Settling of an isolated spherical particle in a yield stress shear thinning fluid

Settling of an isolated spherical particle in a yield stress shearthinning fluid

A. M. V. Putz,1,a� T. I. Burghelea,1,b� I. A. Frigaard,1,2,c� and D. M. Martinez3,d�

1Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver,British Columbia V6T 1Z2, Canada2Department of Mechanical Engineering, University of British Columbia, 6250 Applied Science Lane,Vancouver, British Columbia V6T 1Z4, Canada3Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall,Vancouver, British Columbia V6T 1Z3, Canada

�Received 16 May 2007; accepted 7 January 2008; published online 12 March 2008�

We visualize the flow induced by an isolated non-Brownian spherical particle settling in a shearthinning yield stress fluid using particle image velocimetry. With Re�1, we show a breaking of thefore-aft symmetry and relate this to the rheological properties of the fluid. We find that the shape ofthe yield surface approximates that of an ovoid spheroid with its major axis approximately five timesgreater than the radius of the particle. The disagreement of our experimental findings with previousnumerical simulations is discussed. © 2008 American Institute of Physics.�DOI: 10.1063/1.2883937�

I. INTRODUCTION

The focus of the present work is an experimental studyof the flow induced by the motion of an isolated non-Brownian glass sphere settling in a yield stress fluid at lowReynolds number; i.e., Re�1. Although the settling processis found in a number of industrial and natural settings, thereare still many fundamental, unanswered questions regardingthe critical force required to initiate motion of particles inthis complex medium. This is mainly due to changes in themolecular organization of the fluid when under stress. Themacroscopic effect of this is the coexistence of yielded �flu-idized� and unyielded �solidlike� zones in the fluid domain.The motivation of this work stems from an interest in twoindustrial applications; namely, the separation of particles ofdifferent densities and the transport of small particles in oiland gas well construction operations.

Understanding the motion of a particle in a complex orstructured fluid is difficult. Insight into this phenomenon canbe gained by first examining the simpler case of the motionof particles settling slowly in a Newtonian fluid. For a singledense particle falling at low Reynolds number �Re�1� in aless dense solution, particles settle at the Stokes velocity1,2

Us =2��R2

9�g , �1�

where R is the sphere radius, ��=�p−� f is the differencebetween the density of the particles and that of the fluid, � isthe fluid viscosity, and g is the acceleration due to gravity. Itis clear from the abundant literature in this area that the flowaround the particle is symmetrical and that the sphere expe-riences a drag force that is proportional to the terminal

velocity.3,4 With yield stress fluids, this problem is morecomplex: the presence of a yield stress implies that settlingcan only occur if the net gravitational force is greater thanthe resistive force due to the yield stress of the material.Despite the simplicity of the problem, the mechanism bywhich the particle settles is poorly understood. Clearly, acritical force is required to initiate the motion of the particle.This force is proportional to the magnitude of the yield stressand related to the shape of the yielded envelope, but its exactvalue remains an open question for general particle shapes.

Before summarizing the existing literature, let us definethe problem under consideration mathematically. We willconsider the motion of a spherical particle of radius R anddensity �p settling in an quiescent fluid with a characteristicvelocity U. The fluid behaves �somewhat� like a Herschel–Bulkley fluid, which is traditionally characterized by threeparameters: the consistency k, a power law index n, and theyield stress �y. The density of the fluid � f is constrained suchthat �p�� f. When scaled using these parameters, the equa-tions of motion for both the fluid and the particle become�see Refs. 5 and 6 and references contained therein�

Re� �ui

�t+ uj

�ui

�xj� =

��ij

�xj−

�p

�xi,

�ui

�xi= 0, �2a�

Red�Up�i

dt= Re Ri�1 − �q�qi + �q�

�P

�ijnjdS , �2b�

Re� ��

�t− �J=� � � = �q�

�P

r� ��= n� �dS , �2c�

where u� denotes the fluid velocity; p the pressure and �= theextra stress tensor; U� p denotes the velocity of the center ofmass of the particle, � the angular rotation around the centerof mass, J= the inertia tensor, and r� denotes a material point ofthe particle. The usual Cartesian summation convention is

a�Electronic mail: [email protected]�Electronic mail: [email protected]�Electronic mail: [email protected]�Electronic mail: [email protected].

PHYSICS OF FLUIDS 20, 033102 �2008�

1070-6631/2008/20�3�/033102/11/$23.00 © 2008 American Institute of Physics20, 033102-1

used. The magnitude of the terminal velocity is denoted byU

p*. Using the characteristic scales and the material param-

eters, the Herschel–Bulkley constitutive model can be writ-ten as a two-parametric model

�̇ij = 0, if �=II � Bn,

�3�

�ij = ��̇=II

n−1 +Bn

�̇=II��̇ij, if �=II � Bn,

where M= IIª�MijMij denotes the second invariant of the

second-order tensor M= . This constitutive law introduces asingularity in the ��ij /�xj terms of the Navier–Stokes equa-tions, i.e., Eq. �2a�, as the rate of strain tensor �̇ij approacheszero, effectively prescribing an infinite viscosity in the un-yielded region. Four dimensionless groups are evident in Eq.�2�:

Re =�lR

nU2−n

k, Bn =

�yRn

Unk, �q =

� f

�p, Ri =

gR

U2 , �4�

where Re is the Reynolds number describing the ratio be-tween inertial and viscous forces; Bn is the Bingham orOldroyd number, which is the ratio between the yield stressand the viscous stresses; �q is the ratio of the densities of thefluid to that of the particle; and Ri is the Richardson number,representing the ratio between potential and kinetic energies.

