# Elasto-Inertial Focusing Mechanisms of Particles in Shear-Thinning Viscoelastic Fluid in Rectangular Microchannels

^{*}

## Abstract

**:**

**~**O(1), both the elastic force and secondary flow effects push particles towards the channel center. However, at a high flowrate, Wi

**~**O(10), the elastic force direction is reversed in the central regions. This remarkable behavior of the elastic force, combined with the enhanced shear-gradient lift at the high flowrate, pushes particles away from the channel center. Additionally, a precise prediction of the focusing position can only be made when the shear-thinning extent of the fluid is correctly estimated in the modeling. The shear-thinning also gives rise to the unique behavior of the inertial forces near the channel walls which is linked with the ‘warped’ velocity profile in such fluids.

## 1. Introduction

_{W}) and shear-gradient (F

_{S}) lift forces. While F

_{W}pushes the particles away from the channel walls, F

_{S}generates a wall-directed force due to the parabolic nature of the velocity profile, and the final equilibrium position of particles is determined by the interplay of these two forces. These forces are present in flows with low but finite Reynolds number [41], which is a dimensionless parameter that is the ratio of inertial force to viscous force in the flow and is defined as Re = $\mathsf{\rho}\mathrm{V}$D

_{H}/$\mathsf{\mu}$ where $\mathsf{\rho}$ is the fluid density, V is the characteristic velocity, D

_{H}is the hydraulic diameter and $\mathsf{\mu}$ is the fluid viscosity. In the case of a square channel, particles occupy four focusing positions along the channel walls at the center of each face, while in a rectangular channel, two focused streams are formed near centerline of the wider channel walls. Thus, single-stream focusing for cytometry or size-based separation is not feasible in simple, straight microchannels using Newtonian fluids and instead requires more complex channel geometries [33,42,43].

_{E}) attributed to the unique properties of viscoelastic flows, namely the first and second normal stress differences (N

_{1}and N

_{2}). Studies show that F

_{E}drives particles towards regions with the lowest N

_{1}(Figure 1a), while N

_{2}gives rise to the formation of secondary flows in the cross-section of the channel (Figure 1b), which can in turn affect particle lateral migration [45] by imposing a transversal drag force, F

_{SFD}. Additionally, shear-thinning properties will lead to decrease in the local fluid viscosity near the channel walls and its increase near the channel center. This gives rise to the unique ‘warped’ velocity profile in such flows, yielding velocity gradient that is more flattened at the center and sharper near the channel walls (Figure 1c)

**.**Viscoelasticity of the fluid can thus be characterized by the Weissenberg (Wi) dimensionless number, defined as the ratio of elastic force to viscous force, Wi = $\lambda \dot{\gamma}$, where $\lambda $ is the relaxation time of the fluid, and $\dot{\gamma}$ is the shear rate.

**~**O(1–10). A summary of the related published works is given in Table 1. These studies provide invaluable insight into the physics behind viscoelastic focusing mechanisms in a wide range of channel cross-sectional shapes and flow regimes. Nevertheless, a more detailed study on the elasto-inertial focusing mechanisms of shear-thinning fluids in rectangular microchannels is indeed needed at higher Wi numbers, since most numerical investigations in Table 1 are limited to Wi < 3 due to convergence issues arising at higher Wi numbers [47]. In addition, in most of these works, the contributions of shear-thinning and elastic forces were studied separately and the combined effects were not systematically explored. Moreover, most of these studies are either purely experimental or computational studies, and systematic comparison between experiments and models for elasto-inertial focusing is lacking.

_{2}-induced secondary flow modify the inertial and elastic forces and give rise to unique focusing patterns across channel centerlines. We then propose a general focusing mechanism based on our experimental and numerical observations. Ultimately, our results show that precise prediction of focusing positions, which has crucial impact on the efficiency of particle separation, is possible, but only when realistic constitutive equations and parameters are utilized in the simulations. Lastly, we show how the ‘warped’ velocity profile in shear-thinning fluids impacts the inertial force behavior near the channel walls.

