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J. Non-Newtonian Fluid Mech. 158 (2009) 132–141

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

imulation of viscoplastic flow past cylinders in tubes

van Mitsoulis ∗, Spyros Galazoulaschool of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou 15780, Athens, Greece

r t i c l e i n f o

rticle history:eceived 9 May 2008eceived in revised form 7 October 2008ccepted 10 October 2008

a b s t r a c t

Drag flow past circular cylinders concentrically placed in a tube filled with viscoplastic fluids, obeyingthe Herschel–Bulkley model, is analyzed via numerical simulations with the finite element method. Thepurpose it to find limiting drag values for cessation of motion of the object in steady flow. Different aspectratios have been studied ranging from a disk to a long cylinder. For the simulations, the viscoplastic model

eywords:ield stressiscoplasticityerschel–Bulkley modellow past cylindersrag coefficient

is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in theflow field in both yielded and practically unyielded regions. The extent and shape of yielded/unyieldedregions are determined along with the drag coefficient for a wide range of Bingham numbers. The sim-ulation results are compared with previous experimental values [L. Jossic, A. Magnin, AIChE J. 47 (2001)2666–2672] for cessation of flow. They show that the values of the drag coefficient are lowest for the diskand highest for the long cylinder. Discrepancies are found and discussed between the simulations and the

ulati

fevm[t

�

wson[ccm

m

essation of flow experiments, with the sim

. Introduction

An important class of non-Newtonian materials exhibits a yieldtress, which must be exceeded before significant deformation canccur. A list of several materials exhibiting yield was given in a sem-nal paper by Bird et al. [1]. The models presented for such so-callediscoplastic materials included the Bingham, Herschel–Bulkley andasson. Analytical solutions were provided for the Bingham plas-ic model in simple flow fields. Since then a renewed interest haseveloped among several researchers to study these materials inon-trivial flows both numerically and experimentally. A majoreview article on the subject of yield stress appeared in 1999 byarnes [2], where the reader can find a wealth of references (about60) and the latest (at the time) conclusions.

One important aspect of yield-stress (or viscoplastic) fluids isheir ability to stop motion of objects suspended in them, whenhe yield stress forces balance the gravity forces. This phenomenonas been termed (perhaps incorrectly) “stability” of objects in auspending viscoplastic fluid. What is really meant is a limitingondition (criterion) for cessation of motion of an object in steadyow. The significance of such phenomena can be found in diverse

elds of chemical engineering, from the stability of cosmetics to

ood texture during chewing and swallowing, to drilling muds, toandslides.

∗ Corresponding author.E-mail address: mitsouli@metal.ntua.gr (E. Mitsoulis).

Bsefcob[

377-0257/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2008.10.006

ons providing lower values in the limit of very high Bingham numbers.© 2008 Elsevier B.V. All rights reserved.

Early studies to find such a criterion focused on the simplestorm of flow past a sphere or a cylinder, which have been studiedxtensively in the fluid mechanics community [2,3]. The limitingalue for cessation of sphere motion in an infinite Bingham plasticedium was found numerically in a pioneering work by Beris et al.

4] to be �∗y = 0.143, where �∗

y is a dimensionless yield stress relatedo the limiting drag force F by

∗y = 2�y�R2

F(1)

here �y is the yield stress of the plastic medium and R is thephere radius. The corresponding results for the planar problemf drag around a cylinder including wall effects were also foundumerically by Mitsoulis [5]. The experiments by Jossic and Magnin3] offer results for a number of objects, including spheres andylinders, but also other objects, which do not move symmetri-ally (cone) or cause a full three-dimensional flow (cube, cylindersoving perpendicular to their axis).New experiments and simulations in the last decade have made

ajor contributions to the field. Experimentally, studies by deruyn, Coussot, and co-workers [6–10] and others [11,12] havehown the difficulty of measuring accurately the yield-stress prop-rties of viscoplastic fluids and of determining the limiting criterion

or flow around a sphere found numerically by Beris et al. [4]. A suc-essful agreement between theory and experiments was achievednly recently [9]. On the other hand, the material of choice (Car-opol) may not be as ideal a viscoplastic fluid as once thought of12].

wtonian Fluid Mech. 158 (2009) 132–141 133

cdcHdwpowgp

2

2

toai

FtTd

Table 1Rheological characteristics for the Carbopol gels used in the experiments by Jossicand Magnin [3] and the corresponding parameters for the Herschel–Bulkley model(Eq. (4)).

Viscoplasticmaterial

Yield stress �y

(kPa)Consistency indexK (kPa sn)

Power-lawindex n (−)

GGGG

csmvfl

∇

E. Mitsoulis, S. Galazoulas / J. Non-Ne

For the present work, we study the creeping flow of circularylinders moving symmetrically parallel to their axis inside a cylin-rical vessel filled with viscoplastic media. Using an appropriateontinuous viscoplastic model, new results will be provided forerschel–Bulkley fluids for different aspect ratios ranging from aisk to a long cylinder. Finally, the corresponding drag coefficientsill be calculated for a wide range of Bingham numbers, with theurpose of finding the limiting conditions for cessation of motionf the cylindrical object. The simulation results will be comparedith the experiments by Jossic and Magnin [3] for the viscoplastic

els used in their study, and conclusions will be drawn regardingossible discrepancies.

