^{1}, R. H. Rangel

^{1}and D. D. Joseph

^{1,2}

### Abstract

The forces acting on two fixed spheres in a second-order uniform flow are investigated. When , where and are fluid parameters related to the first and second normal stress coefficients, the velocity field for a second-order fluid is the same as the one predicted by the Stokes equations while the pressure is modified. The Stokes solutions given by Stimson and Jeffery [Proc. R. Soc. London, Ser. A111, 110 (1926)] for the case when the flow direction is along the line of centers and Goldman *et al.* [Chem. Eng. Sci.21, 1151 (1966)] for the case when the flow direction is perpendicular to the line of centers are utilized and the stresses and the forces acting on the particles in a second-order fluid are calculated. For flow along the line of centers or perpendicular to it, the net force is in the direction that tends to decrease the particle separation distance. For the case of flow at arbitrary angle, unequal forces are applied to the spheres perpendicularly to the line of centers. These forces result in a change of orientation of the sedimenting spheres until the line of centers aligns with the flow direction. In addition, the potential flow of a second-order fluid past two fixed spheres in a uniform flow is investigated. The normal stress at the surface of each sphere is calculated and the viscoelasticeffects on the normal stress for different separation distances are analyzed. The contribution of the potential flow of a second-order fluid to the force applied to the particles is an attractive force. Our explanations of the aggregation of particles in viscoelastic fluids rest on three pillars; the first is a viscoelastic “pressure” generated by normal stresses due to shear. Second, the total time derivative of the pressure is an important factor in the forces applied to moving particles. The third is associated with a change in the normal stress at points of stagnation which is a purely extensional effect unrelated to shearing.

This work was sponsored by National Science Foundation under Grant Nos. CBET-0302837 and OISE-0530270. The first author acknowledges the Zonta International Foundation for an Amelia Earhart fellowship.

I. INTRODUCTION

II. THEORETICAL DEVELOPMENT

III. FORCES ACTING ON TWO NONROTATING FIXED SPHERES IN A SECOND-ORDER FLUID

A. Free stream along the line of centers

B. Free stream perpendicular to the line of centers

C. Two spherical particles in a free stream at an arbitrary angle

IV. FORCES ACTING ON TWO FREELY ROTATING FIXED SPHERES IN A FREE STREAM OF A SECOND-ORDER FLUID

V. VISCOUS POTENTIAL FLOW

VI. CONCLUSIONS

### Key Topics

- Viscoelasticity
- 24.0
- Potential flows
- 21.0
- Torque
- 12.0
- Rotating flows
- 10.0
- Stokes flows
- 9.0

## Figures

Two spherical particles in a arbitrary-direction free stream.

Two spherical particles in a arbitrary-direction free stream.

Schematic of forces acting on two particles. (a) Newtonian fluid. (b) Second-order fluid.

Schematic of forces acting on two particles. (a) Newtonian fluid. (b) Second-order fluid.

Forces acting on particles in a second-order fluid while the free stream is along the particles line of centers.

Forces acting on particles in a second-order fluid while the free stream is along the particles line of centers.

Stresses on the surface of particle I in a second-order fluid while the free stream is along the particles line of centers: ; ; .

Stresses on the surface of particle I in a second-order fluid while the free stream is along the particles line of centers: ; ; .

Schematic of the forces acting to two particles.

Schematic of the forces acting to two particles.

Force acting on the particle I in a second-order fluid while the free stream is perpendicular to the particles line of centers.

Force acting on the particle I in a second-order fluid while the free stream is perpendicular to the particles line of centers.

Torque acting on sphere I in a second-order fluid while the free stream is perpendicular to the particles line of centers.

Torque acting on sphere I in a second-order fluid while the free stream is perpendicular to the particles line of centers.

Stresses on the surface of particle I in a second-order fluid when the flow is perpendicular to the line of centers: , ; ; .

Stresses on the surface of particle I in a second-order fluid when the flow is perpendicular to the line of centers: , ; ; .

Stresses on the surface of particle I in a second-order fluid when the flow is perpendicular to the line of centers. (*) shear stress, (○) normal stress and pressure in second-order fluid: , ; , .

Stresses on the surface of particle I in a second-order fluid when the flow is perpendicular to the line of centers. (*) shear stress, (○) normal stress and pressure in second-order fluid: , ; , .

Schematic of the forces acting on two particles.

Schematic of the forces acting on two particles.

Forces acting on particle I in a second-order fluid in a free stream at an arbitrary angle. (a) vs particle separation distance when . (b) vs the free stream angle when .

Forces acting on particle I in a second-order fluid in a free stream at an arbitrary angle. (a) vs particle separation distance when . (b) vs the free stream angle when .

Force acting on the particle I while the spheres are rotating in a quiescent second-order fluid.

Force acting on the particle I while the spheres are rotating in a quiescent second-order fluid.

Torque acting on sphere I while the spheres are rotating in a quiescent second-order fluid.

Torque acting on sphere I while the spheres are rotating in a quiescent second-order fluid.

Rotation rate of freely rotating spheres in a free stream of a second-order fluid.

Rotation rate of freely rotating spheres in a free stream of a second-order fluid.

The distribution of shear rate and Stokes pressure on the surface of sphere I for nonrotating and freely rotating spheres: ; ; ; .

The distribution of shear rate and Stokes pressure on the surface of sphere I for nonrotating and freely rotating spheres: ; ; ; .

The distribution of and pressure on the surface of sphere I for nonrotating and freely rotating spheres: ; ; ; .

The distribution of and pressure on the surface of sphere I for nonrotating and freely rotating spheres: ; ; ; .

The distribution of normal and shear stresses on the surface of sphere I for nonrotating and freely rotating spheres: ; ; ; .

The distribution of normal and shear stresses on the surface of sphere I for nonrotating and freely rotating spheres: ; ; ; .

Forces acting on sphere I for nonrotating and freely rotating spheres.

Forces acting on sphere I for nonrotating and freely rotating spheres.

Two spherical particles in a free stream.

Two spherical particles in a free stream.

Two spherical particles in a free stream at , , , and . Normal stress at surface of sphere I is shown.

Two spherical particles in a free stream at , , , and . Normal stress at surface of sphere I is shown.

Two spherical particles in a free stream at , cm, , and . Normal stress at surface of sphere II is shown.

Two spherical particles in a free stream at , cm, , and . Normal stress at surface of sphere II is shown.

Two spherical particles in a free stream at and .

Two spherical particles in a free stream at and .

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