Note on the method of matched-asymptotic expansions for determining the force acting on a particle

TL;DR

This paper reviews and refines the method of matched asymptotic expansions for calculating forces on particles in fluid flows, introducing an alternative matching procedure using series expansions of the far-field solution.

Contribution

It presents a new matching procedure based on series expansions of the far-field solution using generalized functions, enhancing the theoretical framework for particle force calculations.

Findings

Introduces an alternative matching procedure for asymptotic expansions.

Provides a simple example illustrating the new method.

Enhances the theoretical approach to particle-fluid interaction analysis.

Abstract

This paper is an addendum to the article by Candelier, Mehaddi & Vauquelin (2013) where the motion of a particle in a stratified fluid is investigated theoretically, at small Reynolds and P\'eclet numbers. We review briefly the method of matched asymptotic expansions which is generally used in order to determine the force acting on a particle embedded in a given flow, in order to account for small, but finite, inertia effects. As part of this method, we present an alternative matching procedure, which is based on a series expansion of the far-field solution of the problem, performed in the sense of generalized functions. The way to perform such a series is presented succinctly and a simple example is provided.