ORIGINAL CONTRIBUTION

Non-Newtonian end effects in falling ball viscometryof concentrated suspensions

Patrick T. Reardon & Alan L. Graham & Shihai Feng &

Vibha Chawla & Rahul S. Admuthe & Lisa A. Mondy

Received: 6 February 2006 /Accepted: 29 August 2006 / Published online: 10 October 2006# Springer-Verlag 2006

Abstract In a Newtonian fluid contained in a cylinder, asmall ball initially at rest released just below the surfacewould accelerate to achieve a steady-state velocity withinone cylinder diameter. After traversing the center sectionof the cylinder, the ball would begin slowing down withinone cylinder diameter of the bottom. This behavior is alsoobserved in suspensions where the size of the suspendedparticles is small relative to the containing cylinder.However, in concentrated suspensions of larger suspendedparticles, balls released near the upper surface travel fasterthan the steady state velocity. In addition, the length ofthe upper surface end effect, where the falling balldecelerates to the steady state velocity, and the lowerend effect zone, where the ball decelerates to rest at thebottom, is many times longer than in a Newtonian single-phase liquid. These non-Newtonian end effects arereduced if the suspended particles are polydisperse intheir size distribution.

Keywords End-effects . Suspensions . Non-Newtonian

Introduction

At steady state, a solid ball settles in an unbounded,incompressible viscous Newtonian fluid at a velocity suchthat the combination of form and friction drag due to themotion of the fluid around the sphere exactly balances thenet buoyant force on the ball (see, for example, Bird et al.2002). The relationship between the terminal settlingvelocity and the diameter and density of the settling balland the density and viscosity of an unbounded fluid hasbecome known as Stokes’ Law (Stokes 1851). Wall effectsfor balls settling in cylindrical vessels filled with New-tonian liquids far from the ends of the containing cylinderhave been studied extensively (see, for example, Phillips etal. 1992; Haberman and Sayre 1958; Bohlin 1960). Happeland Brenner (1983) provide an excellent summary andanalysis of much of this earlier work. More recently,Graham et al. (1989) and Ilic et al. (1992) have extendedthis work to larger relative ball sizes and other containinggeometries. Bohlin’s wall correction was shown to be validfor 0<af/Rcyl<0.6 (where af is the radius of the falling balland Rcyl that of the containing cylinder) in recent experi-ments by Kaiser and coworkers (2003).

Several authors, including Mena et al. (1987), Gottlieb(1979), Chhabra and Uhlherr (1988), Feng et al. (2006),and Butcher and Irvine (1990), have undertaken studies ofnon-Newtonian fluids in which the ball settles along theaxis of the cylinder. For low-shear-rate experiments andsmall settling balls, these authors found that the wall effectsare essentially Newtonian. Under these conditions, New-tonian wall corrections such as those described above willrecover the low shear-rate viscosity of the fluid in questionand can sometimes provide additional rheological proper-ties of the fluid (see, for example, Gottlieb 1979; Chhabraand Uhlherr 1988). Ataide et al. (1999) studied the effect of

Rheol Acta (2007) 46:413–424DOI 10.1007/s00397-006-0138-7

P. T. Reardon :A. L. Graham : S. Feng (*)Los Alamos National Laboratory,P.O. Box 1663, MS-C930,Los Alamos, NM 87545, USAe-mail: sfeng@lanl.gov

V. Chawla : R. S. AdmutheDepartment of Chemical Engineering, Texas Tech University,Lubbock, TX 79409-3121, USA

L. A. MondySandia National Laboratories,Albuquerque, NM 87185-0834, USA

confining walls on the free settling of spherical particlesalong the axes of cylindrical tubes in Newtonian and non-Newtonian liquids. He did not report specifically endeffects but dealt with the effects of the sidewalls. Machacand Lecjaks (1995) investigated the effect on the terminalfalling velocity of a spheremoving through power-law liquidscontained in a rectangular duct in the creeping flow region.

Other investigators have used falling-ball rheometry toprobe the microstructure and average properties of suspen-sions of particles. Mondy and coworkers (1986) performedexperiments in which the ball settles down the centerline ofthe containing cylinder, showing that suspensions canexhibit wall effects that are non-Newtonian in suspensionsof large particles in Newtonian fluids. Papers by Milliken etal. (1989) and Mor et al. (1996) further detail these effects.

Information on the effect of the ends of the containingvessels is more limited. Tanner (1963) used the method ofreflections to estimate velocities of small balls approachingthe bottom of the containing cylinder. His calculationsindicated that, for small balls, the end effects becamesignificant at about one cylinder radius from the bottomfixed surface and the top free surface for a Newtonian fluid.He verified these predictions experimentally. Sonshine et al.(1966) investigated Stokes’ translation of a particle ofarbitrary shape along the axis of a circular cylinder filled toa finite depth with viscous liquid and found similar results.Sutterby’s experimental study (Sutterby 1973) was inagreement with Tanner’s (1963) predictions. Sutterby foundthat, for L/Dcyl=2 and af/Rcyl<0.125, and L/Dcyl=1.025 andaf/Rcyl<0.047 (where Dcyl is the diameter of the containingcylinder and L is the length of the cylinder), there were noobservable end effects throughout the middle third of thecylinder. However, prominent end effects were much largerin experiments where L/Dcyl=0.627, and L/Dcyl=0.469.

Graham and coworkers (1989) combined finite-elementcalculations, theoretical predictions, and experimentalobservations to analyze wall and end effects for a widerange of ball sizes. The numerical results confirmed that,for small settling balls, the end effects at the bottom of thecylinder can be neglected if z/Rcyl>1, where z is the verticaldistance from the cylinder bottom. Ilic et al. (1992) usedboundary element method (BEM) calculations to study thesettling velocities of spherical particles in circular andsquare conduits. Their results show that the region affectedby the bounding surface extends further into the fluid as theball size increases. These theoretical and experimentalstudies helped establish a rule-of-thumb that falling-ballexperiments must be conducted in Newtonian liquids atleast one diameter from the upper and lower surfaces to befree of end effects.