A number of researchers have attempted to solve thissystem of equations, and we divide the simulation methodsinto three groups. In the first category are regularizationmodels, which avoid the singularity by regularizing the ef-fective viscosity as �̇

=II approaches zero �see Ref. 7 for a

detailed analysis of different types of regularization�. Onecan show that these regularization methods converge to thecorrect flow field, but due to their nature the exact position ofthe yield surface is difficult to recover; more specifically,there is no guaranteed convergence of the stress field. Bench-mark computations of this type have been conducted byBlackery and Mitsoulis8 and the convergence of a regular-ized method in this flow situation was studied by Liu et al.9

Sedimentation in a Herschel–Bulkley fluid can be found inBeaulne and Mitsoulis.10 A nice steady state treatment ofmultiple particles has appeared recently by Jie et al.,11 focus-ing on drag reduction. However, they do not explicitly solvefor the particle motion but instead prescribe the velocitiesand the relative positions of the spheres.

The second category of methods involves domain map-ping of the sheared region about the particle, thus removingany ambiguity about stress convergence. In this categoryfalls the work of Beris et al.,12 which we consider to be thebenchmark paper in this area. Combining a regularizedmodel with an intricate mapping of the yield surfaces onto astandard domain, they were able to calculate the position ofthe yield surface of a sedimenting sphere very accurately. Bydoing so they advanced the argument that two yield surfacesare evident: a kidney shaped surface in the far field and twosomewhat triangular cusps attached to the leading and trail-ing edges of sphere �Fig. 1�.

In the third category are simulations conducted usingaugmented Lagrangian schemes.13–15 Saramito et al.16 com-

bined the augmented Lagrangian method with anisotropicgrid refinement to obtain very accurate numerical results andhave applied this method successfully to the flow aroundsedimenting cylinders.

We now turn our attention to the experimental literature.An extensive summary of settling and sedimentation experi-ments in different media, including viscoplastic fluids, can befound in the book by Chhabra �Ref. 17, pp. 52–87�. Thisreview was published in 1993 and focuses generally on thecalculation of the drag coefficient and the terminal velocity;i.e., engineering properties useful for design purposes. A re-cent study by Tabuteau et al.18 focuses on the yielding crite-rion and the drag force and confirms the numerical and the-oretical predictions in Refs. 12 and 10. In a related paper,19

the appearance of shear waves generated by the settlingsphere is observed and the development of shocks and aMach cone under supercritical conditions is studied in detail.Recently there has been renewed interest in this problem dueto the availability of �digital� particle image analysis �PIV�.Gueslin et al.,20 for example, used this technique to measurethe flow field of a spherical particle settling in Laponite®, anextremely thixotropic yield stress fluid. The objective of thiswork was to study the aging properties of this fluid.

To summarize, what is clear from this body of work isthat during settling the flow is confined in the vicinity of theparticle within an envelope the size of which is related to theyield stress of the material.

The numerical simulations are limited to understandingthe steady state case with Re→0 and disregard elasticity andthixotropy. With regards to experimental work, there is lim-ited work available attempting to characterize the yield sur-face. Most studies report the total drag acting on the sphereat its terminal velocity. As a result, the objective of this studyis to visualize the motion of an isolated sphere settling in ashear thinning yield stress fluid in an attempt to characterizethe yield surface. We study the low Reynolds number sedi-mentation of a sphere at comparatively high values of theBingham number, and try to extract an estimate of the yieldsurface from the experimental data.

U

(1)

(2)

-g

FIG. 1. �Color online� Schematic illustrating the topology of yielded �region�1�: white� and unyielded flow regions �region �2�: blue�, according to thenumerical results by Beris �Ref. 12�.

033102-2 Putz et al. Phys. Fluids 20, 033102 �2008�

II. EXPERIMENTAL SETUP AND METHODS

Our apparatus is similar to the experimental systemsused by Gueslin et al.20 The experiments were performed ina 7.62 cm diameter Plexiglas tube with approximately threeliters of solution �see Fig. 2�. Atapattu et al.21 studied theinfluence of the container walls to the settling process indetail and they reported a critical radius beyond which noinfluence can be found. The tube diameter of our experimentwas chosen large enough to fulfill this criterion. The Plexi-glas tube was immersed in a rectangular tank �GC� filledwith water, as it has nearly the same index of refraction asthe Plexiglas. This minimized optical distortions.

Spherical particles were released through a slightly im-mersed funnel �DM� located on the central axis of the tube.Only one particle was released through the funnel and al-lowed to settle to the bottom before the next particle wasreleased. No mechanism was in place to hinder particle ro-tation during release. This simple deploy mechanism pro-duced only minimal variations in the radial position of theparticle. Two sets of glass spheres with diameters of3.9� 0.1� mm and 6.4� 0.1� mm were employed. The glassspheres had a density of approximately 2300 kg /m3. The useof glass minimized the shadow created from illumination.The main fluid used in this study was an aqueous solution ofCarbopol®-940 �Noveon�. The solutions were prepared bydissolving a known mass of Carbopol®-940 in distilled waterand then neutralizing it using 0.15 g / l NaOH under vigorousagitation for several hours. A glycerine-water solution, i.e., aNewtonian fluid, was also used for comparative purposes. Intotal five cases were conducted as outlined in Table I. TheNewtonian case is indicated by “Control” in this table.

Our visualization system consisted of a progressive scanDCAM charge coupled device �CCD� camera �Prosilica,EC750, 8 bit gray scale and a spatial resolution of 752480 with a maximum framing rate of 60 frames per sec-ond�, mounted with an F-mount Micro-NIKKOR 50 mmlens. A cross section of the tube was illuminated by a lasersheet �LS� created from a 100 mW laser �Photop Suwtech,DPGL 2100� emitting a 532 nm beam, which was expandedinto a laser sheet. The laser sheet was created by a telescopicarrangement of cylindrical lenses �CO�, where the first lens�a thin 1 mm diameter glass rod� expands the laser beam intoa vertical sheet and a second lens compresses the width ofthe laser sheet to approximately 0.2 mm in the beam waistregion. The laser sheet illuminated a plane intersecting thecentral vertical axis. The fluid velocity was measured using astandard particle image velocimetry �PIV� technique. Thefluid was seeded with 20 �m diameter particles at200 ppm �wt. % �. The seeding particles remained suspendedthrough the yield stress of fluid. With the optical setup de-scribed, the imaged area is 1321 mm, with its center lo-cated at 300 mm above the bottom of the tube. The camerawas interfaced to a computer and controlled by the DCAMcapturing and controlling tool CORIANDER, which saves thevideo stream as individual image files with an accurate time-stamp � 0.001 s�. These files and the time-stamps were thenanalyzed using MATLAB. Before any other image analysiswas conducted, the averaged background was subtractedfrom the original images and the range of image intensitieswere scaled to vary in the range �0, 1�. MATPIV,22 a MATLAB

