Chemical Engineering Science 60 (2005) 1859–1873

www.elsevier.com/locate/ces

Flow around individual Taylor bubbles rising in stagnant CMC solutions:PIV measurements

R.G. Sousaa, M.L. Riethmullerb, A.M.F.R. Pintoa, J.B.L.M. Camposa,∗aCentro de Estudos de Fenómenos de Transporte, Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto,

Rua Dr. Roberto Frias, 4200-465 Porto, Portugalbvon Karman Institute for Fluid Dynamics, 72 Chaussée de Waterloo, 1640 Rhode-Saint-Genèse, Belgium

Received 29 April 2004; received in revised form 22 November 2004; accepted 24 November 2004Available online 22 January 2005

Abstract

The flow around single Taylor bubbles rising in non-Newtonian solutions of Carboxymethylcellulose (CMC) polymer was studied usingsimultaneously particle image velocimetry (PIV) and shadowgraphy. This technique made it possible to determine the correct position ofthe bubble interface. Solutions of polymer with weight percentage varying from 0.1 to 1.0 wt% were used to cover a wide range of flowregimes. The rheological fluid properties and pipe dimension yielded Reynolds numbers between 4 and 714 and Deborah numbers for thehigher concentration solutions between 0.001 and 0.236. The shape of the bubbles rising in the different solutions was compared. Theflow around the nose of the bubbles was found to be similar in all the studied conditions. Velocity profiles in the liquid film around thebubble were measured and different wake structures were found. With increasing solution viscosity, the wake flow pattern varied fromturbulent to laminar, and a negative wake was observed for the higher polymer concentration solutions.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Taylor bubble; Multiphase flow; Non-Newtonian fluids; Viscoelasticity; Particle image velocimetry (PIV); Liquid films

1. Introduction

Slug flow is a two-phase flow regime found when a gasand a liquid flow simultaneously in a pipe over certain flowrate ranges. It is characterised by elongated gas bubbles(Taylor bubbles or slug bubbles) almost filling the pipecross-section and liquid flowing around and between thebubbles. This flow pattern is found in several industrial pro-cesses as is the case of geothermal, oil and gas transporta-tion in wells, fermentation, polymer devolatilisation andair-lift reactors among others. In some chemical processes,slug flow is induced to increase the reaction rate due to themixing action mainly promoted by the wake of the Taylorbubbles.

The velocity of the bubble depends, among other param-eters, on the velocity of the liquid flowing ahead (Moissis

∗ Corresponding author. Fax: +351 225081449.E-mail address:jmc@fe.up.pt(J.B.L.M. Campos).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.11.035

and Griffith, 1962). When two bubbles rise together in apipe, the wake of the leading one influences the velocity ofthe following, if the distance between the bubbles is smallerthan the minimum stable length (Taitel et al., 1980). In thiscase, two bubbles can merge (coalescence) giving place toa larger bubble and reducing the number of mixing zones.Coalescence is therefore a problem to avoid and the min-imum stable distance is strongly dependent on the flow inthe wake of the leading bubble.

Concerning Newtonian liquids, among other studies,Campos and Guedes de Carvalho (1988)studied the wakestructure for different liquid flow regimes.Pinto and Cam-pos (1996)studied the interaction between two consecutivebubbles and established minimum stable lengths for differ-ent operation conditions. Particle image velocimetry (PIV)measurements were performed to characterise the flowfield around individual Taylor bubbles in Newtonian fluids(Bugg and Saad, 2002; van Hout et al., 2001; Nogueiraet al., 2000; Polonsky et al., 1998).

1860 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

The study of Taylor bubbles rising in non-Newtonian liq-uids is still scarce, although this two-phase flow is frequentlyfound in industry. Due to the complex liquid rheology, thegas–liquid flow patterns have completely different character-istics, and so different bubbles interaction behaviour. There-fore, there is a need for extending the slug flow research to-wards non-Newtonian liquids. Velocities and shapes of smallbubbles flowing in non-Newtonian liquids have been stud-ied by different authors (Astarita and Apuzzo, 1965; Leal etal., 1971; Acharya et al., 1977). Hassager (1979)described,for the first time, a negative wake behind unconfined smallbubbles rising in non-Newtonian liquids. Bubbles coales-cence in non-Newtonian liquids was studied byAcharya andUlbrecht (1978), Kee et al. (1990)and Li et al. (2001).Funfschilling and Li (2001)used PIV and birefringencevisualisation to study the flow of non-Newtonian liquidsaround small bubbles.

Concerning Taylor bubbles flow in non-Newtonian flu-ids, the effects of power law rheology and pipe inclina-tion on slug bubble velocity were studied byCarew et al.(1995). Otten and Fayed (1976)studied the pressure dropand the friction drag reduction in non-Newtonian slug flow.Rosehart et al. (1975)measured the void fraction, slug ve-locity and frequency for co-current slug flow of air in highlyviscous non-Newtonian liquids.Terasaka and Tsuge (2003)performed gas hold-up measurements for gas slugs rising inviscous liquids with a yield stress.Kamıslı (2003)derived aone-dimensional flow equation for the motion of a long bub-ble rising steadily in a vertical or inclined tube filled with apower-law fluid. The author concluded that the thickness ofthe liquid film flowing around the bubble increases with de-creasing power-law index. It is also mentioned the disagree-ment found by several authors between the predicted and theexperimental liquid film thickness for non-Newtonian fluidsand that, viscoelastic effects tend to reduce the film thick-ness when compared with predictions from purely viscoustheory.

In the present work, the flow around single Taylor bubblesrising in stagnant non-Newtonian Carboxymethylcellulose(CMC) solutions is studied using PIV and shadowgraphytechniques simultaneously. The results obtained contributeto a better understanding of the non-Newtonian flow fieldaround the bubbles and the coalescence mechanism.

