Mass transfer from a cylindrical body in a linear ambient velocity distribution

We consider the two-dimensional problem of mass transport from a cylindrical body of circular cross section which is immersed in a fluid whose ambient velocity varies linearly with position. Such a flow is quantified by a single parameter $\mathrm{\Omega }$ representing the ratio of its associated vorticity to its characteristic rate-of-strain magnitude, $\mathrm{\Omega }=\pm 1$ corresponding to simple shear. Using matched asymptotic expansions to analyze the limit of small Péclet numbers, $\mathrm{Pe}\ll 1$, we find that the leading-order Nusselt number is $2/\left[log(1/\mathrm{Pe})+\lambda \left(\mathrm{\Omega }\right)\right],$ wherein the function $\lambda \left(\mathrm{\Omega }\right)$ is provided in terms of simple quadratures. No steady solutions exist for $\left|\mathrm{\Omega }\right|>1$, where the streamlines of the ambient flow are closed. The case of simple shear, analyzed by Frankel and Acrivos [Phys. Fluids 11, 1913 (1968)], is accordingly a borderline one. Using conformal mappings, the more general problem of arbitrary cross-sectional shape is recast as the above transport problem about a circle, with the Péclet number appropriately modified. While the more general problem is unsteady in the case of a freely suspended cylinder, the associated Nusselt number is independent of time.

Figure 1

Variation with $\mathrm{\Omega }$ of $\lambda$.