# Effect of inertial lift on a spherical particle suspended in flow through a curved duct

## TL;DR

This paper models the inertial lift forces on spherical particles in curved duct flow at low Reynolds numbers, predicting stable equilibrium positions and how they vary with particle size, bend radius, and duct shape.

## Contribution

It extends existing inertial lift models to curved ducts, incorporating effects of duct shape and curvature on particle migration.

## Key findings

- Stable equilibrium positions depend on particle size and duct curvature.
- Focusing position scales with a dimensionless parameter involving particle and duct dimensions.
- Cross-sectional shape influences particle migration dynamics.

## Abstract

We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrates. The Navier-Stokes equations determine the hydrodynamic forces acting on a particle. A leading order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales, specifically the particle radius, duct height, and duct bend radius. Additionally, a trapezoidal shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles.