Research on the Inertial Migration Characteristics of Bi-Disperse Particles in Channel Flow

Abstract

The inertial focusing effect of particles in microchannels shows application potential in engineering practice. In order to study the mechanism of inertial migration of particles with different scales, the motion and distribution of two particles in Poiseuille flow are studied by the lattice Boltzmann method. The effects of particle size ratio, Reynolds number, and blocking rate on particle inertial migration are analyzed. The results show that, at a high blocking rate, after the same scale particles are released at the same height of the channel, the spacing between the two particles increases monotonically, and the change in the initial spacing has little effect on the final spacing of inertial migration. For two different size particles, when the smaller particle is downstream, the particle spacing will always increase and cannot remain stable. When the larger particle is downstream, the particle spacing increases firstly and then decreases, and finally tends to be stable.

1. Introduction

The precise focusing process of particles is a critical step of separation and screening techniques [1,2,3] and is widely used in the fields of biochemical engineering and medical areas. In recent decades, microfluidic technology has been widely used, owing to its unique advantages in many fields and rapid development, among which inertial microfluidic technology is one of the typical representatives [4]. Inertial microfluidic, as a passive method, has attracted wide attention for its advantages of simple operation, simple structure, being harmless to cell viability, and high-throughput.

The suspended particles naturally gathered together and migrated to the equilibrium position, under specific flow fields and geometric conditions, confirming, by Segré and Siberberg [5] with extensive experiments, that the particles gathered into a concentric ring on the cross section of the channel and continued to move in the flow direction. This focusing was thought to be a balance of shear-induced, wall-induced, and rotation-induced lift forces, and the underlying mechanisms have been extensively investigated [6]. Remarkably, Segré and Siberberg [5] proposed that the particles not only migrated by inertia to the equilibrium position of the cross section, but also ordered longitudinally along the main flow direction to form particle trains that may result from particle–particle interactions.

Matas et al. [7] observed that, after particles randomly enter the pipe, the particles would be arranged along the flow direction to form a multi-particle train structure, defined as the particle train [7,8] when three or more particles are aligned with a regular inter-particle spacing. Gao et al. [8] found that increasing the Reynolds number was beneficial to the formation of particle trains, but it reduced the number of particles in the particle trains. The distance between adjacent particles decreases with the increase in Reynolds number. Hu et al. [9] found in numerical simulations that, during inertial migration, owing to the force between particles and the fluid, with the effect of inertia, the particles moved to the equilibrium position one by one, forming a line of evenly spaced particle structures. However, the particle spacing increases slowly with the increase in the inertial migration length; the particle inertial migration length decreases with the decrease in the power-law exponent, and the shear thinning effect will increase the formation of ordered particle trains; the particle spacing decreases with the increase in Reynolds number and blocking ratio. Hu et al. [10] showed that, for single-line particle pairs, the inter-particle spacing increased to a larger value for further downstream, while two lines of 12 particles would self-organize the staggered particle trains, which were dependent on the Reynolds number and blockage ratio.

However, the orderly arrangement technology of particle trains is difficult to apply to cell populations because of size dispersion and deformability of cells [11]. The particle size not only affects the equilibrium position of the particles [12], but also affects the migration speed of the particles [13] and the arrangement of the flow direction [14]; therefore, label-free enrichment or separation of particles in polydisperse suspensions can be achieved. However, most of these works only focus on the separation efficiency, and there is little research on the physical mechanism of particle inertial migration by the size dispersion of particles. Up to now, a large number of theoretical, numerical, and experimental studies have focused on exploring the physical mechanism of particle migration in a closed environment. Only a few works have studied the effect of the interaction between particles of different sizes in multi-scale situations. The effect of particle multiscale on inertial behavior is still unclear and needs further study. In order to explore the effect of particle multiscale on inertial migration, in this paper, the lattice Boltzmann method is used to study the characteristics of inertial migration of two scale particles in channel flow, and the influence of important parameters such as particle size ratio and Reynolds number on the equilibrium position of the particles and the inter-particle spacing is discussed.

2. Materials and Methods

2.1. Lattice Boltzmann Method

The lattice Boltzmann method (LBM) comes from lattice gas automata, which is a numerical simulation method based on the discrete Boltzmann equation. LBM is considered to be an effective numerical simulation method for studying multiphase flow. It is a kind of mesoscopic model that has the advantages of both macroscopic and microscopic models.

