The determination of flow-induced equilibrium positions in pressure-driven flows in microchannels is of great practical importance in particle manipulation. In the computational analysis presented in this paper, the inertial ordering of neutrally buoyant rigid spheres in shear-thinning fluid flow through a hydrophobic microchannel is investigated. The combined effect of the viscosity index *n* of a power-law fluid and fluid slippage at the wall on the lateral focusing of microspheres is examined in detail. Using the finite element method, the Eulerian flow field between partially slipping parallel walls is simulated, and the Lagrangian movement of particles is continuously tracked. The Navier slip model is used to ensure a finite fluid velocity at the wall, and it is tuned by modifying the slip-length. It is observed that inertial particles concentrate at a standard equilibrium position of 0.6 times the channel half-width *H*, irrespective of fluid slip due to the symmetry of the flow field. However, this equilibrium position shifts closer to the walls as the viscosity index increases; for instance, when *n* = 0.5, particles stabilize at 0.75*H*. As a consequence of asymmetry in hydrodynamic behavior due to different fluid slippages at the upper and lower walls, the particle migration path is altered. In a channel with a no-slip upper wall and a partially slipping lower wall (*β*/*H* = 0.4), particles settle closer to the lower wall at 0.8*H*. Most importantly, the lateral movement of a particle released at a given vertical position can be altered by tailoring the wall hydrophobicity and viscosity index, thus enabling multiple equilibrium locations to be achieved.

## I. INTRODUCTION

Efficient manipulation and separation of rigid and deformable particles by active and passive means are critical in many biomedical applications involving microfluidic technologies.^{1–3} However, passive methods are preferred over active ones, owing to their lower energy requirements and less complex designs.^{3} Typically, pinched flow fractionation, inertial migration, Dean flow fractionation, microvortex manipulation, trilobite pillars, deterministic lateral displacement, and hydrodynamic filtration are the major passive particle separation strategies.^{1,3} Among these, inertial migration plays a pivotal role in many lab-on-chip applications, owing to its simple construction, versatility, and efficacy.^{4–6} Lateral migration of rigid particles results in their focusing at specific locations in a microchannel in flows with moderate Reynolds number (1 < Re < 100). Generally, inertial focusing is cross-stream migration of randomly distributed particles toward a specific equilibrium position in a pressure-driven flow.^{4,7,8} Indeed, inertial migration, based on size-dependent hydrodynamic effects in a microchannel, has great potential as a high-volume-throughput sample processing method.^{5,6,8,9} Moreover, lateral focusing is a widely used and reliable technique to manipulate particles in a fluid flow by passive means and therefore finds a wide range of applications in microbiology, biochemistry, and biotechnology.^{1,5,8,9} Counting and separation of red blood cells and leukocytes in a blood sample can be performed by using inertial migration in a straight channel.^{10,11} The same technique can be used to differentiate red blood corpuscles from plasma.^{12} Furthermore, Mach and Di Carlo^{13} created a passive microfluidic device that uses inertial migration to extract dangerous pathogens from a diluted blood sample.

Segré and Silberberg^{7} were the first to identify and report the phenomenon of inertial migration in their observations of the ordering of particles dispersed randomly at the inlet of a tube. They noticed that these randomly oriented rigid particles migrated to a distance of 0.6 times the tube radius while traversing a certain space along the pipe in what is known as the “tubular-pinch effect.” The shear-induced lift and wall-induced lift cancel each other at this equilibrium point, and the lateral shift of particles stops at this location. Later, Asmolov^{14} studied the inertial lift force on small rigid spheres in Poiseuille flow at high Reynolds numbers and discovered that wall-induced lift is significant near the wall, whereas the lift force due to the velocity profile dominates farther away from the wall. The net lift force acting on the particle in plane Poiseuille flow is given by $FL=(\rho Um2D4/Dh2)fc(Rc,xc)$, where *f*_{c} is the coefficient of lift. Matas *et al.*^{15} carried out an experimental study to gain an understanding of the phenomenon of inertial migration in a high-Reynolds-number range of 67 ≤ Re ≤ 1700 and found that the annular equilibrium position moved nearer to the wall as Re increased. In another important study, Bhagat *et al.*^{16} demonstrated high preferential particle focusing along the vertical sidewalls of a rectangular microchannel at low Re, thereby separating particles of size 590 nm from a mixture of particles of sizes 1.9 *μ*m and 590 nm. Krishnaram and Kumar Ranjith^{17} studied migration in a microchannel at Re = 60 and a blockage ratio of 0.25 and demonstrated that particles from the center and near-wall positions in the entrance region required 23.8% and 7.8% more time to reach equilibrium compared with fully developed flow.

