Previous studies of the equilibrium property and catastrophic behavior of coronal magnetic flux ropes assumed that the background field is potential and the background plasma is in static equilibrium. This paper abandons these assumptions and considers the effect of the background solar wind. For simplicity, we take a polytropic process approximation in the energy equation and confine ourselves to axisymmetrical problems in spherical geometry. Steady solutions are obtained by time-dependent simulations for a system consisting of a magnetostatic helmet streamer astride the equator with a flux rope in it and a steady solar wind outside. The coronal flux rope is characterized by its azimuthal and annular magnetic fluxes and total mass. It is shown that the system possesses a catastrophic behavior. As the azimuthal or annular flux of the rope increases or the total mass in the rope decreases, the flux rope and the helmet streamer expand, the field lines in the outermost part of the streamer are gradually opened, and the originally static plasma there flows outward along the newly opened field lines and joins the solar wind. The magnetic energy of the system increases during this process. We find an energy threshold that is larger than the corresponding open field energy by about 8% if the field in the rope is close to force-free. The gravity raises the threshold by an amount that approximately equals the magnitude of the excess gravitational energy in the rope. The flux rope sticks to the solar surface in equilibrium if the magnetic energy of the system is below the threshold, whereas it erupts otherwise. In the latter case, the flux rope is gradually accelerated and reaches a velocity that is slightly larger than the background solar wind velocity. We argue that such a catastrophe may serve as a physical mechanism for slow coronal mass ejections. A brief comparison is also made between the present case and the magnetostatic case in the catastrophic behavior of the system.