^{1}, I. M. Klink

^{1}and R. J. Phillips

^{1,a)}

### Abstract

Axisymmetric sphere–wall and two-sphere interactions were examined in a viscoelastic solution composed of polyisobutylene polymer in tetradecane. The Reynolds and Stokes numbers were small, so that inertia played at most a minor role, while the Deborah numbers De were in the range 0.4 < De < 3.5. When single spheres fell away from the solid top of the containing vessel, or toward the bottom, the range of the sphere–wall interaction was reduced in the viscoelastic fluid relative to a Newtonian fluid. The reduced range was more pronounced at higher settling speeds. In addition, the interaction in the viscoelastic fluid was reversible, in that the effect of the wall on a sphere moving away from it was the same as that on a sphere moving toward it. In the experiments with two equal spheres sedimenting along their line-of-centers, the spheres moved together until they touched. The rate at which they moved together was measured for sphere–sphere separations ranging from contact to many diameters, and for separations greater than 2 diameters was in fair agreement with analytic predictions. The average sedimentation rate of the two spheres increased as they moved together, by an amount that was in good agreement with the Newtonian solution at low Reynolds number.

This work was funded by NSF-CBET Award 0851552.

I. INTRODUCTION

II. MATERIALS AND METHODS

A. Experimental procedure

B. Properties of particles and fluids

C. Time scales and dimensionless groups

III. EXPERIMENTAL RESULTS AND DISCUSSION

A. Sphere–wall interactions

B. Sphere–sphere interactions

C. Discussion

IV. CONCLUSION

### Key Topics

- Viscoelasticity
- 20.0
- Shear thinning
- 17.0
- Non Newtonian flows
- 12.0
- Sedimentation
- 12.0
- Elasticity
- 11.0

## Figures

Steady shear viscosity of the Newtonian fluid and shear-thinning PIB/PB/TD polymer solution. Solid curve is from Eq. (1) with η0 = 7.0 Pa s, λC = 0.36 s, and n = 0.59.

Steady shear viscosity of the Newtonian fluid and shear-thinning PIB/PB/TD polymer solution. Solid curve is from Eq. (1) with η0 = 7.0 Pa s, λC = 0.36 s, and n = 0.59.

Storage (G′, upper) and loss (G″, lower) moduli for the PIB/PB/TD polymer solution. Solid curves show results from the fit to Eqs. (2) and (3) with the parameters given in Table II .

Storage (G′, upper) and loss (G″, lower) moduli for the PIB/PB/TD polymer solution. Solid curves show results from the fit to Eqs. (2) and (3) with the parameters given in Table II .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres moving away from a wall. The suspending fluid is Newtonian. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres moving away from a wall. The suspending fluid is Newtonian. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres moving toward a wall. The suspending fluid is Newtonian. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres moving toward a wall. The suspending fluid is Newtonian. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres away from a wall. The suspending fluid is the PIB solution. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres away from a wall. The suspending fluid is the PIB solution. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres moving toward a wall. The suspending fluid is the PIB solution. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Sphere velocity versus sphere-to-wall separation for (a) titanium, (b) stainless steel, and (c) tungsten carbide spheres moving toward a wall. The suspending fluid is the PIB solution. The solid curve is the solution to Stokes' equations derived by Brenner (1961) .

Vector plots of the velocity field around a 0.318 cm tungsten carbide sphere moving away from the top, solid surface at when the distance from the sphere center to the top is (a) 3.6 diameters, (b) 5.8 diameters, and (c) 12.6 diameters. The suspending fluid is the PIB solution. The top surface is located at the top edge of the image.

Vector plots of the velocity field around a 0.318 cm tungsten carbide sphere moving away from the top, solid surface at when the distance from the sphere center to the top is (a) 3.6 diameters, (b) 5.8 diameters, and (c) 12.6 diameters. The suspending fluid is the PIB solution. The top surface is located at the top edge of the image.

Vector plots of the velocity field around a 0.318 cm tungsten carbide sphere moving toward the bottom of the container when the distance from the sphere center to the bottom is (a) 11.3 diameters, (b) 5.7 diameters, and (c) 3.9 diameters. The suspending fluid is the PIB solution. The bottom surface is located at the bottom edge of the image.

Vector plots of the velocity field around a 0.318 cm tungsten carbide sphere moving toward the bottom of the container when the distance from the sphere center to the bottom is (a) 11.3 diameters, (b) 5.7 diameters, and (c) 3.9 diameters. The suspending fluid is the PIB solution. The bottom surface is located at the bottom edge of the image.

Rate of approach ΔU = U1 − U2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.318 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances.

Rate of approach ΔU = U1 − U2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.318 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances.

Average velocity 〈U〉 = (U1 + U2)/2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.318 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances. Solid curve is the low-Reynolds-number Newtonian result of Stimson and Jeffery (1926) .

Average velocity 〈U〉 = (U1 + U2)/2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.318 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances. Solid curve is the low-Reynolds-number Newtonian result of Stimson and Jeffery (1926) .

Rate of approach ΔU = U1 − U2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.476 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances.

Rate of approach ΔU = U1 − U2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.476 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances.

Average velocity 〈U〉 = (U1 + U2)/2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.476 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances. Solid curve is the low-Reynolds-number Newtonian result of Stimson and Jeffery (1926) .

Average velocity 〈U〉 = (U1 + U2)/2 for two (a) stainless steel and (b) tungsten carbide spheres with diameter d = 0.476 cm separated by distance s in the shear-thinning PIB solution. Vertical error bars are standard deviations of velocities, and horizontal error bars are standard deviations of separation distances. Solid curve is the low-Reynolds-number Newtonian result of Stimson and Jeffery (1926) .

Rate of approach as shown in Figs. 9(a) and 11(a) , renormalized as shown in Eq. (12) . Solid curve is Eq. (13) , from Phillips and Talini (2007) .

Rate of approach as shown in Figs. 9(a) and 11(a) , renormalized as shown in Eq. (12) . Solid curve is Eq. (13) , from Phillips and Talini (2007) .

Vector plot of the axisymmetric velocity field around two 0.476 cm tungsten carbide spheres sedimenting in the PIB solution, when center-to-center separation is (a) 4 diameters and (b) 1 diameter.

Vector plot of the axisymmetric velocity field around two 0.476 cm tungsten carbide spheres sedimenting in the PIB solution, when center-to-center separation is (a) 4 diameters and (b) 1 diameter.

## Tables

Properties of spherical particles.

Properties of spherical particles.

Relaxation spectrum.

Relaxation spectrum.

Terminal velocities and Deborah numbers.

Terminal velocities and Deborah numbers.

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