2. Experiments

2.1. Experimental techniques

The flow field around the Taylor bubbles rising in stag-nant liquids was obtained applying PIV and shadowgraphysimultaneously. This technique was first applied byLindkenand Merzkirch (2001)to bubbly flow andNogueira et al.(2003) adapted it to slug flow. The technique is fully de-scribed inNogueira et al. (2003)andSousa et al. (2004)andconsists in placing a board of light emitting diodes (LEDs)

behind the test section pulsing simultaneously with the lasersource, so that a CCD camera acquires an image containingboth the PIV particles and the bubble shadow. Two lensesof 35 and 50 mm of focal length were used to obtain respec-tively the flow field in the nose/wake region and a closeview of the liquid film around the bubble. The LEDs boardis composed by 350 LEDs. Fluorescent particles (an orangevinyl pigment with 10�m of mean size) were used as seed-ing, emitting light at 590 nm. A Nd:YAG laser was used tocreate a vertical laser sheet of about 1 mm thickness pass-ing through the axis of the column. The laser, the cameraand the LED’s board were all triggered by the same signalgenerator making it possible to acquire the PIV image andbubble shadow in the same frame.

Two laser diodes were installed near the outside columnwall, pointing into two photocells placed in the opposite sideof the column. The signal yielded by the photocell is pro-portional to the light received, so when the Taylor bubbleis passing between a laser diode and a photocell, the laserbeam is deflected and the photocell signal drops abruptly.The Taylor bubble velocity was determined dividing the dis-tance between the two photocells by the time delay betweentheir dropped signals.

2.2. Facility

The present study was performed in the experimentalsetup represented inFig. 1 and fully described inNogueiraet al. (2003)andSousa et al. (2004). The facility is mainlycomposed by an acrylic cylindrical column of 6 m heightand 0.032 m of internal diameter, opened at the top to theatmosphere, two pneumatic valves, two storing tanks anda pump. The test section was located near the top of thecolumn, to avoid entrance effects and to assure a stabilisedflow. A box with plane faces surrounding the test section(0.5 m × 0.12 m× 0.11 m) was filled with the studied liq-uid in order to minimise the optical distortion. IndividualTaylor bubbles were injected at the bottom of the column,by operating valves A and B. Opening valve B, the liquidtrapped between the valves falls into the reservoir being sub-stituted by air. Closing valve B and opening valve A, theair between the valves starts rising up in the form of a longtubular bubble (Taylor bubble). The volume of the bubbleswas controlled by setting the time during which valve B wasopened. For each solution, the study was focused on bub-bles with sufficient length to have a developed liquid filmflowing around but not enough length to generate wave in-stabilities in this liquid film. The PIV instrumentation wasbasically composed by a PCO (SensiCam) CCD camera andan acquisition and data processing system. A Nd:YAG laserwith 400 mJ of pulse power was used to illuminate the mea-sured plane. The laser wavelength was 532 nm and the pulseduration 2.4 ns. A LED array emitting light at 650 nm wasplaced behind the test section with a diffuser paper to ob-tain the Taylor bubble shadow at the same time as the PIV

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1861

LASEROPTICS

PHOTO CELLS

LASERDIODES

CCDCAMERA

TESTSECTION

A

B

PNEUMATICVALVES

FRONT VIEWTEST SECTION DETAIL

(top view)

PUMPDATAACQUISITION

SYSTEM

LED’S BOAR D

CCDCAMERA

VERTICALLASERSHEET

90°

Fig. 1. Representation of the experimental setup with a detailed view of the test section.

image. A red filter, opaque below 550 nm was placed infront of the PCO CCD camera to block the intense greenreflections of the laser and to allow the passage of the lightemitted by the fluorescent particles and by the LEDs. Twothermocouples were placed below and above the test sectionto measure the working temperature and to check a possibletemperature gradient along the column. The fluid rheologywas determined in a AR 2000 DTA Instruments Rheometer.

3. Data processing

The images acquired with the simultaneous PIV andshadowgraphy technique contain both the PIV particlesimages and the shadow of the bubble in the same frame.The PIV processing method is fully described byNogueiraet al. (2003). The flow field was obtained using the cross-correlation algorithm window displacement iterative multi-grid (WIDIM), developed by Scarano and Riethmuller(1999). In this work, initial interrogation windows had20 pixels× 40 pixels and, after the first vector estimative,final windows had 10 pixels× 20 pixels. An interrogationareas overlap of 50% was used. Spurious vector identifica-tion was used; vectors with a signal to noise ratio (SNR)less than 1.5 (about 5–7% of the total vectors) were substi-tuted by the neighbours average value. The time betweenPIV images was adjusted according to the liquid velocityand varied between 500 and 2000�s for the wake and noseregions and 80–400�s for the liquid film flowing around

the bubble. Tests made with known particle displacementsshowed a maximum uncertainty of 0.2 pixels on the win-dows displacement. The time between images was setaccording to the liquid velocity, so the maximum particledisplacement was about 5–7 pixels. This means that fora typical displacement of 5 pixels, the maximum relativeerror in the velocity is 4%.

With the bubble shadow it is possible to identify the posi-tion of the gas–liquid interface and so to overcome the prob-lems described byNogueira et al. (2000)due to reflectionand refraction of particles images at the interface. Erroneousvectors created inside the Taylor bubble are then eliminated.The image processing to determine the shadow of the bub-ble is described in previous works (Nogueira et al., 2003;Sousa et al., 2004) and consists of several sequential steps.A median filter is applied to the original image to eliminatethe seeding particles. A background reference image is thensubtracted to the filtered image to eliminate the column andthe background resulting the shadow of the bubble. The pro-cess is completed by defining a gray level threshold, whichbinaries the image and by filling the interior of the Taylorbubble.