. Mathematical modelling

.1. Governing equations

The problem of the creeping flow of a viscoplastic material under

he influence of a drag force (Couette flow) past a circular cylinderf diameter D and length L moving symmetrically parallel to itsxis inside a cylindrical vessel of diameter Dres and infinite lengths schematically depicted in Fig. 1. There is equivalence between

ig. 1. Schematic representation of creeping axisymmetric drag flow of a viscoplas-ic medium past a cylinder of diameter D inside a cylindrical vessel of diameter Dres .he inner cylinder is considered stationary and the vessel is moving with a constantrag velocity U giving rise to a plug velocity profile away from the cylinder.

0

wt

ges

�

�

wKw(btiatp

FH

el 1 0.0173 0.0166 0.32el 2 0.0358 0.0310 0.19el 3 0.0259 0.0155 0.23el 4 0.0514 0.0268 0.22

onsidering either the cylinder moving with a speed U and the ves-el walls stationary, or the cylinder stationary and the vessel wallsoving with a speed U. The flow is governed by the usual conser-

ation equations of mass and momentum. For an incompressibleuid under isothermal, creeping flow conditions we have

· v̄ = 0 (2)

= −∇p + ∇ · � (3)

here v̄ is the velocity vector, � is the extra-stress tensor, and p ishe scalar pressure.

To model the stress-deformation behaviour of the viscoplasticels used in the experiments, the Herschel–Bulkley constitutivequation was found adequate by Jossic and Magnin [3]. In simplehear flow it takes the form [1]:

= �y + K�̇n for|�| > �y (4a)

˙ = 0 for|�| ≤ �y (4b)

here � is the shear stress, �̇ is the shear rate, �y is the yield stress,is a consistency index and n is the power-law index. Note thathen the shear stress � falls below �y, a solid structure is formed

unyielded). For the four Carbopol gels, rheological characterizationy Jossic and Magnin [3] has determined these constants, which are

abulated in Table 1 and shown in Fig. 2. To avoid the discontinuitynherent in any viscoplastic model, Papanastasiou [13] proposed

modification by introducing a material parameter, which con-rols the exponential growth of stress (basically a “regularization”arameter). In this way any viscoplastic model is valid for both

ig. 2. Experimental data (symbols) and their fit (lines) with the viscoplasticerschel–Bulkley model (Eq. (4)) with the parameters of Table 1 for Carbopol gels.

134 E. Mitsoulis, S. Galazoulas / J. Non-Newton

F(m

ya

�

wsital

�

w

�

aw

|

wyte

Th

i

B

wcsad

HB

�

wt

|

�

M

Tb

2

a�atud

C

Tvp

F

wdasttido

ig. 3. Shear stress vs. shear rate according to the modified Herschel–Bulkley modelEq. (6)) for several values of the stress growth exponent m. For n = 1, the Bingham

odel is obtained.

ielded and unyielded areas. Papanastasiou’s modification, whenpplied to the Herschel–Bulkley model, becomes:

= �y[1 − exp(−m�̇)] + K�̇n (5)

here m is the stress growth exponent (with units in seconds). Ashown in Fig. 3, this equation mimics the constitutive law of thedeal viscoplastic fluid (for m ≥ 1000 s). The exponential modifica-ion has now been a standard way to treat viscoplastic materials,nd a plethora of papers has appeared in the last 10 years in theiterature applying this model to different types of flows [2,14].

In full tensorial form the constitutive Eq. (5) can be written as:

= ��̇ (6)

here � is the apparent viscosity given by:

= K |�̇ |n−1 + �y

|�̇ | [1 − exp(−m|�̇ |)] (7)

nd |�̇ | is the magnitude of the rate-of-strain tensor �̇ = ∇ v̄ + ∇ v̄T ,hich is given by

�̇ | =√

12

II�̇ =[

12

{�̇ : �̇}]1/2

(8)

here II�̇ is the second invariant of �̇ . To determineielded/unyielded regions, we shall employ the criterion thathe material flows (yields) only when the magnitude of thextra-stress tensor |�| exceeds the yield stress �y, i.e.,

yielded:

|�| =√

12

II� =[

12

{� : �}]1/2

> �y (9a)

unyielded:

|�| ≤ �y (9b)

he results are usually given as a function of a dimensionless Bing-am number, Bn [1]. For Herschel–Bulkley fluids, a generalized Bn

Mc

do

ian Fluid Mech. 158 (2009) 132–141

s defined by

n = �y

K(U/HLEN)n = �y

K

(L

U

)n

(10)

here U is the drag speed or fluid velocity far from the object-ylinder and HLEN is the characteristic length of the object undertudy, equal to V/A, where V is the object volume and A its frontalrea along a plane perpendicular to the flow. In the case of cylin-rical objects moving parallel to their axis HLEN = L.