Here we describe a series of experiments and numericalcalculations to compare the effect of the ends of thecontaining vessel on the settling velocity of a falling ball in

pure Newtonian fluids to that in model suspensions. Theinstantaneous velocity of the settling balls is determinedalong the centerline of the cylinders by measuring time anddistance from the upper surface as the balls settle towardthe bottom of the cylinder. The Reynolds number, Re, forthe settling balls is always less than 0.1 and in most casesmuch less. These results are compared with numericalsimulations and experiments of end effects in pure New-tonian fluids. The suspensions are well-characterized, two-phase systems of neutrally buoyant spheres of radius as inviscous Newtonian liquids. The primary experimentalparameters are Rcyl/as, af/Rcyl, and the volume fraction ofparticles, φ.

The length of the bottom end-effect zone, ZBEE, is defined

as the point at which the settling-ball velocity begins toslow appreciably from the constant velocity observed in themiddle of the cylinder. In this investigation, this is eitherthe point where the measured velocity is 98% of the steady-state velocity or the measurement zone where the firstdistinct reduction in the steady-state velocity is observed.Careful measurements show that for a suspension in whichRcyl/as=24, the length of the end-effect zone increases fromapproximately one Dcyl from the bottom to more than 12Dcyl for the same-sized falling ball as φ increases from 10to 50%. This increase in the length of the end effect zonedepends only on φ and the relative geometry of the systemand is independent of the density of the falling balls.Additional experiments have been performed to determinethe effect the diameter of the cylinder relative to thesuspension-particle size has on ZB

EE. There is a pronouncedpeak in ZB

EE at about Rcyl/as=24 that decreases as Rcyl/asincreases until Newtonian-like behavior is observed forRcyl/as>100.

Velocity measurements were made along the entirelength of the columns. The settling balls near the freesurface at the top of the column filled with a suspensionwhere φ≤0.5 were observed to move much faster than thesteady-state velocity. The length of this upper end-effectzone is denoted as ZB

EE. When lids are added to the columnsthat touch the suspension surface, creating a fixed bound-ary, the settling velocity is approximately constant at thesteady-state value near the upper free surface, and nodramatic increase in settling velocity is observed near theupper surface. Note that this behavior contrasts to thatobserved in pure Newtonian fluids. Our numerical simu-lations in pure fluids show for both stick and traction-freeboundary conditions on the upper surface that falling ballsmove slower than the steady-state velocity near the uppersurface and achieve their steady-state velocity within onecylinder diameter for both stick and traction-free uppersurfaces.

The next section describes in detail the suspensioncomponents, the experimental methodology, and the

414 Rheol ActaRheologica Acta (2007) 46:413–424

experimental apparatus used in this study. The followingsection presents the results of the experiments anddiscusses the effect of the primary experimental parame-ters on the end effects. The final section summarizes theresults and discusses the implications of these findingswith respect to earlier works in this area.

Experiments

Materials

The suspending fluid used for these experiments matchesthe density and refractive index of the particles. This choiceproduces a neutrally buoyant suspension that is transparentand allows optical techniques to be used for suspensions oflarge particles. The suspending fluid is a solution of 1,1,2,2,tetrabromoethane (TBE, Eastman Kodak, Rochester, NY,USA), polyethylene glycol (90,000 UCON oil, UnionCarbide, Danbury, CT, USA), and alkylaryl polyetheralcohol (Triton X-100, J.T. Baker, Phillipsburg, NJ, USA).The relative proportions of each are 14.07, 35.66, and50.27% by weight, respectively. The fluid also containsTinuvin 328™ (Ciba-Geigy, Ashley, NY, USA), which wasadded to the TBE (0.1% by weight of the TBE) as anantioxidant to prevent the discoloration of the fluid causedby dissociation of the TBE due to exposure to ultravioletlight or contact with certain metals, particularly iron alloys.Details of the characterization of this Newtonian fluid arediscussed by Abbott and coworkers (1991).

The suspending particles were polymethyl methacrylate(PMMA) spheres obtained from three separate manufac-turers, as no single company had the wide range of sizesused in the experiments. All the particles were large enoughthat electrostatic or Brownian forces did not exert anappreciable effect on the suspension behavior (Abbott etal. 1991). Particles with a mean diameter of 140 μm and astandard deviation of 29 μm were obtained from LuciteInternational (Cordova, TN, USA). The distribution ofparticle sizes was determined using a Beckman Coultercounter particle-size analyzer (Miami, FL, USA). Largerparticles, approximately 800 μm in diameter with astandard deviation of 15 μm, were obtained from Esschem(Linwood, PA, USA). Scanning electron micrographimages show that the particles were essentially sphericaland free from internal defects. Even larger, individuallyground particles with diameters of 1.588 and 3.175 mmwere obtained from Engineering Laboratories (Oakland,NJ, USA). Manufacturer-stated tolerances of the mono-diperse particles were ±0.05 mm variation in diameter and±0.025 mm in sphericity. Suspensions of these particles inthe fluid described above were prepared with solid fractionsof φ=0.10, 0.20, 0.30, 0.40, and 0.50.