## 2. Methods

#### 2.1. Experimental Methods

#### 2.1.1. Device Fabrication

^{®}, Midland, MI, USA) were mixed with a ratio of 10:1. The mixture was cast on the dry film wafer after degassing for 30 min. The PDMS slab containing the replicated microchannel was peeled after 2 h curing on an 80 °C hotplate. A biopsy punch with an outer diameter of 1.5 mm (Ted Pella Inc., Redding, CA, USA) was used to manually punch the inlet and outlet holes in the PDMS channel, which was subsequently bonded to 1″ × 3″ glass slides (Fisher Scientific, Hampton, NH, USA) after O

_{2}surface plasma treatment (PE-50, Plasma Etch Inc., Carson City, NV, USA) for 20 s.

#### 2.1.2. Particle Sample Preparation

^{6}g/mol (Sigma-Aldrich, Burlington, MA, USA) was dissolved into distilled water to create a viscoelastic carrier fluid. The concentration of PEO was 0.1% (wt/wt) for the 1000 ppm solution. Fluorescent polystyrene particles with 4.16 µm and 7.32 µm diameter were suspended into the PEO solutions at a volume fraction of 0.01% to avoid potential particle-particle interactions. A drop (1% v/v) of Tween 80 (Sigma-Aldrich, Burlington, MA, USA) was added to the suspensions to reduce particle aggregation. According to Li et al. [48], the zero-shear viscosity (${\mathsf{\mu}}_{0}$) for the 1000 ppm PEO solution is 2.3 mPa·s. The corresponding effective relaxation time and the density of the PEO solution are 6.8 ms, and 1000 kg/m

^{3}, respectively.

#### 2.1.3. Flow Experiments

^{®}, Air-Tite Co Inc, Virginia Beach, VA, USA), which was connected to 1/16″ Tygon

^{®}tubing (Cole-Palmar, Vernon Hills, IL, USA) using proper fittings (IDEX Health & Science LLC, Northbrook, IL, USA). The other end of the tubing was secured to the device inlet. A syringe pump (Legato 200, KD Scientific Inc, Holliston, MA, USA) was used to sustain various stable flow rates (1 & 5 µL/min).

#### 2.1.4. Data Acquisition and Analysis

^{®}(NIH, Bethesda, MD, USA).

#### 2.2. Numerical Methods

#### 2.2.1. Direct Numerical Simulation (DNS)

_{C}= 150 µm as a segment of the microchannel and a spherical hole with diameter d is embedded inside the domain representing the particle (Figure 2a). Viscoelastic Flow interface (vef) in COMSOL Multiphysics 5.6

^{®}(COMSOL, Inc., Stockholm, Sweden) is used to solve the momentum and mass conservation equations which can be written as:

**T**is the extra elastic stress tensor.

_{e}**L**is the velocity gradient tensor. Fluid density and solvent viscosity is set to 1000 kg/m

^{3}and 1 mPa·s, respectively. Periodic flow condition with appropriate pressure difference values is used to generate the flow inside the channel, and quadratic and linear shape functions are chosen for the discretization of the velocity and pressure fields, respectively. Translational velocity (U

_{P}) of the particle in the downstream direction is modeled by applying velocity of −U

_{P}to the channel walls, while rotation is simulated using appropriate moving wall boundary condition on the particle surface to account for the rotational velocity—${\varphi}_{x},{\varphi}_{y},\mathrm{and}{\varphi}_{z}$—about all axis.