In dimensionless form, the viscosity function for theerschel–Bulkley–Papanastasiou model is written as a function ofn and the power-law index, n:

∗ = |�̇∗|n−1 + Bn[1 − exp(−M|�̇∗|)]|�̇∗| (11)

here the dimensionless viscosity �*, magnitude of rate-of-strainensor |�̇ |∗, and exponent M are given by

�̇ |∗ = |�̇ |U/L

(12a)

∗ = �

K(U/L)n−1(12b)

= mU

L(12c)

hen the criterion that separates yielded from unyielded regionsecomes: |�| = Bn.

.2. Dimensionless groups

An object moving in a viscoplastic fluid at very low velocity cre-tes stresses in its immediate vicinity of the order of the yield stressy. The drag force Fd exerted upon it may therefore be expressed asdimensionless quantity by the product A�y in which A represents

he frontal area of the object in question along a plane perpendic-lar to the flow, that is, the surface opposing the flow. Hence, therag coefficient

∗d = Fd

A�y(13)

his expression is only meaningful in the field of infinitely lowelocities where inertia and viscous effects are negligible in com-arison with yield-stress effects.

For an object in free fall, the buoyancy force Fb is given by

b = �D3eq��g

6(14)

here �� is the difference between the fluid density � and theensity of the object-cylinder �c, g is the acceleration of gravity,nd Deq = (6 V/�)1/3 is the equivalent diameter of a sphere of theame volume V as the object-cylinder. Thus, in order to calculatehe stability criterion, cylinders of different shapes are consideredo be equivalent to spheres of equal volume. This definition, whichnvolves taking a sphere as reference, means that it is possible toetermine how well the stability criterion is evaluated for a randombject on the basis of the value obtained in the case of a sphere.

oreover, it has the advantage of enabling different shapes to be

ompared at constant volume.At very small velocities, the buoyancy force Fb is balanced by the

rag force Fd due to the yield stress, which is in theory independentf the velocity. The drag force can be expressed as a function of yield

E. Mitsoulis, S. Galazoulas / J. Non-Newtonian Fluid Mech. 158 (2009) 132–141 135

Table 2Cylindrical objects used in the experiments by Jossic and Magnin [3] and their dragcoefficient: D = diameter, L = length.

Objects Dimensions (cm) Position C∗d

(smooth) C∗d

(rough)

Cylinder 1 (disk) L = 0.5, D = 3.5 Vertical 12.3 12.0

CC

s

F

w

b

Y

Tot

Y

a

�

a

Y

0wwtE

dp

3

bpb(anwscacasa

Fvic

•••

•

••

raeLd

mtplngBiwBvts(ia

Bfrom Eq. (10) and set for the current run. The solution process startsfrom the Newtonian solution (�y = Bn = 0), which is used to obtain afirst approximation. Picard iterations are performed (direct substi-tution) until convergence of the solution for the current U-speed (orBn number) is achieved. This is usually considered to be the case

L/D = 0.14ylinder 2, L/D = 1 L = 2.0, D = 2.0 Vertical 9.4 27.8ylinder 3, L/D = 5 L = 5.0, D = 1.0 Vertical 17.0 32.0

tress by

d = C�y�D2

eq

4(15)

here C is a dimensionless coefficient.The ratio of yield-stress effects to gravity effects is represented

y the dimensionless gravity-yield number YG defined as [3]:

G = �y

gHLEN��(16)

he gravity-yield number is used in stability studies of fallingbjects as the number beyond which the object will not fall (cessa-ion of motion).

Equating the two forces leads to the stability criterion, YG,max

G,max = �y

gDeq��= 2

3C(17)

The dimensionless yield stress �∗y (Eq. (1)) can then be written

s

∗y = 2�R2

eq�y

Fd= 2

C(18)

nd according to the above definitions can be related to YG by

G = �∗y

3(19)

In all cases, the Newtonian fluid corresponds to �∗y = Bn = YG =

. However, at the other extreme of an unyielded solid, Bn → ∞,hile �∗

y reaches a dimensionless value of �∗y,max = 0.143, beyond

hich the cylinder will not fall in an infinite medium (in analogyo the sphere problem, see Beris et al. [4]). For this case and usingq. (19), YG,max = 0.0477.

The experimental results associated with this work for the cylin-ers moving parallel to their axis are tabulated in Table 2 and areart of the subject of the present investigation.