Computational studies have shed more light on the physics of the lateral shifting of particles in pressure-driven flows. The intricate hydrodynamic interactions responsible for the lateral ordering of particles in a channel flow have also been revealed by numerical simulations.^{8,18} Using direct numerical modeling, Zeng *et al.*^{19} investigated the wall-induced lift on rigid spheres moving parallel to a wall for Re < 100. They noted that the lift coefficient decreases as the Reynolds number and distance from the wall increase. In addition, particle properties such as size, shape, and deformability govern the equilibrium positions. Hood *et al.*^{20} employed a 3D asymptotic expansion technique in a numerically investigation of how particle size and Reynolds number affect the lateral force generation and migration of particles. Later, Wang *et al.*^{21} used data fitting to establish a formula for the particle focusing position based on particle size and Reynolds number. Liu and Wu,^{22} using the lattice Boltzmann method (LBM), investigated the influence of particle concentration on the inertial separation of neutrally buoyant particles in two-dimensional Poiseuille flow with Re = 20, 40, and 100 and established a classification of particle focusing regimes. Moreover, it has been observed that whereas particles in a circular tube migrate to a single equilibrium point, those in a rectangular tube migrate to multiple locations.^{23,24} In addition, particle train formation in a square microchannel has been observed after particles have reached an equilibrium state, and the interparticle distance depends on the particle Reynolds number.^{25} However, the migration pattern in a trapezoidal microchannel is significantly different from that in a rectangular channel, owing to asymmetry in the velocity profile.^{26} Unlike neutrally buoyant particles, which migrate to the channel’s geometrical centerline, the lateral focusing of non-neutrally buoyant particles depends strongly on the fluid–particle density difference.^{27} Another parameter that influences transverse focusing is the deformability of the particles.^{28} The shape of a deformable particle can change owing to surface tension forces or material elasticity; common examples include droplets and bubbles. In this case, the continuous phase closely interacts with the dispersed phase, with boundary layer fluctuations, until a stable condition is reached. Furthermore, owing to multiparticle interactions, the equilibrium position is also altered, and a staggered particle train is formed.^{29} Recent control force studies to predict the fate of particles undergoing inertial migration^{30,31} have highlighted the practical relevance of understanding the physics involved.

It should be noted that in much of the work described above, particles have been assumed to translate in a Newtonian fluid, although many fluids of biological significance exhibit non-Newtonian behavior in microconstrictions.^{32–36} In particular, biological and chemical fluids, such as blood,^{37} saliva,^{38} and polyacrylamide,^{39} exhibit shear-thinning behavior. Importantly, the inertial migration phenomenon has been widely studied experimentally in the context of shear-thinning fluids.^{33} Inertial ordering of single and multiple particles in pressure-driven flow of power-law fluids at Re = 20, 40, and 60 was numerically investigated by Hu *et al.*,^{40} who observed that the stable equilibrium position shifted toward the centerline for a high power-law index. In addition, the inertial focusing length has also been noted to be directly proportional to the power-law index. Importantly, the viscosity of the fluid has a dominant influence on the inertial focusing length.^{41} Later, Chrit *et al.*^{42} used three-dimensional LBM simulation to study the cross-stream migration of rigid and elastic neutrally buoyant spherical particles in the flow of power-law fluids at Re = 10 and 100. They reported that the equilibrium position shifts toward the centerline for a shear-thickening fluid and toward the channel wall for a shear-thinning fluid. Recently, Hu *et al.*^{43} studied the migration of spheroids in the flow of a power-law fluid through a square channel and found that shear-thinning fluids with strong inertial effects promote faster migration. Trofa *et al.*^{44} numerically studied the influence of symmetrical wall slip on migration in viscoelastic flow in a hydrophobic channel, focusing on the relative velocity between the particle and the fluid. In general, the particle migration dynamics are complex for viscoelastic fluid and particularly for shear-thinning fluids.^{35,36}

Meanwhile, hydrophobic microchannels have attracted much recent attention, owing to the low hydrodynamic resistance that they offer as a consequence of the presence of entrapped air in their microtextured boundaries.^{45–48} Another technique for drag reduction is the use of liquid/lubricant-infused surfaces.^{49} In comparison with a conventional hydrophilic channel, a hydrophobic channel enables larger flow rates for the same pressure drop.^{47,50} Indeed, fluid slip at the wall provides another mechanism for manipulating hydrodynamic behavior and the associated effects, including mixing and separation.^{51,52} Further, by altering the wettability of different boundaries, it is easy to create an asymmetric the flow field, which may be utilized for various applications.^{47,48,52} The significance of microchannels with varying wall slip conditions has been described in detail in Refs. 53–56. Often, in real-world scenarios, manufactured microchannels possess asymmetrical slip on their walls as a consequence of difficulties in maintaining hydrophobicity on both upper and lower walls.^{57,58} Manipulation of microparticles in superhydrophobic microchannels has also been investigated, with the aim of understanding the mechanisms of lift creation and subsequent cross-stream particle movement.^{59,60} It has been observed that particle ordering and equilibrium positions are also influenced significantly by fluid slippage.