#### 2.2.2. Constraint Particle Rotational and Angular Velocities

#### 2.2.3. Mesh Dependence and Model Validation

^{4}tetrahedral mesh elements (Figure 2b) with two levels of refinement to ensure accuracy of the results. Extra fine triangular mesh with the average element size of 0.1 d is employed on the particle surface and four stretching boundary layer mesh are used around the particle. This guarantees capturing of sharp stress and velocity gradients in the vicinity of the particle (Figure 2c,d). They specifically help with the convergence of the model at high Wi numbers where these sharp gradients are expected. Equations solved under this mesh configuration will generate about 85 × 10

^{4}degrees of freedom which meets our computational capacity when solving multiple cases in parallel as an attempt to optimize computational time while maintaining sufficient accuracy. The simulation scheme used here was previously validated by Di Carlo et al. [40]. Independence of the simulation results on the number of mesh elements is shown in Figure S1.

#### 2.2.4. Giesekus Constitutive Equation

**D**is the strain rate tensor and is defined as $\frac{1}{2}(\mathit{L}+{\mathit{L}}^{\mathrm{T}})$. Equation (6) is the upper convective derivative of the elastic stress tensor in the steady state. $\mathsf{\lambda}\mathrm{and}{\mathsf{\mu}}_{\mathrm{p}}$ are the relaxation time and polymer part of the viscosity of the fluid and are set to 6.8 ms and 1.3 mPa·s, respectively, and $\mathsf{\alpha}$ is the mobility factor. The total viscosity of the fluid is defined as $\mathsf{\mu}=\text{}{\mathsf{\mu}}_{\mathrm{s}}\text{}+\text{}{\mathsf{\mu}}_{\mathrm{p}}.$ Note that the extent of shear-thinning behavior of the fluid can be fine-tuned by adjusting the $\mathsf{\alpha}$ constant in the Giesekus model. Based on experimental measurements by Rodd et al. [58], the normalized viscosities ($\mathsf{\mu}/{\mathsf{\mu}}_{0}$) of 1000 ppm PEO solution at the working conditions of our experiments, Wi = 3.6, and 18, are 0.83 and 0.71, respectively, where ${\mathsf{\mu}}_{0}$ is the initial total viscosity at zero shear rate. This shear-thinning behavior can be reasonably estimated by setting $\mathsf{\alpha}=0.2$ in the Giesekus equation (Figure 3). Therefore, this value of the mobility factor will be used in our simulations to capture the shear-thinning behavior of the viscoelastic fluid. The shear-rate dependent viscosity of the PEO solution might also be modeled using the fractal derivative rheological model recently introduced by Zuo and Liu [60]. This model can correctly predict the non-linear velocity distribution of SiC paste.

## 3. Results and Discussion

#### 3.1. Experimental Results

#### 3.2. Simulation Results

#### 3.2.1. Force Analysis along Y-Midline

_{SFD}is caused by the N

_{2}-induced secondary flow which is a viscoelastic phenomenon, it will impact the non-elastic stress component due to the inertial nature of the drag force. Therefore, fluid viscoelasticity will impact particle migration directly due to F

_{E}, and indirectly through F

_{SFD}.

_{1}) regions [61], with the maximum force magnitudes occurring closer to the channel walls where stronger N

_{1}is present, as seen in Figure S3. This behavior of the elastic force was obtained at Q = 1 µL/min (Wi = 3.6) for both particle sizes and is presented in Figure 5a,b. On the other hand, at Q = 5 µL/min (Wi = 18), unexpected negative elastic force values were obtained near the channel center. The wall-directed migration of particles at high flowrates is typically attributed to the shear-thinning effects only. While this new finding suggests that the migration of particles away from the channel center at higher flowrates can also be due to the direct effect of the elastic force. This counter-intuitive result can be explained by the possible disturbance caused to the N

_{1}distribution on the surface and at the vicinity of the particle at higher Wi numbers.

