Flow around individual Taylor bubbles rising in stagnant polyacrylamide (PAA) solutions

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1 J. Non-Newtonian Fluid Mech. 135 (2006) Flow around individual Taylor bubbles rising in stagnant polyacrylamide (PAA) solutions R.G. Sousa a, M.L. Riethmuller b, A.M.F.R. Pinto a, J.B.L.M. Campos a, a Centro de Estudos de Fenómenos de Transporte, Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias Porto, Portugal b von Karman Institute for Fluid Dynamics, 72 Chaussée de Waterloo, 1640 Rhode-Saint-Genèse, Belgium Received 7 July 2005; received in revised form 5 December 2005; accepted 10 December 2005 Abstract The flow around single Taylor bubbles rising in stagnant non-newtonian solutions of polyacrylamide (PAA) polymer was studied using a technique employing simultaneous particle image velocimetry (PIV) and shadowgraphy. Solutions with different weight percentages of polymer, varying from 0.01 to 0.80 wt.%, were used to cover a wide range of flow regimes. The rheological fluid properties and pipe dimension yielded Reynolds numbers between 2 and 1160, and Deborah numbers up to 115. The shape of the bubbles rising in the different solutions was compared and quantified by fitting correlations. The flow around the nose of the bubbles was found to be similar for all conditions studied. Velocity profiles were measured and analysed in the liquid film around the bubbles. A comparison of bubble wake flow patterns was made. For the 0.10 and 0.20 wt.% PAA solutions, long wakes with a recirculation region were observed. Below the wakes, a flow of stretched liquid was found. Negative wakes were also observed for the more concentrated solutions Elsevier B.V. All rights reserved. Keywords: Taylor bubble; Multiphase flow; Non-Newtonian fluids; Viscoelasticity; Particle image velocimetry (PIV); Films 1. Introduction Slug flow is a two-phase flow regime, found when gas and liquid flow simultaneously in a pipe over a specific range of flow-rates. This regime is characterised by elongated gas bubbles (Taylor bubbles or gas slugs) which almost fill the pipe cross-section, causing the liquid to flow around and between the bubbles. This type of flow is found in various geothermal, fermentation and polymer devolatilisation processes, pipeline transport in oil and gas wells, air-lift reactors, and many others. Slug flow is used in some chemical processes to increase the reaction rate by taking advantage of the mixing action of the wake of the Taylor bubbles. The flow of Taylor bubbles in Newtonian liquids has been deeply studied since the 1940s. Dumitrescu [1] and Davis and Taylor [2] were among the pioneers studying the shape and velocity of these elongated bubbles. Since then, several studies of Taylor bubbles have also dealt with the liquid film region surrounding the bubble [3 8], and the bubble wake region [8 13]. Corresponding author. Fax: address: jmc@fe.up.pt (J.B.L.M. Campos). Recently the flow around Taylor bubbles rising in Newtonian liquids was studied using particle image velocimetry (PIV), and the most relevant studies are described in [14 17]. The flow of Taylor bubbles rising in non-newtonian liquids has not been so deeply studied, despite being frequently found in industrial processes. Due to the complex liquid rheology, the gas liquid flow patterns have several unusual characteristics. The effects of power law rheology and pipe inclination on slug bubble rise velocity were studied by Carew et al. [18]. Otten and Fayed [19] studied the pressure drop and friction drag reduction in two-phase non-newtonian slug flow. Rosehart et al. [20] measured the void fraction, slug velocity and frequency for co-current slug flow of air bubbles in highly viscous non- Newtonian fluids. Terasaka and Tsuge [21] made gas hold-up measurements for gas slugs rising in viscous liquids with a yield stress. Kamışlı[22] derived a one-dimensional flow equation for the motion of a long bubble rising steadily in vertical and inclined tubes filled with a power-law fluid. Sousa et al. [23] applied PIV and shadowgraphy simultaneously to describe the flow of Taylor bubbles rising in stagnant carboxymethylcellulose (CMC) solutions. The authors described the wake flow patterns found in solutions with /$ see front matter 2006 Elsevier B.V. All rights reserved. doi: /j.jnnfm