Although there have been a number of separate investigations of the effects of hydrophobic microchannel walls and non-Newtonian fluids, respectively, on inertial migration, their combined effects remain largely unexplored and require further examination. To the best of our knowledge, particle migration in shear-thinning flows with asymmetric slip wall boundary conditions has yet to be explored, despite its potential practical importance. In the present study, a two-dimensional numerical simulation is carried out to investigate the inertial migration of neutrally buoyant spherical particles in a shear-thinning flow within a hydrophobic microchannel. Given that the separate effects of particle size, flow rate, and geometrical parameters have already been thoroughly examined, the focus of this study is on the effect of the asymmetric velocity profile generated by fluid slippage at the hydrophobic wall in a shear-thinning fluid on inertial migration. It is pertinent to note that inertial migration has been extensively studied at moderate Reynolds numbers, such as Re = 20, 40, 60, and 100.^{22,40,42} Here, we consider primarily a Reynolds number of 60, given that this value is particularly relevant to observed natural phenomena. Thus, the scope and main objectives of the present study can be summarized as follows:

Understanding the impact of nonidentical wall slip on inertial migration in a Newtonian flow.

Investigating the effect of identical slip conditions in shear-thinning laminar flow and the influence of the viscosity index on inertial motion of particles in a hydrophobic channel.

Evaluating the influence of a nonidentical slip distribution on particle migration in shear-thinning laminar flow.

## II. NUMERICAL METHODOLOGY

In this section, the details of the computational procedure for the modeling of inertial migration of neutrally buoyant spherical particles in shear-thinning flow through a hydrophobic microchannel are presented. A finite element procedure is adopted to solve the governing equations using COMSOL Multiphysics software. The numerical simulation is executed in two steps. First, a steady-state flow solution is obtained by solving the governing balance equations in an Eulerian manner to ensure a consistent baseline flow field. Second, the spherical particles are injected into the flow, and their trajectories are calculated using a force balance equation on the particle motion in a Lagrangian framework. This approach ensures an accurate representation of both the fluid dynamics and the complex behavior of the particles within the system. Details of the computational procedure, including the problem description, governing equations, boundary conditions, and validation are elaborated here.

### A. Problem description

A schematic representation of the problem under consideration is provided in Fig. 1. A neutrally buoyant rigid spherical particle is depicted in transit through a pressure-driven fluid flow, confined between two infinite hydrophobic parallel plates. These plates are characterized by differing degrees of hydrophobicity. A pressure gradient is exerted in the positive *x* direction, causing the fluid to move in a deterministic manner. To facilitate analysis, the nondimensionalized *x* and *y* directions are defined as *χ* = *x*/2*H* and *ζ* = *y*/2*H*, where *H* is the channel half-height, serving as a critical scaling factor. The particle selected for this investigation is a neutrally buoyant sphere, with specific physical properties including a density *ρ*_{p} = 1178 kg/m^{3} and a confinement ratio *D*/2*H* = 0.25, where *D* is the diameter of the particle.

### B. Flow-field governing equations

**u**is the velocity vector field,

*ρ*is the fluid density,

*p*is the pressure field,

**K**is the viscous stress tensor, and

**I**is the identity tensor. The viscous stress tensor for the power-law fluid model is defined as Refs. 42 and 61 $K=\mu app(\gamma \u0307)$, where

*μ*

_{app}is the apparent viscosity, given by $\mu app=m|\gamma \u0307|n\u22121$, and $\gamma \u0307$ is the rate of shear deformation (or shear rate), given by $\gamma \u0307=2(\epsilon :\epsilon )$, with the strain rate tensor $\epsilon =12(\u2207u+\u2207uT)$. In the above formulations, the constants

*m*and

*n*are the constitutive parameters known as the flow consistency coefficient and power-law index. The power-law index essentially categorizes the fluid behavior, with

*n*> 1 corresponding to a shear-thickening fluid,

*n*< 1 to a shear-thinning fluid, and

*n*= 1 to a conventional Newtonian fluid. In this study, the viscosity index

*n*of the fluid is varied from 0.5 to 1, with the fluid consistency index

*m*and fluid density

*ρ*kept constant at 0.214 Pa s and 1178 kg/m

^{3}, respectively. Although alternative models such as the Carreau–Yasuda, cross power law, Herschel–Bulkley, and Oldroyd-B models are available, we adopt the power-law model because of its ability to accurately mimic non-Newtonian fluid behavior with the minimal number of tunable parameters, namely, the viscosity and consistency indices. This model also provides a satisfactory representation of the inertial migration trends within the shear-thinning regime in which we are interested.