based PIV analysis suite, was used to obtain the velocity

II II

I

III

FC

GC

LS

CCD

COL

DM(a) (b)FIG. 2. �Color online� �a� Schematicof the experimental setup. The nota-tion in the image is as follows:L—laser, CO—cylindrical optics,DM—deploy mechanism, CCD—camera, LS—laser sheet, GC—glasscontainer, FC—fluid container. �b�Schematic illustrating the relevantflow regions: �I� frontal region, �II� re-circulation zones, and �III� wake re-gion. The downwards arrow indicatesthe direction of gravity and the solidcircle represents the moving sphere.

TABLE I. The experimental conditions, the rheology, and the derived dimensionless quantities.

CaseR

�mm�c

�%�U

p*

�mm/s��y

�Pa�k

�Pa sn� n Rea Bn Ri �q

1 3.2 0.07 80 0.5 1.8 0.28 1.4100 0.11 4.8 0.43

2 1.95 0.07 17 0.5 1.8 0.28 8.710−2 0.15 66 0.43

3 3.2 0.08 3.3 1.4 5.5 0.22 2.010−3 0.25 2.9103 0.43

4 1.95 0.08 2.1 1.4 5.5 0.22 7.910−4 0.25 4.3103 0.43

Control 1.95 0 14 0 0.5 1 6.310−2 0 97 0.55

aThe terminal velocity Up* was set as the characteristic velocity U.

033102-3 Settling of an isolated spherical particle Phys. Fluids 20, 033102 �2008�

field and provided the filtering and smoothing algorithms forthe obtained velocity data.

The rheological properties of yield stress fluids includingCarbopol® solutions have been discussed by a number ofdifferent groups; e.g., by Kim et al.,23 Møller et al.,24 andrecently by Piau.25 A study of the microrheology ofCarbopol® can be found in Ref. 26 comparing the microscaleresults with the bulk rheology. It is widely accepted thatwhen neutralized, these solutions are thixotropic and have ayield stress, at least over the length and time scales of ourexperiment. A controversial discussion of the existence of a�by some definition true� yield stress can be found in Refs.27 and 28. In addition, when fully yielded they behave mac-roscopically as a shear thinning fluid. The rheological prop-erties of these solutions were determined using a Bohlin�now Malvern� C-VOR rheometer at 25 °C. To eliminatethixotropic effects, we applied 60 s of preshear followed by a60 s resting period before we started any rheological mea-surements. In order to minimize wall slip effects a vane toolgeometry was used, which, according to Stokes andTelford,29 does not exhibit any wall slip effects. Additionallywe compared the vane tool data with measurements usingrough, serrated plates. The datasets acquired with both ge-ometries were quite similar, displaying precisely the samedistinct flow features �see the discussion below�, except for aslight stress underestimation in the range of small appliedstresses.

As we are interested in slow flows close to the yieldingregime, our main interest lies in the transition from fluidliketo gel-like behavior and vice versa. We determined the strainrate-stress dependence �flow curve� of the Carbopol®-940 so-lutions by operating the rheometer in the controlled stressmode �as our sedimentation experiment is in a sense a stresscontrolled experiment; i.e., the stress is set by the buoyancyof the sphere and its surface area�. We set the range of thestresses from 0.01 Pa up to a value far enough above theyield stress to ensure a good fit with a Herschel–Bulkleyconstitutive model. Corresponding to each value of the ap-plied stress, the rate of strain, averaged over 6 s, has beenmeasured for both increasing �circles in Fig. 3� and decreas-ing �open squares in Fig. 3� values of the stress. As illus-trated in this representative figure for C=0.08% Carbopol®,we delineate three regimes:

�1� With 0.01 Pa���1 Pa, the strain increases linearlywith the stress �see the inset in Fig. 3� according to

� � � . �5�

We interpret this as an elastic response of the material inthis range of applied stresses.

�2� With 1 Pa���3.8 Pa, an abrupt increase of the rate offluid deformation is clearly visible. The response of thefluid in this regime is less understood, but we assumethat it is dominated by both shear banding and a compe-tition between breakage and rejuvenation of thematerial.18,19 A hysteresis is evident in regions �1� and�2�.

�3� Above these limits, i.e., in region �3�, the data are con-sistent with a Herschel–Bulkley model and the three pa-

rameters were estimated through regression �see TableI�. No hysteresis was observed in this region.

A similar observation can be made for the lower concentra-tion solution; however, the regions occur at different thresh-olds.

In addition to the controlled stress ramps, we have con-ducted oscillatory stress measurements for different values ofthe stress amplitude and measured the elastic and viscousmoduli of the material �see Fig. 4�. We note that the sameflow regimes as illustrated in Fig. 3 are visible in Fig. 4 aswell. For low stress amplitudes, the elastic modulus is largeand approximately independent on the stress value. This cor-responds to the elastic solid flow regime discussed above. Asthe stress amplitude is gradually increased, the elastic modu-lus decreases sharply but it is still larger than the viscousmodulus. In this range of stresses, both elastic and viscouscontributions are significant and this regime corresponds to

FIG. 3. �Color online� Flow curve of the 0.08% Carbopol® solution. ���Increasing stress; ��� decreasing stress, positive rates of strain; ��� decreas-ing stress, negative rates of strain. The solid �blue� line denotes theHerschel–Bulkley fit, the dashed black line denotes the linear fit. The fitvalues are summarized in Table I. The inset shows the dependence of thestrain � on the applied stress. The full line is a guide for the eye; �=0.042�.

