^{1,a)}

### Abstract

Wall effect on the hydrodynamic interaction among particles is important for their transport in many applications such as filtration. We investigate an axisymmetric creeping flow caused by one or two spherical particles migrating towards a circular orifice or disk. A boundary integral/element method is used to solve for the flow field and calculate the drag force. A crucial advantage of this approach is its capability of tackling a problem with more than one particle in the close vicinity of a solid wall. In the absence of a second particle, our results for the particle drag force agree more favorably with asymptotic behaviors than those from a superposition/collocation method. For cases with two particles driven by a constant external force, a relative motion between them arises from the hydrodynamic friction of the solid wall, leading to a decrease in the evolved separation distance or even occurrence of coagulation.

This work was financially supported by the National University of Singapore via Grant No. R-279-000-352-112.

I. INTRODUCTION

II. METHOD

III. RESULTS AND DISCUSSION

A. Single particle near a disk

B. Single particle near an orifice

C. Two particles near an orifice

IV. CONCLUSION

### Key Topics

- Hydrodynamics
- 13.0
- Boundary element methods
- 9.0
- Integral equations
- 7.0
- Lubrication
- 7.0
- Solid surfaces
- 6.0

## Figures

Schematic of two identical particles moving axisymmetrically along the z axis towards a circular orifice (a) and a circular disk (b).

Schematic of two identical particles moving axisymmetrically along the z axis towards a circular orifice (a) and a circular disk (b).

Normalized z-component stress as a function of location on the disk surface with d/a = 1.5.

Normalized z-component stress as a function of location on the disk surface with d/a = 1.5.

Normalized z-component stress as a function of location on the sphere surface with d/a = 1.5. The curves from top to bottom are for a/b = 0.25, 0.5, 1, and 2, respectively.

Normalized z-component stress as a function of location on the sphere surface with d/a = 1.5. The curves from top to bottom are for a/b = 0.25, 0.5, 1, and 2, respectively.

Normalized drag force as a function of particle location around an orifice.

Normalized drag force as a function of particle location around an orifice.

Normalized z-component stress as a function of location on the sphere surface in the vicinity of an orifice with a/c = 0.5.

Normalized z-component stress as a function of location on the sphere surface in the vicinity of an orifice with a/c = 0.5.

Normalized z-component stress as a function of location on the sphere surface in the vicinity of an orifice with a/c = 0.85.

Normalized z-component stress as a function of location on the sphere surface in the vicinity of an orifice with a/c = 0.85.

Normalized velocities vs location for two particles (d 2 − d 1 = 8a) moving towards an infinite plane under a constant external force; BE results (solid curves) and Eq. (13) (dashes curves).

Normalized velocities vs location for two particles (d 2 − d 1 = 8a) moving towards an infinite plane under a constant external force; BE results (solid curves) and Eq. (13) (dashes curves).

Variation of normalized drag force with location for two particles (z 1 − z 2 = 3a) moving at an identical velocity in the vicinity of an orifice; particle 1 (solid curves) and particle 2 (dashed curves).

Variation of normalized drag force with location for two particles (z 1 − z 2 = 3a) moving at an identical velocity in the vicinity of an orifice; particle 1 (solid curves) and particle 2 (dashed curves).

Drag force of particle 1 normalized by the value in the absence of particle 2 as a function of location in the vicinity of an orifice. Solid curves from top to bottom denote a/c = 0.9, 0.6, and 0.3 for z 1 − z 2 = 3a, while dashed curves correspond to z 1 − z 2 = 2.5a.

Drag force of particle 1 normalized by the value in the absence of particle 2 as a function of location in the vicinity of an orifice. Solid curves from top to bottom denote a/c = 0.9, 0.6, and 0.3 for z 1 − z 2 = 3a, while dashed curves correspond to z 1 − z 2 = 2.5a.

Evolution of the shortest surface-to-surface distance between two particles moving subject to a constant external force in the vicinity of an orifice with particle 1 being initially located at z 1/a = −10 and s 0/a = 1 (a), 0.5 (b), and 0.2 (c).

Evolution of the shortest surface-to-surface distance between two particles moving subject to a constant external force in the vicinity of an orifice with particle 1 being initially located at z 1/a = −10 and s 0/a = 1 (a), 0.5 (b), and 0.2 (c).

## Tables

Normalized drag force for a particle moving towards a circular disk with varying number of elements.

Normalized drag force for a particle moving towards a circular disk with varying number of elements.

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