The main idea of LBM is to numerically solve the discrete Boltzmann equation to simulate the macroscopic motion of fluids. LBM models usually include discrete velocity models, equilibrium distribution function, and evolution equation for the distribution function. The discrete velocity model adopted in this paper is the D2Q9 model [15], which applies to two-dimensional space and contains nine discrete velocity directions [16]:

$$e_i=\left\{\begin{array}{l}(0,0)i=0\\ c\left(\cos \left[(i-1)\frac{\pi }{2}\right],\sin \left[(i-1)\frac{\pi }{2}\right]\right)i=1,2,3,4\\ \sqrt{2}c\left(\cos \left[\left(2i-1\right)\frac{\pi }{4}\right],\sin \left[\left(2i-1\right)\frac{\pi }{4}\right]\right)i=5,6,7,8\end{array}\right.,$$

(1)

where i represents the component of the function in the i direction; e_i represents the direction vector of the discrete velocity in the i direction; and c = Δx/Δt, where Δx, and Δt are the grid step size and the unit lattice time, respectively, in the standard LBM Δx = 1, Δt = 1.

The equilibrium distribution function is as follows:

$$\begin{array}{l}{f}_i^{eq}=\rho w_i\left[1+\frac{{\mathit{e}}_{\mathit{i}}\cdot \mathit{u}}{c_s^2}+\frac{{({\mathit{e}}_{\mathit{i}}\cdot \mathit{u})}^{{}^2}}{2c_s^4}-\frac{{\mathit{u}}^2}{2c_s^2}\right]\\ c_s=\frac{1}{\sqrt{3}},w_i=\left\{\begin{array}{l}4/9i=0\\ 1/9i=1,2,3,4\\ 1/36i=5,6,7,8\end{array}\right.\end{array},$$

(2)

where c_s is the speed of sound; w_i is the weight factor; and ρ and u represent the fluid density and velocity, respectively. The evolution equation of the distribution function with the external force term is as follows:

$$f_i\left(\mathit{x}+\Delta t{\mathit{e}}_i,t+\Delta t\right)=f_i\left(\mathit{x},t\right)+\frac{1}{\tau }\left[f_i^{eq}\left(\mathit{x},t\right)-f_i\left(\mathit{x},t\right)\right]+\Delta t\cdot F_p,$$

(3)

where τ is the relaxation time, f_i(x,t) is the distribution function at the position x and time t, and F_p is the external force term that drives the Poiseuille flow. Using the external force model with good stability proposed by He et al. [17],

$$F_p=\left(1-\frac{1}{2\tau }\right)\frac{\left({\mathit{e}}_{\mathit{i}}-\mathit{u}\right)\cdot {\mathit{F}}_b}{c_{\mathrm{s}}^2}{f}_i^{eq}\left(\mathit{r},t\right),$$

(4)

where F_b is the body force. The fluid density and velocity are calculated as follows:

$$\rho ={\displaystyle \sum _{}{f}_i,}\mathit{u}=\frac{1}{\rho }{\displaystyle \sum f_i}{\mathit{e}}_i+\frac{\Delta t}{2\rho }{\mathit{F}}_b.$$

(5)

Navier–Stokes equations can be derived through the Chapman–Enskog expansion and have second-order accuracy in both time and space [18].

The hydrodynamic force experienced by the particles is calculated by means of momentum exchange, and the specific process is as described in [10].

2.2. Repulsive Force Model

When the distance between particles or between the particles and the wall is less than a lattice length, the hydrodynamic force on the particles cannot be calculated. Therefore, the following repulsive force model is introduced to analyze the repulsive force between the two particles or walls that will collide.

When the distance between the centre of the particle and the wall is less than 2Δx [19], the repulsive force is introduced:

$$f_r=\left\{\begin{array}{cc}\frac{C_{\mathrm{m}}}{\epsilon }{\left(\frac{d-d_{\min }-\Delta r}{\Delta r}\right)}^2{\mathit{e}}_r,& d\le d_{\min }+\Delta r\\ (0, 0),\hfill & d>d_{\min }+\Delta r\end{array}\right.,$$

(6)

where C_m = MU²/a, M is the particle mass, U is the velocity, and a is the particle radius. Ε = 10⁻⁴ is a positive coefficient, d is the distance between the center of the particles or the direction from the center of the particle to the wall, and e_r is the direction vector; d_min = 2a, Δr = 2Δx represents two lattices when repulsive force exists in the simulation.

