The hydrodynamic force experienced by a small spherical particle undergoing an arbitrary time-dependent motion in a weakly density-stratified fluid is investigated theoretically. The study is carried out under the Oberbeck-Boussinesq approximation and in the limit of small Reynolds and small Péclet numbers. The force acting on the particle is obtained by using matched asymptotic expansions. In this approach, the small parameter is given by a/ℓ, where a is the particle radius and ℓ is the stratification length, as defined by Ardekani & Stocker (2010), which depends on the Brunt-Väisälä frequency, on the fluid kinematic viscosity and on the thermal or the concentration diffusivity (depending on the case considered). The matching procedure used here, which is based on series expansions of generalized functions, slightly differs from that generally used in similar problems. In addition to the classical Stokes drag, it is found that the particle experiences a memory force given by two convolution products, one of which involves, as usual, the particle acceleration and the other one, the particle velocity. Owing to the stratification, the transient behaviour of this memory force, in response to an abrupt motion, consists of an initial fast decrease followed by a damped oscillation with an angular-frequency corresponding to the Brunt-Väisälä frequency. The perturbation force eventually tends to a constant which provides us with correction terms that should be added to the Stokes drag to accurately predict the settling time of a particle in a diffusive stratified-fluid.

We compute the drag force on a sphere settling slowly in a quiescent, linearly stratified fluid. Stratification can significantly enhance the drag experienced by the settling particle. The magnitude of this effect depends on whether fluid-density transport around the settling particle is due to diffusion, to advection by the disturbance flow caused by the particle, or due to both. It therefore matters how efficiently the fluid disturbance is convected away from the particle by fluid-inertial terms. When these terms dominate, the Oseen drag force must be recovered. We compute by perturbation theory how the Oseen drag is modified by diffusion and stratification. Our results are in good agreement with recent direct-numerical simulation studies of the problem at small Reynolds numbers and large (but not too large) Froude numbers.Key words: Recent direct-numerical simulation studies of the problem (Yick et al. 2009;Zhang et al. 2017) explored how the drag depends on the importance of diffusivity versus advection, and upon the degree of density stratification. Our goal is to explain their results by perturbation theory, assuming that both Re and Ri are small but finite.Chadwick & Zvirin (1974b,a) analysed this question, but for a sphere moving horizontally in a quiescent non-diffusive stratified fluid, along surfaces of constant fluid density.Here we study the settling problem, where the particle settles vertically along the fluiddensity gradient, so that it crosses the surfaces of constant density. The two problems are quite different: when the particle moves horizontally, the streamlines of the flow tend to encircle the sphere in the horizontal plane. When the sphere moves vertically, by contrast, light fluid is pushed down into regions of larger fluid density, giving rise to complex disturbance-flow patterns (Ardekani & Stocker 2010).Neglecting effects of convective fluid inertia, the difference between horizontal and vertical motion was compared earlier. When density transport is entirely diffusive, the additional drag due to stratification is five times larger in the vertical than in the horizontal direction (Candelier et al. 2014). When density advection dominates, the vertical drag is seven times larger than the horizontal one (Zvirin & Chadwick 1975).Despite these qualitative and quantitative physical differences, the horizontal and vertical problems share an important mathematical property: regular perturbation expansions fail to describe the effects of convective fluid inertia and buoyancy due to stratification even if these perturbations are weak. Therefore so-called 'singular-perturbation' methods are required to solve the problem. We use the standard method of asymptotic matching (Saffman 1965), where inner and outer solutions of the disturbance problem are matched, describing the disturbance flow close to and far from the particle.We parameterise the effect of convective inertia and stratification in terms of length scales: the particle radius a, the Oseen length o = a/Re, and the stratification length s ...