An Ubbelohde viscometer (Cannon Instrument, StateCollege, PA, USA) was used to determine the viscosity ofthe pure suspending fluid to be 3.36 Pa s at 21.5 C. Weestimate the error in these measured viscosities to be about1%. Density of the fluid, 1.806 g/cm3 at 21.5 C, wasdetermined using a Mettler-Toledo density meter. Themanufacturer states the limit of error is 0.0001 g/cm3

(±0.003%).Falling balls of three different materials were obtained

from Salem Specialty Ball (West Simsbury, CT, USA). Thespheres were Anti-Friction Bearing Manufacturer’s Associ-ation grade 200 brass (specific gravity 8.5, +/−0.005 mm insphericity and +/−0.025 mm in diameter), grade 1,000aluminum (specific gravity 2.7, +/−0.025 mm in sphericityand +/−0.127 mm in diameter), and grade 25 tungstencarbide (specific gravity 15, +/−0.0006 mm in sphericityand +/−0.0025 mm in diameter) ball bearings.

The cylinders were either custom built from PMMAtubes to achieve the desired length-to-diameter ratio orcommercially available glass cylinders when the aspectratio was appropriate. The cylinders ranged in length from216 to 1,707 mm. The inside diameter of the cylinders wasdetermined using a Mitutoyo Absolute Digimatic microm-eter (Aurora, IL, USA) with a tolerance of ±0.04 mm. Eachcylinder had a lid, in which a hole was machined at thecenter. In the free surface experiments, the liquid was filledto just short of the lid. A short guide tube extended throughthe hole in the lid to the surface of the suspension to ensurethe balls were dropped along the centerline of thecontaining cylinder.

Experimental apparatus and procedure

The balls were kept in a water bath until drop time, insertedin the guide tube, and allowed to slowly settle beneath thesurface at the centerline of the cylinder. The spheres settlingin the transparent suspension were timed with stopwatchespurchased from Fischer Scientific (Hampton, NH, USA).The stopwatches had stated tolerances of ±0.001%. Lineswere inscribed on each cylinder at intervals equal to onecylinder diameter. The mean velocity, V, for the falling ballsat the midpoint between adjacent lines was calculated fromthe interval time and known distance. Each interval velocitywas then averaged over 5–10 ball drops according to thetechnique described by Milliken and coworkers (1989). Aset of velocities from several trials was averaged and thenstatistically compared with another set of an equal numberof trials. If a two-sided student t test showed that thedifference between the two averages was statisticallyindistinguishable at the 95% confidence limit described byMiller and Freund (1977), the average of the combinednumber of trials was accepted as correct. If there was astatistical difference, more measurements were taken until

Rheol Acta (2007) 46:413–424 415

the condition mentioned above was met. The suspensionwas slowly and thoroughly stirred before each ball drop torandomize the microstructure and to maintain uniformconcentrations in each experiment.

Several suspensions were created using the very smallPMMA particles. A sonic cleaner and surfactant solutionwas used to wet sieve the particles and to eliminate the verysmall submicron particles in the polydisperse batches.Sample preparation is described in greater detail by Reardon(2003). Because of the very large number of interfaces,these suspensions of small particles were translucent, andthe visual technique of measuring the falling ball velocitycould not be used. The measurements were done with aneddy-current apparatus to time the settling ball throughseveral zones of the cylinder. Powell and coworkers (1989)discuss the eddy-current apparatus in greater detail.

Figure 1 illustrates the cylinder geometry and thecoordinate system utilized in this study. The origin is atthe bottom of the cylinder. Measurements were taken as theball traversed from top to bottom. The position of thefalling ball was normalized with the cylinder diameter(z/Dcyl).

Experimental design

Experiments were designed to investigate end effects as afunction of φ, Rcyl/as, and af/Rcyl. In each experiment, theaxial velocity profile for the entire column was determined.

The first set of numerical and physical experimentsdetermined the effect of af on the length of the end-effectzone in pure Newtonian fluids. In the next series ofphysical experiments, one falling-ball size was dropped insuspensions prepared with one suspending particle size in asingle cylinder, and only φ was varied. The solid fractionsexamined were φ=0.1, 0.2, 0.3, 0.4, and 0.5. The finalphase of experiments was to change Rcyl/as while main-taining the af/Rcyl ratio. Trials were done both withindividually ground, single-sized (monodisperse) particlesand with polydisperse particles. The effect of a free surfacerelative to a solid upper surface on the containing cylinderwas also determined.

Results

Numerical experiments

A BEM simulation for a settling ball in a pure Newtonianfluid was performed to firmly establish the Newtonianbaseline for end-effect functionality at Re=0, as this has notbeen adequately developed in enough detail elsewhere inthe literature. In the BEM, the fundamental singularsolutions of the governing differential equations arecontinuously distributed over the boundaries of the prob-lem, and the boundary conditions then lead to integralequations for the densities of the fundamental solution.

Fig. 1 a Schematic of the cyl-inder used in the falling-ballexperiments with the index-matched and density-matchedfluid. The falling ball does notsettle in a straight line as itinteracts with suspended par-ticles on the way down. Thecoordinate system is located atthe center of the bottom surfaceof the cylinder. b The chart onthe right shows a typical plot ofinterval position vs the time ittakes to traverse that interval asthe falling ball settles (normal-ized by the cylinder diameter)

416 Rheol ActaRheologica Acta (2007) 46:413–424

Thus, the solution of a differential equation in n dimensionsis reduced to the solution of an integral equation in n−1dimensions. In addition to reducing the dimensions of theproblem, this method is attractive for Stokes’ problemsbecause it is a very general approach independent of thebody geometry and the form of the external flow field. Theboundary element formulation used here has been docu-mented and extensively benchmarked elsewhere for three-dimensional dynamic calculations of immersed particlesmoving in Stokes’ flows (see, e.g., Dingman 1992;Dingman et al. 1992; Ingber and Mondy 1993). The detailsof the calculation technique, convergence tests, and otherconditions pertaining to the results below can also be foundelsewhere (Feng 2003).

