Free convection from a point source of heat, and heat transfer from spheres at small Grashof numbers

A numerical description, based on the Boussinesq equations, is given for the steady free convection flow due to a point source of heat and heated spheres. We begin with the non-dimensional formulation of the problem for the point source giving the numerical solution for a wide range of values of the Prandtl number, the remaining parameter. The analytical description of the temperature and flow fields close to the point source includes constants that are evaluated numerically and are used to obtain the flow field around heated spheres for small Grashof numbers, and the correction of the Nusselt number, due to free convection, from its heat conduction value. The numerical solution of the free convection problem over a sphere at moderate and small Grashof numbers is carried out for Pr=0.72 and 7. Based on the small and large Grashof number descriptions, a correlation expression is proposed for the laminar flow Nusselt number, covering all Grashof numbers.