Journal of Nanotechnology

Volume 2018, Article ID 2615404, 10 pages

https://doi.org/10.1155/2018/2615404

## Two-Dimensional Numerical Study on the Migration of Particle in a Serpentine Channel

Institute of Fluid Mechanics, China Jiliang University, Hangzhou, China

Correspondence should be addressed to Deming Nie; moc.liamg@zhniein

Received 2 February 2018; Accepted 4 April 2018; Published 17 May 2018

Academic Editor: Martin Seipenbusch

Copyright © 2018 Yi Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work, the momentum exchange scheme-based lattice Boltzmann method is adopted to numerically study the migration of a circular particle in a serpentine channel for the range of 20 ≤ Re ≤ 120. The effects of the Reynolds number, particle density, and the initial particle position are taken into account. Numerical results include the streamlines, particle trajectories, and final equilibrium positions. Close attention is also paid to the time it takes for the particle to travel in the channel. It has been found that the particle is likely to migrate to a similar equilibrium position irrespective of its initial position when Re is large. Furthermore, there exists a critical solid-to-fluid density ratio for which the particle travels fastest in the channel.

#### 1. Introduction

Solid particles immersed in a viscous fluid lead to a two-phase flow problem, which is very common in nature and in many industrial processes, including atmospheric currents, aerosol deposition, fluidized beds, and so on. The motion and dynamics of particles suspended in a fluid is fundamental to understanding suspension hydrodynamics. Over the past decade, great progress has been made for microfluidic devices because of several benefits over conventionally sized systems, such as small volume of sample and reagent, low energy consumption, high efficiency, and enhanced analytical sensitivity. In microfluidic devices, manipulation and separation of particles are usually necessary in the processes of enzymatic analysis, DNA analysis, and sample separation. However, it is necessary to focus the samples in a tight stream before separation, sorting, or analysis in order to ensure that these samples are passing through the microchannels quickly. Therefore, understanding the behavior and characteristics of particle suspensions in microfluidics is helpful to provide insight into the design of microfluidic channels [1].

There are several methods of particle focusing that have been developed and used in microfluidic systems. Inertial focusing is usually adopted to align the particles along a tight stream, which has the advantage over other methods because it does not require external forces or multiple streams to focus particles. The most famous phenomena of inertial focusing may be the Segré–Silberberg effect [2], which is a fluid dynamic separation effect where a dilute suspension of neutrally buoyant particles flowing in a tube equilibrates at a distance ∼0.6*R* (tube radius) from the tube center. Later a full analytical solution of the forces that dominate particles in Poiseuille flow was provided by Ho and Leal [3]. They [3] showed that particles migrate from the center of a channel towards the wall due to shear-induced lift forces and are rejected from the channel perimeter by wall-induced lift forces creating a stable equilibrium at a distance of 0.6*R* from the center of the channel. Feng et al. [4] were the first to observe the Segré–Silberberg effect for the motion of a single circular particle in plane Poiseuille flow by using the finite-element method. Pan and Glowinski [5] simulated the motion of multiple circular particles in plane Poiseuille flow. Their results [5] showed that the collisions between particles have a significant influence on the inertial migration of particles. Chun and Ladd [6] investigated the inertial migration of neutrally buoyant particles in a square duct. For the case of elliptical particles, Yi et al. [7] reported that the particles fluctuate about an averaged position at low Reynolds numbers while they converge to an equilibrium position on each side of the channel center at high Reynolds numbers. Chen et al. [8] studied the motion of two spherical particles in tube flow through the lattice Boltzmann method (LBM). They reported an oscillatory state of motion for two spherical particles with different radii in opposite sides. Similarly, Abbas et al. [9] simulated the motion of a spherical particle in a square channel flow. They demonstrated that there exist two states for the migration of particle which are cross-streamline stage and cross-lateral stage, respectively. In addition, they [9] showed that the former stage is much faster than the latter one. Recently, Jiang et al. [10] investigated the migration of particles in a symmetrical serpentine channel through a three-dimensional LBM. They focused on the influence of the Dean flow on particle focusing in a serpentine channel. Their results showed that the alternation of the Dean flow direction has special hydrodynamic effects to focus or separate particles of different sizes as the flow intensity becomes stronger.

