-
J. Non-Newtonian Fluid Mech. 166 (2011) 118132
Contents lists available at ScienceDirect
Journal of Non-Newtonian Fluid Mechanics
journa l homepage: www.e lsev ier .co
Hydrod ubin shea
J. Rodrigo ama Instituto de In 70-360b Department o 1, USAc
Department o 6T 1Z
a r t i c l
Article history:Received 13 JuReceived in reAccepted 4 No
Keywords:Bubble pair inNon-NewtoniaShear-thinnin
in shical snts o
wercess:ds tolity olts supair a
1. Introdu
Bubble interactions have received considerable study in the
lit-erature owing to its inherent scientic interest and
importancein applications such as gasliquid contactors [13]. The
study oftwo-bubblephenomenawas how rissue: onewis highly dwas the
stucoalescenceascending iditions) is emake a revtions. In
parNewtonian
For the ccally predicof the sepa[11], two bacquire a hvelocity
inceld does napproachin
CorresponE-mail add
l datapplies as well to settling particles or rising
bubbles.
When the inertia is small but nite (Re0.25) the vorticityaround
a spherical body looses its fore-aft symmetry and the trail-ing
body acquires a higher velocity than the leading one, reaching
0377-0257/$ doi:10.1016/j.interaction was initially motivated by
the coalescencedue to its impact in bubble columns efciency.
That
esearchers took two different paths concerning thisas the
studyof the coalescencemechanism itself,which
ependent on the liquid composition [46]; the otherdy of the
trajectories that two bubbles take before[79]. In this work, the
interaction of two bubbles
n shear-thinning inelastic uids (non-coalescing
con-xperimentally and numerically studied. First, we williew of the
current literature of hydrodynamic interac-ticular, to put our work
in perspective, we focus on thecase.ase of creeping ow, Stimson and
Jeffery [10] analyti-ted the velocity of two spheres moving in-line
in termsration distance between them. As widely described byodies
moving in this way in the creeping ow regimeigher velocity than
that attained by a single body; thereases as the separation
distance decreases. As the owot have inertia, the bodies keep their
distance withoutg each other. This trend is in agreement with
experi-
ding author. Tel.: +52 55 5622 4593; fax: +52 55 5622 4602.ress:
[email protected] (R. Zenit).
the latter after some time. Crabtree and Bridgwater [7] were
therst ones to approximate the trailing bubble velocity as the
sumof the terminal velocity of the single bubble plus its wake
velocityat the distance where the trailing bubble is found. This
hypoth-esis was referred to as a superposition principle by Bhaga
andWeber [13]. In later works [1315] this hypothesis was tested
andconrmed for two in-line bubbles rising with Reynolds numbersup
to O(100). Crabtree and Bridgwater also reported the
curiousphenomenon (not fully explained yet) in which the trailing
bubbleexperiences a signicant deformation (fromoblate to prolate
form)moments before touching the leading one. Such deformation
wasalso reported and photographed byNarayanan et al. [14]. These
lastauthorsworkedwithdifferent bubble sizes producingbasically
twodifferent wake structures: one forming a thin trailing wake and
theother forming a wake with a stable toroidal vortex. For the
formerit was observed that the Stimson and Jefferys equation
describedwell the rise velocities of the trailing bubbles even
though it wasformulated for creeping ows and spherical bodies. For
the secondcase, a superposition principle similar to the one
proposed by [7]was used. Bhaga and Weber [13] also worked with
bubbles form-ing a wake with a toroidal vortex (Reynolds80,
Etvs70). Theexperimental measurements of the wake velocity were in
agree-mentwith thevelocity calculatedusing the
superpositionprinciple.MangaandStone [16,17] studied theeffectsof
bubbledeformability
see front matter 2010 Elsevier B.V. All rights
reserved.jnnfm.2010.11.003ynamic interaction between a pair of
br-thinning inelastic uids
Vlez-Corderoa, Diego Smanoa, Pengtao Yueb, Jvestigaciones en
Materiales, Universidad Nacional Autnoma de Mxico, Apdo. Postalf
Mathematics, Virginia Polytechnic Institute and State University,
Blacksburg, VA 2406f Chemical and Biological Engineering,
University of British Columbia, Vancouver, BC, V
e i n f o
ly 2010vised form 3 November 2010vember 2010
teractionn
g
a b s t r a c t
The interaction of two bubbles risingresults were complemented
by numertechnique. Different initial alignmeferences with the
Newtonian uidsdraftingkissingtumbling (DKT) prophase does not occur
and the pair tenthe amount of inertia and deformabiThe experimental
and numerical resuan important role in the speed of the
ction mentam/locate / jnnfm
bles ascending
es J. Fengc, Roberto Zenita,
, D.F. 04510, Mexico
3, Canada
ear-thinning inelastic uids was studied. The
experimentalimulations conducted with the arbitrary
LagrangianEulerianf the bubble pair were considered. Similarities
and dif-e found. The most noticeable difference is the so-calledfor
the case of bubbles rising in thinning uids, the tumblingform a
stable doublet. The DKT process is also inuenced by
f the individual bubbles and the initial angle between
them.ggest that the thinning wake formed behind the bubbles playsnd
the formation of clusters in thinning uids.
