^{1,2}, Mingxiang Luo

^{1}, Joëlle Fréchette

^{1}and German Drazer

^{2}

### Abstract

We present a numerical study of the effect that fluid and particle inertia have on the motion of suspended spherical particles through a geometric constriction to understand analogous microfluidic settings, such as pinched flow fractionation devices. The particles are driven by a constant force in a quiescent fluid, and the constriction (the pinching gap) corresponds to the space between a plane wall and a second, fixed sphere of the same size (the obstacle). The results show that, due to inertia and/or the presence of a geometric constriction, the particles attain smaller separations to the obstacle. We then relate the minimum surface-to-surface separation to the effect that short-range, repulsive non-hydrodynamic interactions (such as solid-solid contact due to surface roughness, electrostatic double layer repulsion, etc.) would have on the particle trajectories. In particular, using a simple hard-core repulsive potential model for such interactions, we infer that the particles would experience larger lateral displacements moving through the pinching gap as inertia increases and/or the aperture of the constriction decreases. Thus, separation of particles based on differences in density is in principle possible, owing to the differences in inertia associated with them. We also discuss the case of significant inertia in which the presence of a small constriction may hinder separation by reducing inertia effects.

We thank Professor A. J. C. Ladd for making the LB code – *Susp3d* – available to us. This work is partially supported by the National Science Foundation Grant Nos. CBET-0933605, CMMI-0748094, and CBET-0954840. This work used the resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

I. INTRODUCTION

II. SYSTEM DEFINITION AND PARAMETER SPACE

III. HARD-CORE MODEL FOR NON-HYDRODYNAMIC INTERACTIONS

A. Stokes regime and the critical offset

B. Inertia effects on the critical offset

IV. RESULTS AND IMPLICATIONS OF THE HARD-CORE MODEL

A. Effect of pinching in the Stokes regime

B. Effect of inertia in the absence of pinching

C. Effect of pinching and inertia

V. SUMMARY

### Key Topics

- Particle trajectory
- 11.0
- Hydrodynamics
- 9.0
- Trajectory models
- 7.0
- Kinematics
- 4.0
- Microfluidics
- 4.0

##### B81B

## Figures

(a) System, simulation box, and boundary conditions. The schematic is to scale, i.e., the box is 16 obstacle/particle radii in length. The top wall creates the pinching gap. In the absence of walls, we use the periodic boundaries for the box, and the obstacle is at the center of the box. (b) The part of the simulation box indicated with dashed lines in (a) is enlarged here. The initial offset b in , the final offset b out , and the aperture of the pinching gap D are indicated. The dashed circle depicts the position of the particle when the particle-obstacle separation is minimum (in the absence of inertia). (c) Enlarged view of the particle (dashed) and the obstacle (solid) surfaces showing the minimum separation aξ min between them.

(a) System, simulation box, and boundary conditions. The schematic is to scale, i.e., the box is 16 obstacle/particle radii in length. The top wall creates the pinching gap. In the absence of walls, we use the periodic boundaries for the box, and the obstacle is at the center of the box. (b) The part of the simulation box indicated with dashed lines in (a) is enlarged here. The initial offset b in , the final offset b out , and the aperture of the pinching gap D are indicated. The dashed circle depicts the position of the particle when the particle-obstacle separation is minimum (in the absence of inertia). (c) Enlarged view of the particle (dashed) and the obstacle (solid) surfaces showing the minimum separation aξ min between them.

Trajectories corresponding to the three possible types of particle-obstacle “collisions,” adapted from Ref. 9 . The particle travels from left (incoming half of the trajectory) to right (outgoing half of the trajectory). The dashed circle represents the excluded volume inaccessible to the particle centers.

Trajectories corresponding to the three possible types of particle-obstacle “collisions,” adapted from Ref. 9 . The particle travels from left (incoming half of the trajectory) to right (outgoing half of the trajectory). The dashed circle represents the excluded volume inaccessible to the particle centers.

In presence of significant inertia (in this particular case, particle inertia, St = 1.11), the trajectories are asymmetric even in the absence of non-hydrodynamic interactions. Such asymmetry leads to two critical offsets, viz., b c,in and b c,out as shown.

In presence of significant inertia (in this particular case, particle inertia, St = 1.11), the trajectories are asymmetric even in the absence of non-hydrodynamic interactions. Such asymmetry leads to two critical offsets, viz., b c,in and b c,out as shown.

Various trajectories with b in = 2.0, exhibiting the effect of pinching in the Stokes regime (Re = 5 × 10−4 and St = 1.11 × 10−4). The vertical dashed lines at x ≈ ±7a represent the initial and final x-coordinates of the particle, respectively.

Various trajectories with b in = 2.0, exhibiting the effect of pinching in the Stokes regime (Re = 5 × 10−4 and St = 1.11 × 10−4). The vertical dashed lines at x ≈ ±7a represent the initial and final x-coordinates of the particle, respectively.

Various trajectories with b in = 2a exhibiting the effect of inertia in the absence of pinching. The vertical dashed lines at x ≈ ±7a represent the initial and final x-coordinates of the particle, respectively.

Various trajectories with b in = 2a exhibiting the effect of inertia in the absence of pinching. The vertical dashed lines at x ≈ ±7a represent the initial and final x-coordinates of the particle, respectively.

The dimensionless surface-to-surface separation between the particle and the obstacle as a function of the dimensionless x-coordinate of the particle. The plot shows the combined effect of particle and fluid inertia on the dimensionless separation along particle trajectories for b in = 2a in the absence of pinching. The vertical dashed line depicts the symmetry plane of the system at x = 0.

