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

Whittaker, RJ ORCID: https://orcid.org/0000-0001-9598-5938 and Lister, JR (2006) Steady axisymmetric creeping plumes above a planar boundary. Part 1. A point source. Journal of Fluid Mechanics, 567. pp. 361-378. ISSN 0022-1120

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## Abstract

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 $(zz_0)^{1/2} [\ln (z/z_0)]^{-1/4}$.

Item Type: Article Faculty of Science > School of Mathematics http://dx.doi.org/10.1017/S0022112006002... Robert Whittaker 10 Mar 2011 12:48 03 Oct 2022 02:31 https://ueaeprints.uea.ac.uk/id/eprint/20893 10.1017/S0022112006002370