FIG. 4. �Color online� Stress dependence of the elastic �squares� and viscous�circles� moduli for an oscillating stress measurement for 0.08% Carbopol®

at 2 Hz. Empty symbols refer to increasing stresses and filled symbols todecreasing stresses.

033102-4 Putz et al. Phys. Fluids 20, 033102 �2008�

the second flow region illustrated in Fig. 3 and discussedabove. When the stress amplitude becomes even larger, theviscous modulus exceeds the elastic one and flow is consis-tent with the third regime illustrated in Fig. 3. We also notethat a hysteresis similar to that illustrated in Fig. 3 is presentin Fig. 4 as well.

Another issue that deserves a brief discussion is thesteadiness of the rheological regimes �1� and �2� illustrated inFig. 3. Creep tests �data not shown here� at constant stressvalues corresponding to the second flow region discussedabove indicate that a steady rheological state �which we as-sess by linearity of the angular displacement with the creeptime� is reached for creep times of order of tss�15 s, that ismore than twice as large than the averaging time per stressvalue in the flow curve. These estimates clearly indicate thatthe rheological states presented in the flow curve above wereactually unsteady in the first two flow regimes. Next, weassess the steadiness of our sedimentation experiments bycomparing the characteristic time of each sedimentation ex-periment �which we estimate as tc=R /U

p*�. According to

Table I, the largest characteristic time is tc�0.97 s, which isagain smaller than tss. Thus, we can conclude that from thepoint of view of steadiness, the sedimentation data seems tolie in the same regime as our rheological data presented inFig. 3. This will further justify our attempt to understand thesedimentation flow patterns in the context of the rheologicalflow curves.

III. EXPERIMENTAL RESULTS AND DISCUSSION

A. Reproducibility and steady state velocities

Before proceeding to the main findings of this work, it isinstructive to begin by commenting on the precision of ourexperimental protocol. To do so we examine two aspects;namely, estimating the reproducibility of the terminal veloc-ity of the particles and then comparing our results to esti-mates of the settling rate in a Newtonian fluid �see Table I,control case�. We examined the reproducibility of the experi-ments in two different yet complementary experiments. Inthe first set of experiments we measured the settling time fora sequence of particles released at prescribed time intervals.We do so and find that Carbopol® ages and the reproducibil-ity of sedimentation experiments seems to be poor in thisfluid.17 Atapattu et al.30 advance the argument that the ex-periments can indeed become reproducible after a certainnumber of precursor spheres have been released. Harihara-puthiran et al.31 followed this in more detail by releasingseveral sequences of four spheres and with increasing timeintervals between the sets. For the purpose of our study, werepeat this work to delineate where this region exists. To doso we varied the time interval �t between releases from30 s to 30 min and measured the time for a sequence of par-ticles to fall 30 cm �see Fig. 5�. As shown, after a sequenceof approximately eight particles, the fall time was found tobe essentially constant. Significant healing was observedwith a time interval of several hours and the initial state canbe recovered after approximately 48 h. Tabuteau et al.18,19

claim that this thixotropic behavior can be eliminated using acareful and intensive mixing procedure.

In addition to this, we examine the change in position ofthe center of the particle as a function of time �see Fig. 6�.The results of the linear fit for the terminal velocity U

p* for

each case are presented in Table I. To ensure that we conductthe PIV measurements within the steady sedimentation re-gime, we repeat the experiment several times and comparethe slopes of space-time diagrams. As soon as we are satis-fied that these slopes are identical, we accept these runs forour PIV analysis.

Finally, to ensure that our measurement system is accu-rate, we repeated this prestudy with a Newtonian fluid andfound the flow field to be symmetrical �see Fig. 7� and that asteady state had been achieved with a terminal velocity U

p*

=14 mm s−1. This measured settling velocity is in goodagreement with the Stokes velocity Us=16 mm s−1 stated inEq. �1�.

FIG. 5. �Color online� Time tf for a sequence particles, released at pre-defined time intervals to fall a distance of 30 cm in a 0.07% Carbopol®

solution prepared with method 1. ��� First sequence of particles, ��� secondsequence 24 h later.

FIG. 6. �Color online� Distance d of the center of the spheres to thereference position at t=0 for the different cases as stated in Table I. The dataare fitted to a linear function of the form d=U

p*t with a terminal settling

velocity Up*.

033102-5 Settling of an isolated spherical particle Phys. Fluids 20, 033102 �2008�

B. PIV velocity fields

We now turn our attention to the main findings of ourwork; namely, the flow field around the sphere. One subtletyof the PIV method is that the quality of the measured veloc-ity field depends on the size of the interrogation windowsand the time between subsequent particles. The relativelyhigh sampling rate of 60 fps allows us to discretely vary thetime between image pairs by choosing pairs of the form�j , j+�j� for the PIV algorithm and hence allows us tochoose the optimal frame step �j for a given size of theinterrogation window. The optimal frame step �j has beenchosen adaptively �in relation to the flow speed� in order tokeep the mean particle displacement within the optimalrange of 5–15 pixels. However, due to high velocities, espe-cially in front of the particle and the lower density of seedingparticles close to the front part of the spheres, the resolutionof the velocity field in this region is notoriously poor, and weuse standard filtering techniques to remove those values.Here, we averaged approximately 100 images, and in Fig. 8we have plotted the averaged data at each spatial locationrelative to the center of the sphere.