4. Results

In this work, the flow around Taylor bubbles ris-ing in stagnant solutions of CMC (molecular mass of300 000 kg kmol−1, grade 7H4C from Hercules) with

1862 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

Shear rate (1/s)

Vis

cosi

ty (

Pas

)

10-2 10-1 100 101 102 103 10410-3

10-2

10-1

100

101

1.0% CMC (22.0°C)0.8% CMC (22.0°C)0.6% CMC (19.0°C)0.5% CMC (25.0°C)0.4% CMC (19.0°C)0.3% CMC (21.5°C)0.1% CMC (20.3°C)Carreau-Yasuda

Fig. 2. Viscosity as a function of the shear rate for the studied solutionsand Carreau–Yasuda model fitted curves.

different weight percentages was studied. The shear viscos-ity of the different solutions was measured in a AR 2000DTA Instruments Rheometer at experimental temperature.All the solutions exhibited a shear-thinning behaviour, ascan be seen in the viscosity-shear rate plot onFig. 2.

The Carreau–Yasuda viscosity model is expressed by thefollowing equation:

� = �∞ + (�0 − �∞)(1 + (��)a)(n−1)/a, (1)

where� is the measured viscosity,� the applied shear rate,�0 the viscosity limit when� → 0, and�∞ the viscosityat infinite shear rate, which was set as the solvent viscosity.The parameter� has units of time,n anda are dimensionlessparameters.

This model was fitted to the experimental data andrepresented inFig. 2 by full lines. The values of theCarreau–Yasuda model parameters are listed inTable 1.

The first normal stresses difference,N1, was measuredfor the higher concentration solutions and is represented inFig. 3 as a function of the shear rate. In order to quantify

Table 1Carreau–Yasuda viscosity model parameters for the CMC solutions

CMC (wt%) T (◦C) �0 (Pa s) �∞ (Pa s) � (s) a n � (s−1)

0.1 20.3 0.0091 0.001 0.0214 0.8497 0.8711 1–40000.3 21.5 0.0510 0.001 0.0566 0.7124 0.7212 0.7–40000.4 19.0 0.1102 0.001 0.1099 0.8087 0.6751 0.125–40000.5 25.0 0.2203 0.001 0.0631 0.5654 0.5095 0.25–40000.6 19.0 0.3602 0.001 0.1828 0.8317 0.5745 0.08–40000.8 22.0 1.0497 0.001 0.2214 0.6610 0.4330 0.04–40001.0 22.0 2.9899 0.001 0.3653 0.6683 0.3997 0.04–4000

the viscoelastic effects, the fluid relaxation time was deter-mined for the fluids with measurable first normal stressesdifference.Leider and Bird (1974)defined the fluid relax-ation time,�f (Eq. (2)), using the fact that both shear stress,�, andN1 can be well approximated by power functions ofthe shear rate over the range of conditions of interest, i.e.,� = m(�)r andN1 = m1(�)s .

�f =(m1

2m

)1/(s−r)

. (2)

The characteristic shear rate of the flow, defined as�f =Ugb/D, whereUgb is the Taylor bubble velocity andD thepipe internal diameter, is between 5 and 6 s−1 for the higherconcentration solutions. To obtain the relaxation time, theshear stresses and the first normal stresses difference wereapproximated by power functions of the shear rate in therange 0.04–10 s−1. The relaxation time,�f , and the corre-sponding Deborah number,De = �f �f , are represented inTable 2for the three more concentrated solutions.

The values of the Deborah number depend on the pro-posed relaxation time definition, however fromTable 2it isclear a change in its order of magnitude, from the less to thehigher concentration solutions.

The Reynolds number,Re = �UgbD/�, where� is theliquid density, was also determined, using the viscosity at thecharacteristic flow shear rate,�f . Values of the experimentaltemperature, gas bubble velocity and Reynolds number arepresented inTable 3for all the studied solutions.

The Taylor bubble shape and the velocity field around thebubbles rising in the different solutions are described in thenext sections.

4.1. Taylor bubble shape

The Taylor bubbles are elongated bubbles characterised bya prolate spheroid leading edge, whose curvature is higherfor higher viscosities. The bubble nose shapes, obtained fromthe bubble shadows, are represented inFig. 4 for some ofthe studied solutions, in the form of dimensionless distanceto the nose(z/D) versus the dimensionless bubble radius(r/D).

The bubble radius increases withz/D, until it reachesa maximum value at a certain distance from the nose. The

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1863

Shear Rate (1/s)

N1

(Pa)

10-2 10-1 100 101 102 103 104100

101

102

1031.0% CMC (22.0°C)0.8% CMC (22.0°C)0.6% CMC (19.0°C)

Fig. 3. First normal stresses difference in function of the shear rate forthe higher concentration solutions.

Table 2Relaxation time and Deborah number for the higher concentration solu-tions

CMC (wt%) �f (s) De

0.6 0.0002 0.0010.8 0.0295 0.1661.0 0.0472 0.236

Table 3Experimental temperature, bubble velocity and Reynolds number

wt% T (◦C) Ub (m/s) Re

0.1 20.3 0.199 7140.3 21.5 0.198 1440.4 19.0 0.195 700.5 25.0 0.192 410.6 19.0 0.187 240.8 22.0 0.180 101.0 22.0 0.160 4

values of the maximum bubble radius,(r/D)max, for eachsolution are presented inTable 4.

An attempt to obtain a general equation for the bubbleshape was successfully performed. The bubble shapes arewell described by

r/D = k1 tanh(k2(z/D)k3), (3)

wherek1 is the dimensionless maximum bubble radius pre-sented inTable 4, which is function of the Reynolds number:

(r/D)max = k1 = 0.019 ln(Re) + 0.342 (4)

and k2, k3 were obtained by fitting the experimental val-ues. These parameters are function of Reynolds number

r/D

z/D

0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1 wt% CMC0.4 wt% CMC0.8 wt% CMC1.0 wt% CMC

Fig. 4. Dimensionless distance to the bubble nose,z/D as a function ofthe dimensionless bubble radiusr/D.