2 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Nomenclature a 1,a 2 dimensionless parameters of Carreau Yasuda viscosity model D tube internal diameter k p1,k p2,k p3 dimensionless parameters L b bubble length m 1,m 2 dimensionless parameters N 1 first normal stress difference Q A flow rate ahead of Taylor bubble in a frame of reference moving with the bubble Q B flow rate in the liquid film in a frame of reference moving with the bubble r radial distance to the tube axis r maximum bubble radius R tube internal radius s 1,s 2 dimensionless parameters T temperature U b bubble velocity V exp liquid velocity far ahead of the bubble V r radial component of liquid velocity V z axial component of liquid velocity z axial distance to the bubble nose z stable axial distance to the bubble nose where the liquid film becomes fully developed z axial distance to the bubble trailing edge Greek letters γ shear rate γ f characteristic flow shear rate = U b /D η viscosity η f viscosity at characteristic shear rate η 0 viscosity at zero shear rate η viscosity at infinite shear rate λ time constant of Carreau Yasuda viscosity model ν η/ρ ρ liquid density σ(x) standard deviation of x σ yx shear stress τ fluid relaxation time different CMC weight percentage, covering a range of Reynolds numbers between 4 and 714 and Deborah numbers up to For the higher Reynolds numbers, the wake flow patterns were similar to those found in Newtonian solutions. For the lower Reynolds numbers and higher Deborah numbers a negative wake was found and a change in the trailing edge shape was observed. These findings are fully described in [24]. Some studies have also been done on small unconfined bubbles rising in non-newtonian liquids. Bubble velocities and bubble shapes in non-newtonian liquids can be found in different studies [25 27]. A negative wake behind small unconfined bubbles rising in non-newtonian liquids was described for the first time by Hassager [28]. Studies on bubble coalescence in non- Newtonian liquids can be found in [29 31]. Funfschilling and Li [32] used PIV and birefringence visualisation to study the flow of non-newtonian fluids around small bubbles. Li et al. [33,34] described the influence of the time gap between the injection of consecutive bubbles on the bubble velocity, showing the accumulation of residual stresses after the passage of a chain of small bubbles. The authors observed an increase of the bubble velocity for shorter time gaps. In this study the flow around single Taylor bubbles rising in stagnant non-newtonian solutions of polyacrylamide is analysed, using PIV and shadowgraphy simultaneously. The results obtained contribute to a better understanding of Taylor bubble flow in non-newtonian liquids, particularly in viscoelastic liquids. The main difference from the work presented in [23] is a much higher viscoelasticity of the polyacrylamide solutions, when compared with the carboxymethylcellulose solutions. This higher viscoelasticity is responsible for significant differences in the liquid flow pattern around the bubble, particularly in the wake and near wake regions as it will be observed. 2. Experiments The experiments were performed using the experimental setup sketched in Fig. 1. The main column was 6 m long with a 32 mm internal diameter. Two pneumatic valves in a lateral pipe were used to inject the Taylor bubbles. A transparent square box with plane faces encased the test section, and was filled with the working liquid so as to reduce the optical distortion. A Nd:YAG laser with 400 mj of pulse power was used to illuminate a vertical plane across the axis of the column. The laser wavelength was 532 nm and the pulse duration 2.4 ns. Fluorescent particles (an orange vinyl pigment with a mean size of 10 m) were used as seeding, emitting light at 590 nm. Behind the test section, a board with 650 LEDs emitting light at 650 nm illuminated the background through a diffusive paper. Perpendicular to the laser sheet, a PCO (SensiCam) CCD camera with a resolution of pixels 2 was mounted behind a red optical filter (opaque below 550 nm) to photograph the illuminated particles and the bubble shadow created by the LEDs. From the acquired images, the grey level differences in the illuminated background and the shadow of the bubble allowed identification of the gas liquid interface with simple image processing. The bubble velocity was measured using signals from two pairs of photocells/laser diodes placed just below the test section. More details about the experimental setup, techniques and data processing can be found in [35,24,23]. 3. Results 3.1. Rheology In this section, the flow around Taylor bubbles rising in stagnant solutions of polyacrylamide (molecular weight of kg kmol 1 ) with different weight percentages is described. In Fig. 2, the viscosities of the different solutions are represented as a function of the shear rate, measured with an AR 2000 DTA Instruments Rheometer.

