velez 2011

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J. Non-Newtonian Fluid Mech. 166 (2011) 118–132 Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm Hydrodynamic interaction between a pair of bubbles ascending in shear-thinning inelastic fluids J. Rodrigo Vélez-Cordero a , Diego Sámano a , Pengtao Yue b , James J. Feng c , Roberto Zenit a,a Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Apdo. Postal 70-360, D.F. 04510, Mexico b Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA c Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, V6T 1Z3, Canada article info Article history: Received 13 July 2010 Received in revised form 3 November 2010 Accepted 4 November 2010 Keywords: Bubble pair interaction Non-Newtonian Shear-thinning abstract The interaction of two bubbles rising in shear-thinning inelastic fluids was studied. The experimental results were complemented by numerical simulations conducted with the arbitrary Lagrangian–Eulerian technique. Different initial alignments of the bubble pair were considered. Similarities and dif- ferences with the Newtonian fluids were found. The most noticeable difference is the so-called drafting–kissing–tumbling (DKT) process: for the case of bubbles rising in thinning fluids, the tumbling phase does not occur and the pair tends to form a stable doublet. The DKT process is also influenced by the amount of inertia and deformability of the individual bubbles and the initial angle between them. The experimental and numerical results suggest that the thinning wake formed behind the bubbles plays an important role in the speed of the pair and the formation of clusters in thinning fluids. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Bubble interactions have received considerable study in the lit- erature owing to its inherent scientific interest and importance in applications such as gas–liquid contactors [1–3]. The study of two-bubble interaction was initially motivated by the coalescence phenomena due to its impact in bubble columns efficiency. That was how researchers took two different paths concerning this issue: one was the study of the coalescence mechanism itself, which is highly dependent on the liquid composition [4–6]; the other was the study of the trajectories that two bubbles take before coalescence [7–9]. In this work, the interaction of two bubbles ascending in shear-thinning inelastic fluids (non-coalescing con- ditions) is experimentally and numerically studied. First, we will make a review of the current literature of hydrodynamic interac- tions. In particular, to put our work in perspective, we focus on the Newtonian case. For the case of creeping flow, Stimson and Jeffery [10] analyti- cally predicted the velocity of two spheres moving in-line in terms of the separation distance between them. As widely described by [11], two bodies moving in this way in the creeping flow regime acquire a higher velocity than that attained by a single body; the velocity increases as the separation distance decreases. As the flow field does not have inertia, the bodies keep their distance without approaching each other. This trend is in agreement with experi- Corresponding author. Tel.: +52 55 5622 4593; fax: +52 55 5622 4602. E-mail address: [email protected] (R. Zenit). mental data [12] and with other analytical expressions [11], and applies as well to settling particles or rising bubbles. When the inertia is small but finite (Re 0.25) the vorticity around a spherical body looses its fore-aft symmetry and the trail- ing body acquires a higher velocity than the leading one, reaching the latter after some time. Crabtree and Bridgwater [7] were the first ones to approximate the trailing bubble velocity as the sum of the terminal velocity of the single bubble plus its wake velocity at the distance where the trailing bubble is found. This hypoth- esis was referred to as a ‘superposition principle’ by Bhaga and Weber [13]. In later works [13–15] this hypothesis was tested and confirmed for two in-line bubbles rising with Reynolds numbers up to O(1 0 0). Crabtree and Bridgwater also reported the curious phenomenon (not fully explained yet) in which the trailing bubble experiences a significant deformation (from oblate to prolate form) moments before touching the leading one. Such deformation was also reported and photographed by Narayanan et al. [14]. These last authors worked with different bubble sizes producing basically two different wake structures: one forming a thin trailing wake and the other forming a wake with a stable toroidal vortex. For the former it was observed that the Stimson and Jeffery’s equation described well the rise velocities of the trailing bubbles even though it was formulated for creeping flows and spherical bodies. For the second case, 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 (Reynolds 80, Eötvös 70). The experimental measurements of the wake velocity were in agree- ment with the velocity calculated using the superposition principle. Manga and Stone [16,17] studied the effects of bubble deformability 0377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2010.11.003

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  • 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

  • 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