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

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. *e 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.


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.e 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.erefore, understanding the behavior and characteristics of particle suspensions in microfluidics is helpful to provide insight into the design of microfluidic channels [1].
ere 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.
e 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 ∼0.6R (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].ey [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 0.6R 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 finiteelement method.Pan and Glowinski [5] simulated the motion of multiple circular particles in plane Poiseuille flow.
eir 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).ey 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.ey 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.ey focused on the influence of the Dean flow on particle focusing in a serpentine channel.eir 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][12][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 expansioncontraction 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.e 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.erefore, 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.is 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].e 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).e 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.

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 f i (x, t) and f (eq) i (x, t) are the distribution functions and corresponding equilibrium distribution functions associated with the ith discrete velocity direction c i .Δt is the time step and τ is the relaxation time, respectively.In the two-dimensional nine-velocity lattice (D2Q9), the model proposed by Qian et al. [17] is adopted here, e lattice speed is c � Δx/Δt, where Δx is the lattice spacing.f (eq) i (x, t) for this lattice is where ρ f is the fluid density, c s is the sound speed, and w i is the weight coefficient given by e fluid density ρ f and velocity u can be calculated by the following formula: ( e Navier-Stokes equations can be obtained from the lattice Boltzmann equation (LBE) through a Chapman-Enskog expansion proposed by He and Luo [18].
In order to avoid the in uence of the inlet on the particle motion, the particle is always placed 20 d from the inlet, which is released after the ow is fully developed.In this work, we focus on the e ect of the Reynolds number, the particle initial position, as well as the density of particle on the nal equilibrium position of particle, and its migration trajectory.

Validation
In order to validate the computational model in this work, the migration of a circular particle in a straight channel is numerically tested.We compare the present results with the previous results proposed by Chen et al. [8], as shown in Figure 2. e parameters used here are the same as those in [7].Explicitly, the diameter of the circular particle is d 22 and the width of the straight channel is H 200, which leads to the size ratio K d/H 0.11.e equilibrium position of the particle Y eq , which is the vertical position of the particle to the width of the channel, at di erent Reynolds numbers (Re 20, 40, 60, 80, 100, 120, 160, 180, and 200) is shown in Figure 2. e Segré-Silberberg e ect is realized, with the equilibrium position a little outside the midpoint between the wall and the channel centerline.In addition, the nal equilibrium position of particle is closer to the channel centerline when increasing the Reynolds number.
In Table 1, we present the relative errors between our results and those in [11] at di erent Reynolds numbers, which shows a good agreement.

Results and Discussion
4.1.Steady Flow Field.In the simulations, the particle is released after the ow eld is fully developed.To better illustrate the ow eld in the serpentine channel, we present the steady streamlines at di erent Reynolds numbers (Re 20, 70, and 120) in Figure 3.Only the local enlargement of steady ow eld is shown because of large computational domain.
It is hard to observe the recirculation zones when the Reynolds number is small, such as Re 20, as shown in Figure 3(a).As a result, the distortion of the streamlines is not signi cant.However, the recirculation zones are becoming more notable for larger Re, as one can see in Figures 3(b) and 3(c).Totally speaking, there are two types of corners in the present serpentine channel: corners with inwards right angle and those with outwards right angle, which lead to two types of recirculation zone.It is observed that there always exists a recirculation zone for each corner when Re is large.Furthermore, the rotation of recirculation zones on the upper wall is always counterclockwise while the opposite is true for  Journal of Nanotechnology those on the bottom wall.Due to the recirculation zones, the distortion of the streamlines becomes signi cant when the Reynolds number is large, such as Re 120, which is expected to considerately in uence the migration of particle in the channel.