Table 4Maximum bubble radius for the different studied solutions

Re (r/D)max

714 0.455144 0.43870 0.42741 0.41824 0.40610 0.3934 0.353

according to the following equations

k2 = 3.983Re−0.111, (5)

k3 = 0.648Re−0.068. (6)

The maximum deviation in the bubble shapes obtained withcorrelation (3) is 4% and occurs only for the limit viscosities.

The bottom region of the bubble is where the main dif-ferences in the bubble shape are present. Some PIV images

1864 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

Fig. 5. Bubble trailing edge images for (a)Re = 714, (b)Re = 144, (c)Re = 70, (d) Re = 24, (e)Re = 10, (f) Re = 4.

showing the Taylor bubble trailing edge are represented inFig. 5. For the higher Reynolds number (Fig.5a and b), thebottom surface is unstable and oscillates three-dimensionallyas the bubble rises. As the viscosity increases, the trailingedge becomes stable and with a concave shape (Fig. 5c andd). For the lower Reynolds numbers (Fig. 5e and f), thetrailing edge loses its concavity and gets a lacrimal shape.For these fluids, there is a sharp increase in the value of theDeborah number, as seen in Section 4, which means that thechange in the bubble trailing edge shape should be relatedto viscoelastic effects.

4.2. Flow field in the bubble nose

The flow fields around the Taylor bubbles were obtainedusing PIV. The vector fields represented in the next sectionsare in a vertical plane containing the central axis of the pipe.

The flow pattern around the nose of the bubbles is sim-ilar for every studied solution, varying only in the velocitymagnitude.Fig. 6 shows the flow field around the nose ofa Taylor bubble rising in a 0.5 wt% CMC solution. The ve-locity vectors are relative to a fixed reference frame and theTaylor bubble is moving upwards. FromFig. 6 it is possibleto see that as the bubble rises, the liquid in front of the bub-ble is pushed forward and away from the centre and startsfalling around the bubble forming a thin liquid film.

In Fig. 7 two instantaneous axial velocity profiles alongz = 0 for the limit viscosities studied are represented. FromFig. 7, it is possible to see that the velocity profiles atz=0 forthe limit cases are very similar, varying slightly the velocitymagnitude due to different bubble velocities and shapes. Thevelocity profiles of the other solutions studied are betweenthese two, so they were not plotted for a better visualisation.

r/D

z/D

-0.50 -0.25 0.00 0.25 0.50

-0.20

0.00

0.20

0.40

0.1m/s

Taylor Bubble

Fig. 6. Flow around the nose of the Taylor bubble in a 0.5 wt% CMCsolution.

4.3. Liquid film velocity profile

The liquid flowing around the bubble nose forms a fallingliquid film between the gas–liquid interface and the pipewall. The thickness of the liquid film decreases for highervalues ofz according to the bubble radius, until it reachesa minimum and stable value.Fig. 8 shows the vertical vari-ation of the axial component of the liquid velocity at thecentre of the liquid film for three of the studied solutions.Analysing the velocity in the liquid film and the radius ofthe bubble, alongz/D, they both reach a maximum valuearound a certain value ofz. In Fig. 9, the dimensionless shapeof the bubble and the dimensionless velocity at the centre

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1865

r/D

V (

m/s

)

-0.50 -0.25 0.00 0.25 0.50

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.1 wt% CMC1.0 wt% CMC

z/D = 0

Fig. 7. Instantaneous vertical component of the liquid velocity alongz=0for 0.1 and 1.0 wt% CMC solutions.

z/D

V (

m/s

)

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

1.20.3% CMC0.5% CMC1.0% CMC

Fig. 8. Instantaneous axial component of the velocity at the centre of theliquid film, as a function ofz/D for 0.3, 0.5 and 1.0 wt% CMC solutions.

of the film for the 0.5 wt% CMC solution are represented. Amedian value of the velocity was taken from the 20 neigh-bours (back and forwards) points to reduce the scattering ofthe data.

From Fig. 9, it is possible to see that the bubble radiusand the liquid film velocity both reach a maximum value atapproximatelyzstable= 3.5D. The point at which the liquidfilm becomes fully developed was obtained for all the solu-tions and is represented inTable 5.

Average velocity profiles in the developed liquid film weredetermined, by averaging the velocities at every radial posi-tion, betweenzstable/D (Table5) until the end of the liquidfilm. More than 1000 velocity profiles were used for eachCMC solution, obtained from multiple PIV images of dif-ferent bubbles with the same length.

z/D

V/V

(r/D

) max

, (r/

D)/

(r/

D) m

ax

0 1 2 3 4 5 6 7 8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.5 wt% V/V (r/D)max

0.5 wt% MedianV/V (r/D)max

0.5% (r/D)/ (r/D)max

Fig. 9. Dimensionless bubble radius and dimensionless instantaneousvelocity at the centre of the liquid film for the 0.5 wt% CMC solution.

As seen inFig. 8, as the viscosity decreases, the scatteringof the velocity values increases. The standard deviation ofthe averaged velocity profile in the developed liquid filmwas computed in pixels and is plotted inFig. 10for 0.1 and0.8 wt% CMC solutions. The time between the PIV imageswas set to 80 and 355�s for 0.1 and 0.8 wt% CMC solutionsrespectively.

The standard deviations presented inFig. 10 reach upto 4 times the estimated PIV error (0.2 pixels) mentionedin Section 3 and the same behaviour is found for all thestudied solutions. This means that the higher scattering inthe velocity data presented inFig. 8 for the less viscousCMC solutions is due to the decreasing time gap betweenPIV images.