2010 Elsevier B.V. All rights reserved.
a [12] and with other analytical expressions [11], and
-
J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 119
in the interactions among them. They found that bubble
alignmentand coalescence is enhanced when the buoyancy forces are
muchlarger than the restoring forces due to the interfacial tension
on thebubble surface.
The intethe subjectworkers [1rising in poof approachsion
otherwformed betis larger. Onwhen theythe dynami
The behathat observthe behaviorepulsive foattraction iexplained
ineach bubblin a horizonear a vertiof the trailinasymmetrythat the
twdistance be(repulsive)ing bubblebubblewaktwobubblefact
numerisuchanequrising in difmentally, anclean bubblRe
numbertance,whicnumber [23from the ve
Severalof two bubbthe construows [13,25rising in-linto a
Reynolimating theappears orejected fromthe leadingproperties
otaminated wthen turnsone and nato as the dcommonlythe case of
tthey alwaysRe>200, bedistance [22
Regardinthe verticalis less thanof the twothe ow [23bles rising
s
experimental measurements of the drag coefcients in xed
rigidparticles [31], together with the numerical works of [26,22],
it hasbeen found that two horizontal bubbles will experience less
dragthan the single one for low Reynolds ows (Re
-
120 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
Table 1Bubble diamet
SmallMediumLarge
2. Experim
2.1. Column
TheexpeIt consists oan inner crotest liquid uThree capilsizes.
In ordume, thehyHence, to ginserted in tthe ones th(vertical aliof
the columcenter of thtally, a secoan elbow aninitial horizble size
thadiameters.pump (KDScompletelyin betweennotablediffone after
th
011
102
103
(mPa
s)
low c.85, (ubbles
uids
empsolu
thinnlawterisech
hedeeologents
65DuNnswraturns arnd th
Table 2Physical propebubbles. USI anof the xanthan
Fluids
Newtonian:0.02% Xanth0.1% XanthaFig. 1. Experimental setup.
ers obtained by the capillaries (I.D.: inner diameter of the
capillaries).
Capillary I.D. Bubble diameter
0.2mm 2.1mm0.6mm 2.8mm1.2mm 3.6mm
ental setup
and bubble generation
rimentswere carriedoutwith the setup shown in Fig. 1.f a
rectangular columnmade of transparent acrylicwithss section of 510
cm2. The column was lled with thep to a level of 100 cm measured
from the base plate.
lary diameters were used to produce different bubbleer to avoid
the generation of gas jets with variable vol-draulic resistance
through the capillarieshas tobe large.enerate individual bubbles, a
capillary of 0.2mm was
110
Fig. 2. F() n=0by the b
2.2. Fl
Wetonianshear-powercharaccencemit [4]. TThe rhInstrumgap ofwith
asolutiotempesolutioFig. 2 ahe capillarieswith the larger
diameters, the latter beingat formed the bubbles. To generate
in-line bubble pairsgnment), one capillarywas inserted through the
bottomn, using a sealed feedthrough (Spectite Series PF), at
the
e base plate. To generate bubble pairs aligned horizon-nd
capillary was inserted through the side wall, usingd another
feedthrough connector. In this manner the
ontal separation couldbevaried. Table 1 shows thebub-t were
obtained with capillaries having different innerAir was injected
through the capillaries with a syringecientic 100L). To ensure that
the polymer solution wasat rest, a time interval of approximately
5min was leftexperiments. Regarding this point, we did not observe
aerence in the terminal velocities of twobubbles releasede other
for periods above one minute.
quency valumoduli curis about twvalueof thements conthe uids
unegligible f
2.3. Bubble
The bub(MotionScopDC motor (DC power simage sequ
rties of the uids: , density; , surface tension; , viscosity; k,
consistency index; n, d db are the terminal velocity and diameter
of the single bubble, respectively. The percegum solutions in
weight terms.
kg/m3 mN/m
83% glycerin/water 1214.6 61.9an gum in 75% glycerin/water
1193.1 63.0n gum in 60% glycerin/water 1152.1 65.0100 101 102
103shear rate (1/s)
urves of the test uids. : apparent viscosity, (- - -) Newtonian
uid,) n=0.55. The vertical lines demarcate the shear rate range
produced.
