Diffusion-driven flow about an impermeable sphere or cylinder in a stably stratified liquid: the singular limit of small Rayleigh numbers

When an impermeable spherical particle is suspended in stably stratified ambient salinity gradient, the curving of the constant-density surfaces gives rise to non-conservative body forces which cannot be supported by pressure gradients. As a result, fluid motion is engendered. Considering small particles, we here address the Stokes-flow limit and assess the role of solute advection, characterised by the Rayleigh number. For a purely diffusive salt transport, the velocity field attains an orientation-dependent non-decaying profile at large distances from the sphere, in contradiction to the fundamental requirement of velocity attenuation. The weak advection limit is non-uniformly valid, with advection of the ambient salinity profile entering the dominant balance at large distances. This limit is accordingly resolved via use of matched asymptotic expansions, with the fluid domain conceptually decomposed into two regions, one on the particle scale and the other on the stratification scale. The stratification-scale problem is solved using Fourier transforms, with the solution shown to trivially match the diffusive fields on the particle scale. Simplification of the stratification-scale solution on the symmetry axis reveals that the velocity decays inversely with the eighth power of distance, while the excess concentration undergoes a transition from a diffusive inverse-square scaling to a faster decay, inversely with the sixth power of distance. In the comparable two-dimensional problem the non-conservative body force instigates a velocity field that diverges at large distances, thus exacerbating the weak advection conundrum. In this problem, use of matched asymptotic expansions is necessary to uniquely determine the flow on the obstacle scale.