As the particle is moving, we had to calculate the posi-tion of the particle for each frame. Several attempts to auto-mate this process using standard edge detection algorithmsfailed due to the inconsistent light intensity levels at theboundary of the sphere. In the end, the positions were calcu-lated semi-automatically with a simple MATLAB script. In-stead of edge detection, an approach based on a Mumford–Shah type energy minimization and image registration �seeRefs. 32 and 33� was used for single images, with greatsuccess, but has not yet been incorporated into an automaticroutine. This information allowed us to transform the coor-dinates of an individual frame to a set of coordinates with itsorigin in the center of mass of the particle, thus creating acommon set of coordinates for all images. To ensure that allthe discrete velocity data points coincided, we created a largemaster grid and interpolated all the individual frame dataonto the master grid. We recorded the number of operationsat each grid point of the master grid and used this informa-tion to calculate the weighted average of all the images of thesequence. In addition, the known time-stamps allowed us tocalculate the sphere velocity exactly and we replaced themeasured PIV field inside the particle with this velocity �see

Fig. 6�. If the particle has moved n cells in the original sta-tionary frame of reference, the number of cells on the mastergrid has to be increased by at least n cells, effectively enlarg-ing the field of view of the camera. As a final step, we inter-polated the velocity data onto a finer grid and used standardMATLAB routines to calculate the streamlines.

The first observation that can be made from the PIVfields �Fig. 8� is that the fore-aft symmetry is obviously bro-ken. This becomes especially clear if we transform the Car-tesian velocity field into its radial and azimuthal componentswith respect to the sphere as we display for case 4 �Fig. 9�.For all cases tested, we observe an extended flow region infront of the sphere which includes a pluglike zone. At theback of the sphere we observe a negative wake. These twofeatures are labeled as �I� and �III� in this figure. This findingis not evident in viscoplastic simulations based upon a Bing-ham or a Herschel–Bulkley model, especially at the smallReynolds number limit of our experiments.

The next observation that can be made is that a recircu-lation zone is evident in all cases tested, with its centerslightly downstream of the midplane of the sphere. This islabeled as point �II� in this figure. From the radial velocityfield the asymmetry of the vortex itself is apparent. The zerocontour line going through the center of the vortex clearlychanges slope, leading to high rates of strain in the part ofthe vortex close to the sphere.

Within the resolution of our PIV measurements, we didnot notice any systematic mismatch between the unfilteredand uninterpolated flow velocities near the sphere and thevelocity of the sphere �obtained separately by tracking thesphere in subsequent frames�. Therefore, if wall slip waspresent in our sedimentation experiments, it was small rela-tive to our instrumental resolution. We would like to pointout that the nature of the surfaces coming into contact duringour sedimentation experiments �Carbopol® solution versussmooth glass� is similar to the contact surfaces in our rheo-logical measurements: Carbopol® solution versus smoothstainless steel. Thus, even if nonzero, an apparent slip in oursedimentation experiments should be present in our rheologi-cal measurements as well. This is why we believe that asemiquantitative comparison between our flow data and therheological measurements is motivated.

C. Asymmetry and negative wake

The most striking feature of these flow patterns is theirapparent asymmetry. One possible explanation for the asym-metry would be the influence of inertia. We disregard thisexplanation, as the Reynolds number tested in all cases issmall and for the corresponding Newtonian case �Fig. 7�, athigher Reynolds number, inertial effects are not evident. Fi-nally, based on Newtonian studies, we would expect inertialasymmetries to manifest differently �see Ref. 3�.

We speculate that the negative wake results from elasticeffects—a phenomenon clearly present in the flow curves atlow strain rates in the regions �1� and �2� of the flowrate �seeFig. 3�. In related literature, numerical simulations using vis-coelastic constitutive models have shown the existence of anegative wake; i.e., an upward flow of fluid in the wake of

−5 0 5−4 −3 −2 −1 1 2 3 4

0

1

-1

2

-2

3

−0.012

−0.008

−0.004

0

FIG. 7. �Color online� Average PIV velocity field for the Newtonian case.Note: Particles move from right to left. The color map refers to the modulusof velocity and the full lines are streamlines.

033102-6 Putz et al. Phys. Fluids 20, 033102 �2008�

the particle. We would like to point out that the fluids in-volved in these studies had no apparent yield stress. A studyof the influence of the flow conditions has been presented byKim et al.34 Recent simulations using a lattice Boltzmannapproach combined with a Maxwell model can be found inRefs. 35 and 36. Harlen37 used the finite element method tosimulate the flow around a sphere using the Peterlin�FENE-P� and alternatively the Chilcott and Rallison�FENE-CR� closure approximations to the Finite ExtendibleNonlinear Elastic model �see Refs. 38 and 39�. He concludedthat the shape of the downstream velocity wake is governedby the competition of two forces. The decay of the velocity islengthened by high extensional stresses which are opposedby an elastic recoil of the shear stress. It is the latter forcethat is claimed to be responsible for the appearance of thenegative wake.

To help characterize the magnitude of the asymmetry, weplot the velocity component along the centerline of thesphere �see Fig. 10�. From the velocity profile we can con-struct a similar picture to the flow curve. The stresses closeto the sphere are greater than the yield stress except in thesmall region in front of the sphere and we are in the fullyyielded state of the material �region �3��. Far away from thesphere, at values far below the yield stress, only a pure elas-tic contribution is felt by the material and the material is in apurely unyielded state. In region �2�, we claim to be in thetransition region of the flow curve and we claim that stressrelaxation is responsible for the negative wake. This alsoallows us to draw a parallel between the flow around a set-tling sphere and the flow in the rheometer. The increasingstress curve of the flow curve corresponds to the downstreampart of the flow of the sphere and consequently the decreas-

FIG. 8. �Color online� Flow for cases �1�–�4�. Note:Particles move from right to left. The color map refersto the modulus of velocity and the full lines are stream-lines. For clarity, we display only a fraction 1 /25 of thetotal velocity vectors.

