The hydrodynamical problem of free convection flow due to a point source of heat in an infinite medium is considered with a view to finding a global flow field approximations by the method of matched asymptotic expansions.
In the process of formulating these expansions, it is shown that the Boussinesq approximation is equivalent to certain other assumptions in the plume region, the most crucial of these being that the asymptotic forms of the solutions be "similar".
Estimates on the range of validity of these "similar" expansions are given.
For the two dimensional problem, the matching of the inner and outer expansions is formulated. The outer expansion is completely determined. The existence of a solution to the equations of the first order approximation in the plume is demonstrated, and some bounds on the axial velocity and temperature are given. Exact solutions of these equations are known for Prandtl numbers 5/9 and 2, and a first approximation to the solution for Prandtl numbers near to 5/9 is known in closed form. A similar first approximation is obtained for Prandtl numbers near to 2. Previous work on these equations is reviewed. A non-linear transformation is used to study the equations for the case of inviscid flow.
The problem in three dimensions is also briefly considered. The inner and outer expansions are formulated, and the first few terms of the outer expansion are found explicitly. The first stage of the matching is completed, and the difficulties associated with the matching of higher order terms presented.
An appendix briefly discusses heat transfer from a cylinder by free convection at small Grashof numbers.