The problem of interest is a heated sphere in a fluid of infinite extent that is otherwise not flowing. We expect that if the sphere is heated it will be at a higher temperature than the surrounding fluid, energy will be transferred to the fluid and the resulting temperature difference within the fluid will lead to a density gradient. If the sphere is in a uniform gravity field, you did not want pay to put your sphere on Mir, or you could not convince NASA to give you money to put it on the space shuttle, then the density gradient will lead to a body force acting on the fluid around the sphere. The region of low density fluid will be forced upward and replaced by higher density fluid. The question is, will the fluid flow?
Dimensional analysis for this problem would identify a Grashof number that represents the ratio of buoyancy force that would be causing flow over the viscous force that is resisting the flow. For slightly-heated road tar, we expect a low Grashof number and thus no flow. For a lighted light bulb in air, the Grashof number can be calculated as
this will certainly cause a strong flow.
For the case of low Gr, heat removal will be by conduction and radiation. If you define a Nusselt number in the usual way, hD/k, the value will be two. This is the pure condition limit. Since the Nusselt number gives the ratio of total heat transfer to pure condition (we are not considering radiation), we expect that as the Grashof number gets large, there will be a significant fluid flow and thus significant convective enhancement.
![[Graphics:../Images/thermal_bl_gr_2.gif]](../Images/thermal_bl_gr_2.gif)