033102-7 Settling of an isolated spherical particle Phys. Fluids 20, 033102 �2008�

ing stress curve corresponds to the upstream velocity field.To gain qualitative insight, we can compare the ratio �r be-tween the rates of strain of the increasing stress test versusthe rates of strain of the decreasing stress test �see Fig. 11�a��to the ratio � f between the upstream and downstream veloc-ity along the vertical particle axis �see Fig. 11�b��. For a fullysymmetric situation, both these ratios should be constant andequal to one. In our case, the ratio based on the velocityprofile �� f� follows a pattern similar to the ratio based on theflow curve �r. Very close to the sphere, where we suspect thehighest stresses, the ratio is close to 1 �region 1 in Fig. 11�.With increasing distance from the sphere, the stress in thefluid decreases and we enter the hysteresis regime �region 2�.We observe the same kind of singularity both in �r and � f.

Arigo and McKinley40 suggest an exponential decay ofthe velocity in the upstream direction, close to the sphere.We use such a fit in Fig. 10 and found it to be

u

Up= e−r/�0.20 0.02�R, �6�

with a corresponding reduced �2 error of 0.01. The weightsfor the calculation of the reduced �2 error have been chosento be equal to one. A similar decay law can be applied to thefront region. An exponential scaling for the frontal region iswithout strict theoretical foundation, but a least-squares fit to

u

Up= e−�r+�1.14 0.07��/��1.24 0.06�R�, �7�

leads to a reduced �2 error of 0.003.For the experimental case 4, enough raw data points

were available to use cubic B-spline interpolation to the datapoints. To accommodate the pluglike appearance in the frontregion, use can be made of the C2 continuity of the B-splineby adding artificial data points inside the sphere. Both fitsallow the exact calculation of their derivative, and hencedirect access to the rate of strain along the centerline �seeFig. 12�. We note the fast decay of the rate of strain in theupstream region from a value which lies clearly in theHerschel–Bulkley zone of the flow curve to a rate of strain of0.1 s−1, where we claim the onset of hysteresis and elasticeffects in the corresponding flow curve for case 3 and case 4.This fast decay contrasts to the slower decay in the down-stream section of the sphere, indicating that the fore yielded�Herschel–Bulkley� zone is considerably larger than the aftone.

In case 4 we can compare the exponential fits directlywith the spline interpolation of the data. In the upstreamregion of the sphere we find a good agreement close to thesphere, but the exponential fit alone cannot describe the elas-tic recoil, which is clearly visible in the spline. However, inthe downstream section, the spline fit differs greatly from theexponential fit. Firstly, we find a change of dominating slopebetween the hysteresis and elastic region below 0.1 s−1 andthe Herschel–Bulkley dominated region above this value. Wealso note that the rate of strain reaches the same maximalvalue as the exponential fit, but at half a radius distanceupstream of the sphere, and decays to zero from there, indi-cating the existence of a low shear of even pluglike regionattached to the upstream part of the sphere.

D. Yielded and unyielded regions

The final analysis of our data is to attempt to extract ayield surface from the PIV velocity fields. By definition ofthe term “yield surface” �see Beris et al.12 and Duvaunt andLions41�, we would call the boundary of an unyielded region,i.e., a region where the second invariant of the rate of sheartensor is zero, a yield surface. In principle, we can calculatethis quantity from our PIV data �Fig. 13� simply by calculat-ing the extremal zero contours, but we would not expect tosatisfy this exact criterion because of the noise level in ourPIV measurements. However, if we take our rheological situ-ation into account, we do not expect this criterion based on

(1) (3)

(2) (1)

FIG. 10. �Color online� Scaled velocity modulus u along the centerline ofthe particle vs the scaled radial distance r from the sphere center. Theshaded region in the graph denotes the area of poor PIV resolution in frontof the sphere and inside the sphere itself. The numbers correspond the rheo-logical states defined in Fig. 3. Inset: Magnification of the region behind thesphere. The solid line denotes an exponential fit. Case 1 ���, case 2 ���,case 3 ���, and case 4 ���.

0 2 4 6

0

1

1

0

0

0 2 4 6

0

1

1

00

0

I III

II

III

(a)

(b)

0

0

FIG. 9. �Color online� Polar components of the velocity field for case 4: �a�radial component and �b� azimuthal component

033102-8 Putz et al. Phys. Fluids 20, 033102 �2008�

the Herschel–Bulkley material law to hold, as we find a morecomplicated behavior of our fluid close to the yield stress.Instead of the exact zero level set, we would like to detectthe transition between the flow regime �3� and the hysteresis/elastic regime �1/2�. Insofar as the flow situation in the rhe-ometer is transferable to the more complex flow around thesphere, this would correspond to a level set of approximately0.1 s−1. This level set is well resolved in case �4� as thevelocities corresponding to these small rates of strain arewithin the optimal PIV resolution range. In the data of case3, the level set 0.1 s−1, are outside the range of the PIVmeasurements.

As shown in Fig. 13, the shape of the yield surface ap-proximates that of an ovoid spheroid with its major axisapproximately five times greater than the particle radius. Wealso note, that there exists a small island of relatively lowrates of strain immediately downstream of the sphere, wherewe observe extensional flow. This suggests the existence of apluglike zone attached to the pole of the sphere. No suchzone is observed in the upstream section.

IV. CONCLUSIONS

We have presented a detailed experimental investigationof the flow fields around a spherical object falling freely in ashear thinning yield-stress fluid. The following are our mainexperimental findings:

In all the cases we have explored �except the test casewhere a Newtonian fluid was used�, the flow fields display astrong fore-aft asymmetry. We present a first attempt at un-derstanding the underlying physical reasons of this asymme-try by establishing a direct connection between experimen-tally measured flow fields and the rheological properties ofthe fluid. Our controlled stress rheological measurementsdisplay three distinct flow regimes. The first one, correspond-ing to low rates of deformations in the material, displays thecharacteristics of an �elastic� solidlike region. Increasing theapplied stresses further, the yielding of the material does notoccur at once: there exists a range of applied stresses wherefluidized regions coexist with solidlike regions.

