^{1}and John F. Brady

^{1}

### Abstract

The low-Reynolds-number motion of a single spherical particle between parallel walls is determined from the exact reflection of the velocity field generated by multipoles of the force density on the particle’s surface. A grand mobility tensor is constructed and couples these force multipoles to moments of the velocity field in the fluid surrounding the particle. Every element of the grand mobility tensor is a finite, ordered sum of inverse powers of the distance between the walls. These new expressions are used in a set of Stokesian dynamics simulations to calculate the translational and rotational velocities of a particle settling between parallel walls and the Brownian drift force on a particle diffusing between the walls. The Einstein correction to the Newtonian viscosity of a dilute suspension that accounts for the change in stress distribution due to the presence of the channel walls is determined. It is proposed how the method and results can be extended to computations involving many particles and periodic simulations of suspensions in confined geometries.

I. INTRODUCTION

II. THEORY AND METHODS

A. The grand mobility tensor

B. General solution to the Stokes equations between parallel walls

C. Single particle mobility in a channel

D. Stokesian dynamics

III. RESULTS AND DISCUSSIONS

A. Components of the single particle mobility in a channel

B. Sedimentation of a particle between parallel walls

C. Brownian drift of a particle in a channel

D. Einstein viscosity for a dilute suspension between parallel walls

IV. EXTENSIONS AND CONCLUSIONS

### Key Topics

- Tensor methods
- 65.0
- Stokes flows
- 24.0
- Suspensions
- 24.0
- Hydrodynamics
- 20.0
- Boundary value problems
- 17.0

## Figures

A single spherical particle of radius in a channel of width . The vector is centered on the particle which lies a fractional distance across the channel.

A single spherical particle of radius in a channel of width . The vector is centered on the particle which lies a fractional distance across the channel.

The components of the translation-force coupling in the directions parallel and perpendicular to the walls, respectively.

The components of the translation-force coupling in the directions parallel and perpendicular to the walls, respectively.

The components of the translation-torque coupling. The contribution is not singular and therefore makes no contribution to the single wall problem.

The components of the translation-torque coupling. The contribution is not singular and therefore makes no contribution to the single wall problem.

The components of the translation-stresslet coupling corresponding to couples between translation parallel to the walls and the stresslet and translation perpendicular to the walls and stresslets with components parallel to the walls as well as translation perpendicular to the wall and the stresslet via superposition.

The components of the translation-stresslet coupling corresponding to couples between translation parallel to the walls and the stresslet and translation perpendicular to the walls and stresslets with components parallel to the walls as well as translation perpendicular to the wall and the stresslet via superposition.

The components of the rotation-torque coupling about the axes parallel and perpendicular to the walls, respectively.

The components of the rotation-torque coupling about the axes parallel and perpendicular to the walls, respectively.

The components of the rotation-stresslet coupling which relates rotation of a particle about the axes parallel to the walls to the stresslet.

The components of the rotation-stresslet coupling which relates rotation of a particle about the axes parallel to the walls to the stresslet.

The components of the rate of strain-stresslet coupling. Between two walls, there are only three independent components of the tensor corresponding to the necessary Stokes flow symmetries and the anisotropy caused by the wall.

The components of the rate of strain-stresslet coupling. Between two walls, there are only three independent components of the tensor corresponding to the necessary Stokes flow symmetries and the anisotropy caused by the wall.

The components of the exact translation-force coupling and the translation-force coupling determined using Oseen’s superposition approximation as well as the relative error between this and the Stokesian dynamics results.

The components of the exact translation-force coupling and the translation-force coupling determined using Oseen’s superposition approximation as well as the relative error between this and the Stokesian dynamics results.

The fall speed, , and rotation rate, , of a particle sedimenting along a channel. The fall speed and rotation rate are normalized by the Stokes velocity of the same particle subject to the same force in an otherwise unbounded fluid (i.e., and ).

The fall speed, , and rotation rate, , of a particle sedimenting along a channel. The fall speed and rotation rate are normalized by the Stokes velocity of the same particle subject to the same force in an otherwise unbounded fluid (i.e., and ).

The fall speed, , of a particle sedimenting along a channel normalized by the Stokes velocity of the same particle subject to the same force in an otherwise unbounded fluid (i.e., ).

The fall speed, , of a particle sedimenting along a channel normalized by the Stokes velocity of the same particle subject to the same force in an otherwise unbounded fluid (i.e., ).

The fraction of the channel over which a particle sediments at 95% of its midchannel fall speed.

The fraction of the channel over which a particle sediments at 95% of its midchannel fall speed.

The drift velocity of a single Brownian particle in channel of width plotted as a function of height above the lower channel wall.

The drift velocity of a single Brownian particle in channel of width plotted as a function of height above the lower channel wall.

The additional contribution to the viscosity of a dilute suspension, , is plotted against the separation between the channel walls. The superposition approximation due to Guth and Simha is also plotted.

The additional contribution to the viscosity of a dilute suspension, , is plotted against the separation between the channel walls. The superposition approximation due to Guth and Simha is also plotted.

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