. Method of solution

The constitutive equation for the Herschel–Bulkley fluids muste solved together with the conservation Eqs. (2) and (3) and appro-riate boundary conditions. Fig. 4 shows the solution domain andoundary conditions. Because of the creeping flow assumptionRe = 0), there is symmetry in both the r- and z-directions in anxisymmetric cylindrical coordinate system (r,z,�). It is thereforeecessary to consider only one quadrant of the flow domain, as

as done previously by Beris et al. [4] in the case of flow around a

phere. Note that this is not the case for flows with inertia or vis-oelastic flows, where the flow field is not symmetric before andfter the sphere/cylinder. The benchmark problem of flow past aylinder considers the cylinder stationary while the vessel wallsnd the viscoplastic material at the outflow are moving with a con-tant velocity U (drag or Couette flow). The boundary conditionsre therefore (see Fig. 4):

TDgu

D

481

ig. 4. Schematic representation of the system of a cylinder inside a cylindricalessel filled with a viscoplastic medium and moving under the effect of drag veloc-ty U. Definition of variables and relevant boundary conditions. The shaded regiononstitutes the computational field.

symmetry along AE (vr = 0, ∂vz/∂r = 0),symmetry along CB (vr = 0, ∂vz/∂z = 0),fixed axial velocity along the outer cylinder wall CD (vz = U, vr =0),plug axial velocity profile along the outflow boundary DE (vz = U,vr = 0),no slip at the cylinder surface along BA (vz = vr = 0),while the reference pressure is set to zero at one point (here pointC).

The length of the domain Lf has been chosen judiciously to cor-espond to a long finite distance, beyond which the flow is safelyssumed fully developed (plug velocity profile). Using previousxperience with flows around a cylinder [5,15], we have chosenf/Rres = 20/8 = 2.5, and verified that the centerline velocity profileoes level off with distance for all simulation runs.

The numerical solution is obtained with the finite elementethod (FEM), employing as primary variables the two veloci-

ies and pressure (u-w-p formulation) as explained in our previousapers [16,17]. The finite element grids employed in the simu-

ations are shown in Fig. 5, while their characteristics regardingumber of elements, nodes and unknown degrees of freedom areiven in Table 3. Simulations are carried out for a wide range ofingham numbers (0 ≤ Bn < ∞). To simulate the experiments, this

s done – for a given geometry and gel – by solely decreasing theall speed U. Because of the very small velocities used (e.g., forn = 1000, U = 10−9 cm/s), we found that in the modified apparent-iscosity Eq. (7) a value for the exponent m = 1000 s was not enougho give results independent of m even for an integrated quantityuch as the drag coefficient. For this reason, we have used in Eq.11) a dimensionless stress exponent M = mU/L = 1000, as suggestedn other works (e.g., Smyrnaios and Tsamopoulos [18]). Then thectual m values are much higher.

A zero-order continuation scheme has been used to increase theingham number. For a given Bn number, the wall speed U is found

able 3escription of the finite elements grids used for the current computations. For eachrid are given the number of elements, nodes and the unknown degrees of freedom-w-p.

res/D L/D Elements Nodes DOF

.55:1 0.14 1052 4343 9499:1 1 1540 6327 138916:1 5 1698 6971 15305

136 E. Mitsoulis, S. Galazoulas / J. Non-Newtonian Fluid Mech. 158 (2009) 132–141

a) 4.55

wwmBaa

4

hynit8(f

4

fttctri

ac

trsssatttdiu

rtacuFil

Fig. 5. Full view of the finite elements grids used in the simulations: (

hen both the norm-of-the-error and the norm-of-the-residualsere less than 0.001. The new solution is then used as an initial esti-ate for a lower wall speed value to obtain a solution for a higher

ingham number. The number of iterations increased substantiallys the speed U was decreased (departure from a Newtonian fluidnd approach to a plastic solid with very high Bn values).

. Results

Numerical simulations with the Herschel–Bulkley modelave been carried out with the purpose of finding out theielded/unyielded regions, especially in the range of Binghamumbers (1 ≤ Bn ≤ 10) used by Jossic and Magnin [3] in their exper-

ments. The simulations were performed for all four gels and thehree geometries corresponding to Dres/D diameter ratios of 4.55:1,:1 and 16:1 (see Tables 2 and 3). A full presentation of the resultsalso for the theoretical case of the Bingham plastic model) can beound in [19].

.1. Yielded/unyielded regions

We begin by showing the simulated yielded/unyielded regionsor three Bingham numbers, namely Bn = 1, 3, and 10. Fig. 6 showshe extent of the yielded (white) and unyielded (black) regions with

he model constants corresponding to gel 1 and the 3 cylindri-al objects, while Fig. 7 shows blow-ups of these regions aroundhe cylinders. Increasing Bn leads to an increase of the unyieldedegions. It is observed that the geometry of the unyielded regionss dictated by the geometry of the “slenderness factor” L/D, which

sb

aB

:1, (b) 8:1, (c) 16:1. Blow-up view of the grids for (d) 8:1 and (e) 16:1.

lso determines the object geometry (disk, square cylinder, oblongylinder).