2.3. Problem Definition

The migration of particles in the Poiseuille flow is shown in Figure 1. Periodic boundary conditions are selected in the x direction, the upper and lower walls adopt the standard rebound boundary, and the boundary conditions of the particle surface adopt the moving wall rebound format.

During the simulation, when the calculation domain length changes from 1500Δx to 3500Δx, the result obtained does not change much, so the length of 1500Δx and 2000Δx is adopted, and the channel height H = 150Δx. The particle density is equal to the fluid density, the diameter D = 18~45Δx, and the blocking ratio k = D/H. Re = ρU_maxH/m, ρ and U_max are fluid density and maximum velocity, respectively.

3. Validation

3.1. Velocity Profile in a Poiseuille Flow

Figure 2 shows the comparison between the velocity profile of the Poiseuille flow simulated by the numerical simulation and the analytical solution (7), which shows that the method is reliable.

$$\begin{array}{l}{u}_y=0\\ u_x=U_{\max }\left[1-{\left(\left|1-\frac{2y}{H}\right|\right)}^2\right]\mathrm{f}\mathrm{o}\mathrm{r}0\le y\le H\end{array}$$

(7)

3.2. Trajectory of a Particle in Shear Flow

Figure 3 shows the trajectory of a particle in the shear flow of Newtonian fluid. The flow domain is 2000 × 80Δx, the particle diameter is 20Δx, Re = 40, and the upper and lower walls move in opposite directions at a speed of U_W = 1/120. Under the same simulated conditions, it can be seen that the numerical simulation results are very consistent with the results given by Feng and Michaelides [20]. Hu et al. [9] also verified this result with the IB-LBM under the same experimental conditions.

3.3. Trajectory of a Particle in Poiseuille Flow

Figure 4 shows the trajectory of particles after they are released at different initial vertical heights. The flow domain is set to 1000 × 120Δx and the particle diameter is 36Δx. It can be seen that, after the particles are released at y₀ below the channel centerline, they always migrate to the same equilibrium position y^eq, which is consistent with the Segré and Silberberg effects.

4. Results and Discussion

4.1. A Pair of Single-Scale Particles

4.1.1. The Effect of Initial Spacing

Figure 5 shows the variation of the spacing l with the migration distance of a pair of single-scale particles in the migration process under different grid resolutions. The figure shows the preliminary process of the migration. It can be seen that the evolution of inter-particle spacing under different initial spacing is relatively similar. Comparing (a) and (b), it can be seen that the results are almost the same in different flow domains and different particle diameters.

Figure 6 shows the change in the distance l with the migration distance after the particles have migrated for a long distance. It can be seen that the spacing of the particle pair increases rapidly at first, and then keeps increasing slowly, especially for the particle pair with a small initial spacing, which is consistent with Hu et al. [10] and Lee et al. [6]; the final spacing of the particle pair under different initial spacing reached the same value. It shows that, at a high blocking ratio, the initial particle spacing has little effect on the final particle spacing.

4.1.2. Effect of Blocking Ratio

Figure 7 shows the change in the distance between a pair of single-scale particles with different blocking ratios calculated by the 2000 × 150Δx flow domain during the migration process. The initial distance is l = 2D and Re = 20. It can be seen that, the smaller the blocking rate, the larger the horizontal spacing between particles, which is consistent with the numerical results of Hu et al. [10]; when the blockage ratio is small, the restriction of the pipeline to the particles is weaker, and the spacing of the particles cannot be kept stable.

4.2. A Pair of Dual-Scale Particles

In order to study the effect of the interaction between particles of different sizes, the following provides the results of the influence of size ratios of particles, Reynolds number, and blocking ratio on particle inertial migration. The flow field and the position of the particles are shown in Figure 8. The particles migrate from left to right.