3 18 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 1. Sketch of the experimental setup with a detailed view of the test section. The Carreau Yasuda viscosity model, Eq. (1), was fitted to the experimental data and is represented in Fig. 2 by the solid lines η( γ) = η + (η 0 η )(1 + (λ γ) a 1 ) (a 2 1)/a 1 (1) The parameters of the Carreau Yasuda viscosity model are presented in Table 1. The first normal stress difference is shown in Fig. 3 for the solutions studied, except for the less concentrated ones where the values were below the measurement range of the equipment. The fluid relaxation time, τ, was estimated using the same approach mentioned in [36]. Supposing that both N 1 and σ yx can be well approximated by power functions of the shear rate over the range of conditions of interest, i.e., N 1 = m 1 ( γ) s 1 and σ yx = m 2 ( γ) s 2, the fluid relaxation time can be given by ( ) 1/(s1 s m1 2 ) τ = (2) 2m 2 The relaxation times for the most viscous solutions and the fitting parameters for Eq. (2) are represented in Table 2. The parameters of Eq. (2) were obtained in the range of shear rate where both Fig. 2. Representation of fluid viscosity vs. shear rate for the PAA solutions studied. The solid lines show the Carreau Yasuda viscosity model fitting to the experimental data. Fig. 3. Representation of the first normal stress difference vs. the shear rate for the studied PAA solutions.

4 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Table 1 Parameters of the Carreau Yasuda viscosity model for the PAA solutions PAA (wt.%) T ( C) η 0 (Pa s) η (Pa s) λ (s) a 1 a 2 γ (s 1 ) Table 2 Relaxation times for PAA solutions and fitting parameters for Eq. (2) PAA (wt.%) γ (s 1 ) m 1 s 1 m 2 s 2 τ (s) < N 1 and σ yx are a function of shear rate by a power law. For the 0.10 wt.% PAA solution, the relaxation time determined by this method is already negligible. The experimental conditions are summarised in Table 3. The Deborah number is defined as De = τ γ f and the Reynolds number as Re = ρu b D/η f, where γ f = U b /D is the characteristic shear rate of the flow and η f the viscosity of the liquid at that shear rate. The maximum bubble length was limited by the volume of the tube between the pneumatic valves, used to release the bubbles. The bubble velocity presented in Table 3 is the average velocity obtained from a large number of replications. Due to the high relaxation times, for the more concentrated solutions the bubble velocity was strongly dependent on the time-gap between consecutive bubbles, which led to a high standard deviation of the bubble velocity data Taylor bubble shape The Taylor bubble shape was determined from the shadow of the bubbles. In Fig. 4, the shapes of the Taylor bubbles rising in some PAA solutions are presented. This figure shows that the bubble shapes are very similar, with a concave prolate spheroid leading edge, and increasing bubble radius as the distance from the nose increases. The main differences between the bubble shapes are the maximum bubble radius and the distance from the nose at which this radius is reached. These values are presented in Table 4 as a function of the Reynolds number. For the less concentrated solution (0.01 wt.%), the stabilisation of the bubble radius was not observed; the bubble length was 9.4D. Fig. 4. Representation of the bubble shapes: axial distance to the nose, z/d, vs. the bubble radius, r/d. Table 3 Experimental temperature, T, average bubble velocity, U b, and corresponding standard deviation, σ(u b ), dimensionless bubble length, L b /D, Reynolds number, Re and Deborah number, De wt.% T ( C) U b (m/s) σ(u b ) L b /D Re De < < Fig. 5. Representation of the normalised liquid axial velocity at z/d = 0, as a function of the normalised radial position, r/d, for 0.01 and 0.80 wt.% PAA solutions and for 0.10 wt.% CMC solution.