So far, the inertial focusing has numerous applications in microparticle manipulation ranging from microfluidic cell sorting to particle separation and ordering [11–13]. However, due to the compact size of microfluidic devices, the majority of the biomedical processes of inertial focusing are carried out in the curved channels, such as expansion-contraction array channel, spiral channel, and serpentine channel. In comparison with the straight channel, the flow conditions are very different in the curved channels. For instance, it is naturally expected that the boundary layer separation will take place in the corners of a serpentine channel, which usually affects the flow structure and of course the motion of particles. The centrifugal force in the curved channels is also a key factor which may dominate the migration of particles and should be taken into account [14]. Furthermore, it is possible that the hydrodynamic interactions between particles in the curved channels are more complex which are usually responsible for the aggregation of particles in the finite-Reynolds-number regime. Unlike the straight channel, it is very difficult to derive a detailed mathematical description of the forces that dominate particles in the curved channels due to the complex nature of flow. At present, much of the development in the curved channels has followed an empirical approach which usually fails to predict the equilibrium position of particles. Therefore, a more complete understanding of the migration of particles in these channels is needed to provide help with the design of microfluidic channels and to further enhance the focusing of particles. However, attempts to study the flow characteristics as well as the motion of particles in a curved channel are rarely reported in the past, most of which are involving experimental work. Little effort has been paid to the study of the migration of particles in a curved channel from a numerical aspect. This motivates the present work.

Among all the types of curved channel, the serpentine channel with linear structure is an optimal choice due to its small footprint and easy parallelization. Furthermore, experimental work has shown that the serpentine channel can achieve focusing and separation within a much shorter length due to the assistance of secondary flow [15]. The similar behavior was also demonstrated in our recent work [16], which presents some primitive results involving particle migration in a serpentine channel. On basis of this, we aim to further present a thorough two-dimensional numerical study of a single particle migrating in a serpentine channel through direct numerical simulations (DNS). The effects of the Reynolds number as well as the initial position of particle on the particle migration are studied. Numerical results include the streamline, particle trajectory, and the equilibrium position of particle. Close attention will be paid to the time it takes for the particle to travel in the serpentine channel. We hope the simulation results would be helpful for the designing of microfluidic devices.

#### 2. Numerical Model

##### 2.1. Lattice Boltzmann Method

In this work, the lattice Bhatnagar–Gross–Krook (LBGK) Boltzmann method proposed is used to solve the fluid flow [17]:where and are the distribution functions and corresponding equilibrium distribution functions associated with the discrete velocity direction . is the time step and is the relaxation time, respectively. In the two-dimensional nine-velocity lattice (D2Q9), the model proposed by Qian et al. [17] is adopted here,

The lattice speed is , where is the lattice spacing. for this lattice iswhere is the fluid density, is the sound speed, and is the weight coefficient given by

The fluid density and velocity can be calculated by the following formula:

The Navier–Stokes equations can be obtained from the lattice Boltzmann equation (LBE) through a Chapman–Enskog expansion proposed by He and Luo [18].

##### 2.2. Problem Description

In this work, we aim to numerically study the migration of a circular particle in a serpentine channel. The physical model is shown in Figure 1. The width of the channel is denoted as *h*. Other parameters such as *L*_{1}, *L*_{2}, *L*_{3}, and *H* are shown in Figure 1, which are set to be *L*_{1} = *L*_{2} = *L*_{3} = 5*h* and *H* = 3*h*. The diameter and density of the particle are expressed by *d* and , respectively. At the inlet, a parabolic flow with the maximum velocity of is applied, while the fully developed condition is applied at the outlet. No slip boundary is used on all the channel walls. In the simulations, the parameters are chosen as follows: (in lattice unit).