### C. Particle tracking model

*m*

_{p}and

**v**

_{p}are the mass and velocity of the rigid spherical particle, and

**F**

_{t}is the cumulative force experienced by the particle. This force is given by

**F**

_{D}is the drag force and

**F**

_{L}is the lift force. In the case considered here, the density of the particle is equal to that of the fluid, and therefore the buoyancy force is neglected. Combining Eqs. (3) and (4), we obtain

*k*+ 1 and

*k*indicate the values corresponding to times

*t*+ Δ

*t*and

*t*, respectively.

#### 1. Drag and lift forces

*r*

_{p},

*u*, and

*u*

_{p}are the particle radius, fluid velocity, and particle velocity, respectively. Equation (8) is accurate for low Re, but a correction factor is needed for high Re.

^{62}However, the relative magnitude of

*F*

_{D}with respect to

*F*

_{L}is very low, and so Stokes’s equation is sufficient to predict the lateral migration of rigid spheres in an effective manner.

^{63,64}

^{65,66}and can be expressed mathematically as

**n**is the unit wall-normal vector,

*s*is the ratio of the distance between the particle and the channel wall, and

*G*

_{1}and

*G*

_{2}are dimensionless functions of the normalized wall distance.

^{64,67}

### D. Boundary conditions

To solve the governing equations, an accurate mathematical representation of the computational domain is essential and is illustrated in Fig. 1. At the outlet, a pressure boundary condition is implemented, with gauge pressure set to zero. Note that the Navier slip condition is utilized at the channel walls, allowing the adjacent fluid to attain a finite tangential velocity, characterized by the slip-length *β*, representing the intrinsic property of the hydrophobic wall. Mathematically, this is defined as *u*_{wall} = *β*[*∂u*/*∂y*]_{wall}, where *u*_{wall} is the tangential velocity at the wall. The upper and lower channel walls are assigned specific slip-lengths *β*_{1} and *β*_{2}, central to characterizing the flow dynamics as depicted in Fig. 1. The contact angle determines the formation of a liquid–gas interface at the surface roughness elements of a hydrophobic microchannel and the consequent fluid slippage. However, in the theoretical investigations here, the bounding walls are assumed to be mathematically smooth, and the velocity slip boundary condition is employed.^{68} The microscopic effect of surface contact angle is macroscopically manifested as the effective slip-length of the channel.^{49,69,70}

### E. Numerical technique

This study employs the finite element method (FEM) to handle the governing equations of non-Newtonian fluid flow, modeling velocity with second-order elements and pressure with linear ones. Particle tracing is performed separately, using a unidirectional coupling, with the particle position defined by the local pressure and velocity distributions, combined with the Newtonian formulation. The partial differential equations are solved using the parallel space direct solver (PARDISO) and the generalized minimum residual solver (GMRES) for time-dependent particle tracing with residual and relative tolerances of 10^{−3} and 10^{−6}, respectively. A structured rectangular mesh populates the domain, and the centerline velocity converges with limited mesh elements. A grid-independence study conducted with mesh elements ranging from 6274 to 113 613 revealed marginal variations in the coefficient of lift, confirming the model’s robustness and minimal error; see Table I. Subsequent simulations are executed with 113 613 mesh elements, since this offers an optimal balance between precision and computational efficiency, with no significant alteration in value discernible with finer meshes. Moreover, the meshing strategy ensures that the domain mesh size remains smaller than the particle size.