In particular, for the cylindrical disk (L/D = 0.14) and for Bn = 1,he unyielded region appears as an ellipse, with its main axis in the-direction and the secondary axis in the z-direction. On the diskurfaces around the radial axis we observe unyielded regions in thehape of small hills, much like the polar caps found in flow around aphere [4]. For Bn = 3, the yielded region shrinks and appears agains an ellipse but with a reversal of its axes with the main one inhe flow z-direction. The hills on either side of the disk extend tohe edges, while small islands of unyielded material appear nearhe sides parallel to the z-axis. These islands are apparent and are airect consequence of the value of M used here (M = 1000). A further

ncrease of the Bingham number leads to the whole region beingnyielded (Bn = 10), as claimed by Jossic and Magnin [3].

Turning to the square cylinder (L/D = 1) and for Bn = 1, the yieldedegion (white) appears as a circle following the cylinder geome-ry. In this case the hills on the sides around the radial axis have

smaller extent than in the previous case. For Bn = 3, the cir-le becomes an ellipse. The unyielded hills extend, while largernyielded islands appear near the cylinder sides around the z-axis.or Bn = 10 (also shown as a blow-up in Fig. 7a), the yielded regions situated along the cylinder side parallel to the flow and appearsarger at the cylinder corners, areas of higher strain rates and hence

tresses. In Fig. 7a we can better observe the extent of the yieldedoundary layer. Its maximum height is in the middle of the side.

For the oblong cylinder (L/D = 5) and for Bn = 1, the yielded regionppears as an ellipse following the geometry of the cylinder. Forn = 3 neither hills nor islands are observed. The unyielded region

E. Mitsoulis, S. Galazoulas / J. Non-Newtonian Fluid Mech. 158 (2009) 132–141 137

F aded)

aestu

ig. 6. Simulation results showing the progressive growth of the unyielded zone (sh

ppears squeezing the yielded one along the cylinder, while thextent of the yielded region has shrunk appreciably. For Bn = 10 (alsohown as a blow-up in Fig. 7b), the yielded region is situated alonghe cylinder and appears larger around the cylinder bases, wherenyielded hills also appear.

anur

with model constants corresponding to gel 1 for different Bn numbers (M = 1000).

From these figures we conclude that increasing L/D leads todecrease in the unyielded regions. Characteristic of this phe-

omenon is that for Bn = 10, the disk shows the whole regionnyielded, while for the other two cylinders there are yieldedegions, which are bigger for the oblong cylinder.

138 E. Mitsoulis, S. Galazoulas / J. Non-Newtonian Fluid Mech. 158 (2009) 132–141

FL

cddcgwsstrvfbc(b

hc

Fass

dngtse(ettys

4

ia

F

wsawd

C

w

D

ig. 7. Blow-up view of the unyielded zones (shaded) for Bn = 10: (a) L/D = 1 and (b)/D = 5 (M = 1000).

It is noted in Fig. 7a that the black (unyielded) regions touch theylinder. This at first sight is puzzling and contradictory, becauseifferent velocities are imposed in the simulation (0 on the cylin-er and U on the outer tube wall, see Fig. 4), so the cylinder wallannot have the outer wall velocity. In reality a very small yieldedap exists, and had an exact solution method been possible, thereould indeed be a small region of yielded fluid separating the plugs

urrounding the cylinder and that attached to the wall. Presumablyuch a yielded region vanishes asymptotically as Bn → ∞. However,here is no contradiction in the results presented, since with theegularized model used here the fluid in the shaded regions has aery small strain rate, depending on the values of Bn and M. There-ore, touching of the unyielded zones of the cylinder should note used as a criterion for cessation of motion (critical static flowonditions). Instead, extrapolation from the bulk fluid properties

e.g., drag) is a much better method for this, and it is used here (seeelow).

The use of a continuous viscoplastic model, such as the one usedere, to determine the exact shape of yielded/unyielded regionsan be criticized (see Liu et al. [20]), since the model predicts

vo

f

ig. 8. Drag coefficient C∗d

for cylinders moving vertically in a cylindrical vessel asfunction of L/D. Comparison of experiments (Jossic and Magnin [3]) with present

imulations for the experimental range of Bn numbers with model constants corre-ponding to gel 1.

eformation (albeit extremely small) for all values of the expo-ent m (M). A careful examination of the velocities and velocityradients in disputed regions, such as the islands and the stagna-ion regions, showed that there the velocity gradients are extremelymall (but not identically zero), and these regions are really appar-ntly unyielded regions (AUR), as opposed to truly unyielded regionsTUR) that do exist whenever a plug velocity profile occurs. How-ver, in the present simulations we do not differentiate betweenhe two, and the separating line has been drawn as the contour ofhe magnitude of the extra-stress tensor having a value equal to theield stress, as in our previous publications [5,15–17,21,22] and inimulations by others [18,23,24].

.2. Drag coefficient

The numerical simulations give results for the drag force Fd byntegrating the total normal and shear stresses at the cylinder wallsccording to

D = −2

∫ �

0

[zz cos � + rz sin �]r=RRd� (20)

here zz = −p + �zz is the total normal stress, and rz = �rz is thehear stress on the cylinder surface. The drag force can be expresseds a dimensionless quantity by dividing with the product A�y,here A is the frontal area of the object, perpendicular to the flowirection. Hence, the drag coefficient is defined as:

∗d = Fd

A�y= 4Fd

(�D2eq)�y

(21)

here the Deq for the cylinders at hand is:

eq =(

6V

�

)1/3=

(32

D2L)1/3

(22)

The expression (21) has meaning in a flow field of very smallelocities, where inertia is negligible compared with the influencef the yield stress (Re ≈ 0).

