Mathematical Modelling of Inertial Migration in Periodic Microcentrifuge Devices Using Regular Perturbation

Mathematical Modelling of Inertial Migration in Periodic Microcentrifuge Devices Using Regular Perturbation

Abstract

Cavity-based microcentrifuges use recirculating flow beside a microchannel to separate and retain suspended particles and cells, and to exchange the fluid surrounding them. Experiments demonstrate size-selective capture, but a two-dimensional vortex image cannot predict whether a particle will enter a cavity or remain there. The background flow is three-dimensional, and finite particles can move relative to the fluid. This project combines three-dimensional background-flow calculations with a finite-particle model to study these processes.

The work completed to date includes steady finite-element solutions and a topological analysis of the background flow in a periodic channel-cavity cell. The periodic formulation is the principal model. A fully developed Poiseuille inlet provides an inlet-outlet reference, while a quadratic inlet provides a secondary sensitivity case. All three use the same target upstream mean velocity and flux. Fixed-depth critical points, their continued branches, local eigenvalues and selected three-dimensional fluid trajectories reveal changes in flow structure with Reynolds number.

Chapter One Introduction

One point one. Motivation

Controlling the motion of suspended particles and cells is important in microfluidic applications, including blood-cell separation and the isolation of rare circulating tumour cells. Microfluidic devices can use fluid motion and channel geometry to manipulate these suspensions within compact systems. Understanding how particles respond to the flow is therefore important for predicting and improving device operation.

Inertial microfluidics uses fluid inertia effects to manipulate particles. One such effect is lateral migration in pressure-driven channel flow: fluid inertia can generate a cross-stream hydrodynamic force on a neutrally buoyant rigid sphere, causing it to migrate towards preferred positions within the channel. The resulting migration depends on particle size and hydrodynamic interactions with the surrounding flow and channel walls. This dependence provides a physical basis for size-selective particle manipulation and motivates mathematical models of particle motion in confined flows.

Devices containing repeated expansions and contractions can create recirculating flow regions alongside the main channel. Hur et al. demonstrated preferential capture of larger particles and cells in expansion reservoirs under suitable operating conditions, and Mach et al. used this principle for automated cellular sample preparation. These applications motivate questions about particle transport: what determines whether a particle passes a cavity, enters it, remains inside, or returns to the main channel? Answering these questions requires understanding how the cavity flow interacts with a particle of finite size.

The mathematical challenge is to relate the three-dimensional fluid flow to the translation and rotation of a finite particle. The flow varies spatially through each expansion and contraction, so a moving particle encounters changing hydrodynamic conditions even when the particle-free flow is steady. A predictive model must therefore assess trapping, motion relative to the surrounding fluid, and acceleration within a consistent formulation. This motivates the present numerical investigation of the three-dimensional background flow and the development of a stationary-device-frame particle model. Together, these provide a basis for investigating how flow strength, particle size and initial position influence migration, cavity entry, retention and release.