Grid . | No. of mesh elements . | Lift coefficient . |
---|---|---|

G_{1} | 6 274 | −0.248 |

G_{2} | 11 118 | −0.678 |

G_{3} | 25 396 | −0.0346 |

G_{4} | 61 302 | −0.0054 |

G_{5} | 113 613 | −0.0054 |

Grid . | No. of mesh elements . | Lift coefficient . |
---|---|---|

G_{1} | 6 274 | −0.248 |

G_{2} | 11 118 | −0.678 |

G_{3} | 25 396 | −0.0346 |

G_{4} | 61 302 | −0.0054 |

G_{5} | 113 613 | −0.0054 |

A time step of 5 × 10^{−3} s is selected for the particle simulation to confine the maximum Courant number to ∼0.3, a strategic constraint to guarantee the accuracy of the computation in tracking the particle. Indeed, the determination of this time-independent interval is connected with the flow Reynolds number as well.^{66,71}

### F. Model validation

*n*> 1), where the viscosity increases with increasing rate of shear, Newtonian (

*n*= 1), where the viscosity is constant, and shear-thinning (

*n*< 1), where the viscosity decreases with increasing shear rate. Here, no-slip boundary conditions are adopted, and pressure is applied in the streamwise direction. A comparative analysis is conducted, focusing on the computed

*x*-velocity profile along the cross-section for each case. The following analytical solution

^{40}serves as the reference:

*U*

_{m}is the mean fluid velocity. The

*x*component of the velocity is nondimensionalized as

*U*=

*u*/

*U*

_{m}. A comparison of simulated and analytical results demonstrates that they are in good agreement, as can be seen in Fig. 2(a). Thus, the current strategy is capable of modeling the hydrodynamics of both Newtonian and non-Newtonian fluids. Indeed, the Reynolds number Re emerges as a pivotal nondimensional parameter for surveying the properties of the non-Newtonian flow, defined

^{61}as $Re=\rho Um2\u2212n(2H)n/m$.

Next, a simulation of particle lateral movement in Newtonian and non-Newtonian pressure-driven flows is performed. First, the numerical predictions of migration distance obtained are compared against benchmarks provided by Tachibana’s experimental data^{72} and Ho and Leal’s analytical results^{65} for a Newtonian fluid. The particle trajectory in a Poiseuille flow with Re = 32 and a blockage ratio *D*/2*H* = 0.159 is simulated numerically. A particle introduced at *ζ* = 0.1 in the inlet is found to migrate to a well-established equilibrium position *ζ*_{e} = 0.2. The computed trajectory is in excellent agreement with the previously reported experimental and analytical results, as depicted in Fig. 2(b). Furthermore, the model’s precision and accuracy are tested by simulating lateral particle movement in a shear-thinning flow. The equilibrium positions of particles exhibiting blockage ratios of 0.05, 0.06, 0.08, and 0.1 at a Reynolds number Re = 100 are estimated from simulations and compared against the findings of Chrit *et al*.^{42} This comparison confirms that the present numerical scheme captures the physics of the fluid flow in an excellent manner.

## III. INERTIAL MIGRATION IN NEWTONIAN FLUID

Inertial migration in a Newtonian fluid is simulated for three different scenarios: (i) no slip on either wall; (ii) identical slip on both walls; (iii) different slips on the two walls. Figure 3 shows both the fluid velocity profiles and particle trajectories for these three cases. From the normalized velocity profiles, it is observed that the hydrodynamic behavior is symmetrical with respect to the geometrical center when both walls have identical slip-lengths [Figs. 3(a) and 3(b)]. However, the velocity profile is asymmetric when the two walls have different slip-lengths [Fig. 3(c)]. Owing to the fluid slippage at the wall, both the velocity gradient over the width and the shear-induced lift are lowered, which affects the lateral force generation as well.

Rigid particles are then released at different locations in the developed flow, and their time-dependent trajectories are recorded. These neutrally buoyant particles are released at six different locations along the width of the channel. Of these six positions, three are below and three above the centerline, at *ζ*_{1} = 0.05, *ζ*_{2} = 0.3, *ζ*_{3} = 0.47, *ζ*_{4} = 0.53, *ζ*_{5} = 0.7, and *ζ*_{6} = 0.95. The particle trajectories are monitored for all three configurations. When no-slip conditions are applied on both walls $(\beta 1\u2032=\beta 2\u2032=0)$, the velocity profile is symmetric and a particle released in either the top or bottom half migrates to a constant distance from the center at *ζ* = 0.8 and 0.2 (or at 0.6*H*), as shown in Fig. 3(d). Similarly, when both walls are hydrophobic $(\beta 1\u2032=\beta 2\u2032=0.4)$, particles released at the same locations also move laterally to the same equilibrium locations at *ζ* = 0.8 and 0.2, as shown in Fig. 3(e). However, the distance traversed by a particle before reaching its equilibrium position is greater than in the no-slip case, as is evident in Fig. 3. By contrast, when the top and bottom walls have different hydrophobicities ($\beta 1\u2032=0$ and $\beta 2\u2032=0.4$), the velocity profiles are asymmetric with respect to the geometrical center, and the equilibrium locations are also altered owing to changes in shear-induced lift. In the top half, where the wall is no-slip, the particle equilibrium position is shifted toward the centerline, *ζ*_{e} = 0.77, whereas the lower equilibrium position is shifted toward the slipping wall, *ζ*_{e} = 0.17 [Fig. 3(f)]. These results are also in line with previous observations.^{59}

## IV. INERTIAL MIGRATION IN SHEAR-THINNING FLUID

The results of simulations of inertial migration of rigid spherical particles in shear-thinning-fluid flowing through hydrophobic microchannels are discussed in this section. The effects of fluid slippage and non-Newtonian behavior on lateral ordering are examined in comparison with the particle behavior in a Newtonian fluid. Further, the effect of fluid slip on the equilibrium position is examined for bounding walls with identical and different slip-lengths.