_{W,}and a wall-directed shear-gradient force, F

_{S}. In viscoelastic fluids however, addition of F

_{SFD}and shear-thinning effects can impact the traditional inertial force behavior. The inertial force distribution for the 7.32 µm particle at the Y-midline is shown in Figure 5c. At Q = 1 µL/min, inertial force was positive along the whole midline except for the near the wall region (Y

_{P}< 10 µm). The positive force at the central region could be caused by the presence of the relatively strong center-directed N

_{2}-induced secondary flow along the Y-midline of the channel. The formation of secondary flow in the cross-section of the channel is shown in Figure S4. Conceivably, shear-gradient lift (F

_{S}) is simply not strong enough at this flowrate to counteract the transversal drag force caused by the secondary flow (F

_{SFD}). Conversely, at the near the wall region, negative inertial force values were obtained. This behavior is in agreement with the observations by Raffiee et al. [54] near the wall for Giesekus fluid, and in contrast with the expected behavior of inertial force near the wall in Newtonian and non-shear-thinning fluids such as Oldroyd-B fluid, where positive, wall-repulsive force (F

_{W}) is expected to push the particles away from the wall [39,41,51].

_{SFD}on the net inertial force distribution, we plotted the secondary flow magnitude as a function of flowrate (Figure 6). It is observed that the magnitude of the secondary flow is weakly proportional to the flowrate, as the maximum velocity was only doubled when increasing the flowrate from 1 µL/min to 5 µL/min. The N

_{2}-induced transversal drag force can be approximated by the Stokes’ drag formulation [62], where the force is linearly proportional to the velocity of the fluid. Therefore, F

_{SFD}is weakly proportional to the increase in flowrate. At Q = 5 µL/min, inertial force was negative for the 7.32 µm particle in the whole midline. The magnitude of the negative F

_{S}, is proportional to the second power of the flowrate [63]. As a result, F

_{S}overcomes F

_{SFD}at high flowrate and generates negative net inertial force. Conversely, a different behavior was obtained for the 4.16 µm particle. At Q = 5 µL/min, inertial force took positive values near the wall and negative values near the center (Figure 5d). This behavior is identical to the inertial force distribution on a particle in Newtonian fluid.

_{P}< 8 µm. Under this strong wall-directed force, the particle is either trapped in the proximity of the wall, or hits the wall and bounces back toward the center. Therefore, we term it the “attraction-repulsion” region.

_{P}≈ 14 µm (also Y

_{P}≈ 36, due to symmetry). Therefore, simulations predict the experimental focusing positions of the 7.32 µm particle almost perfectly at both 1 µL/min and 5 µL/min. The attraction-repulsion region near the wall exists at the high flowrate as well.

_{p}≈ 17 µm (also Y

_{P}≈ 33 µm, due to symmetry). Experimental line scan data for the 4.16 µm particle at 5 µL/min peaked at Y ≈ 29 µm (due to the non-uniform input, we only captured one stream near the bottom sidewall) (Figure S2). Therefore, simulation results perfectly predict the focusing position of the 4.16 µm particles at 1 µL/min. However, the predicted value of the equilibrium position at 5 µL/min is ≈ 4 µm different from that observed in the experiments. This could potentially be due to the channel not being long enough for the 4.16 µm particles to complete their migration towards the stable focusing positions.

#### 3.2.2. Force Analysis along Z-Midline

_{P}< 9, but changed sign and became positive near the center (Figure 7c). For the 4.16 µm particle, inertial force behaved similarly near the wall as in the Y-midline and exerted a positive, center-directed force to the particles. The force curves then started to change sign and took negative values in the intermediate region between the channel walls and the channel center. However, similar to the 7.32 µm particles, they experienced a positive force in the proximity of the channel centerline (Figure 7d). This positive trend near the center for both particle sizes was only observed in the Z-midline. However, it cannot be attributed to the difference in the blockage ratio of particles in the two midlines. The Z-midline blockage ratio of the 4.16 µm particle, $\frac{4.16}{25}$ = 0.16, is almost equal to the Y-midline blockage ratio of the 7.32 µm particle, $\frac{7.32}{50}$ = 0.15. However, the positive trend was not observed for the 7.32 µm particle in the Y-midline.