Corresponding to this flow region, the rheological dataacquired at increasing stresses does not overlap with thatacquired at decreasing stresses. This suggests that, in thisrange of applied stresses, yielding of the material and itsrelaxation to the initial �elastic� solid state follow a differentsequence of intermediate states. We have investigated furtherthis rather unexpected behavior by conducting a sequence ofoscillation measurements for several values of the appliedstress amplitude. The same three flow regimes are identifiedfrom the oscillatory measurements as well as the nonrepro-ducible character of the transition among them. We note herethat the transition between different flow regimes in a physi-cal gel is interesting by itself, and certainly deserves a par-ticular focus. However, a detailed investigation of theCarbopol® rheology around the solid-fluid transition is be-yond the scope of the current work; a more complete accountof our findings will be published elsewhere.

By comparing the typical values of the rate of strain inboth rheological measurements and flow fields, we concludethat this second flow region is of direct relevance to thestrong fore-aft asymmetry. Indeed, we find a striking consis-tency between the magnitude of the hysteresis measuredfrom the rheological flow curves �Fig. 3� and the magnitudeof the fore-aft asymmetry measured from the flow curves�Fig. 11�. This consistency can be explained as follows. Thefore region of the flow should be associated with the increas-ing stress part of the flow curve. Indeed, a sedimentationexperiment is actually a “stress-controlled” experiment, inthe sense that a gradually increasing stress is applied to thematerial located in front of the moving object. The regionbehind the moving object is complementary: the fluid relaxes

r

FIG. 11. �Color online� �a� Plot of theratio �r between the increasing stresstest vs the decreasing stress test. �b�Plot of the velocity ratio � f betweenthe front and wake velocity of thesphere vs the distance r from the cen-ter normalized by particle radius R.Case 1 ���, case 2 ���, case 3 ���,and case 4 ���.

FIG. 12. �Color online� Rates of strain calculated from the exponential�dashed line� or the B-spline �solid line� fit for case 4 in Table I.

033102-9 Settling of an isolated spherical particle Phys. Fluids 20, 033102 �2008�

and the applied stresses decay, which is consistent with thedecreasing stress part of the flow curve.

Consistent with previous experiments, e.g., Ref. 20, weobserve the formation of a negative wake behind the movingobject. Although we did not quantitatively verify the crite-rion of formation of the negative wake as Arigo andMcKinley did,40 based on stress relaxation measurements�data not shown here�, we tend to believe that the explana-tion proposed by Harlen,37 is correct. However, this may bethe object of a future study, where extensional viscosity mea-surements are correlated with direct measurements of theflow fields.

We present a first experimental assessment of the posi-tion of the yield surface around the moving object. Our resultis not consistent with the numerical result by Beris et al.; theprimary reason for this inconsistency is that the rheologicalmodel used by Beris et al., namely, the Bingham model, doesnot properly describe the hysteresis in the flow curve wediscussed, being valid only for large values of the appliedstresses �region 3�. Based on our data, the relevant flow re-gions behind and in front of the moving object, correspond tothe second region of the flow curve. A central conclusion ofour work is that a future theoretical/numerical work aimingto properly characterize the flow fields we have characterizedexperimentally should use a more sophisticated rheologicalmodel which properly accounts for the hysteresis of the flowcurve. Closely related to that would be a better understand-ing of the solid-fluid transition in a viscoplastic yield stressfluid.

1G. G. Stokes, “On the effect of the internal friction of fluids on the motionof pendulums,” Trans. Cambridge Philos. Soc. 9, 8 �1851� �reprinted in G.Stokes, J. Larmor, and J. Rayleigh, Mathematical and Physical Papers�Cambridge University Press, Cambridge, 1880–1905�, Vol. 3, p. 1�.

2E. Guyon, J.-P. Hulin, and C. D. Mitescu, Physical Hydrodynamics �Ox-ford University Press, New York, 2001�.

3G. K. Batchelor, An Introduction To Fluid Dynamics �Cambridge Univer-sity Press, New York, 2000�.

4L. Landau and E. Lifschitz, Fluid Mechanics �Pergamon, Oxford, 1987�.5R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph, and J. Periaux, “Adistributed Lagrange multiplier / fictitious domain method for the simula-tion of flow around moving rigid bodies: Application to particulate flow,”Comput. Methods Appl. Mech. Eng. 184, 241 �2000�.

6Z. Yu, A. Wachs, and Y. Peysson, “Numerical simulation of particle sedi-mentation in shear-thinning fluids with a fictitious domain method,” J.Non-Newtonian Fluid Mech. 136, 126 �2006�.

7I. Frigaard and C. Nouar, “On the usage of viscosity regularisation meth-ods for visco-plastic fluid flow computation,” J. Non-Newtonian FluidMech. 127, 1 �2005�.

8J. Blackery and E. Mitsoulis, “Creeping motion of a sphere in tubes filledwith a Bingham plastic material,” J. Non-Newtonian Fluid Mech. 70, 59�1997�.

9B. T. Liu, S. J. Muller, and M. Denn, “Convergence of a regularizationmethod for creeping flow of a Bingham material about a rigid sphere,” J.Non-Newtonian Fluid Mech. 102, 179 �2002�.

10M. Beaulne and E. Mitsoulis, “Creeping motion of a sphere in tubes filledwith Herschel–Bulkley fluids,” J. Non-Newtonian Fluid Mech. 72, 55�1997�.

11P. Jie and K.-Q. Zhu, “Drag force of interacting coaxial spheres in visco-plastic fluids,” J. Non-Newtonian Fluid Mech. 135, 83 �2006�.

12A. N. Beris, J. A. Tsamopoulos, R. C. Armstrong, and R. A. Brown,“Creeping motion of a sphere through a Bingham plastic,” J. Fluid Mech.158, 219 �1985�.

13R. Glowinski, Numerical Methods For Nonlinear Variational Problems�Springer-Verlag, New York, 1983�.

