Results of numerical calculations of the buoyancy-induced flow adjacent to a vertical isothermal surface with a leading edge are presented. Coordinate transformations have been used to allow efficient calculation of both the boundary layer flow and the extensive ambient entrainment in-flow. A motion pressure field, which develops owing to the motion pressure deficit at the surface, drives this entrainment. Increased boundary layer mass flow results, compared with classical boundary layer theory. Conduction into the upstream entrainment region is appreciable, ahead of the leading edge. This flow, below the surface, arises as a mass sink for the distant entraining flow. Motion pressure effects determine the flow in this leading edge entrainment region and in the entrainment flow arising along such a heated surface. Modeling of both these regions is of increasing importance in many technologically important applications, such as electronics packaging.