Collision phenomena in free-convective flow over a sphere

Two types of singularity that occur at the upper pole of a heated sphere in a fluid at rest when the Grashof number is large are discussed. The first is a property of a limit solution of the unsteady boundary-layer equations and is indicative of the fact that the boundary layer growing from the lower pole does not remain empty for all time but erupts into a plume above the sphere. The second arises from a solution of the steady boundary-layer equations and illustrates the phenomenon of an axisymmetric boundary layer converging at a point, with a velocity component parallel to the sphere that is non-zero over the major part of the boundary layer. An analysis is prcscnted for each situation and comparison made with a numerical integration of the appropriate equations.