14R. Glowinski and P. Le Tallek, Augmented Lagrangian and OperatorSplitting Methods in Nonlinear Mechanics, in Studies in Applied and Nu-merical Mathematics �Society for Industrial & Applied Mathematics,Philadelphia, 1987�.

15M. Fortin and R. Glowinski, Augmented Lagrangian Methods �North-Holland, Amsterdam, 1983�.

16N. Roquet and P. Saramito, “An adaptive finite element method for Bing-ham fluid flows around a cylinder,” Comput. Methods Appl. Mech. Eng.192, 3317 �2003�.

0.2

1

20.

25

0.5

0.75 1

1.25

1.25

0.75 2

−4 −2 0 2 4

0

1

2

−6 −4 −2 0 2 4 6

0

2

4

2

1

0

0.1

0.25

0.5

0.75

11

0.5 2

1.25

1.25

(b)

(a)

FIG. 13. �Color online� Sample calculation of the sec-ond invariant �a� for case 3 and �b� for case 4.

033102-10 Putz et al. Phys. Fluids 20, 033102 �2008�

17R. P. Chhabra, Bubbles, Drops, And Particles in Non-Newtonian Fluids�CRC, Boca Raton, FL, 1993�.

18H. Tabuteau, P. Coussot, and J. R. de Bruyn, “Drag force on a sphere insteady motion through a yield-stress fluid,” J. Rheol. 51, 125 �2007�.

19H. Tabuteau, D. Sikorski, and J. R. de Bruyn, “Shear waves and shocks insoft solids,” Phys. Rev. E 75, 012201 �2007�.

20B. Gueslin, L. Talini, B. Herzhaft, Y. Peysson, and C. Allain, “Flow in-duced by a sphere settling in an aging yield-stress fluid,” Phys. Fluids 18,103101 �2006�.

21D. D. Atapattu, R. P. Chhabra, and P. H. T. Uhlherr, “Wall effect forspheres falling at small Reynolds number in a viscoplastic medium,” J.Non-Newtonian Fluid Mech. 38, 31 �1990�.

22J. K. Sveen, “An Introduction to MatPIV v.1.6.1,” Eprint No. 2, ISSN0809-4403, Department of Mathematics, University of Oslo �2004�,http://www.math.uio.no/~jks/matpiv.

23J.-Y. Kim, J.-Y. Song, E.-J. Lee, and S.-K. Park, “Rheological propertiesand microstructures of Carbopol gel network system,” Colloid Polym. Sci.281, 614 �2003�.

24P. C. F. Møller, J. Mewis, and D. Bonn, “Yield stress and thixotropy: onthe difficulty of measuring yield stresses in practice.” Soft Mater. 2, 274�2006�.

25J. Piau, “Carbopol gels: Elastoviscoplastic and slippery glasses made ofindividual swollen sponges: Meso- and macroscopic properties, constitu-tive equations and scaling laws,” J. Non-Newtonian Fluid Mech. 144, 1�2007�.

26F. K. Oppong and J. R. de Bruyn, “Diffusion of microscopic tracer par-ticles in a yield-stress fluid,” J. Non-Newtonian Fluid Mech. 142, 104�2007�.

27K. Barnes and H. A. Walters, “The yield stress myth?” Rheol. Acta 24,323 �1985�.

28H. A. Barnes, “The yield stress—a review or ‘����� ���’-everythingflows?” J. Non-Newtonian Fluid Mech. 81, 133 �1999�.

29J. Stokes and J. Telford, “Measuring the yield behaviour of structured

fluids,” J. Non-Newtonian Fluid Mech. 124, 137 �2004�.30D. D. Atapattu, R. P. Chhabra, P. Raj, and P. H. T. Uhlherr, “Creeping

sphere motion in Herschel–Bulkley fluids: Flow field and drag,” J. Non-Newtonian Fluid Mech. 59, 245 �1995�.

31M. Hariharaputhiran, R. Shankar Subramanian, G. A. Campell, R. P.Chhabra, “The settling of spheres in a viscoplastic fluid,” J. Non-Newtonian Fluid Mech. 79, 87 �1998�.

32M. Droske and W. Ring, “A Mumford–Shah level set approach for geo-metric image registration,” SIAM J. Appl. Math. 66, 2127 �2006�.

33S. Angenent, E. Pichon, and E. Tannenbaum, “Mathematical methods inmedical image processing,” Bull., New Ser., Am. Math. Soc. 43, 365�2006�.

34J. M. Kim, C. Kim, C. Chung, K. H. Ahn, and S. J. Lee, “Negative wakegeneration of FENE-CR fluids in uniform and Poiseuille flows past acylinder,” Rheol. Acta 44, 600 �2005�.

35X. Frank and Z. L. Huai, “Negative wake behind a sphere rising in vis-coelastic fluids: A lattice Boltzmann investigation,” Phys. Rev. E 74,056307 �2006�.

36X. Frank, M. Kemiha, S. Poncin, and H. Li, “Origin of the negative wakebehind a bubble rising in non-Newtonian fluids,” Chem. Eng. Sci. 61,4041 �2006�.

37O. G. Harlen, “The negative wake behind a sphere sedimenting through aviscoelastic fluid,” J. Non-Newtonian Fluid Mech. 108, 411 �2002�.

38B. R. Bird and O. Hassager, Fluid Mechanics, in Dynamics of PolymerLiquids Vol. 1, 2nd ed. �Wiley-Interscience, New York, 1987�,

39R. G. Owens and T. N. Phillips, Computational Rheology �Imperial Col-lege Press, London, 2002�.

40M. T. Arigo and G. H. McKinley, “An experimental investigation of nega-tive wakes behind spheres settling in a shear-thinning viscoelastic fluid,”Rheol. Acta 37, 307 �1998�.

41G. Duvaunt and J. L. Lions, Inequalities in Mechanics and Physics�Springer-Verlag, New York, 1976�.

033102-11 Settling of an isolated spherical particle Phys. Fluids 20, 033102 �2008