_{P}< 9.5 for both flowrates, where due to any minimal disturbance, the particles were either pushed towards a stable equilibrium position in the vertical centerline of the channel (Z

_{P}= 12.5), or to an attraction-repulsion region near the channel walls. However, for the 4.16 µm particle, the attraction-repulsion region was only observed at Q = 1 µL/min. At the high flowrate (Q = 5 µL/min), only one, stable focusing position was obtained at the vertical centerline of the channel. The center stable focusing positions obtained in the simulations are in agreement with the experimental observations. The line scan data showed that the fluorescent intensity peaks at the channel vertical center (Z = 12.5 µm) for both particle sizes at 1 µL/min and 5 µL/min (Figure S2).

#### 3.3. General Focusing Mechanisms

_{S}), N

_{2}-induced secondary transversal drag (F

_{SFD}), and the elastic (F

_{E}) forces balance particles laterally, leading to their final, stable, equilibrium positions. These conclusions are based on the Y-midline force curves. Wall-induced force (F

_{W}) has little to no effect on the focusing mechanisms because the focusing occurs considerably far from the channel walls in all cases.

_{S}along the whole midline, which was not strong enough to overcome the positive F

_{SFD}, and therefore, net inertial lift force remained positive as seen in the Y-midline inertial force curves (Figure 5c,d)

**.**This positive net inertial lift, alongside with the positive F

_{E}pushed the particles towards the center of the channel (Figure 8a). At Q = 5 µL/min, particles near the center experienced negative elastic, and net inertial lift forces and hence moved away from the center. This wall-directed migration persisted until the strong positive F

_{E}near the walls counteracted negative net inertial force and pushed back the particles towards the equilibrium line where negative F

_{S}counterbalanced positive F

_{E}+ F

_{SFD}(Figure 8b). This mechanism can also explain the reason behind the 4.16 µm particles focusing closer to the channel center compared to the 7.32 µm particles at Q = 5 µL/min. Based on the above conclusion, decreasing particle size decreases the net positive lift force (F

_{SFD}+ F

_{E}) more significantly than the negative component of the lift (F

_{S}). This occurs because both F

_{E}and F

_{S}scale with ∼d

^{3}[41,61], while F

_{SFD}which can be simply derived by the stokes drag formulation scales with ∼d. As a result, the 4.16 µm particle is focused closer to the channel center.

_{S}was weakened due to the more flattened velocity profile at higher alpha, and (2) F

_{SFD}got stronger with increasing $\alpha $. The relationship between alpha and secondary flow magnitude, which is linearly proportional to F

_{SFD}, is illustrated in Figure 10 and is in agreement with observations by Villone et al. [53]. Conversely, when $\alpha $ = 0.1, stronger F

_{S}overcomes opposing F

_{SFD}(now weakened) and pushes the particle all the way to the channel walls, leading to no stable focusing position along the Y-midline. If D

_{C}is the distance of the particle measured from the channel center, the error percentage for the predicted focusing position can be defined as: E = $\frac{|\text{}({\mathrm{D}}_{\mathrm{C}})simulation-({\mathrm{D}}_{\mathrm{C}})experiment|}{({\mathrm{D}}_{\mathrm{C}})experiment}\times 100$. When $\alpha $ = 0.2, average E for the two focused bands was ≈ 4.15%. However, in the case of overestimation of the mobility factor, $\alpha $ = 0.4, average E ≈ 21.3%. On a side note, it is worth mentioning that the behavior of particles in a rectangular channel flow of viscoelastic fluid can be different from their behavior in a square channel. In a square channel, eight vortices are formed in the cross-section of the channel [53], pushing the particles towards the channel walls in all orthogonal directions, while direction of the secondary flow is towards the channel center on the Y-midline of a rectangular channel. Therefore, F

_{SFD}and F

_{S}might compete in some cases, while they always have synergetic effects in square channels.

