Sedimentation of a small sphere in stratified fluid

We report an extensive theoretical and numerical study of gravitational settling of small particles in a stratified fluid at rest. The particle motion creates a flow that advects the stratified agent, which in turn impacts the velocity by buoyancy. A peculiar insulation flow also exists when a particle and fluid at infinity are at rest, screening the particle from the outside gradients. We derive an integral representation for the linear coupled flow that holds at low Reynolds and Péclet numbers. The representation demonstrates that the far flow contains a previously unnoticed term that is produced by the thermal component of the fundamental solution. We obtain nontrivial symmetry relations that constitute a form of the Onsager reciprocal relations. Stratification is a singular perturbation: however small it is, far from the particle the flow is fast decaying and does not obey the slow power-law decay of the Stokes flow at zero stratification. We derive the enhancement of the drag force due to the stratification by adopting the integral equation on the surface traction. This allows one to find the force directly, circumventing finding the flow whose calculation demands the matched asymptotic expansions that complicated previous studies. In this way we are able to extend the domain of validity of the drag formula by orders of magnitude. We confirm the predictions by numerical simulations at low Reynolds numbers. We also performed simulations for a range of non-small Reynolds numbers where the flow is nonlinear, but the Reynolds number is not too large. The considered range of parameters demonstrates that inclusion of stratification might be necessary for correct prediction of the sedimentation velocity of plankton particles and small organic particles of marine snow. We provide the settling velocity of Volvox as an example. We sum up our observations by deriving an empirical fitting formula for the drag coefficient. Finally, we discuss the observability of the insulating flow around a floating body.