The history force on a small particle in a linearly stratified fluid

TL;DR

This paper theoretically investigates the hydrodynamic force on a small particle in a stratified fluid, revealing a memory force with oscillatory transient behavior influenced by stratification and providing correction terms for particle settling predictions.

Contribution

It introduces a novel matched asymptotic expansion method to derive the particle force, including a memory effect, in stratified fluids under specific approximations.

Findings

Memory force involves convolution with particle acceleration and velocity.

Transient response shows initial decrease followed by damped oscillations.

Provides correction terms to Stokes drag for better settling time predictions.

Abstract

The hydrodynamic force experienced by a small spherical particle undergoing an arbitrary time-dependent motion in a 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 P\'eclet numbers. The force acting on the particle is obtained by using matched asymptotic expansions in which the small parameter is given by a/l where a is the particle radius and l is the stratification length defined by Ardekani & Stocker (2010), which depends on the Brunt-Vaisala 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 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-Vaisala 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.