Steady Diffusion-Driven Flow in a Tilted Square Container

Abstract

This paper considers ‘diffusion-driven flows’ for which a temperature flux condition on a sloping non-slip surface in a stratified fluid generates a slow steady upwards flow along a thin ‘buoyancy layer’. The principles for this steady-flow phenomenon are well understood in a semi-infinite fluid and more recently have been applied to steady flows within a contained fluid under more general conditions, including where buoyancy layers expel or entrain fluid from their outer edge. In this paper, additional features of these flows are described in the context of a two-dimensional flow in a tilted square box, and some of the finer details of flow structure are elucidated. In particular, the key regions of the asymptotic structure are considered when the Wunsch–Phillips parameter R is small, and the leading-order solutions are derived in most of those regions. To illustrate the theory, three simple case studies of diffusion-driven motion are solved, and the results compared with accurate numerical solutions for R = 10−4. In some cases, a ‘corner-induced’ motion is found to occur, extending across the width of the container. The details of that feature are examined using both higher-order solutions of the outer flow and an integral treatment across the corner regions, and a solution for the corresponding ‘R1/3 layer flow’ is proposed. Theoretical and numerical results are compared as the angle of inclination β is varied.