loyed two shear-thinning uids and a reference New-tion. These
uids were also used in [33]. Theseing uids based on xanthan gum
solutions follow abehavior with negligible elasticity in a wide
range oftic ow times. As we are not interested in the coales-anism
itself,we also added0.04MofMgSO4 to suppresstails of
thepreparationof the solutionsaregiven in [33].ical measurements
were conducted in a rheometer (TAAR1000N) with a cone-plate
geometry (60mm, 2, a
m). The surface tension measurements were performedouy ring
(diameter of 19.4mm, KSV Sigma 70). All theere stirredbefore the
surface tensionmeasurement. Thee of the room was 23 C. The physical
properties of thee summarized in Table 2, the ow curves are shown
ine oscillatory measurements in Fig. 3. Note that the fre-e
(inverse of the relaxation time) at which the dynamic
ves intersect in the shear thinning (xanthan) solutionso orders
of magnitude higher than the correspondingviscoelastic (PAAm)
reference solution. Thesemeasure-rm that the relaxation time is
comparatively smaller in
sed in this work, which indicate that elastic effects areor our
study.
size and velocity measurement
ble pairs were followed by a high speed camerae PCI 8000 s)
mounted on a vertical rail activated by a
Fig. 1). The velocity of the motor was regulated with aupply. A
recording rate of 60 frames/s was used. Theence obtained with the
camera was binarized using a
ow index; , shear rate range, estimated as 2USI/db , achieved by
thentages of liquid mixtures are given in volume terms, the
percentages
or k n 0.4
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J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 121
101102
100
102
104
G,G
(Pa)
Fig. 3. Dynamuli G , empty s(PAAm) referethe estimation
thresholdprovided bcalculated uprojections
db = (d2MAXdwhere dMAbubble diam
As the pused a pattelate the disdisplacemethe absolutof two
partselected fropattern is idis calculatethe rst imand their edure
is repeuncertaintyingproblemThevelocitytral differencomputed f
3. Comput
The arbinumericallyangles of apto solve theand modiein 2D and
atechnique afeatures of
The techtions of theelement mefollow the mIn the interfollow
theferential eq
includes a remeshing tool that generates a new mesh upon
detect-ing elements with unacceptable distortion. When this
happens, aprojection scheme is also invoked to project the ow eld
obtainedon the old mesh onto the new one. The continuity and
momen-
uatit GaNico
0
+ (u
the v12
u iscosi thtainthetive ed tobouslipnditiy comuid se tanary c
I +
e tan isrvatbble+g(ightdia
.01consdomand2D100 101 102 103frequency (1/s)
icmoduli of the shear thinning solutions. Filled symbols:
elasticmod-ymbols: loss moduli G , () n=0.85, () n=0.55, () a
polyacrylamidence solution (0.04% in 80/20 glycerinwater with 0.04M
MgSO4). Forof G and G the procedure followed by [42] was used.
value computed according to the Otsus methody Matlab. The
equivalent bubble diameter db wassing the short and long diameters
of the elliptic bubble:
MIN)1/3
(1)
X is the larger bubble diameter and dMIN the shortereter.ictures
were taken by a moving reference frame, wern of circles, glued on
the side of the column, to calcu-
placement of the camera in between frames; once thent of the
camera was known is was possible to computee displacement of the
bubbles. This procedure consisteds: rst a pair of consecutive
images (j and j1) arem the sequence. Then the same circle of the
referenceentied in each image and the position of its centroid
d. After doing this, the reference position is taken fromage;
hence the location of the bubbles is determinedvolution from one
frame to the next one. This proce-ated for the entire image
sequence. In this manner, thein locating the bubble position is
minimized eliminat-s due to vibrationof the camera and changes in
lighting.of thebubbles at the image jwascalculatedusinga cen-ce
scheme. The distance between bubbles was directly
tum eqelemenCrank
u =
[ut
where
= k[
wherethe visindex,was obing toderivaX afx
Thethe noslip covelocitthe liqand thboundtion:
n (p
with thHere,face cuthe bupb =p0the hebubblesure (1pbVb =
Thethe 2Dfor therom the images.
ational technique
trary LagrangianEulerian (ALE) technique was used tostudy the
interaction of bubbles pairs with different
proach. This technique was developed by Hu et al.
[43]motionofparticles in two-and three-dimensionalowsd by Yue et
al. [44] to study bubble and foam problemsxisymmetric geometries. A
detailed description of thisnd its algorithm can be found in
[45,44]. The generalthe code will be described briey.nique combines
an Eulerian and Lagrangian descrip-ow and bubble motion using an
unstructured nitesh. This means that the boundary nodes of the
meshotion of the bubbles and the walls with possible slip.
ior of the domain, however, the mesh motion does notuid ow but
is computed from an elliptic partial dif-uation which guarantees a
smooth variation. The code
axisymmetdimensionathe followinvelocity anminal velocand
gravityues of therespectiveltemequatiosubspace itminimum ra
computatwood.iam.uless than alimit for thethe single aaxisymmetor
with othsolver.ons were spatially discretized using the standard
nitelerkin formalism and temporally discretized using thelson
scheme. The conservation equations are:
(2)
um) u]
= p + [u + (u)T] + g (3)
iscosity is dened by the power-law model:
( : )
]n1(4)
the liquid velocity, the density, p the pressure, ty, g the
gravity, k the consistency index, n the owe shear rate tensor and
um the mesh velocity, whiched from the displacement of the mesh
nodes accord-xed computational coordinates. The referential time/
t= / t|xX is made using the Lagrangian coordinatethe moving
mesh.