e E ect of the Reynolds Number.
In this section, we study the e ect of the Reynolds number on the particle migration in the serpentine channel shown in Figure 1. e density of the particle is xed at ρ p 1.0, suggesting that the particle is neutrally suspended.Here, a parameter q is introduced to describe the initial lateral position of the particle in the channel which is de ned by the distance of particle to the upper channel wall normalized by the channel width h.As a result, the value of q 0.5 indicates that the particle is initially placed on the channel centerline.In what follows, we focus on the particle trajectory as well as the nal equilibrium position of the particle under di erent channel Reynolds numbers.Figure 4 shows the particle trajectories at Re 20, 70, and 120 for di erent initial particle positions.As one can see, the e ect of Re on the particle migration in the channel is signi cant.To some extent, the particle trajectories are similar to the streamlines shown in Figure 3.In comparison with the result of Re 20, the particle is driven farther away from the channel wall after passing through each bend when increasing Re, which is more signi cant for small values of q, as shown in Figure 4. is is due to the fact that the larger the Reynolds number, the larger the centrifugal force experienced by the particle.
e e ect of q on the particle trajectories is shown in Figure 5. e particle trajectories of di erent q are almost parallel to each other for each Reynolds number.It is observed that the dependence of particle migration on the value of q is stronger when Re is smaller.In other words, the nal equilibrium position of particle is more sensitive to the value of q for smaller Reynolds number.For instance, no visible di erence is observed for all results except that of q 0.5 when Re 120, the largest Reynolds number studied in the work, as one can see in Figure 5(c).In order to provide a better understanding, we summarize the dependence of the nal equilibrium positions of particle on Re as well as q in Table 2, which further demonstrates that the e ect of q on the particle migration is more signi cant at smaller Re.It is also clear that the particle eventually stays on the channel centerline for the range of Re studied when q 0.5.
High e ciency is occasionally required in the microuidic devices such as fast separation.In these cases, the particles are expected to travel fast in the serpentine channel.As a result, much attention should be paid to the time needed for the particles to travel in the channel.We carried out a preliminary study upon this issue.e dependence of T * on the value of q is shown in Figure 6 for di erent Re.It should be stated that T * is normalized by T 0 which is determined through T 0 d/U 0 .As shown in Figure 6  Journal of Nanotechnology results decrease as q increases when q < 0.5, while the opposite is true when q > 0.5. is leads to the fact that the time T * reaches its minimum value when q 0.5 for all cases, suggesting that the particle will travel fastest if it is initially placed on the channel centerline.In addition, the results are more likely to be symmetrical when Re is smaller, such as Re 20, which is due to the symmetrical ow conditions.However, the recirculation zones resulting from corners become signi cant as increasing Re, which destroys the ow symmetry.Furthermore, it is found that the time needed for the particle to travel in the channel has a quadratic-like relationship with the value of q for small Re (Re 20): To further study the migration behavior of the particle in the serpentine channel, we also pay close attention to the evolution of particle orientation θ which has an initial value of π/2, as shown in Figures 7 and 8, which present the corresponding results of q 0.1-0.3 and q 0.5 at di erent Reynolds numbers, respectively.As we can observe in Figure 7, the particle always rotates counterclockwise when traveling in the channel if q < 0.5 irrespective of Re.In addition, the smaller the value of q is, the faster the particle rotates. is is because the particle experiences larger gradient of uid velocity if it is closer to the channel wall.
Figure 8 shows a di erent pattern of particle motion in the serpentine channel when q 0.5.Instead of rotating counterclockwise, the particle will oscillate if it is initially

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Journal of Nanotechnology placed on the channel centerline for the range of Re in this work.Especially, it is observed that the particle is oscillating around θ π/2 when traveling in the channel for large Reynolds numbers, such as Re 70 and 120, as shown in Figure 8.