In the present simulations, a spherical ball was initiallyplaced just under the upper surface of a Newtonian fluidcontained in a cylinder with radius Rcyl and length 22.5Rcyl. The ball was either at rest or with an initial nonzerovelocity at the beginning of each simulation. No-slipboundary conditions were applied to the wall and bottomof the cylinder, as well as the ball surface. The ball wasfalling under gravity, with a density difference between thefluid and the falling ball equal to 1 (in any consistent set ofunits). To model the upper free surface of the fluid, atraction-free boundary of the fluid surface was implementedby setting the tractions to be zero on the tangentialdirections of the surface and the velocities to be zero onthe normal direction of the surface. In other simulations, theeffect of lid on the upper surface of the fluid was modeledby setting zero velocities on all three directions of the upperfluid surface.

Figure 2a shows the predicted velocity along the lengthof a cylinder, after a ball is released from rest just under thetop surface, under conditions of af/Rcyl=0.085. The upperboundary is traction-free. As shown in this figure, thepredicted end-effect zones were less than one cylinderdiameter from either top or bottom, and there was a largeconstant-velocity zone between the end effects.

Although not shown in the figures, the simulationspredicted that a falling ball moves significantly faster nearan upper surface with a traction-free boundary than with astick boundary condition. For example, for a falling ball ina cylinder with af/Rcyl=0.1 and L/Dcyl=2.5 located veryclose to the top surface [(L−z)/Dcyl=0.02], the ball ispredicted to have a velocity V almost twice as fast near theslip surface (V/Vss=0.413) as it does near the stick surface(V/Vss=0.213). Here, Vss denotes the steady-state velocityfar from the ends. Note that in neither case, with the ballstarting from rest, did the falling ball’s early velocityexceed that observed in the steady-state region in themiddle of the containing cylinder. In addition, falling ballswith either boundary condition achieved more than 99% oftheir steady-state velocity within one cylinder diameter of

the upper surface. In other words, the distance required toachieve this steady-state velocity is fairly insensitive towhether the top boundary is stick or traction-free.

Next we examined the case where the ball was given aninitial velocity, at just below the liquid surface, equal to thevelocity it would acquire in air after freely falling 0.02 m(24.7 Vss). Results are also shown in Fig. 2a. After travelingabout 0.2 Rcyl distance in the liquid, the initial velocityeffect has completely disappeared and the velocity of theball is indistinguishable from the velocity it would haveacquired if initially at rest. At this point in the cylinder, theball is still accelerating to its final Vss.

Experimental reproducibility

Because the balls gained some momentum in the guide tubebefore hitting the surface of the liquid, initial experimentswere performed in the pure, Newtonian, suspending fluid totest whether this initial momentum affected the initialvelocity of the ball and to compare with the numericalresults above. Here, Dcyl=37.2 mm, L=834 mm, and6.35 mm<af<1.5875 mm. Within the resolution of ourapparatus (not within two cylinder diameters from the topsurface), we could detect no difference in the calculatedviscosity of the fluid when using balls of different densities(2.7 g/cm3<ρ<15 g/cm3) and sizes. We also tested thefluids under the conditions in which the top surface was incomplete contact with the lid. Our experiments were notsensitive enough to distinguish a difference in the viscosityin the first measurement zone of the cylinder betweenmeasurements done under the two conditions.

Next, a set of experiments was performed in a highlyconcentrated suspension (φ=0.5) held in a column with anaspect ratio of 5.9 and a suspension with Rcyl/as=24(Dcyl=76.2 mm, as=3.175 mm, and af=6.35 mm). Underthese conditions, the brass falling ball never reached aconstant velocity. This can be seen in Fig. 2d. A very largecylinder was prepared, also, for the same highly concen-trated suspension. The height of the cylinder was 1,707 mmand the liquid height was about 1,704 mm, giving an aspectratio of 22.4. With the same type of falling ball, initialmeasurements showed a constant-velocity zone could befound in this tall column. However, this large cylinder tooksuch a large amount of suspension and a large amount ofeffort to stir the suspension to maintain a uniformdistribution of the particles that we next tested a dimen-sionally similar cylinder in which all the dimensions,including the size of the suspending particles and thefalling ball, were approximately halved. Here, Dcyl=37.2 mm and the height of the liquid was about 834 mm.The experiments for φ=0.5 were repeated to ensure that thesame result for the velocity profile would be obtained forthe larger column. The initial experiments showed that the

Rheol Acta (2007) 46:413–424 417

ball in the taller column reached steady state at about 12.80diameters from the bottom with an error of +/−0.32. Thehalf-sized ball in the smaller but dimensionally similarcolumn reached steady state at 12.89 diameters from thebottom with an error of +/−0.74. We compared the meansand variance of the steady-state viscosities in the twocolumns. At 95% confidence level with a student t test, wecan conclude that viscosities in the two columns werestatistically indistinguishable. Figure 2d compares theresults in the smaller, but larger-aspect-ratio, cylinder withthose in the small-aspect-ratio cylinder.

Subsequent experiments in this series were done in thesmaller cylinder with an aspect ratio of 22.4. In this

cylinder, the suspension could be more easily maintainedat a constant temperature and stirred between experimentswithout excessive effort. The results are reported in Fig. 2and will be discussed in the next section.

