Mass transfer dominated by thermal diffusion in laminar boundary layers

Abstract

The concentration distribution of massive dilute species (e.g. aerosols, heavy vapours, etc.) carried in a gas stream in non-isothermal boundary layers is studied in the large-Schmidt-number limit, Sc[Gt ]1, including the cross-mass-transport by thermal diffusion (Ludwig–Soret effect). In self-similar laminar boundary layers, the mass fraction distribution of the dilute species is governed by a second-order ordinary differential equation whose solution becomes a singular perturbation problem when Sc[Gt ]1. Depending on the sign of the temperature gradient, the solutions exhibit different qualitative behaviour. First, when the thermal diffusion transport is directed toward the wall, the boundary layer can be divided into two separated regions: an outer region characterized by the cooperation of advection and thermal diffusion and an inner region in the vicinity of the wall, where Brownian diffusion accommodates the mass fraction to the value required by the boundary condition at the wall. Secondly, when the thermal diffusion transport is directed away from the wall, thus competing with the advective transport, both effects balance each other at some intermediate value of the similarity variable and a thin intermediate diffusive layer separating two outer regions should be considered around this location. The character of the outer solutions changes sharply across this thin layer, which corresponds to a second-order regular turning point of the differential mass transport equation. In the outer zone from the inner layer down to the wall, exponentially small terms must be considered to account for the diffusive leakage of the massive species. In the inner zone, the equation is solved in terms of the Whittaker function and the whole mass fraction distribution is determined by matching with the outer solutions. The distinguished limit of Brownian diffusion with a weak thermal diffusion is also analysed and shown to match the two cases mentioned above.