e E ect of Particle Density.
e particle inertia is central to the behavior of the particle suspended in uids because it determines how the velocity of the particle decays due to uid drag.As is known to all, the particle inertia strongly depends on its density.Consequently, we focus on the e ect of the particle density on the migration behavior of the particle in the serpentine channel in this section.e solid-to-uid density ratio is de ned as α ρ p /ρ f .All the results in the previous section treat the particle density ρ p equal to the uid density ρ f , that is, α 1.To save the computational resources, some parameters are xed at ρ f 1, Re 20, and q 0.3 in this section.
We rst consider the cases for which the uid is heavier than the particle, that is, α < 1. Figure 9 shows the particle trajectories for di erent values of α.It is interesting to nd that no visible di erence is observed for all the results shown, suggesting that the e ect of the density ratio on the migration behavior of the particle in the serpentine channel is negligible for α < 1.However, things are very di erent for Figure 8: e orientation of the particle for q 0.5 at di erent Reynolds numbers (also appeared in [16]).8 Journal of Nanotechnology α > 1, as shown in Figure 10.e particle is more likely to maintain its motion for larger density ratio, leading to the fact that the particle approaches closer to the channel wall after traveling in the straight section of channel. is trend is very signi cant when the particle is heavy such as α 10 and 20, as shown in Figure 10(b).Similarly, we also present the time needed for the particle to travel in the channel upon di erent values of α in Figure 11.
e e ect of density ratio on T * is signi cant.It is obvious that T * decreases as α increases initially and then increases afterwards.In addition, T * reaches its minimum value at α ≈ 7, as we can observe from Figure 11. is suggests that it may take the shortest time for the particle to travel in the channel if it has a particular density.It is also interesting to nd that for α < 1, the value of T * decreases as α increases despite that the particle trajectories (Figure 9) are almost identical, as shown in Figure 11.Finally, it is found that a quintic polynomial function like (8) ts the results quite well, T * −0.0006123α 5 + 0.03686α 4 − 0.8384α 3 + 9.486α 2 − 51.62α + 571.7.
Finally, it should be stated that the present work only involves two-dimensional numerical simulation, which is di erent from the existing experimental work.For instance, the secondary ow may play an important role on the migration of particles as well as the interaction between particles in channel ows when the uid inertia is significant, which cannot be taken into account in the present two-dimensional study.However, it is generally agreed that the two-dimensional simulations can be served as a tool for studying threedimensional ow characteristics when the Reynolds number is not large.As mentioned above, this work is the rst step of investigation.We hope it casts a light on some features of particle migration in a serpentine channel.Certainly, the motion of spherical particles as well as their interactions in a threedimensional channel ow would be our future work.

Conclusion
In this work, the lattice Boltzmann method based on the momentum exchange scheme has been adopted to numerically study the migration of a particle in a serpentine channel.We focus on the effects of the Reynolds number (Re) as well as the initial position of the particle (q) on the migration behavior of the particle in the channel.e Reynolds number ranges from 20 to 120.
(1) e effect of Re on the final equilibrium position of the particle is significant, which is found to be more sensitive to the initial position of the particle when Re is small.For Re � 120, the largest Reynolds number studied in this work, the particle has a similar equilibrium position irrespective of its initial position for q < 0.5 or q > 0.5.Interestingly, results show that the particle almost stays on the channel centerline for the range of Re studied when q � 0.5.(2) It has been found that the particle is driven farther away from the channel wall after passing through each bend while increasing Re, which is more significant for small values of q. is is due to the fact that the larger the Reynolds number, the larger the centrifugal force experienced by the particle.(3) e solid-to-fluid density ratio α is central to the migration of the particle in the channel.It has been found that the time it takes for the particle to travel in the channel decreases as α increases initially and then increases afterwards.ere exists a critical value of density ratio for which the particle travels fastest in the channel.

Figure 1 :Figure 2 :
Figure 1: Physical model of the present work.

Figure 6 :
Figure 6: Time needed for the particle to travel in the channel for di erent values of q.
In this work, we aim to numerically study the migration of a circular particle in a serpentine channel.e physical model is shown in Figure1.e width of the channel is denoted as h.Other parameters such as L 1 , L 2 , L 3 , and H are shown in Figure1, which are set to be L 1 L 2 L 3 5h and H 3h. e diameter and density of the particle are expressed by d and ρ p , respectively.At the inlet, a parabolic ow with the maximum velocity of U 0 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: ρ 2 Journal of Nanotechnology 2.2.Problem Description.f 1, d 10, h 8d, and U 0 0.05 (in lattice unit).e channel Reynolds number Re in this work is de ned as Re U 0 h ] .

Table 1 :
Relative errors of di erent Reynolds numbers.
, all the

Table 2 :
Dependence of the nal equilibrium position on Re as well as q.