Effect of solids fraction

Experiments were designed to investigate the end-effectzones as a function of ϕ. The fractions examined wereϕ=0.0, 0.1, 0.2, 0.3, 0.4, and 0.5. A single cylinder, onesuspending-ball size, and the same-sized settling ball wereused in all of the trials. Solids concentration was therefore theonly variable affecting the end-effect zone location. In these

z/Dcyl

0 5 10 15 20

V/V

ss

0.01

0.1

1

10=0%, BEM ball initially at rest

=0%, Experimental Results

=0%, BEM ball with high initial velocity

z/Dcyl

0 5 10 15 20

V/V

ss

0.0

0.2

0.4

0.6

0.8

1.0

1.2

φφ

=0%, Simulation Results=20%, Experimental Results

Z BEE

φ

z/Dcyl

0 5 10 15 20

V/V

ss

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

φφ

φφφ

=30%, Experimental Results

=0%, Simulation Results

Z BEE

ZTEE

z/Dcyl

0 5 10 15 20

V/V

ss

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6φφφ

=50%, Experimental Results (L/Dcyl=22.5) =0%, Simulation Results=50%, Experimental Results (L/Dcyl=5.9)

ZBEE ZT

EE

a

c

b

d

Fig. 2 a The axial velocity profile of a settling ball in a pureNewtonian fluid based on a BEM simulation and experimental balldrops. b A suspension of neutrally buoyant spheres with φ=0.2. Thevelocity profile for this moderately concentrated suspension stillexhibits near-Newtonian behavior. c A suspension of solid particleswith φ=0.3 relative to the Newtonian fluid compared with the BEMsimulation. This more concentrated suspension exhibits definite non-Newtonian behavior. The end effect begins influencing the falling ball

about seven diameters from the bottom and the top of the cylinder.d A suspension of solid particles with φ=0.5 relative to the Newtonianfluid, compared with the BEM simulation. The end effects in theseconcentrated suspensions are asymmetric and begin influencing thefalling ball about 13 diameters from the bottom of the cylinder andthree diameters from the top. In these experiments, each concentrationexamined had a steady-state velocity zone, and this steady-statevelocity, Vss, was used to normalize the results

418 Rheol ActaRheologica Acta (2007) 46:413–424

experiments, af/Rcyl=0.085, L/Dcyl=22.4, and Rcyl/as=24.Figure 2a shows the results in the suspending liquidcompared to the previously reported numerical calculations.The effects of the bottom surface cannot yet be seen atz/Dcyl=19, which agrees well with the numerical calcula-tions. By the time we can measure the velocity at the top,the ball is already at a steady-state viscosity, also consistentwith the numerical results.

The φ was increased systematically from 0.1 to 0.5.Samples of the velocity profiles are shown in Fig. 2b–d.The bottom end-effect zone, ZB

EE, was defined originally asthe halfway point between the last data point of constantvelocity and the first data point where the average velocitywas less than its steady-state value by at least 2%. This isshown in each figure. The cutoff value of 2% is a somewhatarbitrary choice given that the scatter in the data led touncertainty bars larger than this. However, this allowed adirect comparison to the computations, and seemed alogical starting point to study end effects. Figure 2b showsthe results for a moderately concentrated suspension(φ=0.2). The experimental results show that ZB

EE has grownto about 2.5 cylinder diameters, using this definition of ZB

EE.If, instead, we defined ZB

EE as halfway between the last datapoint of constant velocity and the first data point where theaverage velocity was different statistically from those thatmake up the steady-state value, ZB

EE would be approximate-ly the same as in the suspending Newtonian liquid alone.

Figure 2c,d clearly shows that more highly concentratedsuspensions (φ=0.3 and 0.5) produce non-Newtonian endeffects, no matter which definition of ZB

EE we use. Keepingwith the original definition using the 2% rule, the bottomend-effect zone grew to about 13 cylinder diameters fromthe bottom for the φ=0.5 case.

Continuing to keep the original definition of ZBEE, Fig. 3

shows the dependence of ZBEE on φ. Note that a pure

Newtonian fluid had ZBEE equal to about one cylinder

diameter. Suspensions with φ=0.1 showed near-Newtonianbehavior. In these suspensions with Rcyl/as=24, the lengthof the end-effect zone increased rapidly as ϕ increased. Theoverall trend in these results does not change substantiallyif the alternate definition of ZB

EE is used instead: here thecurve would start rising around φ=0.2 to a maximum ofabout 5 at φ=0.5. Because the trends are the same at thehigher concentrations, no matter which way we define ZB

EE,we will continue to use the original definition in theremainder of the paper.

It is interesting to note the contrast in the velocity profileat the top and bottom of the cylinder. The moderatelyconcentrated suspensions (φ=0.2) showed no end-effectzone at the top within our measurement range (past the firsttwo cylinder diameters). The more highly concentratedsuspensions (φ≥0.3) showed an end-effect zone at the topwhere the falling ball moves faster than its steady-state

value (Fig. 2c). Although not statistically distinguishablefrom the steady-state values, the data were consistentlyfaster in the zone denoted by ZT

EE, which extended to about7 Dcyl from the top. This observation is in marked contrastwith the numerical results for a Newtonian fluid in whichthe ball, even when initially at a high velocity, slows veryquickly (by about 0.1 Dcyl from the top) and thenaccelerates to the steady-state velocity within about 0.3Dcyl from the top. In the experiments with the Newtoniansuspending liquid, the initial deceleration of the ball fromits velocity in air to its steady-state velocity in the fluidmust also occur, albeit in a zone too small for us to detect.In the more concentrated suspensions, however, it appearsthat the deceleration occurs in a much larger zone, as ifthere is less resistance at the top of the suspension thanfurther down the cylinder. Also, it is interesting to note thatthis zone is much larger in the suspensions with φ=0.3 thanwith φ=0.5 (Fig. 2d). Also, ZT

EE is smaller than ZBEE in the

most highly concentrated suspension, although in thesuspension with φ=0.3, the two zones are approximatelythe same size. Like the bottom end effect, this enhanced topend effect was quite repeatable, with different suspensions,different cylinders, and with different densities of fallingballs; therefore, we do not believe that it could be caused byany slight settling of the suspended particles.