ndary conditionswere the following: at the bottomwallcondition
was applied; at the right and left walls theon in the ydirection
was applied and the horizontalponentwas set to zero; stress
conditionwas applied at
urface, both on the normal component (yy =pp0 = 0)gential
component (xy =0); on the bubble surface theondition was obtained
from the YoungLaplace equa-
) = (pb + K)n (5)
ngential components also set to zero (no surfactants).the normal
vector to the bubble surface, K the sur-ure, the surface tension
and pb the pressure inside. The initial pressure inside the bubble
is given byHh) + 2/r where H is the height of the domain, hin which
the bubble was released, r the half of themeter dened by Eq. (1)
and p0 the reference pres-105 Pa). In the simulations, pb is
updated according to
t, where Vb is the bubble volume.ain size was 16r50r. This size
was the same foraxisymmetric calculations; therefore, 16r is the
widthdomain and also the diameter of the cylinder of theric
geometry; 50r is the height in both geometries. Non-lized
variableswere introduced to the code consideringg scales: r for the
length, r/USI for the time, USI for the
d U2SI for the pressure and stress, USI being the ter-ity of the
single bubble. The viscosity, surface tensionwere
non-dimensionalized using the experimental val-Re (USIr/), We
(U2SIr/) and Eo (Eq. (13)) numbers,y, and the scales mentioned
above. The non-linear sys-ns are solvedbyNewtonsmethod
togetherwithKryloverative solvers such as the preconditioned
generalizedesidual (GMRES). The simulations were conducted inional
grid located in Canada (glacier.westgrid or drift-bc). A typical
job consisting of 25,000 elements takesday to complete a run with
10,000 time steps. An uppertime step is given by t=0.0005t*, where
t* = r/USI. Fornd in-line bubbles (Sections 4.1 and 4.2) we used
theric geometry while for the bubbles rising side-by-sideer angles
(Sections 4.3 and 4.4) we employed the 2D
-
122 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
101100
101
102C d
Fig. 4. Drag coExperimentaltions: () NewHadamard pre
4. Results
4.1. Single b
The expfunction ofvalues are aCd were de
Re = USIdb
where
= k(
2USIdb
and
Cd =4dbg
3U2SI
We founand Stokesthe experima Reynoldsapproachesthe Stokes
playabovebodition resulsame Re. Nonumerical vFig. 4), indicdomain
sizethe numeriin the n=0the n=0.55of the Carreference is d(Eq.
(7)), orto attain a ration or becathe bubblefollowing wical
results,simulations
.4
1.5
1.6
rag coRe
-
J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 123
term that occurs as n decrease (this component of the total drag
isactually negligible for bubbles without surface active
agents).
We conducted other simulationswith higher Reynolds numbers(Re8);
the corresponding Y(n) values are also shown in Fig. 5. Inthis
case, sincannot be uues [4648Y(n) we divby its Newtbut
changincosity valu[52] have cids, maintahowever, keguaranteed
Y(n) = CdtCdN
The resudrag increaa similar coties obtainethe comparlar to
whatviscosity vaThese authothinning u
4.2. Two-bu
Let us ridentical spcenter sepaones used inthe simulatwith the
upration distathe trailingtonian andresults forFig. 6. In thcloser
to eaY(n) valuesrising one atance than tto the wayously
said,characteristthinning anbubble, thebeing that ithe
bubblesbubble andthat the comtially denestudy singlethe
apparensingle bubbcal procedutwo bubbleume). The scan be visuelds
are shshear rate afour times)
0.6 0.7 0.8 0.9 1.9
5
1
5
1.1
0.5r
0.1r
flow index (n)rag coefcient ratio Y(n) as a function of the ow
index for the in-lineair; lled symbols: trailing bubble, empty
symbols: leading bubble; ()ration distance of 0.1r, () gap
separation distance of 0.5r. As the bubblewas obtained from the
average between the velocities at the top
andbottomthebubbleboundary, theminimumseen in the leadingbubble for
0.1r couldo bubble surface deformation.
ith the higher shear rate surrounds the bubble pair, forminge
jaat thtionositimentcenctiothisusine t*ters.o ancanio re
hear rate contours obtained for a single bubble and an in-line
pair rising in.5 uid. The values were taken at the same time (t* =
5) for both the singleble pair. The initial separation of the
bubbles was 4r. The shear rate wasd using the formula inside the
square brackets of Eq. (4) and normalized byacteristic time
r/USI.ce the Renumber is not small theHadamard predictionsed and a
direct comparison with the theoretical val-] cannot be made.