Fig. 8 presents sample calculations (corresponding to constantsor gel 1) and experimental values for the 3 cylindrical objects for

E. Mitsoulis, S. Galazoulas / J. Non-Newtonian Fluid Mech. 158 (2009) 132–141 139

Fcr

ttmwrdoadwgmt

dtttwaaastowr(

dtt

C

wBt

Fa

vn

cai

g(vtbod

nitit

csaT��

fwcw

ig. 9. Simulation results for the drag coefficient C∗d

vs. Bn number for the threeylindrical objects and model constants corresponding to gel 1. The numbers atight show extrapolated values for Bn → ∞.

he drag coefficient C∗d

as a function of the slenderness factor L/D inhe range 1 ≤ Bn ≤ 10. As in the experiments, all 4 gels give approxi-

ately the same results in such dimensionless plots [19]. However,hile in the experiments a constant value was quoted for C∗

din the

ange 1 < Bn < 10 for each cylinder, in the simulations these valuesepend on Bn, more so as the object becomes slenderer (cf. rangef values for L/D = 5 with those for L/D = 0.14). In the simulationsn increase in the slenderness factor leads to an increase of therag coefficient. This is also found out for the rough-surface objects,hile for the smooth-surface ones the results are inconclusive. In

eneral, we note that the simulations underestimate the experi-ental values but show the same trends with the experiments on

he rough-surface objects.Increasing Bn further causes a reduction of the dimensionless

rag coefficient leading asymptotically to a limiting value for cessa-ion of motion for the objects at hand. This is shown in Fig. 9, wherehe drag coefficient is plotted as a function of Bn for all geome-ries for simulations corresponding to gel 1. The C∗

dvalues decrease

ith increasing Bn, reaching asymptotic values for Bn > 100. Thesesymptotic values given on the graph are found by extrapolationnd correspond to Bn → ∞. In this region there does not appearny viscous flow, and the drag is solely due to the plastic (yield)tresses of the material. This is the region where we can considerhe drag force caused by the motion of the object as independentf the velocity. The asymptotic values are drastically decreasedith decreasing slenderness factor L/D. In the experiments this

eduction is seen for the rough-surface objects, as noted abovesee Fig. 8).

It is interesting to note that the 3 curves of Fig. 9 are shiftedownwards in a parallel fashion as L/D decreases. It is then possibleo obtain a master curve for C∗

das a function of Bn and L/D according

o:

b

∗d = a +

Bn(23)

here a and b are functions of L/D. Then representing C∗d

− a/b vs.n gives a master curve, as shown in Fig. 10. The values of a arehe ones given as Bn → ∞ (given at the RHS of Fig. 9), while the

ld

rb

ig. 10. Master curve from the normalization of the simulation results of Fig. 9ccording to Eq. (23).

alues of b = C∗d|Bn=1 − C∗

d|Bn=1000 = C∗

d|Bn=1 − a to make the graph

ormalized between 0 and 1.From the above it becomes evident that the geometry of the

ylindrical object, and in particular its reduced length L/D, playssignificant role on the drag coefficient. Its increase leads to an

ncrease of the surface on which the shearing forces are exerted.Jossic and Magnin [3] calculated the maximum dimensionless

ravity-yield number YG,max from the limiting drag coefficient (Eq.17)) based on their experimental findings. Table 4 presents thesealues in comparison with the ones obtained from the simula-ions. There are discrepancies, with the experimental YG,max valueseing lower than the simulated ones for L/D = 0.14 and 1, while thepposite occurs for L/D = 5. These discrepancies are the subject of aiscussion in the following section.

The values obtained from the simulations for the same Binghamumber show that for the disk the contribution of the plastic forces

s bigger than in the other two geometries. This is also observed inhe results for the yielded/unyielded regions (Fig. 6). Correspond-ngly, the square cylinder (L/D = 1) gives bigger values of YG,max thanhe oblong one (L/D = 5).

Fig. 11 shows the shear stress distribution along the oblongylinder (L/D = 5) for different Bn numbers. We observe constanthear stress values except at the edges, hence a constant drag force,t the value corresponding to the yield stress of the material, �y.hat is, when U → 0, hence Bn → ∞, the dimensionless yield stress,∗y , tends to an asymptotic value �∗

y,max, which for L/D = (0.14, 1, 5),∗y,max(1.30, 0.67, 0.38).

Fig. 12 presents the drag coefficient as a function of L/D for dif-erent Bn numbers. We observe a reduction of the drag coefficientith increasing Bn for the same L/D and an increase of the drag

oefficient for constant Bn with increasing L/D. At high Bn numberse have a predominance of the plastic effects, as discussed above,

eading to a stabilization of the shearing forces and hence of therag coefficient.