Effect of settling-ball size

The size of the settling ball was also a factor in determiningZBEE. Two studies were done for the pure Newtonian fluid.

φ 0 10 20 30 40 50

0

2

4

6

8

10

12

14

16

Rcyl /as=24

Fig. 3 Measurements of ZBEE, as a function of φ. In all these

experiments, the suspensions consisted of 1.5875-mm particles in a37.2-mm-diameter cylinder (Rcyl/as=24). The falling-ball diameterwas 3.175 mm. The pure fluid, φ=0.0, showed a ZB

EE of less than onecylinder diameter, and error bars are contained within the symbol. Theconcentrated suspension, φ=0.5, showed a bottom end effect of nearly13 diameters for Rcyl/as=24

Rheol Acta (2007) 46:413–424 419

These consisted of a series of numerical simulations usingthe BEM, followed by physical experiments. Thesenumerical experiments established that, in pure Newtonianfluids, the length of the end-effect zone got larger as thesettling-ball size increased. These observations are inagreement with those made earlier by Tanner (1963) andGraham et al. (1989). As shown in Fig. 4, the velocityprofile normalized by the steady-state velocity is, to a firstapproximation, only a function of the gap distance betweenthe ball surface and the cylinder bottom.

In the physical experiments, balls were dropped in thesame cylinder described in the previous section, which is37.2 mm in diameter and 835 mm tall. The balls droppedwere brass and had diameters of 1.5875, 2.3825, 3.1750,4.7625, 6.3500, 7.9375, and 9.5250 mm Reardon (2003).Data were in good agreement with measurements reportedearlier by Graham et al. (1989). An optimized fit of all theavailable data had the form

V=VBohlin ¼ 1� exp �8 z� af� ��

Rcyl

� � ð1Þwhere VBohlin is the steady-state velocity corrected for walleffects (Happel and Brenner, 1983) and z is the verticallocation. This observation is qualitatively similar to thatobserved earlier by Tanner (1963). Note a qualitativecomparison of experimental data to Tanner’s predictions(Tanner 1963) may also be found in the work by Grahamet al. (1989).

In a Newtonian fluid, ZBEE increases for the larger balls

and, in our experiments, was dependent on the gap betweenthe settling-ball surface and the cylinder bottom (Fig. 5).The containing walls create a backflow and reduce theterminal velocity of the settling ball. A smaller ball willtherefore be less affected than a larger ball, even though theball centers are at the same location.

This observation contrasts with the behavior observed inthe concentrated suspension where φ=0.5 and Rcyl/as=24(Fig. 5). The experiments in the suspensions showed thatthe bottom end-effect zone size decreased as the falling-ballsize increased. Again, the same cylinder and balls asdescribed above were used. The length of the bottom end-effect zone dropped from about 14 cylinder diameters tonine as the settling balls got larger. A plateau in ZB

EE seemedto be reached for af/Rcyl≥0.17. As will be shown in the nextsection, these conditions (Rcyl/as=24) produced the maxi-mum end effects for any particular value of af/Rcyl.

Note that additional experiments with 6.35-mm balls ofthe three different falling-ball densities used in thisinvestigation were performed in these same suspensions(φ=0.5 and Rcyl/as=24). The steady-state velocity wasfound to be proportional to the density difference betweenthe settling ball and the suspending fluid. However, thevelocity profiles were similar in all cases and the endeffects were independent of the density of the falling ballsReardon (2003).

(z-af)/Rcyl

0.0 0.2 0.4 0.6 0.8 1.0

V/V

Boh

lin

0.2

0.4

0.6

0.8

1.0

BEM 0.0356BEM 0.1 free surfaceBEM 0.238BEM 0.1 fixed surfaceTanner 0.0356Tanner 0.238V/Vss = 1-exp[-8(z-a

f)/R

cyl]

Experimental

Fig. 4 Comparison of the BEM simulation of various sizes of settlingballs. Included are Tanner’s (1963) results and the experimental dataas the settling balls near the bottom of a cylinder in a Newtonian fluid.The ZB

EE is a function of the gap (z−af) between the ball surface andthe bottom. The larger settling balls have a larger end-effect zone. Thevelocity is normalized with the expected Bohlin’s correction velocity(Bohlin 1960)

af/R

cyl

0.0 0.1 0.2 0.3 0.4

ZB

EE

0

2

4

6

8

10

12

14

16

Rcyl /as=24

Newtonian fluid

Fig. 5 Experimental data showing ZBEE as a function of af/R in a

concentrated suspension (φ=0.5, Rcyl/as=24) and a pure Newtonianfluid. Notice that unlike the pure fluid behavior where ZB

EE increaseswith af, the concentrated suspension shows that as the settling ball sizeincreases, the ZB

EE decreases

420 Rheol ActaRheologica Acta (2007) 46:413–424

Effect of number of suspending particles across the cylinder

Experiments to investigate the effect of the ratio Rcyl/as(the number of suspension particles across the cylinder) onthe location of the end-effect zone were also performed.The number of particles across was varied from 12 to about400, and the settling-ball-to-cylinder-radius ratio (af/Rcyl)was held constant at approximately 0.08. Different cylin-ders, suspending-ball sizes, and falling-ball sizes were usedin an effort to vary Rcyl/as while preserving the af/Rcyl ratio.These are summarized in Table 1. Note that the settlingballs were all brass and the suspensions were eithermonodisperse or polydisperse.

Figure 6 shows the results of these experiments. Thelength of the end-effect zone went through a maximum atabout 24 particles across and decreased to near-Newtonianbehavior at about 100 particles across and greater. For Rcyl/as≥100, both the suspensions of monodisperse and poly-disperse particles showed, within the resolution of theexperiment, similar results to those measured for theNewtonian fluid.