Several hypothesis can be discussed to justify the highscattering level of the velocity data. A standard deviationof the windows displacement independent of the Reynoldsnumber, is a strong argument to conclude that turbulenceshould not be the cause. The bubble shape described in Sec-tion 4.1 contains small disturbances. These disturbances aredue to the presence of particles reflections at the interfacewhich interfere in the data processing used to obtain the bub-ble shape. Even though, if these disturbances were causedby bubble interface oscillations, they would induce, in theworst case, a maximum variation of 4% in the liquid filmthickness and could be responsible for a maximum variationof 8% in the liquid film velocity. This is still half of the vari-ations obtained in all the studied solutions. Different win-dows sizes were tested in the processing of the liquid filmprofiles but 0.8 pixels of maximum standard deviation wasalways obtained. Other tests were made by simply pumpingliquid in the pipe without the presence of the Taylor bub-ble, and taking different flow rates and times between PIVimages to cover a large range of particle displacements. Inthese tests, the liquid velocity is only function of the radialposition, like in the liquid film, and the maximum scattering

1866 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

Table 5Approximate values ofz/D from which the liquid film is fully developed

wt% 0.1 0.3 0.4 0.5 0.6 0.8 1.0

zstable/D 8.0 6.0 4.0 3.5 2.4 1.8 1.6

r/D

0.38 0.40 0.42 0.44 0.46 0.48 0.500

1

2

3

4

5

6

7

0.1% ∆y (pixels)0.1% σy (pixels)0.8% ∆y (pixels)0.8% σy (pixels)

∆y (

pix

els)

, σy

(pix

els)

Fig. 10. Mean windows vertical displacement,�y (pixels), in the liquidfilm and standard deviation,�y (pixels), for 0.1 and 0.8 wt% CMCsolutions.

in the displacement data did not exceed 0.2 pixels even inthe region close to the pipe wall where the liquid film pro-files are taken. The only difference, between these tests andthe experiments, is the presence of the bubble. The compar-ison between the histograms of the gray levels in the liq-uid film and in the same region of an image without thebubble shows that there is a slight increase in the gray lev-els when the bubble is present. One explanation could besome reflection of the laser sheet in the bubble interfaceback to the liquid film region. Although this reflection ofthe laser sheet is much less intense than the laser sheet it-self, the curvature of the bubble interface makes the reflectedlaser sheet larger than the original one. This fact, in con-junction with small undetectable misalignments of the lasersheet/Taylor bubble/column axis could lead inevitably tothe illumination of particles in a plane wider than the orig-inal laser sheet and be responsible for the decrease in themeasurements accuracy. This is the most probable reasonfor the high velocity data scattering in the liquid film, butit is not easy to prove it experimentally. As a conclusion, amaximum error of 0.8 pixels in the windows displacementshould be considered for the liquid film region, which is 4times higher than the 0.2 pixels considered for all the otherregions.

The averaged fully developed velocity profiles are rep-resented inFig. 11. It is possible to verify that the liquidfilm thickness decreases with decreasing viscosity. For the

r/D

V (

m/s

)0.35 0.40 0.45 0.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.0 wt% CMC0.8 wt% CMC0.6 wt% CMC0.5 wt% CMC0.4 wt% CMC0.3 wt% CMC0.1 wt% CMC

Column wallBubble interface

|

Fig. 11. Average of the axial component of the velocity in the liquidfalling film for the different solutions.

lowest viscosity, due to the thinnest liquid film, the liquidvelocity increases significantly, reaching up to 7 times thebubble velocity.

A comparison between the experimental velocity profilesand those previewed by a theoretical analysis was performed.Considering the developed film as a free falling film withconstant pressure at the gas–liquid interface, the weight ofthe liquid is balanced by the shear stresses:

�(�r)�r

− �gr = 0, (7)

where� is the shear stress,r the radial position,� the liq-uid density,g the gravity acceleration and� the liquid filmthickness. Integrating Eq. (7) with the boundary condition� = 0 at r = R − �, the shear stress profile is given by

� = �gr

2

(r − (R − �)2

r

). (8)

Taking the shear stress at each radial position, the correspon-dent shear rate is then obtained from the rheological data:

� = �(�)�, (9)

where�(�) is given by the Carreau–Yasuda model. Fromthis equation, the shear rate,�, cannot be explicitly set as afunction of the shear stress and must be computed numeri-cally for each radial position.

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1867

r/D

V (

m/s

)

0.35 0.40 0.45 0.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.0 wt% Experimental0.8 wt% Experimental0.5 wt% Experimental0.3 wt% Experimental0.1 wt% ExperimentalTheoreticalpipe wall

|

Fig. 12. Comparison of theoretical and experimental average liquid filmprofiles.

With the shear rate profile, the velocity profile is thenobtained from its definition equation

�V

�r= −�(r) (10)

by integrating betweenr=R andr=R−�, with the boundaryconditionV = 0 at r = R.

Once the liquid film thickness,�, is not known a priori,it must be iteratively determined after performing a conser-vative mass balance. In a reference frame moving with thebubble, the liquid film flow rate is given by

Q =∫ R

R−�(V (r) + Ub)2r dr, (11)

which, by mass conservation, must also be equal to thedownward liquid flow rate far ahead of the bubble (referenceframe attached to the bubble) given by

Q = UbR2. (12)

The value of� was iteratively determined matching the flowrate given by Eq. (11) with the one obtained from Eq. (12).

The theoretical and experimental velocity profiles are rep-resented for some of the studied solutions inFig. 12. Fromthis figure it is possible to see that the experimental veloc-ity profiles are very close to the theoretical ones. The ex-perimental flow rates in a reference frame moving with thebubble differ less than 10% from the expected theoreticalflow rates. The theoretical film thickness differs less than 5%from the experimental value even for the higher concentra-tion solutions, indicating a scarce influence of the viscoelas-tic effects. This analysis shows that the developed velocityprofiles in the liquid film around the Taylor bubbles can betheoretically estimated if the bubble velocity and the fluidrheology are known.

