Steady axisymmetric creeping plumes above a planar boundary. Part 1. A point source

Asymptotic solutions are obtained for the rise of an axisymmetric hot plume from a localized source at the base of a half-space filled with very viscous fluid. We consider an effectively point source, generating a prescribed buoyancy flux $B$, and show that the length scale of the plume base is $z_0 \,{=}\, (32\upi \kappa^2\nu/B)$, where $\nu$ and $\kappa$ are the kinematic viscosity and thermal diffusivity. The internal structure of the plume for $z \,{\gg}\, z_0$ is found using stretched coordinates, and this is matched to a slender-body expansion for the external Stokes flow. Solutions are presented for both rigid (no-slip) and free-slip (no tangential stress) conditions on the lower boundary. In both cases we find that the typical vertical velocity in the plume increases slowly with height as $(B/\nu)^{1/2} [\ln (z/z_0)]^{1/2}$, and the plume radius increases as $(zz_0)^{1/2} [\ln (z/z_0)]^{-1/4}$.