Therefore, to calculate the values ofided the drag of the thinning
uid by the value attainedonian counterpart using the same physical
parametersg the ow index value to one, leaving the same vis-
e at the characteristic shear rate USI/r. Other authorsompared
both cases, the Newtonian and thinning u-ining the same Reynolds
number. Using this scheme,eping the same physical data between uids
cannot be. The Y(n) values were thus calculated as:
hinning
ewtonian=(
USINewtonianUSIthinning
)2(12)
lts have the same trend as in the low Re case, that is, theses
with the thinning behavior. Zhang et al. [53] mademparison of their
numerical terminal bubble veloci-d with thinning and Newtonian
uids. In order to makeison they changed the value of the ow index
(simi-was done here) but considering a constant zero-shearlue (we
considered a shear-rate dependent viscosity).rs found that the
terminal velocity of bubbles rising inidswas higher than the
correspondingNewtonian case.
bble interaction: vertical alignment
st consider the numerical results. Initially we place twoherical
bubbles one above another, with a center-to-ration of 4r using the
same physical properties as thethe single bubble simulation with
Re8. Upon start of
ion, both rise and in time the lower bubble catches upper one.
When the bubbles reached a certain gap sepa-nce, two Y(n) values
(Eq. (12)) were calculated, one forbubble (comparing the bubble
velocities of the New-thinning uids) and another for the leading
one. Thetwo separation distances, 0.1r and 0.5r, are shown inis
gure we can observe that as the bubbles becomech other, the trend
seen for single bubbles changes: thefall slightly below one. This
means that two bubblesfter the other in a thinning uid experience
less resis-heir Newtonian counterparts. This behavior is inherentwe
made the thinning uid simulations. As we previ-we xed the apparent
viscosity corresponding to theic shear rate USI/r of the single
bubbles for both, thed Newtonian cases. When for both is added a
secondshear rate is increased in both uids, the difference
n the thinning case a zone with a lower viscosity nearwill
appear. The comparison made between the singlebubble pair is
similar to that made by [40], in the senseparison depends on how
the uid properties are ini-
d and serve as input information. For example, in thisbubble and
bubble pair simulations were done usingt viscosity calculated at
the characteristic time of thele as reference. However, this is an
arbitrary numeri-re since one can also dene the reference viscosity
fors (considered as a single one with the equivalent vol-hear rate
formed around a bubble or a pair of bubblesalized using the
numerical code. In Fig. 7 the shear rateown for the
thinninguidwithn=0.5 (10
-
124 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
6
7a b
20
6
7
Fig. 8. Center-6r. (a) db = 2.1Numerical ressmaller and
mNewtonian uof the article.)
(drafting) ctowards theconstant nestant approincreased (slope
has th(Fig. 8b andare in goodstage (kissislope. The cof the
trailinbles due to tis the onenon-Newtoand mediumn=0.5 uidtogether
afttained a vera certain an(Fig. 9b). Inposition aft0 10 20 30 40
50 600
1
2
3
4
5
c
t*
d*
smallbubbles
00
1
2
3
4
5
d*6
70 20 40 60 800
1
2
3
4
5
t*
d*
largebubbles
to-center dimensionless distance as a function of the
dimensionless time of two bubbles rmm, 0.45
-
J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 125
Fig. 9. Non cocamera. ThebuThe t betwenumber corres
Newtonianwhen long rin that casefraction.
101101
100
101
102
Rey
nold
s
Fig. 10. Clustebles. () hydrothinning uidsthe thinning uid, (+)
pair fbols correspon[33]. The iso-Mincreases from
0.
0.6
0.
0.7
0.8
CdT/
CdL
Fig. 11. Experbles, () medi=4r.
In this wthe Newtonblet (Fig. 8afor the sambubbly owuids at
low[32], i.e., thregime willdifferent sin
The factdoubg inurrois emas nonsecutive snapshots of the bubbles
position taken with the movablebble size is 2.8mm.The size of the
image is approximately 717 cm2.
en shots is 0.5 s. (L) leading and (T) trailing bubbles. The
indicated Reponds to the maximum value reached by the bubbles.
stablemovinbeing sditionhere wows, in [32] it was found that
clustering can also occurange interactions are signicant (low Re
numbers), butthe effect is rapidly hindered by the increase of the
gas
100 101
Mo=103not tumbling
tumbling
sovtoE
r condition formation mapped in a EoRe curve for the single
bub-dynamic conditions where the tumbling stage was not seen in
the, () hydrodynamic conditions where the tumbling stage was seen
inuids, ( ) Mo=103,(*) free bubbles after contact in the
Newtonianormation after contact in the Newtonian uid. The () and ()
sym-d to the cluster and free bubble conditions found in bubble
swarmsorton line was taken from [54]. The Morton number in an EoRe
plottop to bottom.
contradictiofor a givennumber is lform clusteThis
concluNewtonian
Fig. 8 sucess can onthe rate of aindex (Fig.can compuCdT to the
l
CdTCdL
=(
ULUT
where U is tat a separaindex n is sbles) conrless than itsSuch
effectis increasedsubtle.