4.4. Taylor bubble wake

The Taylor bubble wake is the region where the main dif-ferences appear in the flow pattern of the studied cases. Forthe higher values of Reynolds number, the liquid film ve-locity is higher and the annular jet plugging into the wakecauses turbulence, which is in someway responsible for theinstability of the bubble trailing edge.Fig. 13shows an in-stantaneous velocity field in the wake of a Taylor bubblerising in a 0.1 wt% CMC solution(Re=714), in a referenceframe fixed to the pipe and in a reference frame movingwith the bubble. The variablez∗ is the vertical distance tothe centre of the trailing edge of the bubble. FromFig. 13itis possible to identify small vortices in continuous displace-ment in the wake of the bubble. It should be noticed that thevelocity field is obtained from a vertical plane through thecentre of the column and due to the three-dimensional os-cillation of the bubble trailing edge, there is a lot of liquidcrossing this plane. The maximum upward velocity in thewake can reach up to 4 times the bubble velocity, so someliquid is flowing upwards in the wake although it is not thesame fluid that follows the bubble along all the rise, dueto the turbulence and vortices displacement and shedding.When subtracting the bubble velocity to obtain the flow fieldin a reference frame moving with the bubble, there are nosignificant changes due to the high velocities present in thewake, but it is possible to see that at distances higher than1.2D from the bubble trailing edge,there is no liquid goingup with the bubble. Behind the Taylor bubble, due to thegreat instability of the trailing edge, a train of small bubblesis formed, starting at about three column diameters from thetrailing edge and extending along several diameters. Thesesmall bubbles are present during all the bubble rise main-taining approximately the same distance to the bubble trail-ing edge. A similar behaviour was found for all the Taylorbubbles rising in this solution. The presence of the bubblesgreatly increases the distance at which the fluid returns torest. This is expected to have great influence in the coales-cence phenomena.

As the viscosity increases, the liquid film velocity de-creases and the wake instabilities tend to disappear. For the0.3 wt% CMC solution, there is still some instability in thetrailing edge of the bubble but much less than in the 0.1 wt%CMC solution. InFig. 14, an instantaneous flow field in thewake of a Taylor bubble rising in a 0.3 wt% CMC solution(Re=144) is represented. In contrast with the previous case,it is now possible to identify a large recirculation zone be-hind the trailing edge, around which flows the liquid comingfrom the falling film, this recirculation zone is asymmetricand not completely “closed” due mainly to the small am-plitude oscillations that the trailing edge still exhibits. Therecirculation and the liquid film expansion is clearly seen inthe reference frame moving with the bubble (Fig. 14right),and it is also possible to observe the reattachment of the liq-uid flowing around the wake at approximately 1.2D fromthe trailing edge.

1868 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

r/D

z*/D

-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Taylor Bubble

1 m/s r/Dz*

/D

-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Taylor Bubble

1 m/s

Fig. 13. Instantaneous velocity field in the wake of a Taylor bubble rising in a 0.1 wt% solution, in a fixed reference frame (left) and in a referenceframe moving with the bubble (right).

r/D

z*/D

-0.40 -0.20 0.00 0.20 0.40

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Taylor Bubble

0.5 m/sr/D

z*/D

-0.40 -0.20 0.00 0.20 0.40

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Taylor Bubble

0.5 m/s

Fig. 14. Instantaneous velocity field in the wake of a Taylor bubble rising in a 0.3 wt% CMC solution, in a fixed reference frame (left) and in a referenceframe moving with the bubble (right).

In the 0.4 wt% solution(Re=70), the bubble trailing edgeis now stable and no oscillations occur. The result is a sym-metric wake flow within a “closed” recirculation zone (Fig.15). The liquid coming from the falling film smoothly ex-pands after the bubble trailing edge, inducing the formationof a closed donut shaped vortex in the wake rising attachedto the bubble. The reattachment of the liquid film is betterseen through a streamline representation in a reference frame

moving with the bubble. It takes place much closer to thetrailing edge (at about 0.8D) than in the previous solution.The symmetry of the wake flow allows an easy visualisationof the three-dimensional flow field in the wake by rotationof the paper plane around the column axis.

As the Reynolds number decreases, the size of the wakealso decreases. Due to the higher viscosity, the shear stressesin the liquid are higher, and these stresses, by momentum

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1869

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Taylor Bubble

0.5 m/s r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Taylor Bubble

Fig. 15. Instantaneous velocity field in the wake of a Taylor bubble rising in a 0.4 wt% CMC solution, in a fixed reference frame (left) and streamlinesrepresentation in a reference frame moving with the bubble (right).

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Taylor Bubble

0.5 m/s r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Taylor Bubble

Fig. 16. Instantaneous velocity field in the wake of a Taylor bubble rising in a 0.5 wt% CMC solution, in a fixed reference frame (left) and streamlinesrepresentation in a reference frame moving with the bubble (right).

diffusion, slow down the fluid coming from the liquid film.The liquid film expansion takes place closer to the bubbletrailing edge.

The instantaneous flow field in the wake of a Taylorbubble rising in a 0.5 wt% CMC solution(Re = 41) isrepresented inFig. 16and is quite similar to the one foundfor Re = 70. The relevant difference, is the wake length orthe distance at which the liquid film reattaches, which isnow 0.65D. The streamlines in these solutions seem to en-ter through the bubble interface but in fact they are withinthe concave region in the bubble trailing edge, as shownin Fig. 5.

The increase of viscosity decreases the size of the wakeand for the 0.6 wt% CMC solution(Re = 24) the wakerecirculation becomes almost imperceptible. InFig. 17, theflow field in the wake of a Taylor bubble rising in such asolution is represented. In the fixed reference frame (Fig.17 left), the upward velocity in the centre of the wake isvery close to the bubble velocity, in opposition to the pre-vious cases. Therefore, in a reference frame moving withthe bubble (Fig. 17 right), these velocities are very closeto the measurement uncertainty making difficult the plotof the streamlines in that region. The wake length is nowbelow 0.2D.

1870 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

0.0

0.2

0.4

0.6

0.8

1.0

0.5 m/s

Taylor bubble

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

0.0

0.2

0.4

0.6

0.8

1.0

Taylor bubble

Fig. 17. Instantaneous velocity field in the wake of a Taylor bubble rising in a 0.6 wt% CMC solution, in a fixed reference frame (left) and streamlinesrepresentation in a reference frame moving with the bubble (right).