4.2.1. Effect of Size Ratio of Particles

Figure 9 shows the effect of the size ratio of two particles on the distance between the two particles. The particle size ratio is defined β = D_p1/D_p2, which represents the ratio of the downstream particle diameter to the upstream particle diameter, blocking ratio k = 0.125~0.3 (k = D/H), particle diameter D_p = 18.75Δx, 22.5Δx, 30Δx, 37.5Δx, 45Δx. As shown in Figure 9a, when the small particle is downstream, the inter-particle spacing will continue to increase and cannot remain stable; when the large particle is downstream, the inter-particle spacing increases at the beginning and then decreases, and finally stabilizes, Gao et al. [14] found that mixed particle trains often begin with a large particle and end with a small one; two particles of the same size and the spacing increases monotonously, which is consistent with the numerical results of Hu et al. [10] and Lee et al. [6]. The reason is that, when the large particle is downstream, the weak production by the large particle forms a large reversing streamline zone, which has a very large impact on the small particle downstream. The small particle and the large particle form a stable structure; the difference between the velocity of the particles is small, so the spacing between particles can eventually remain stable. In this case, when the small particle is downstream, the reverse streamline area formed by the wake of the small particle has little effect on the large particle. The velocity of the small particle is relatively higher, and the distance between the particles cannot be kept stable.

As shown in Figure 9b, at the initial stage of particle movement, if the large particle is downstream and the small particle is upstream, the inter-particle spacing will first decrease and then increase; if the small particle is downstream and large particle is upstream, the distance between particles will increase monotonously, and there will be no collisions between particles. This is because of the small inertia of the small particle, which shows that the small particle follows well and, at the initial stage, it accelerates faster.

Figure 10 is a schematic diagram of particle movement from t0 to t6, and the time interval is equal. (a) shows the situation of small particle downstream; the spacing between particles keeps increasing and the relative velocity between particles increases first and then tends to level off. (b) shows the situation of large particle downstream; it can be seen that the spacing between particles first increases and then decreases.

4.2.2. Effect of Reynolds Number

Figure 11 shows the effect of Reynolds number on the distance between two particles. It can be seen that, when β = 0.5, that is, when the small particle are downstream, the larger the Reynolds number, the larger the inter-particle spacing, and the particle spacing cannot be kept stable. The larger the Reynolds number, the greater the resistance to the large particle, so the spacing becomes wider and wider. When β = 2.0, that is, when the large particle is downstream, the larger the Reynolds number, the smaller the inter-particle spacing, and the inter-particle spacing can remain stable within the studied Reynolds number range. Gao et al. [14] and Hu et al. [10] used experimental and numerical simulation methods, respectively, to find that the average distance between particles in a single-line particle trains decreased with the increase in Re. The larger the Reynolds number, the greater the resistance to the large particle, and the greater the effect of wake on the small particle downstream, so the spacing remains stable.

4.3. The Equilibrium Position of the Particles

The equilibrium position of a single particle is shown in Figure 12. It can be seen that the particles tend to the equilibrium position quickly. For particles with a low blocking rate, the equilibrium position of the particles is closer to the wall, which is consistent with the numerical results of Hu et al. [10]. The equilibrium positions of two particles of different scales are shown in Figure 13. When small particles are downstream, their equilibrium positions are closer to the center-line of the pipeline, while the equilibrium positions of large particles remain unchanged. The speed of small particles is greater than that of large particles, resulting in a continuous increase in the spacing of particles. When large particles are downstream, their equilibrium position is basically unaffected, while the equilibrium position of downstream small particles is closer to the wall. The speed of small particles is lower than that of large particles, and the distance between the particles remains unchanged after a series of oscillating migration. Figure 14 shows the change in the equilibrium position of the smaller particle when compared with a single particle.

5. Conclusions

In this paper, the LBM method is used to study the inertial migration characteristics of single-scale and dual-scale particles in channel flow. The results show that, in the case of a high blocking ratio, the final particle spacing of two particles of equal diameter released at the same height does not change with the change in the initial spacing of the particles. For two particles of different sizes, the equilibrium position of the large particle is almost the same as that of a single particle, while the equilibrium position of the small particle has changed. When the large particle is upstream, the equilibrium position of the small particles is closer to the wall; when the small particle is upstream, the equilibrium position of the small particle is closer to the center line. When the large particle is downstream, the inter-particle spacing can eventually remain stable, while, when the small particle is downstream, the spacing between the two particles will continue to increase and cannot be stabilized. When the small particle is downstream, the larger the Reynolds number, the larger the inter-particle spacing, and the particle spacing cannot be balanced. When the large particle is downstream, the larger the Reynolds number, the smaller the inter-particle spacing, and the spacing remains stable within the studied Reynolds number range.