The dimensionless surface-to-surface separation between the particle and the obstacle as a function of the dimensionless x-coordinate of the particle. The plot shows the combined effect of particle and fluid inertia on the dimensionless separation along particle trajectories for b in = 2a in the absence of pinching. The vertical dashed line depicts the symmetry plane of the system at x = 0.

Effect of reducing the aperture of the pinching gap (d/D ↑) on the minimum surface-to-surface separation between the particle and the obstacle for different magnitudes of inertia and a fixed initial offset, b in = 2a.

Effect of reducing the aperture of the pinching gap (d/D ↑) on the minimum surface-to-surface separation between the particle and the obstacle for different magnitudes of inertia and a fixed initial offset, b in = 2a.

The final offset b out (corresponding to b in = 2a) as a function of Stokes (top) and Reynolds (bottom) numbers for different apertures of the pinching gap.

The final offset b out (corresponding to b in = 2a) as a function of Stokes (top) and Reynolds (bottom) numbers for different apertures of the pinching gap.

The critical final-offset b c,out corresponding to ε = 0.3, as a function of Reynolds and Stokes numbers in the presence and the absence of a pinching wall.

The critical final-offset b c,out corresponding to ε = 0.3, as a function of Reynolds and Stokes numbers in the presence and the absence of a pinching wall.

The minimum separation ξ min as a function of the initial offset b in , Re and St are negligible.

The minimum separation ξ min as a function of the initial offset b in , Re and St are negligible.

The minimum separation ξ min as a function of the initial offset b in in the absence of pinching. (Left) Effect of particle inertia when fluid inertia is negligible and (right) effect of fluid inertia when particle inertia is negligible.

The minimum separation ξ min as a function of the initial offset b in in the absence of pinching. (Left) Effect of particle inertia when fluid inertia is negligible and (right) effect of fluid inertia when particle inertia is negligible.

The minimum separation ξ min as a function of the initial offset b in showing the effect of significant fluid as well as particle inertia in the absence of pinching.

The minimum separation ξ min as a function of the initial offset b in showing the effect of significant fluid as well as particle inertia in the absence of pinching.

The dimensionless surface-to-surface separation ξ as a function of the dimensionless x-coordinate of the particle. The x-coordinate at which the minimum separation is attained along a trajectory varies with inertia. (Left) Effect of particle inertia and (right) effect of fluid inertia. The minimum separation is attained before the x = 0 symmetry-plane in both cases.

The dimensionless surface-to-surface separation ξ as a function of the dimensionless x-coordinate of the particle. The x-coordinate at which the minimum separation is attained along a trajectory varies with inertia. (Left) Effect of particle inertia and (right) effect of fluid inertia. The minimum separation is attained before the x = 0 symmetry-plane in both cases.

The effect of inertia on the ξ min –b in relationship in the presence of a pinching wall. The aperture of the pinching gap is 3d (i.e., d/D = 0.333). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The effect of inertia on the ξ min –b in relationship in the presence of a pinching wall. The aperture of the pinching gap is 3d (i.e., d/D = 0.333). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The effect of inertia on the ξ min –b in relationship in the presence of pinching. The aperture of the pinching gap is 1.7d (i.e., d/D = 0.588). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The effect of inertia on the ξ min –b in relationship in the presence of pinching. The aperture of the pinching gap is 1.7d (i.e., d/D = 0.588). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

Combined effect of the pinching wall, particle and fluid inertia on the ξ min –b in relationship. (Left) The aperture of the pinching gap is 3d (i.e., d/D = 0.333) and (right) the aperture is 1.7d (i.e., d/D = 0.588).

Combined effect of the pinching wall, particle and fluid inertia on the ξ min –b in relationship. (Left) The aperture of the pinching gap is 3d (i.e., d/D = 0.333) and (right) the aperture is 1.7d (i.e., d/D = 0.588).

The outgoing offset b out as a function of the incoming offset b in . (Left) Effect of particle inertia and (right) effect of fluid inertia.

The outgoing offset b out as a function of the incoming offset b in . (Left) Effect of particle inertia and (right) effect of fluid inertia.

The minimum separation as a function of the final offset. (Left) Effect of particle inertia and (right) effect of fluid inertia.

The minimum separation as a function of the final offset. (Left) Effect of particle inertia and (right) effect of fluid inertia.

Effect of both significant fluid as well as particle inertia on the ξ min –b out relationship in the absence of pinching.

Effect of both significant fluid as well as particle inertia on the ξ min –b out relationship in the absence of pinching.

The individual effect of particle and fluid inertia on the ξ min –b out relationship in the presence of pinching. The aperture of the pinching gap is 3.0d (i.e., d/D = 0.333). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The individual effect of particle and fluid inertia on the ξ min –b out relationship in the presence of pinching. The aperture of the pinching gap is 3.0d (i.e., d/D = 0.333). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The individual effect of particle and fluid inertia on the ξ min –b out relationship in the presence of pinching. The aperture of the pinching gap is 1.7d (i.e., d/D = 0.588). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The individual effect of particle and fluid inertia on the ξ min –b out relationship in the presence of pinching. The aperture of the pinching gap is 1.7d (i.e., d/D = 0.588). (Left) The effect of particle inertia and (right) the effect of fluid inertia.

The effect of inertia on the ξ min –b out relationship in the presence of pinching for two different apertures of the pinching gap. (Left) The aperture of the pinching gap is 3.0d (i.e., d/D = 0.333) and (right) the aperture is 1.7d (i.e., d/D = 0.588).

The effect of inertia on the ξ min –b out relationship in the presence of pinching for two different apertures of the pinching gap. (Left) The aperture of the pinching gap is 3.0d (i.e., d/D = 0.333) and (right) the aperture is 1.7d (i.e., d/D = 0.588).

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