(gD3)1/2 /ν

Lw

/D

0 200 400 600 800 10000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

CMC solutions :Lw /D= 0.2+ 0.00114 (gD3)1/2/νNewtonian fluids: Lw /D= 0.3 + 0.00122* (gD3)1/2/ν

Fig. 18. Wake length (Lw) dependence on(gD3)1/2/ and comparisonwith Campos and Guedes de Carvalho (1988)correlation.

The wake length (Lw) dependence on the dimensionlessparameter(gD3)1/2/, being the cinematic viscosity ofthe liquid, is represented inFig. 18and compared to the cor-relation found byCampos and Guedes de Carvalho (1988)for Newtonian fluids. For a better comparison, the viscosityof the non-Newtonian solutions was determined for a shearrate given by the ratio between the mean velocity in thefalling film and the film thickness.Fig. 18shows a similartendency of the wake length dependence on this parameterfor CMC solutions and Newtonian solutions.

The main differences between the wake flow patterns inNewtonian and non-Newtonian liquids appear in the higherviscosity solutions. For 0.8 and 1.0 wt% solutions, there isa drastic change in the shape of the bubble trailing edge(Fig. 5) and also in the wake flow pattern, which are fully

described bySousa et al. (2004). In Fig. 19, the flow fieldsaround the trailing edges of Taylor bubbles rising in 0.8 wt%(Re = 10) and 1.0 wt%(Re = 4) CMC solutions are rep-resented. The flow in the wake of the Taylor bubbles risingin these solutions follow the previous tendency, i.e., as theviscosity increases, the expansion of the liquid film occurscloser to the trailing edge. The liquid expands immediatelyat the end of the liquid film, following the trailing edgeshape and occupying the place left by the bubble in its as-cending movement. This radial movement of the liquid tothe centre of the column induces, by momentum diffusion,a rotational movement in the liquid that is below. The liq-uid rotates in the downward direction in the centre of thecolumn and upwards away from the centre, creating what iscalled a negative wake, once the velocity in the centre of thecolumn is in the opposite direction to the bubble. In thesecases, there is no liquid transported in the wake, as shownby the streamlines represented inFig. 20.

The main differences between these negative wakes, 0.8and 1.0 wt% CMC solutions, are the trailing edge shapesand the downward velocity magnitude, which is higher inthe 0.8 wt% solution (Sousa et al., 2004).

To compare the wake patterns and the velocity magni-tudes, the instantaneous axial component of the liquid ve-locity is represented atz ∗ = 0.2D in Fig. 21 and atr = 0in function of z ∗ /D in Figs. 22and 23. Fig. 21 puts inevidence the occurrence of the negative wake for 0.8 and1.0 wt% CMC solutions, due to the positive (downward) liq-uid velocity in the centre of the column at 0.2D behind thetrailing edge. It is also possible to see near the tube wall,0.35< r/D < 0.5, the effect of the film expansion in the liq-uid velocity. For the lower viscosities the velocity of theliquid in expansion is high due to the lower shear stresses.On the contrary, for the higher viscosities the fluid alreadysuffered a strong deceleration.

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1871

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Taylor bubble

0.1 m/sr/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.5 m/s

Taylor bubble

Fig. 19. Instantaneous velocity field in the wake of Taylor bubbles rising in 0.8 and 1.0 wt% CMC solutions in a fixed reference frame.

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Taylor bubble

r/D

z*/D

-0.50 -0.25 0.00 0.25 0.50

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

Taylor bubble

Fig. 20. Streamlines in the wake of Taylor bubbles rising in 0.8 (left) and 1.0 wt% (right) CMC solutions in a reference frame moving with the bubble.

r/D

V (

m/s

)

-0.50 -0.25 0.00 0.25 0.50

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.0 wt% CMC0.8 wt% CMC0.6 wt% CMC0.4 wt% CMC0.1 wt% CMC

Fig. 21. Instantaneous axial component of the liquid velocity in a fixedreference frame, along the linez ∗ = 0.2D.

z*/D

V (

m/s

)

0 1 2 3 4 5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.1 wt% CMC0.3 wt% CMC0.4 wt% CMC0.5 wt% CMC0.6 wt% CMC

Fig. 22. Instantaneous axial component of the liquid velocity in a fixedreference frame atr = 0, for positive wakes.

1872 R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873

z*/D

V (

m/s

)

0.0 0.5 1.0 1.5 2.0

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.8 wt% CMC1.0 wt% CMC

Fig. 23. Instantaneous axial component of the liquid velocity in a fixedreference frame, atr = 0, for negative wakes.

In Figs. 22and 23, the extent of the flow perturbationinduced by the wakes for all the studied cases can be com-pared. For 0.1 wt% CMC solution, the high turbulence levelof the flow induces a larger velocity scattering. The men-tioned presence of small bubbles behind the wake does notallow the liquid to get into rest even for distances up to13D behind the trailing edge. For the other “positive” wakes(Fig. 22), the decrease of the perturbed length with increas-ing viscosity becomes clear.Fig. 23puts again in evidencethe negative wakes and the differences in the downward ve-locity magnitude. These different wake flow patterns shouldhave great influence in the coalescence phenomena whichwill be object of a near future study.

5. Conclusions

A correlation that describes the bubble shape from thebubble nose to the end of the liquid film was established.Measurements of the flow field around Taylor bubbles ris-ing in different CMC solutions were made. It was found thatthe flow around the bubble nose is similar for all the studiedcases. In the liquid film surrounding the bubble, the differ-ences in film thickness and in velocity magnitudes were de-scribed and the comparison with theoretical predictions wasperformed. This study shows the different wake flow pat-terns found in a Reynolds number range between 4 and 714,and is essential to understand the coalescence mechanismbetween two slugs and the minimum distance above whichthere is no interaction between Taylor bubbles. From thesestudies it becomes clear that for high values of Reynoldsnumber there should be interaction between two consecutiveslugs even if they are separated by a large distance and thatfor the low values of Reynolds number, where a negativewake was found, it is not expected slugs coalescence.