In the tbecomes altonian andCdT/CdL fordimensionlto the
casetinued travthe drag rat0.5 0.6 0.7 0.8 0.9 16
5
7
5
flow index (n)
2.0
-
126 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
00.5
1
1.5C d
T/CdL
Fig. 12. Exper() Newtoniadb = 2.8mm.
the movemlatory behacatches thement due tohorizontal aseparation
olow to highdecreased.The bubblehence the bble at the
fsimilarperiof settling por four bodperiodic monot beenob[12,28]
normovementobserved inhas very diHence, wedue to non-
Apart frodrafting promost signishows the btion d* for adata
prior ttrend is clogas bubblestion was bainto
accounexperimentprinciple ofdeviation otion at shorthat the
anaterms that c
4.3. Two-bu
As menside-by-sid
00.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
U L/U
T
Velom, N
ian um, n
) the[56].
ressut wasthinnion cs. Intims andl value ca
eswe of sly ane lowulsivses te seearater sel trenwas
ed immenterestimate the lift forces acting on each of their
boundaries.50 100 150t*=t(USI/r)
contact time
imental drag ratio CdT/CdL in terms of the dimensionless time
t*.n uid, (*) n=0.85 uid, () n=0.55 uid. The bubble diameter is
ent of the bubbles forming the pair produces an oscil-vior of
the drag ratio CdT/CdL: when the trailing bubbleleadingone, thepair
turns towards thehorizontal align-the pressure directed against the
bubble motion. Thisrrange is nevertheless not stable because it
leads to thef the bubbles [22], which in turn causes a passage
formviscosity zones since the characteristic shear rate is
A possible conguration is then a diagonal alignment.located at
the front will cause a reduced viscosity path,ubble in the back
will be accelerated, reaching the bub-ront and passing it. The
process is repeated again. Aodicmovementhasbeenobserved for the
caseof groupsarticles formed by three (experimental results, Re2r).
In spite of the data scatter these to the analytical solution of
Rushton and Davies for[56] rising in Newtonian uids. This
analytical solu-
sed on the theoretical study made by [57], which tookt the
velocity of the leading wake. The agreement of theal data with the
theoretical prediction suggest that thesuperposition is also valid
for shear-thinninguids. Thef the experimental values from the
theoretical predic-t separation distances (
-
J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 127
0 20 40 60 80 1005
6
7
8
9
10
11
12a
b
t*
d*
small bubbles
0 20 40 60 80 1005
6
7
8
9
10
11
12
13
t*
d*
large bubbles
Fig. 14. Dimensionless distance in terms of the dimensionless
time of two bubblesreleased side-by-side. (a) db = 2.1, 0.4
-
128 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
y
1
1
2
2
30Newtonian Shear-thinning n=0.55
00
0.5
1
1.5
2
2.5
3
3.5
ad s
Fig. 17. Bubbuid and its Necenters was =
4.4. Two-bu
In this sof the hydraligned horthe productpreservingexample, ato
manipulrather arduent behavion=0.55 andthe initial seanglewas
4bubbles iscan be seenother but thother hand,the leadingkissing
procmotion is nare forcedshear rate ztrend was s
non-tumbling stage). When the two interfaces come too close
toeach other (about 0.03r), the local resolution becomes
insufcientand the code eventually fails to converge. That was where
the sim-
s in the thinning uid ended.exploout
ng tosults witf 4rd froulatNew). Forg ur leso
bubise.eractevedui
tion.ion bcreasximrderputandx5 10 15 20 25 30
5
0
5
0
5
bubble positionb
Newtonian
ulationTo
carriedspondiThe reand 76tance oselectethe simfor
theshownthinninmore othe twotherwthe intis
achin=0.55separaattractThe dethe pro
In owe comn=0.555 10 15 20 25t*
n=0.5
dimensionless distance between bubbles as a func-tion of
time
les position and dimensionless separation distance for the
n=0.55wtonian counterpart. The initial separation distance between
bubble4r, the initial angle was 42 , Re10.
bble interaction: varying the angle of approach
ection we used the ALE code to gain some insightodynamic
interaction of a bubble pair which is notizontally nor vertically.
For the experiments, althoughion of bubbles to be aligned at an
arbitrary angle (andthe same separation distance) is doable
(consider, forstaggered initial arrangement); the necessary
work
ate and change one arrangement for another could beous; hence,
the use of simulations to predict the differ-rs is well justied.
The case of the thinning uid withits Newtonian counterpart is shown
in Fig. 17a, whereparationbetweenbubbles centerswas4r and the
initial2. The variation of the dimensionless distance
betweenplotted against the dimensionless time in Fig. 17b. Itthat
in the Newtonian uid the bubbles approach eachey do notmake contact
and eventually separate. On thein the thinning uid the trailing
bubble catches upwithone and makes contact with it. After this
drafting andess the bubbles do not separate, the so-called
tumblingot observed. This behavior indicates that the bubbles to
stay in a low viscosity region, produced by a highone, rather than
separate from each other (the sameeen experimentally with the
in-line bubbles during the
The results[53] for simues agree wwith theCadifferentwadient by
cothe bubblebubble diamcosity gradiplane is shodifferent diresults
for a
In Fig. 20neous arouwhich extenthe region ozontal plantherefore,
tcases showan angle ofin the n=0.the n=0.85gradient lieids with
higat lower anviscosity grfrom that oas a net drivthe bubbleby the
wak
In Fig. 2with the disthe n=0.55uids is cleratio reachevalue of 3
adepicted inbubble to ere the effect of the degree of
shear-thinning, we alsosimulations with the n=0.85 and 0.76 uids,
corre-the other experimental uids used here and in [33].
are shown in Fig. 18 for three initial angles 0: 42, 61
h respect to the horizontal and for the same initial dis-. The
Eo and Re numbers indicated in this gure werem the single bubble
experiments in order to compareions at a xed value of Eo or Re.
