Steady axisymmetric creeping plumes above a planar boundary. Part 2. A distributed source

Asymptotic solutions are obtained for an axisymmetric plume rising from a localized heat source at the base of a half-space filled with very viscous fluid. Specifically, we consider sources comprising a heated disk with either rigid (no-slip) or free-slip (no tangential stress) conditions on the lower boundary. The boundary layer which forms above the source is solved using stretched coordinates, and then matched to a slender plume which rises above it. At large Rayleigh numbers, the Nusselt number is given by Nu similar to 4.06Ra(1/3) (ln Ra)(-1/3) (free-slip boundary) and Nu similar to 2.90Ra(1/5) (rigid boundary), where the Rayleigh number is based on the radius of the source. Both these expressions have corrections arising from a slender-body expansion in powers of (ln Ra)(-1).