### A. Lift force on particle

When a particle is placed in a pressure-driven flow, it experiences both drag and lift forces and thus moves in both streamwise and cross-stream directions along the length of the channel. Here, we discuss the combined effects of fluid slippage and viscosity index on the lift force generated under (i) symmetric no-slip, (ii) symmetric slip (hydrophobic), and (iii) asymmetric slip conditions. It is well known that shear-induced and wall-induced lift forces play a vital role in determining equilibrium positions and trajectories. To understand the physics of inertial migration in a shear-thinning fluid, we estimate the forces acting on a spherical particle.

The nondimensionalized lift force $FL*=FL/\rho Um2D4/H2$ is plotted in Fig. 4(a) for all three configurations in the case of a Newtonian fluid. There exist three locations where this force decreases irrespective of the boundary conditions. Those are in the upper half, in the lower half, and at the shear-free plane. Often, particles migrate to these locations where the wall-induced and shear-induced lift cancel each other. Indeed, these equilibrium locations are significantly influenced by the wall slip. For bounding walls with identical slips, irrespective of their hydrophobicity, these zero-lift positions are the same, owing to the symmetry of the velocity field. However, once the walls have different slip-lengths, the location of zero lift force changes. For instance, for a hydrophobic microchannel with $\beta 1\u2032=0$ and $\beta 2\u2032=0.4$, the zero-lift point closer to the no-slip wall shifts toward the center in the upper half, whereas the corresponding point in the lower half moves toward the partial-slip surface. This is attributed to the alterations in the velocity profiles due to slip.

When the viscosity index is reduced to *n* = 0.5, the velocity profile attains a uniform profile near the center region, and this further flattens when slip is applied. Consequently, the lift force on the particle decreases even near the wall. It should be noted that for a no-slip surface, a shear-thinning fluid has a large velocity gradient near the walls and its magnitude is more than that of a fluid with *n* = 1, as can be seen in Fig. 4. For slippage at both walls, the sharp velocity gradients near the surfaces disappear, and so the lift force is finite but very close to zero [see Fig. 4(b)], and a greater channel length will be required for a stable position to be reached. Next, a detailed analysis of particle migration due to modifications in velocity profiles resulting from fluid slippage and shear-thinning behavior is presented.

### B. Migration in symmetrically hydrophobic microchannel

First, we discuss the effect of non-Newtonian behavior on inertial migration. To understand the effect of the viscosity index on lateral migration, particles are released at the same location in a no-slip channel for fluids with viscosity indices *n* = 0.5, 0.6, 0.7, 0.85, and 1.0 at Re = 60. The same numerical experiment is then repeated for a hydrophobic microchannel with identical slip-lengths $\beta 1\u2032=\beta 2\u2032$ = 0.4. Figure 5 summarizes the results obtained. Here, the equilibrium positions are not affected by wall slippage, since the velocity profile is symmetric for both the no-slip and hydrophobic channels. Moreover, the velocity gradient present in the hydrophobic channel is less than that in the no-slip channel, and thus the particle traverses a greater distance before reaching a stable position, irrespective of the viscosity index, as can be seen in Figs. 5(a) and 5(b). For instance, when *n* = 0.5, a particle migrates to its equilibrium point at *χ* = 50 for the no-slip microchannel, whereas for the hydrophobic channel, *χ* = 1300. This is attributed to the flattening of the velocity profile due to the combined effects of fluid slippage and a low power-law index.

