On the Maxey-Riley equation of motion and its extension to high Reynolds numbers

TL;DR

This paper reviews the Maxey-Riley equation's development, highlights its limitations at high Reynolds numbers, and proposes an extended formulation for unsteady drag applicable beyond low Reynolds number flows.

Contribution

It introduces a new formulation of the unsteady drag force in the Maxey-Riley equation suitable for high Reynolds number turbulent flows.

Findings

The original Maxey-Riley equation is limited to Re < 1.

A revised unsteady drag formulation is proposed for higher Re.

The new model extends the applicability of particle motion equations in turbulent flows.

Abstract

The inertial response of a particle to turbulent flows is a problem of relevance to a wide range of environmental and engineering problems. The equation most often used to describe the force balance is the Maxey-Riley equation, which includes in addition to buoyancy and steady drag forces, an unsteady Basset drag force related to past particle acceleration. Here we provide a historical review of how the Maxey-Riley equation was developed and how it is only suited for studies where the Reynolds number is less than unity. Revisiting the innovative mathematical methods employed by Basset (1888), we introduce an alternative formulation for the unsteady drag for application to a broader range of particle motions. While the Basset unsteady drag is negligible at higher Reynolds numbers, the revised unsteady drag is not.