Acknowledgements

The authors acknowledge the financial support given byF.C.T., SFRH/BD/3389/2000 and the Von Karman Institutfor the facility. This work was also supported, via CEFT, byPOCTI (FEDER).

References

Acharya, A., Ulbrecht, J.J., 1978. Note on the influence of viscoelasticityon the coalescence rate of bubbles and drops. A.I.Ch.E. Journal 24,348–351.

Acharya, A., Mashelkar, R.A., Ulbrecht, J., 1977. Mechanics ofbubble motion and deformation in non-Newtonian media. ChemicalEngineering Science 32, 863–872.

Astarita, G., Apuzzo, G., 1965. Motion of gas bubbles in non-Newtonianliquids. A.I.Ch.E. Journal 11, 815–820.

Bugg, J., Saad, G.A., 2002. The velocity field around a Taylor bubblerising in a stagnant viscous fluid: numerical and experimental results.International Journal of Multiphase Flow 28, 791–803.

Campos, J.B.L.M., Guedes de Carvalho, J.R.F., 1988. An experimentalstudy of the wake of gas slugs rising in liquids. Journal of FluidMechanics 196, 27–37.

Carew, P.S., Thomas, N.H., Johnson, A.B., 1995. A physical basedcorrelation for the effects of power law rheology and inclination onslug bubble rise velocity. International Journal of Multiphase Flow 21(6), 1091–1106.

Funfschilling, D., Li, H.Z., 2001. Flow of non-Newtonian fluids aroundbubbles: PIV measurements and birefringe visualisation. ChemicalEngineering Science 56, 1137–1141.

Hassager, O., 1979. Negative wake behind bubbles in non-Newtonianliquids. Nature 279, 402–403.

Kamıslı, F., 2003. Free coating of a non-Newtonian liquid onto walls ofa vertical and inclined tube. Chemical Engineering and Processing 42,569–581.

Kee, D.D., Chhabra, R.P., Dajan, A., 1990. Motion and coalescence ofgas bubbles in non-Newtonian polymer solutions. Journal of Non-Newtonian Fluid Mechanics 37, 1–18.

Leal, L.G., Skoog, J., Acrivos, A., 1971. On the motion of gas bubbles ina viscoelastic liquid. The Canadian Journal of Chemical Engineering49, 569–575.

Leider, P.J., Bird, R.B., 1974. Squeezing flow between paralleldisks. I. Theoretical analysis. Industrial and Engineering ChemistryFundamentals 13, 336.

Li, H.Z., Frank, X., Funfschilling, D., Mouline, Y., 2001. Towards theunderstanding of bubble interactions and coalescence in non-Newtonianfluids: a cognitive approach. Chemical Engineering Science 56, 6419–6425.

Lindken, R., Merzkirch, W., 2001. A novel PIV technique formeasurements in multi-phase flows and its application to two-phasebubbly flows. In: Fourth International Symposium on Particle ImageVelocimetry.

Moissis, R., Griffith, P., 1962. Entrance effects in a two-phase slug flow.Journal of Heat Transfer, Transactions of the ASME, Series C 2, 29–39.

Nogueira, S., Dias, I., Pinto, A.M.F.R., Riethmuller, M.L., 2000. LiquidPIV measurements around a single gas slug rising through stagnantliquid in vertical pipes. In: Bound Volume of Selected Papers of theTenth Symposium on Laser Techniques Applied to Fluid Dynamics.Springer, Berlin.

Nogueira, S., Sousa, R.G., Pinto, A.M.F.R., Riethmuller, M.L., Campos,J.B.L.M., 2003. Simultaneous PIV and shadowgraphy in slug flow: asolution for optical problems. Experiments in Fluids 35, 598–609.

R.G. Sousa et al. / Chemical Engineering Science 60 (2005) 1859–1873 1873

Otten, L., Fayed, A.S., 1976. Pressure drop and drag reduction in two-phase non-Newtonian slug flow. The Canadian Journal of ChemicalEngineering 54, 111–114.

Pinto, A.M.F.R., Campos, J.B.L.M., 1996. Coalescence of two gas slugsrising in a vertical column of liquid. Chemical Engineering Science51, 45–54.

Polonsky, S., Barnea, D., Shemer, L., 1998. Image processing procedurefor analysing the motion of an elongated bubble rising in a verticalpipe. Eighth International Symposium on Flow Visualisation, pp.117.1–117.10.

Rosehart, R.G., Rhodes, E., Scott, D.S., 1975. Studies of gas–liquid (non-Newtonian) slug flow: void fraction meter, void fraction and slugcharacteristics. The Chemical Engineering Journal 10, 57–64.

Scarano, F., Riethmuller, M.L., 1999. Iterative multigrid approach in PIVimage processing with discrete window offset. Experiments in Fluids26, 513–523.

Sousa, R.G., Nogueira, S., Pinto, A.M.F.R., Riethmuller, M.L., Campos,J.B.L.M., 2004. Flow in the negative wake of a Taylor bubble rising ina high concentrated shear thinning fluid: PIV measurements. Journalof Fluid Mechanics 511, 217–236.

Taitel, Y., Barnea, D., Dukler, A., 1980. Modelling flow pattern transitionsfor steady upward gas–liquid flow in vertical tubes. A.I.Ch.E. Journal3, 345–354.

Terasaka, K., Tsuge, H., 2003. Gas holdup for slug bubble flow of viscousliquids having a yield stress in bubble columns. Chemical EngineeringScience 58, 513–517.

van Hout, R., Gulitski, A., Barnea, D., Shemer, L., 2001. Experimentalinvestigation of the velocity field induced by a Taylor bubble rising instagnant water. International Journal of Multiphase Flow 28, 579–596.