#### 3.4. Shear-Thinning Effects near the Channel Walls

_{W}) is a center-directed force generated due to the build-up of pressure in the narrow region between the particle and the channel walls if the particle is located close enough to the wall [26]. Conversely, shear-gradient lift (F

_{S}), is a wall-directed force due to the parabolic nature of the velocity profile inside the channel. This parabolic velocity distribution will cause the particle to experience different relative velocities on the two sides that are facing the wall (V

_{W}) and the channel center (V

_{C}), and this will generate a force towards the channel walls in order to minimize this velocity imbalance [26].

_{W}is significantly larger than V

_{S}, creating an amplified shear-gradient force, which is strong enough to overcome the wall-induced force and in turn, net inertial force pushes the particle towards the channel walls. The 4.16 µm particle however, will not experience this enhanced shear-gradient due to its smaller size, and hence, wall-repulsive force is still the dominant of the two. As a result, net inertial force exerted on the smaller-sized particle will be, at least qualitatively, similar to that in a non-shear-thinning fluid. This behavior was also previously observed by D’Avino et al. [50] in a shear-thinning flow of particles in a cylindrical micropipe. They observed that by increasing the blockage ratio, particles tend to focus near tube walls rather than the centerline. Conversely, inertial force behavior observed in an Oldryod-B fluid is identical to that of the Newtonian fluid, as in the study by Raoufi et al. [51], where only center-directed inertial force is observed near the channel walls. This can further support the idea that the strong wall-directed inertial force near the walls in a Giesekus fluid is indeed due to the impact of the shear-thinning on the velocity profile.

## 4. Conclusions

_{E}) and N

_{2}-induced secondary flow transversal drag (F

_{SFD}) push both particle sizes towards the channel center at low flowrate (Wi = 3.6). At high flowrate (Wi = 18), two focusing positions are observed on the Y-midline of the channel due to the balance of F

_{E}, F

_{SFD}, and shear-gradient lift force (F

_{S}). We reported, for the first time, that elastic force can promote wall-directed migration of particles in the central regions at Wi

**~**O(10). Due to the different scaling of the forces responsible for the focusing with the particle diameter, smaller particles equilibrate closer to the channel center. Additionally, we demonstrated that the correct prediction of the focusing positions is directly associated with the correct estimation of the mobility factor or the $\alpha $ constant in the Giesekus equation. Over or underestimation of the shear-thinning extent will lead to substantial error in the predicted focusing positions that can negatively affect the efficiency of particle separation devices. Lastly, we showed that the unique behavior of the inertial force near the channel walls can be explained by exploring the impact of the shear-thinning property of the fluid on the velocity profile. The ‘warped’ velocity gradient present in shear-thinning fluid amplifies the shear-gradient force. This amplified force will overcome the wall-induced lift, causing the net inertial force to be towards the walls in such fluids.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematics of the viscoelastic phenomenon in a rectangular channel flow of Giesekus fluid. (

**a**) First normal stress difference (N

_{1}) distribution in the cross-section; Elastic force drives particles towards the lowest N

_{1}regions. (

**b**) Elasticity-induced secondary flow due to the presence of the non-zero second normal stress difference (N

_{2}). (

**c**) Comparison between the velocity profile in non-shear-thinning fluids (Newtonian, Oldroyd-B, etc.) and the warped velocity profile in shear-thinning fluids (Giesekus, etc.).