A comparison between the simulation and the experimentalesults of Fig. 12 for the rough surface shows the same qualitativeehaviour, since increasing the slenderness factor L/D leads to an

140 E. Mitsoulis, S. Galazoulas / J. Non-Newtonian Fluid Mech. 158 (2009) 132–141

Table 4Comparison between experimental and simulation values for the dimensionless stability criterion YG,max for the three cylindrical objects with L/D = 0.14, 1, 5.

Stability criterion

Experiments [3] Simulations

Object L/D YG,max (smooth) YG,max (rough) YG,max (Bn = 1) YG,max (Bn = 1000)

0.14 0.019 0.0231 0.093 0.0635 0.150 0.075Sphere 0.088 0.064

a Beris et al. [4].

Fig. 11. Simulation results for the shear stress distribution along the cylinder wallfor L/D = 5 and model constants corresponding to gel 1.

Fig. 12. Comparison between the experimental and simulated values for the dragcoefficient C∗

das a function of L/D for all Bn numbers used in the simulations.

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Mmta(fi[flcbiwwe

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0.156 0.4320.061 0.2220.028 0.128– 0.048a

ncrease in the drag coefficient C∗d

as well. Quantitatively, there areiscrepancies, which exceed 100%. The experiments give limitingtability values for the drag C∗

din the range of 12 (disk) to 32 (oblong

ylinder), with the claim by Jossic and Magnin [3] that these valuesccur in the range 1 < Bn < 10. The numerical simulations show thatn this narrow Bn range, there are still large changes in the dragalculation, so that they may not be considered as the limiting val-es for object stability and cessation of motion. More appropriateeem to be values for Bn ≥ 50, since in the range 50 ≤ Bn ≤ 1000, theesults are virtually the same (see Figs. 9 and 10).

If the limiting stability values found from the simulations forarge Bn (≈1000) are considered as correct, then the questionemains how to best approach experimentally these stability limits.t is observed in Figs. 8 and 12 that the surface conditions (roughs. smooth) of the objects play an important role in the experimen-al results. Thus, the observed differences may be partly due to these of additives on or treatment of the object surface to make themough and guarantee a no-slip condition. This may lead to changesn the surface conditions and hence in the measured forces. On thether hand, the simulations consider a no-slip boundary conditionn the surfaces without any changes in their treatment there. Tonclude slip, however, is beyond the scope of the present study, aso experimental data for a slip law are available.

Fig. 12 shows a convergence of experimental and numerical val-es with increasing L/D ratio. This would be expected, since theontribution of any experimental errors is reduced with a biggerbject.

The rheology of Carbopol is another issue. Although Jossic andagnin [3] have been careful in determining the rheology, thereay be some doubt that the experiments represent the computa-

ions. In particular, recent papers by Coussot and co-workers [8–10]nd by Putz et al. [12] have looked at low shear rheology of Carbopolthe latter group during sedimentation). Coussot and co-workersnd a transition between creep and viscous behaviour; Putz et al.12] show particle-image-velocimetry (PIV) data indicating that theow field is non-symmetric. In [3] there is no PIV data, so it is notlear what is happening. In [3] the authors comment a lot on slip,ut this low shear behaviour is another source of uncertainty. Also,

n [3] the authors determine their yield values via extrapolationith no talk of error and very smooth data presented. The recentorks [7–12] show that for Carbopol the yield values are not so

asily specified.Finally, an important factor for finding experimentally the con-

itions of cessation of motion of the objects in viscoplastic fluidsemains the fact that one has to wait an infinite time (correspond-ng to zero velocities). Obviously, this is not possible, and thereforehe numerical simulations are considered better suited for thextraction of accurate results regarding the stability criterion iniscoplastic fluids.

. Conclusions

The present work has simulated experiments on viscoplasticels with the purpose of finding important flow phenomena and for

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E. Mitsoulis, S. Galazoulas / J. Non-Ne

omparisons with experiments. The flow phenomena found fromhe simulations were the extent and shape of unyielded regionsround cylindrical objects with aspect rations L/D = (0.14, 1, 5). Ashe Bn number increased and the material became more plastic, thenyielded regions were extended all around the objects, leading tocessation of motion. The calculated drag coefficients showed that

hey depended on the Bingham number in the range of the experi-ents (1 < Bn < 10). The Bn number has to increase beyond 50 to get

esults independent of Bn, corresponding to the maximum yieldriterion for cessation of motion. The values for the stability crite-ion YG,max were found to be 0.432 for the disk (L/D = 0.14), 0.222or the square cylinder (L/D = 1), and 0.128 for the oblong cylinderL/D = 5).

Although these values are substantially higher that the exper-mental ones, qualitatively they agree with the experiments,howing an increase in the drag coefficient for slenderer cylinders.