First we observed thattonian case, the bubbles separate after
contact (data notthis casewe tested initial angles up to 88. In the
shear-ids, the behavior at initial angle 0 61 (Fig. 18cf) iss the
same as for bubbles in tandem (see Fig. 9). That is,bles will form
a doublet if Mo103 and will separate
For 0 =42 , the greater horizontal separation weakension between
the bubbles such that doublet formationonly for the most
shear-thinning uid in Fig. 18b. Thed in Fig. 18a has a Mo=2103 but
anyway experienceFor the other cases in Fig. 18a and b, there is an
initialetween the twobubbles, but they eventually drift apart.e of
d* in the n=0.55 uid (Fig. 18a) at t* = 20 is due toity of the
wall.to explain more in detail the results observed in Fig. 18,e
the viscosity prole around a single bubble for the0.85 uids having
the same Etvs number (Eo=3).shown in Fig. 19 are similar to the
ones obtained byilar Re numbers (Re10). The maximum viscosity
val-ell with the values given by the rheological data tted
rreaumodel.Weanalyzed these viscosity proles in twoys: rst, we
introduced a denition of the viscosity gra-
mputing the difference between the viscosity value onsurface
(min) and the viscosity value () located at twoeters form the
bubble surface. The curves of such vis-
ent as a functionof the angle formedwith thehorizontalwn in Fig.
20a. Then, we compute the /min ratio forstances from the bubble
surface at a xed angle. Then angle of 20 are shown in Fig. 20b.a we
can see that the viscosity gradient is not homoge-
nd the bubble due to the presence of the bubble wake,ds the
region of non-zero shear rate andhence increasef viscosity
recovery. On the other hand, near the hori-
e, the decay of the shear rate occurs in a smaller region;he
values of the viscosity gradient are larger. For then in Fig. 20a
the maximum viscosity gradient occurs at20 in both uids; however,
the viscosity gradient found55 at this angle is 5 times larger than
the one found inuid. The fact that the higher values of the
viscositynear the horizontal plane could explain why the u-her
thinning behavior can promote bubble clustering
gles, as we saw in Fig. 18. In this sense we think that
theadients produce a stress distribution that is differentf a
Newtonian uid; such stress gradients will not working force but
will reduce the repulsive force created byvortices, causing the
trailing bubble to remain trappede of the leading one.0b we can see
how the viscosity ratio /min increasestance from the bubble
surface. The difference between(clustering condition) and 0.85
(free bubble condition)ar: in the case of the most thinning uid the
viscositys a value of 27, while in the other uid reaches only at
the same ds . The viscosity gradient of the n=0.55 uidFig. 20 acts
as a viscosity holewhich prevents anotherscape from the low
viscosity region.
-
J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 129
3
3.5a b
3
3.5
Fig. 18. Dimen=0.55; initiathe other uid
To compwith the onof two bubare shownlowest viscthe one occthat
the inclowest viscgradients. N0.5
1
1.5
2
2.5
n=0.8
n=0.7
n=0.5
d s d s
0.5
1
1.5
2
2.50 5 10 15 20 25 300
c d
e f
t
Eo = 2.6, o = 42
00
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
n=0.8
n=0.7
n=0.5
t
d s
Eo = 2.6, o = 61
00
0.5
1
1.5
2
2.5
d s
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
n=0.8
n=0.7
n=0.5
t
d s
Eo = 2.6, o = 76
00
0.5
1
1.5
2
2.5
d s
nsionless distances ds between bubble boundaries as a function
of the time t* for severa
l distance, =4r. For the left column Eo=2.6 is xed, for the
right column Re4 is xed. Ts is around 7104.