It is interesting to note that, at 35 particles across for thesuspension of polydisperse spheres (based on the area-averaged particle size) and 33 particles across for thesuspension of monodisperse spheres, polydispersivity in thesolids reduced the length of the bottom end-effect zonefrom 13 to 4.4 cylinder diameters. The suspension withRcyl/as=35 was formulated by combining 0.794-, 1.588-,3.175-, and 6.35-mm suspending particles using volumefractions of 0.03, 0.66, 0.28, and 0.03, respectively. Addinga small fraction of smaller-sized particles with an averagesize of 64 μm and standard deviation of 19 μm (2 wt.%)further reduced the bottom end-effect zone from 4.5 to 3.0.The experiments revealed an extreme sensitivity of thebehavior of highly concentrated suspensions to polydisper-sivity of the particulate phase and the shape of the particle-size distribution, as well as to the size of the container whenRcyl/as≤100 (Admuthe 2003).

Effect of the top surface

Starting from rest just under the surface of a Newtonianliquid contained in a long cylinder, a ball settling along thecenterline increases its velocity as it falls away from thesurface. As discussed earlier, within a cylinder diameter ofthe upper surface, the ball achieves a steady-state velocity.Also discussed earlier, numerical simulations for pureNewtonian fluids reveal that balls fall away from atraction-free interface more rapidly than from a stickinterface. After traversing the column, the ball begins toslow down as it approaches the bottom of the container. Inthe pure fluid, the length of the bottom end-effect zone wasfound to be comparable to that observed at the top of thecylinder. Figure 4 shows the effect of the approach of a ballto either a solid or a fixed surface. However, as shown inFig. 4, again the total length of this end-effect zone is fairlyinsensitive to whether or not the surface is a solid or a freesurface.

Similar behavior at the top of the cylinder was observedfor balls settling through dilute and moderately concentrat-ed suspensions. However, in concentrated suspensions, theballs in the top end-effect zone move significantly fasterthan in the steady-state region if the top of the cylinder is afree surface (Fig. 2c,d). In another series of experiments, acylinder lid manufactured for the containing cylindercontacted just the surface of the suspension, thus creatingan upper surface that was solid. Although not shown in thefigures, the increase in the settling velocity over thatobserved in the steady-state region in the concentratedsuspensions was much less than observed with the freesurface Reardon (2003). The velocity was only slightlyabove its steady-state value near the upper solid surface.This behavior of a ball falling away from a solid surface isin marked contrast to its behavior falling away from an air-suspension interface. Also, in contrast to what is observedin Newtonian liquids and in the more dilute suspensions,the ZT

EE is much larger in the moderately concentrated

Table 1 Parameters forexperiments to study Rcyl/as atφ=0.5

m monodisperse, p polydis-perse

Cylinderdiameter,Dcyl, mm

Cylinderlength,L, mm

Suspending-balldiameter, 2 as,mm

Falling-balldiameter, 2 af,mm

Particlesacross,Rcyl/as

af/Rcyl AverageReynoldsnumber, Re

37.2 835.7 835.7 3.175 12 (m) 0.0851 1.14e-323.4 524.8 524.8 1.981 15 (m) 0.0847 2.49e-537.2 835.8 835.8 3.175 24 (m) 0.0851 1.63e-425.4 524.8 524.8 1.981 33 (m) 0.0780 1.43e-376.2 767.3 767.3 6.350 48 (m) 0.0833 9.09e-276.2 767.3 767.3 6.350 96 (m) 0.083320.3 190.5 190.5 1.588 135 (p) 0.0782 1.86e-558.4 457.2 457.2 4.763 389 (p) 0.0815 7.14e-576.2 855.7 2.151 6.350 35 (p) 0.0833 1.67e-2

Rheol Acta (2007) 46:413–424 421

suspension (φ=0.3) than at a higher concentration (φ=0.5).This is illustrated in Fig. 2c,d.

In suspensions of very small particles (as=75 μm), thesuspensions are opaque and measurements were performedwith the eddy-current apparatus. Within the resolution ofthe apparatus, less than the length of the cylinder radius, theZTEE, and ZB

EE with both stick and traction-free boundaryconditions were indistinguishable from those in pureNewtonian fluids.

Discussion and conclusions

End effects in concentrated suspensions where Rcyl/as<100were markedly different from those observed in Newtonianliquids. The experiments examined how the end effectswere altered by changing φ, af/Rcyl, Rcyl/as, and containing-cylinder geometry. All of these parameters had an influenceon the location of the end-effect zone in highly concentratedsuspensions.

Past studies, such as those performed by Mondy et al.(1986) and Milliken et al. (1989), had assumed that the end-effect zone was not influencing the velocity of the settlingball in the center of a column more than a few cylinderdiameters from the top and the bottom, and the experimentswere confined to the middle third of a column. This wasclearly not the case when we used a cylinder with an aspectratio of 7.8 and a suspension with Rcyl/as<100. In theseexperiments, the highly concentrated suspensions (φ=0.5)

had no region where a constant velocity was observed.Perhaps the reason for the oversight by previous inves-tigators was that the vertical velocity profile has aninflection in the center of the column, and comparing theupper half of the center one-third of the column to the lowerhalf may have produced a difference that was too small tobe statistically significant. The use of columns of insuffi-cient lengths in the previous experiments resulted in anapparently higher viscosity compared with the actual valueobtained if the falling ball were allowed to reach steadystate. For example, for a column that is too short (aspectratio of 6.9) with Rcyl/as=24, the relative viscosity wasfound to be 13.5±1.5 in the middle third of the column. Ataller column (aspect ratio of 22.4) with the same Rcyl/aswas found to have a relative viscosity of 8.1±1.6 in thesteady-state-velocity zone that is free of end effects.