**Figure 2.**Model schematics and mesh configuration. (

**a**) Simulation box consists of a rectangular duct with the cross-sectional dimensions W = 50 µm, H = 25 µm and length of L

_{C}= 150 µm; channel walls move with the speed of −U

_{p}while the particle has angular velocity vector of Ω. (

**b**) Domain mesh (inlet and interior mesh are hidden for easier visualization) (

**c**) cross-section mesh. Y

_{P}and Z

_{P}are the distance of the particle from the side wall and the bottom wall, respectively. (

**d**) Finer mesh on the particle surface and boundary layer mesh surrounding the particle.

**Figure 3.**Shear-thinning behavior of the 1000 ppm PEO solution. Comparison of the normalized viscosity as a function of Wi number between experimental measurements by Rodd et al. [58] and the Giesekus model with the mobility factor $\mathsf{\alpha}$ = 0.2.

**Figure 4.**Experimental observations of focusing and separation of the 4.16 µm and 7.32 µm diameter particles in viscoelastic flow. (

**a**) Fluorescent streak images of top and side views at 1 µL/min (Wi = 3.6) and 5 µL/min (Wi = 18) in 50 µm × 25 µm microchannel. (

**b**) Top view bright-field image confirming distinct focusing positions of the two particle sizes at Q = 5 µL/min. (

**c**) Expansion region at the channel outlet enabling separation of the focused streams. (Scale bar in all images: 25 µm).

**Figure 5.**Horizontal force (F

_{Y}) distribution along the Y-midline of the channel. Elastic force profile for the (

**a**) 7.32 µm and (

**b**) 4.16 µm diameter particles at Q = 1 µL/min (Wi = 3.6) and Q = 5 µL/min (Wi = 18). Inertial force profiles for the (

**c**) 7.32 µm and (

**d**) 4.16 µm diameter particles at the two flowrates. Total force profiles for the (

**e**) 7.32 µm and (

**f**) 4.16 µm diameter particles at the two flowrates.

**Figure 6.**Effects of flowrate on the magnitude of N

_{2}-induced secondary flow orthogonal to the main flow direction along the channel horizontal centerline. Magnitude of the secondary flow is almost doubled when increasing the flowrate from 1 µL/min (Wi = 3.6) to 5 µL/min (Wi = 18) at constant α = 0.2.

**Figure 7.**Vertical force (F

_{Z}) distribution along the Z-midline of the channel. Elastic force profile for the (

**a**) 7.32 µm and (

**b**) 4.16 µm diameter particles at Q = 1 µL/min (Wi = 3.6) and Q = 5 µL/min (Wi = 18). Inertial force profiles for the (

**c**) 7.32 µm and (

**d**) 4.16 µm diameter particles at the two flowrates. Total force profiles for the (

**e**) 7.32 µm and (

**f**) 4.16 µm diameter particles at the two flowrates.

**Figure 8.**General focusing mechanisms of particles in the Y-midline of a rectangular channel in viscoelastic fluid flow. Final stable equilibrium position of particles is determined by the balance of the shear-gradient (F

_{S}), N

_{2}-induced secondary flow transversal drag (F

_{SFD}), and the elastic (F

_{E}) forces at (

**a**) Q = 1 µL/min (Wi = 3.6), and (

**b**) Q = 5 µL/min (Wi = 18). Dashed circles indicate particles migrating towards the equilibrium position; solid circles are particles at final, stable equilibrium position.

**Figure 9.**Effects of fluid shear-thinning on the prediction of focusing position. (

**a**) Experimental results for the streak intensity of the 7.32 µm diameter particles in 1000 ppm PEO at Q = 5 µL/min (Wi = 18). (

**b**) Simulation results for three different alpha constants. The $\alpha $ = 0.2 predicts the focusing positions precisely, while overestimating shear-thinning extent of the fluid ($\alpha $ = 0.4) predicts focusing positions closer to the channel center, and underestimating alpha ($\alpha $ = 0.1) predicts no focusing in the central regions.

**Figure 10.**Effect of the mobility factor (α) on the magnitude of the N

_{2}-induced secondary flow. Higher α values increase the magnitude of the secondary flow orthogonal to the main flow direction along the channel horizontal centerline.