are the viscosity prole obtained for a single bubblee obtained
for a bubble pair, we compute the viscositybles in contact rising
in the n=0.55 uid. The resultsin Fig. 21. In this case we can see
that the region of theosity value (0.05Pa s) comprises a larger
area thanupied in the single bubble case. We may think thenrease of
the cluster size will increase the region of theosity value but at
the same time decrease the viscosityote also that in the rear part
of the bubble pair a region
of high viscvortex, in a
From allfor cluster fdients thatuid reducmakespossble; if the
thof the lift fon=0.8
n=0.7
n=0.55 10 15 20 25t
Re 4, o = 42
5 10 15 20
n=0.8
n=0.7
n=0.5
tRe 4, o = 61
5 10 15 20
n=0.8n=0.7
n=0.5
tRe 4, o = 76
l thinning conditions and initial angles. () n=0.85, () n=0.76,
()he experimental Mo number for the n=0.55 uid is 2103 and for
osity start to form due to the appearance of a toroidalgreement
with [53].these observations we can propose three
mechanismsormation in shear-thinning uids: (i) The viscosity
gra-appear in the wake of a bubble ascending in a thinninge the
repulsive force from the bubble vortices. Thisible for abubble
tobecaught in thewakeof anotherbub-inningbehavior increases, the
critical angle of inversionrce (from repulsion to attraction) is
decreased, leading
-
130 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
Fig. 19. Viscosity contours around a single bubble immersed in
the n=0.55 and 0.85 uids. The vist* = 7.6, Eo=3. To avoid high
viscosity values at low shear rates, an upper limit to the
viscosity was
+ (0 )(1 + []2)(n1)/2
, where 0 is the zero-shear viscosity value, the viscosity at
high she
0 20 40 60 80 1000
10
20
30
40
50
60
70
angle ()
[ m
in]/2
d b (m
Pas/
mm
)
n=0.5
n=0.8
00
5
10
15
20
25
30ba
/ m
in
Fig. 20. (a) Values of the viscosity gradient as a function of
the angle made with the horizontal planefunction of the
dimensionless distance at a xed angle of 20 .
Fig. 21. Viscosity contours around a bubble pair immersed in the
n=0.55 uid. Theviscosity was estimated when the bubbles achieved a
gap separation distance of0.06r, Eo=3. The initial angle was 76 .
The Carreau model was again used as inFig. 19.
to a more ethan103, tformed; (iiilower viscotion of bubwill
decreabehavior) apoint out heasonebigbof its size. Aof the
deforstreamlinesarate bubbltrap additio
The thinforce that rshear stresinterpretedshear, that ito the
negatuxper unithe shear stwe can calc
P = y
(cosity was estimated when the bubbles achieved a steady
velocity,introduced tting the rheological data with the Carreau
model =ar rates, a time constant and n the ow index.
n=0.52 4 6 8d
s*
n=0.8
for the bubbles presented in Fig. 19, d* = 5r. (b) Viscosity
ratio as a
ffective clustering; (ii) if the Morton number is
higherhebubbles donot tumble after contact, a doublet is then) once
two bubbles form a pair, they create a wake withsity which attracts
more bubbles, leading to the forma-ble clusters; however, the
increase of the cluster sizese the viscosity gradient (so attaining
a Newtonian-likend the growth of the clusters will then stop. We
need tore that the cluster deformability (imagining the cluster
ubble)must alsoplayan important role in the increments mentioned
by Manga and Stone [16,17], the increasemability of the bubble
surface leads to the formation ofthat propitiates the alignment and
contact of two sep-es. The velocity eld formed around a cluster can
alsonal bubbles that contribute to its growth.ning property reduces
the hydrodynamic repulsive
esults from the converging streamlines. Let us considers in a
Newtonian uid: = . This relation can beas the momentum transfer
from high to low regions ofs, themomentumper unit area and time is
proportionalive of the velocity gradient [61]. Hence, themomentumt
time and volume, P, can be calculated as the gradient ofress.
Considering now the power law model = knulate P as:
) (18)
-
J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132 131
00
1
2
3
4
5
6
7d s*
2.5db
o=42
Fig. 22. Dimedimensionlessds = 2, () ds =
where = kP = (n where =one, we hav
P
PN=
N+
In the cHowever, iwith n, thabecomes mappear arouwhich, in tuthe
leading
We havthe Newtonof 40 withone. On thecloser to thto the contthat
observbubbles poattraction, wexperiencein terms ofother exper
Finally,alignment,dependentis shown in(n=0.55 anbubbles
surlonger attrahorizontallyThe same tresults suggbe formedcondition
is
5. Conclus
The inteuids (0.55
tories with a movable camera. The bubbles were released by a
pairof capillaries in a vertical and a horizontal alignment. The
exper-iments were complemented by numerical simulations
conductedwith the arbitrary LagrangianEulerian technique. In this
way, a
ge owere
re caviorpulsfromrwishearble aeaseinde
cts.nume lifhighbubbr theu
the ied t-Newed aed be tra
e
-
132 J.R. Vlez-Cordero et al. / J. Non-Newtonian Fluid Mech. 166
(2011) 118132
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Hydrodynamic interaction between a pair of bubbles ascending in
shear-thinning inelastic fluidsIntroductionExperimental setupColumn
and bubble generationFluidsBubble size and velocity measurement
Computational techniqueResultsSingle bubble results and
benchmark simulationsTwo-bubble interaction: vertical
alignmentTwo-bubble interaction: horizontal alignmentTwo-bubble
interaction: varying the angle of approach
ConclusionsAcknowledgementsReferences