In pure Newtonian fluids, the top and bottom end effectswere comparable, and the end-effect zone got larger as thesettling-ball size increased. These findings contrast with thebehavior observed in concentrated suspensions in whichRcyl/as<100. In these suspensions, the length of the end-effect region was much larger than in a pure Newtonianfluid, and the top and bottom end-effect zones were nolonger of comparable length. Also, in contrast with New-tonian behavior, the experiments showed that ZB

EEdecreasedas the falling-ball size increased for suspensions whereφ=0.5 and Rcyl/as=24.

Finally, ZBEE in highly concentrated suspensions, all with

φ=0.5 but consisting of different suspending-particle sizes,was examined. The number of particle diameters across thecolumn was varied from 12 to about 400 using bothmonodisperse and polydisperse particles. The length of ZB

EE

went through a maximum at about Rcyl/as=24 anddecreased to near-Newtonian behavior at about Rcyl/as=100and greater. The length of ZB

EE in a suspension with apolydisperse particle size decreased dramatically from thatin a suspension of monodisperse particles.

Physical arguments consistent with the increased end-effect zone as a function of solids concentration are that (1)an effective hydrodynamic percolation limit exists betweenthe ball and the containing walls of the cylinder and/or (2)small inertial effects lead to restrictions in the recirculationof the suspension off the bottom. In the first case, as theball settles through the suspension, the suspended spheresmove together and form effective “chains” of particlesbetween the falling ball and containing walls. These chainsof suspended particles that would connect the falling ball tothe walls of the containing cylinder would result inadditional resistance on the settling ball. The smallersettling balls would be influenced by this hydrodynamicpercolation limit farther from the bottom than would thelarger balls, because the larger balls could break down thechains more easily. A similar argument was made by Brady

Rcyl/as

10 100 1000

ZB E

E

0

2

4

6

8

10

12

14

16

MonodispersePolydisperseMacropolydisperseMacropolydisperse with fines

Fig. 6 Measurements of ZBEE as a function of Rcyl/as. Notice that ZB

EEpeaks at about Rcyl/as≈24. In addition, suspensions with Rcyl/as≥100exhibit behavior that is indistinguishable from pure Newtonian fluidswithin the accuracy of our measurements. In these experiments, af/Rcyl≈0.08

422 Rheol ActaRheologica Acta (2007) 46:413–424

and Bossis (1985). In their simulation they observed thatparticles in sheared concentrated suspensions form many-particle clusters where stress can be transmitted longdistances by lubrication contacts. At high concentrations,the many-particle clusters create a percolation-like thresholdstructure.

This argument is also consistent with the observations ofthe reduction in the velocity enhancement near the uppersurface when a closed-vs-free surface was used in theexperiments. Note that both the number and effective lengthof these percolation limit chains would be increased as φincreases. Hence, the number of effective chains able toreach the cylindrical walls of the container would increaseas φ increased from 0.3 to 0.5. Thus, the relativecontribution of the chains from the settling ball to a fixedupper surface would be greater at φ=0.3, as there are fewerchains extending radially to the effective cylindrical wallsof the container. Hence, the relative effect of the uppersurface (either fixed or free) would be greater at φ=0.3 thanat φ=0.5, where there is a much greater fraction of chainsable to reach the vertical container surfaces. This finding isconsistent with our experimental observations that ZT

EE isgreater at φ=0.3 than at φ=0.5.

Note that monodisperse chains would be more effectivein transmitting force than would polydisperse chains, asvery small particles could act as spacers. These smallparticles would prevent very close contact of the largerparticles and would disrupt the chains to reduce theireffective strength. This finding is also consistent with ourobservations of the reduction in ZB

EE as the polydispersivityof the suspension increases.

The second argument hypothesizes that small inertialeffects result in the enhanced end effects. All settling ballswould be influenced by a restriction in the recirculation atboth ends of the cylinder because of small, but finite,inertial effects involved in moving finite-sized suspendedparticles. Again, the large falling balls would thereforeexperience less bottom end effects than the smaller ones.The finite-sized suspended spheres would be easier to pushaside at the top of the cylinder with a free surface, leadingto a faster velocity than when further down the cylinder.However, if a lid replaces the free surface, the falling ballcould accelerate only to the steady-state velocity. Thisargument is also consistent with very large end effectsobserved when the falling and suspending balls are ofcomparable size. However, without detailed microstructuralobservations, it would not be possible to confirm or denyeither of these arguments. Such studies are the focus ofongoing experiments in our laboratories.

Acknowledgements The authors are indebted to Allen Kaiser forhis pioneering studies that posed the questions that this work hasattempted to resolve. We also would like to thank Dr. James Abbottfor his guidance and aid throughout this study. Jim Bielenberg is

acknowledged for his insightful conversations concerning the potentialrole of percolation structures in explaining these phenomena. We aregrateful to Gerald Stoker, Sandia National, who designed and built theeddy-current apparatus. Sandia is a multiprogram laboratory operatedby Sandia, a Lockheed Martin company, for the United StatesDepartment of Energy’s National Nuclear Security Administrationunder contract DE-AC04-94AL85000. We also gratefully acknowl-edge the support for this work from Dr. Steven Girrens of Los AlamosNational Laboratory, allowing the authors the freedom to explore theseconcepts. The authors also acknowledge the support of the USDepartment of Energy’s Division of Chemical Sciences, Geosciencesand Biosciences, Office of Basic Energy Sciences; the USDepartment of Energy’s National Energy Technology Laboratory,Environmental Projects Division, National Energy TechnologyLaboratory; and the Advanced Research Program from Texas HigherEducation Coordinating Board.

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