable. Our exploration of non-cooling flows is set in the context of
galaxy/group flows.