**Figure 11.**Schematics of the effect of shear-thinning behavior of the fluid on the inertial force near the channel walls. Both (

**a**) 7.32 µm and (

**b**) 4.16 µm diameter particles experience positive net inertial force (F

_{W}> F

_{S}) in non-shear-thinning fluids. (

**c**) Shear-thinning promotes the shear-gradient lift force (Fs) by imposing stronger relative velocity imbalance on the particle surface for the 7.32 µm particles, causing net inertial force to be negative near the channel walls. (

**d**) The 4.16 µm diameter particles are not large enough to capture this relative velocity imbalance.

**Table 1.**Selected existing studies on the focusing and separation of particles in viscoelastic fluid.

Reference | Approach | Fluid Type Used in Exp. | Flow Regime | Geometry | Considered in Simulations | ||
---|---|---|---|---|---|---|---|

Exp. | Sim. | Shear-Thinning | Secondary Flow | ||||

Li et al., (2016) [48] | ✓ | ✕ | PVP, PAA, PEO | 0.1 < Re < 2 0 < Wi < 30 | Rectangular channel | - | - |

Li and Xuan (2019) [49] | ✓ | ✕ | XG | 0.01 < Re < 30 | Rectangular channel | - | - |

Feng et al., (2019) [45] | ✓ | ✕ | PEO | 10^{−4} < Re < 10^{2}10 ^{−3} < Wi < 10^{3} | Rectangular spiral channel | - | - |

Di’Avino et al., (2012) [50] | ✓ | ✓ | PVP, PEO | Inertia-less De ^{i} < 2 | Circular tube | Yes | Yes |

Raoufi et al., (2019) [51] | ✓ | ✓ | PEO | 0 < Re < 20 ^{ii}1 < Wi < 220 ^{ii} | Square, rectangular, trapezoidal and complex channel | No | No |

Huang & Joseph (2000) [52] | ✕ | ✓ | - | 0 < Re < 56 De < 2.6 | 2-D parallel walls | Yes | No |

Villone et al., (2013) [53] | ✕ | ✓ | - | Inertia-less De < 6 | Square channel | Yes | Yes |

Raffiee et al., (2019) [54] | ✕ | ✓ | - | Re < 30 Wi < 3 | Square channel | Yes | Yes |

Wang et al., (2018) [55] | ✕ | ✓ | - | Re = 1 & 50 Wi < 2.5 | Rectangular channel | Yes | Yes |

Present study | ✓ | ✓ | PEO | Re = 0.2 & 1 Wi = 3.6 & 18 | Rectangular channel | Yes | Yes |

^{i}De is the Deborah number, calculated as De = $\frac{\lambda}{{t}_{p}}$ where ${t}_{p}$ is the characteristic time scale.

^{ii}Re and Wi ranges are calculated based on the given information in the paper.

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**MDPI and ACS Style**

Naderi, M.M.; Barilla, L.; Zhou, J.; Papautsky, I.; Peng, Z.
Elasto-Inertial Focusing Mechanisms of Particles in Shear-Thinning Viscoelastic Fluid in Rectangular Microchannels. *Micromachines* **2022**, *13*, 2131.
https://doi.org/10.3390/mi13122131

**AMA Style**

Naderi MM, Barilla L, Zhou J, Papautsky I, Peng Z.
Elasto-Inertial Focusing Mechanisms of Particles in Shear-Thinning Viscoelastic Fluid in Rectangular Microchannels. *Micromachines*. 2022; 13(12):2131.
https://doi.org/10.3390/mi13122131

**Chicago/Turabian Style**

Naderi, Mohammad Moein, Ludovica Barilla, Jian Zhou, Ian Papautsky, and Zhangli Peng.
2022. "Elasto-Inertial Focusing Mechanisms of Particles in Shear-Thinning Viscoelastic Fluid in Rectangular Microchannels" *Micromachines* 13, no. 12: 2131.
https://doi.org/10.3390/mi13122131