As the viscosity index changes and the fluid becomes more shear-thinning, the zero-lift position moves toward the wall, as shown in Figs. 5(a) and 5(b). Note that as the viscosity index decreases from the Newtonian value of *n* = 1, the velocity profile flattens (see Fig. 2) and the velocity gradient near the wall rises, while that near the center drops. Indeed, the shear-induced lift force, which varies as Ref. 5 $\u223c\gamma \u03071/2$, is affected by the change in viscosity index. Thus, the shear-induced lift is enhanced in fluids with low viscosity indices, and the stable position is shifted toward the wall. A similar observation was made by Chrit *et al.*^{42} in their LBM simulations for the no-slip scenario. Furthermore, the equilibrium locations of spherical particles are estimated separately by releasing particles in the upper and lower halves (*ζ* = 0.55 and *ζ* = 0.025) of a no-slip and a hydrophobic channel, with the results shown in Figs. 5(c) and 5(d). Although the shear-thinning nature of the fluid affects the stable positions of particles, the role of hydrophobicity is insignificant for identical fluid slippage at the walls. The equilibrium positions of channels with identical slippages at the boundaries overlap, as can be seen in Figs. 5(c) and 5(d).

Next, we estimate the rate of particle migration in the transverse direction as given by the nondimensional particle velocity *v*_{p}/*U*_{m}. This lateral particle velocity is estimated in the case of a no-slip channel for different viscosity indices, as shown in Fig. 6. It is observed that the migration velocity approaches zero at the equilibrium position when a particle is released from the center with zero initial velocity, regardless of the value of *n*. It is evident that as the viscosity index decreases, the migration velocity decreases, indicating a lower rate of particle migration. Also, it is evident from Fig. 6 that as the viscosity index decreases, the equilibrium position of the particle is shifted toward the channel walls.

Further, the effect of Reynolds number on inertial migration is investigated first by keeping both walls uniformly hydrophobic, with $\beta 1\u2032=\beta 2\u2032=0.4$. A rigid particle is released at *ζ* = 0.56 in a hydrophobic microchannel having identical slip-lengths, first with a Newtonian fluid (*n* = 1) and then with a non-Newtonian fluid (*n* = 0.5). Subsequently, the flow velocity is changed such that Re = 40, 60, and 80 for both channels, and the particle path is recorded, with the results shown in Figs. 7(a) and 7(b). At higher Reynolds numbers, inertial migration is more rapid, owing to the large velocity gradient resulting from the increased flow rate. Thus, the particle experiences a higher shear-gradient lift force and undergoes migration at a shorter distance. Since the change in Reynolds number has a marginal effect on the equilibrium position, further studies are conducted for a Reynolds number Re = 60 of relevance to microfluidic applications.

Paths traced by a particle released from different initial vertical positions in a non-Newtonian Poiseuille flow with Re = 60 for a channel having identical hydrophobicities are recorded. The trajectories of particles released from two different initial vertical positions are depicted in Figs. 8(a) and 8(b). Both channel walls are assigned identical Navier slip boundary conditions, and the nondimensionalized slip-length is varied from 0 to 0.4. Particles released from *ζ* = 0.56 and 0.895 migrate to an equilibrium position at *ζ*_{e} = 0.874. It is observed that as the hydrophobicity increases, the distance traversed before reaching a stable equilibrium position is also increased [see Fig. 8(a)]. This is attributed to the reduction in velocity gradient due to fluid slippage and the concurrent reduction in shear-induced lift force. It is noteworthy that the particle released closer to the equilibrium location (*ζ* = 0.895), migrates very rapidly within a very short distance, as can be seen in Fig. 8(b).

### C. Migration in asymmetrically hydrophobic microchannel

Next, we consider the effects on inertial migration of particles due to asymmetry in the flow velocity resulting from nonidentical slip-lengths of the upper and lower walls. A fixed no-slip condition is imposed on the upper wall, and the hydrophobicity of the lower wall $\beta 2\u2032$ is varied between 0 and 0.4. The estimated trajectories of spherical particles in a shear-thinning fluid are plotted in Figs. 9(a) and 9(b). Unlike the previous case of identical slip conditions, the different slip-lengths on opposite walls create an asymmetry in the velocity field, and this significantly affects the lateral ordering. Whereas symmetrically hydrophobic walls ensure stable equilibrium positions equidistant from the center, the asymmetrically hydrophobic channel breaks this symmetry of equilibrium locations. It is found that the equilibrium position in the upper half of the channel shifts toward the channel center as $\beta 2\u2032$ increases. For instance, it shifts from 0.874 to 0.85 when $\beta 2\u2032$ increases from 0 to 0.4 [Fig. 9(a)], whereas the equilibrium position in the lower half of the channel shifts from 0.125 to 0.1015 [Fig. 9(b)]. Importantly, in terms of the channel height, this shift is from 0.75*H* to 0.8*H*.