The present work also brought forward the difficulties associ-ted with these experiments, as it is increasingly hard to reach highalues of Bn numbers to compare them with the experiments. Forhis reason, simulations can be an attractive alternative.

cknowledgments

Financial assistance from the General Secretariat for Researchnd Technology (GGET) of the Ministry of Development of Greecend the NTUA is gratefully acknowledged.

eferences

[1] R.B. Bird, G.C. Dai, B.J. Yarusso, The rheology and flow of viscoplastic materials,Rev. Chem. Eng. 1 (1983) 1–70.

[2] H.A. Barnes, The yield stress – a review or ‘�˛v�˛ ���’ – everything flows? J.Non-Newtonian Fluid Mech. 81 (1999) 133–178.

[3] L. Jossic, A. Magnin, Drag and stability of objects in a yield stress fluid, AIChE J.47 (2001) 2666–2672.

[

[

[

an Fluid Mech. 158 (2009) 132–141 141

[4] A.N. Beris, J.A. Tsamopoulos, R.C. Armstrong, R.A. Brown, Creeping motion of asphere through a Bingham plastic, J. Fluid Mech. 158 (1985) 219–244.

[5] E. Mitsoulis, On creeping drag flow of a viscoplastic fluid past a circular cylinder:wall effects, Chem. Eng. Sci. 59 (2004) 789–800.

[6] N.P. Chafe, J.R. de Bruyn, Drag and relaxation in a bentonite clay suspension, J.Non-Newtonian Fluid Mech. 131 (2005) 44–52.

[7] F.K. Oppong, J.R. de Bruyn, Diffusion of microscopic tracer particles in a yield-stress fluid, J. Non-Newtonian Fluid Mech. 142 (2007) 104–111.

[8] H. Tabuteau, F.K. Oppong, J.R. de Bruyn, P. Coussot, Drag on a sphere movingthrough an aging system, EPL 78 (2007) p1–p5, 68007.

[9] H. Tabuteau, P. Coussot, J.R. de Bruyn, Drag force on a sphere in steady motionthrough a yield-stress fluid, J. Rheol. 51 (2007) 125–137.

10] H. Tabuteau, J.C. Baudet, X. Chateau, P. Coussot, Flow of a yield stress fluid overa rotating surface, Rheol. Acta 46 (2007) 341–355.

11] B. Gueslin, L. Talini, B. Herzhaft, Y. Peysson, C. Allain, Flow induced by a spheresettling in an aging yield-stress fluid, Phys. Fluids 18 (2006) p1–p8, 103101.

12] A.M.V. Putz, T.I. Burghelea, I.A. Frigaard, D.M. Martinez, Settling of an isolatedspherical particle in a yield stress shear thinning fluid, Phys. Fluids 20 (2008)p1–p11, 033102.

13] T.C. Papanastasiou, Flow of materials with yield, J. Rheol. 31 (1987) 385–404.14] I.A. Frigaard, C. Nouar, On the usage of viscosity regularization methods for

visco-plastic fluid flow computation, J. Non-Newtonian Fluid Mech. 127 (2005)1–26.

15] Th. Zisis, E. Mitsoulis, Viscoplastic flow around a cylinder kept between parallelplates, J. Non-Newtonian Fluid Mech. 105 (2002) 1–20.

16] S.S. Abdali, E. Mitsoulis, N.C. Markatos, Entry and exit flows of Bingham fluids,J. Rheol. 36 (1992) 389–407.

17] E. Mitsoulis, S.S. Abdali, N.C. Markatos, Flow simulation of Herschel–Bulkleyfluids through extrusion dies, Can. J. Chem. Eng. 71 (1993) 147–160.

18] D.N. Smyrnaios, J.A. Tsamopoulos, Squeeze flow of Bingham plastics, J. Non-Newtonian Fluid Mech. 100 (2001) 165–190.

19] S. Galazoulas, Viscoplastic Flows around Cylindrical Objects, M. Eng. Thesis,School of Mining Eng. & Metallurgy, NTUA, Athens (2003).

20] B.T. Liu, S.J. Muller, M.M. Denn, Convergence of a regularization method forcreeping flow of a Bingham material about a rigid sphere, J. Non-NewtonianFluid Mech. 102 (2002) 179–191.

21] J. Blackery, E. Mitsoulis, Creeping flow of a sphere in tubes filled with a Binghamplastic material, J. Non-Newtonian Fluid Mech. 70 (1997) 59–77.

22] M. Beaulne, E. Mitsoulis, Creeping flow of a sphere in tubes filled withHerschel–Bulkley fluids, J. Non-Newtonian Fluid Mech. 72 (1997) 55–71.

23] D.L. Tokpavi, A. Magnin, P. Jay, Very slow flow of Bingham viscoplastic fluidaround a circular cylinder, J. Non-Newtonian Fluid Mech. 154 (2008) 65–76.

24] B. Deglo De Besses, A. Magnin, P. Jay, Viscoplastic flow around a cylinder in aninfinite medium, J. Non-Newtonian Fluid Mech. 115 (2003) 27–49.