Mathematical modelling of particle-fluid flows in microchannels

Abstract

Flows of fluids and solid particles through microchannels have a very wide range of applications in biological and medical science and engineering. Understanding the mechanism of microflows will help to improve the development of the devices and systems in those applications. The aim of this study is to develop a sophisticated simulation and analysis technique for the study of fluid-particle flow through microchannels. This work involves construction of mathematical models, development of analytical methods and numerical algorithms, and numerical investigation and analysis.The study consists of three parts. The first part of the research focuses on the transient flow of an incompressible Newtonian fluid through a micro-annual with a slip boundary. The flow of the fluid is governed by the continuity equation and the Navier-Stokes equations, and is driven by the pressure field with a timevarying pressure gradient. By using the Fourier series expansion in time and Bessel functions in space, an exact solution is derived for the velocity field. The velocity solution is then used to obtain the exact solutions for the flow rate and the stress field. Based on the exact solutions, the influence of the slip parameter on the flow behaviour is then investigated.The second part of the research focuses on the particle-fluid flow in microchannels. The transport of fluid in the vessel is governed by the continuity equation and the transient Navier-Stokes equations, while the motion of the particles is governed by Newton’s laws. The particle-wall and particle-particle interactions are modelled by the interacting forces, while the particle-fluid interaction is described by the fluid drag force. A numerical scheme based on the finite element method and the Arbitary Lagrangian-Eulerian method is developed to simulate the motion of the particles and the fluid flow in the vessels. The influence of boundary slip on the velocity field in the fluid is also investigated numerically.Based on the work in the second part, the third part of the research focuses onthe control of the movement of particles in the fluid by applying an external magneticfield to the system. Maxwell’s equations are used to model the magnetic fieldgenerated by the external magnetic source, and a finite element based numericalscheme is developed to solve the underlying boundary value problem for the magneticflux density generated. From the computed flux density and magnetic vectorpotential, the magnetic forces acting on the particles are determined. These magneticforces together with the drag force and the particle-particle interacting forcesdominate the behaviour of the particle motion. A numerical scheme, similar to thatfor the second part of the research, is then developed to study the fluid-particle flowin microchannels under magnetic forces, followed by a numerical investigation onthe influence of the magnetic forces on the particle flow behaviour.