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 pi kappa(2)v/B), where v and K are the kinematic viscosity and thermal diffusivity. The internal structure of the plume for 7 >> 70 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/v)(1/2) [ln(z/z(0))](1/2), and the plume radius increases as (zz(0))(1/2) [ln(z/z(0))](-1/4).