A theoretical model has been developed to predict the behavior of a buoyancy-driven upward co-current two-phase flow in an annular channel with uniform gap size that forms between a hemispherical vessel and its surrounding structure. The vessel is fully submerged in water and is heated from within, leading to downward facing boiling on its outer surface. The problem under consideration is relevant to the so-called in-vessel retention (IVR) of core melt, which is a key severe accident management strategy for some advanced pressurized water reactors (APWRs). One available means for IVR is the method of external reactor vessel cooling by flooding of the reactor cavity with water during a severe accident. Design features of most APWRs have the provision for substantial water accumulation in the reactor cavity during numerous postulated accident sequences. With water covering the lower external surfaces of the reactor pressure vessel, significant energy (i.e., decay heat) could be removed from the core melt through the vessel wall by downward facing boiling on the vessel's outer surface. As boiling of water takes place on the vessel outer surface, the vapor generated on the surface would flow upward through the annular channel under the influence of gravity. The vapor motions would entrain liquid water, thus resulting in a buoyancy-driven upward co-current two-phase flow in the channel. While the flow is induced entirely by the boiling process, the rate of boiling, in turn, can be significantly affected by the resulting two-phase flow. As long as the heat flux from the core melt to the vessel wall does not exceed the critical heat flux limit for downward facing boiling, nucleate boiling is the prevailing regime and the vessel wall can be maintained at relatively low temperatures to prevent failure of the lower head. With this scenario in mind, the problem is formulated by considering the conservation of mass, momentum, and energy in the two-phase mixture, along with the use of available information on two-phase frictional drop and void fraction. The resulting governing system is solved numerically to predict the total mass flow rate that would be induced in the channel by the boiling process. Based on the numerical results, the optimal gap size that would maximize the steam venting rate and the rate of downward facing boiling over a range of wall heat fluxes is determined. The effects of system pressure and liquid level in the reactor cavity on the induced mass flow rate have also been identified.