A similar trend is observed under the same conditions for a Newtonian fluid. The equilibrium points *ζ*_{e} for Newtonian and non-Newtonian fluids are plotted by varying the lower wall hydrophobicity $\beta 2\u2032$. For a Newtonian fluid, in the upper half of the channel, the stable location shifts from 0.8 to 0.77, as shown in Fig. 9(c). By contrast, in the lower half, the equilibrium location is moved toward the wall from 0.2 to 0.175, as shown in Fig. 9(d). As the slip-length on the bottom wall increases, the curvature of the velocity profile is reduced in the lower half of the channel, which results in a shift of the zero-lift position toward the slipping wall.

Note that, for a symmetric slip channel, a particle released in either half of the channel migrates only to the equilibrium position in that half. In other words, if both walls have identical hydrophobicities and the particle is released in the upper half, then it will end up at the equilibrium point in the upper half, and, similarly, a particle released in the lower half attains equilibrium in the lower half, as can be seen in Fig. 10. Particles never cross the geometrical centerline in any case. Interestingly, this trend is violated in the asymmetric-slip case, since the geometrical centerline does not then coincide with the shear-free plane, and a particle can move across the centerline into the other half of the channel. To demonstrate this, a spherical particle is released at a vertical distance of *ζ* = 0.44 (lower half) successively under four different conditions. First, the particle is released in a symmetrically hydrophobic channel ($\beta 1\u2032$ = $\beta 2\u2032=0$) containing a Newtonian fluid, and it is found that the particle migrates to an equilibrium position of *ζ*_{e} = 0.2 (or 0.6*H*) in the same half. The same trend is also observed for a shear-thinning fluid as well, and the particle again migrates to the equilibrium position in the same half (*ζ*_{e} = 0.125), as can be seen in Fig. 10(a). By contrast, for an asymmetric slip channel ($\beta 1\u2032=0$, $\beta 2\u2032=0.4$), even when the particle is released in the lower half, it is found to migrate to the equilibrium position in the upper half, as can be seen in Fig. 10, irrespective of the viscosity index. The particle in the Newtonian fluid migrates to *ζ*_{e} = 0.8, while that in the shear-thinning fluid migrates to *ζ*_{e} = 0.874.

Finally, by releasing particles at numerous vertical locations in channels with asymmetrical hydrophobicity in a shear-thinning fluid, a regime map is created, as shown in Fig. 11. A particle released under conditions indicated by the green regions on the map migrates downward to the equilibrium position in the lower half of the channel, whereas a particle released under conditions indicated by the blue regions migrates to the equilibrium position in the upper half. This regime map will aid the design of pressure-driven inertial microfluidic devices based on non-Newtonian fluids.

## V. CONCLUSIONS

This study has involved a comprehensive computational analysis of the inertial migration of neutrally buoyant spherical particles immersed in a pressure-driven shear-thinning fluid flow through a hydrophobic microchannel. The emphasis here has been on the joint effects of fluid slippage and viscosity index on equilibrium particle locations. The computational investigation centered on three important hydrophobic parallel-wall configurations: (i) with both walls having no slip, (ii) with both walls having identical, and (iii) with the two walls having different slip-lengths. The major findings of the study can be summarized as follows:

The lift force on a rigid particle has been estimated for all three cases, and it has been found that each of these microchannels has three distinct zero-lift points along their width for both Newtonian and shear-thinning fluids: in the upper half, in the lower half, and at the shear-free plane. The particle migrates to one of these positions, either in the upper half or in the lower half.

It has been found that the viscosity index plays a significant role in lateral ordering. When both walls are identically hydrophobic, the equilibrium location is shifted toward the centerline as the viscosity index increases. Also, as the power-law index is decreased, the distance required to reach the zero-lift point increases, owing to the flattening of the velocity profile.

The wall hydrophobicity has a significant influence on the equilibrium location for an asymmetric-slip microchannel in which the upper wall is no-slip and the lower wall is hydrophobic. Here, as the hydrophobicity of the lower boundary is increased, the equilibrium location moves toward the center in the upper half and toward the lower wall in the lower half. In particular, a particle in a non-Newtonian fluid that would migrate to 0.75

*H*in a symmetric slip channel is further shifted to 0.8*H*in an asymmetric-slip channel.A particle released at a given vertical position will migrate to different equilibrium positions as the wall hydrophobicity and fluid viscosity index are altered. By tailoring the hydrophobicity of the walls, particles released in the lower half can be made to migrate to the equilibrium position in the upper half after crossing the shear-free plane.

The findings of this study will be of great use in facilitating the efficient design of lab-on-chip devices based on inertial migration principles.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**K. K. Krishnaram**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **K. Nandakumar Chandran**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (equal). **Man Yeong Ha**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). **Ranjith S. Kumar**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Fluid Mechanics and Fluid Power*

*in vitro*models of the circulatory system