5 20 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 6. Representation of the normalised liquid radial velocity at z/d = 0, as a function of the normalised radial position, r/d, for 0.01 and 0.80 wt.% PAA solutions and for 0.10 wt.% CMC solution. Fig. 9. Representation of the axial velocity at r/d = 0.47 as a function of the distance to the bubble nose, for the 0.20 wt.% PAA solution. Fig. 7. Representation of the axial velocity at r/d = 0.47 as a function of the distance to the bubble nose, for the 0.01 wt.% PAA solution. Fig. 10. Average velocity profiles in the stabilised liquid film around Taylor bubbles rising in 0.10 and 0.20 wt.% PAA solutions. The bubble shapes are well fitted by the same type of equation used by Sousa et al. [23] for bubbles rising in CMC solutions, but with different parameters: ( ( z ) ) kp3 k p2 D r D = k p1 tanh (3) The k p1 value coincides with the maximum dimensionless bubble radius and is given by r D = k p1 = ln(re) (4) Fig. 8. Representation of the axial velocity at r/d = 0.47 as a function of the distance to the bubble nose, for the 0.10 wt.% PAA solution. Table 4 Maximum dimensionless bubble radius, r /D, and approximate dimensionless distance from the bubble nose at which r /D is reached, z stable /D Re (r /D) z stable /D 1158 >0.461 >

6 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Table 5 Comparison between the flow rates ahead of the bubble (Q A ) and in the liquid film (Q B ) in a frame of reference moving with the bubble PAA (wt.%) Q A ( 10 4 m 3 /s) Q B ( 10 4 m 3 /s) (Q B Q A )/Q A (%) percentages between 0.01 and 0.80 wt.%. The functionality for solutions of other polymers should be identical [23] however the fitting constants have different values Flow field around the Taylor bubble nose Fig. 11. Average liquid film velocity profile around two bubbles with 10.4D of length rising in a 0.40 wt.% PAA solution. The values of k p2 and k p3 were obtained by fitting Eq. (3) to bubble shape data and they are functions of Reynolds number according to Eqs. (5) and (6), respectively k p2 = Re (5) k p3 = Re (6) The shape of the bubbles, obtained from Eq. (3) in the form of dimensionless bubble radius versus dimensionless distance to the nose, differs less than 5% from the experimental data at every point and for any bubble. These equations are valid for aqueous solutions of polyacrylamide in the range of weight The flow around the Taylor bubble nose is very similar to that of Taylor bubbles rising in CMC solutions [23]. As the bubble rises up the column the flow in front of the nose first moves upwards then radially outwards, and finally falls around the bubble. The small differences between the velocity profiles in the different solutions are due to the different bubble velocities and bubble shapes. In Fig. 5, the axial dimensionless velocity profile at z/d = 0 is represented for the most and least concentrated PAA solutions, and compared with that obtained in a 0.10 wt.% CMC solution. The ordinate axis was inverted in order to have positive velocities in the downward direction. The dimensionless radial velocity profiles at z/d = 0 are represented in Fig. 6 for the same solutions. Positive radial velocities indicate that the fluid is moving from the tube axis to the pipe wall. These figures show that the flow pattern around the nose of the bubbles is practically independent of the liquid rheology. Fig. 12. Asymmetric flow around two Taylor bubbles rising in a 0.80 wt.% PAA solution.

7 22 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 13. Instantaneous flow fields in the wake of a Taylor bubble rising in a 0.01 wt.% PAA solution; fixed frame of reference. Fig. 14. Instantaneous streamlines in the wake of a Taylor bubble rising in a 0.01 wt.% PAA solution; moving frame of reference.

8 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 15. Instantaneous axial velocity profile (fixed frame of reference) at r = 0 as a function of z /D behind a Taylor bubble rising in a 0.01 wt.% PAA solution Flow in the liquid film The liquid film region around Taylor bubbles is a very demanding area to study with PIV technique because of the very high velocity gradients and the thin dimensions of the film (between 1.2 and 3.3 mm in the studied conditions). As mentioned in [23], there is a wide scattering of the velocity data in the liquid film, mainly due to the presence of two interfaces (wall/liquid and liquid/gas) which increase the reflections of the laser light, illuminating particles in a region wider than the laser sheet. The liquid film was only analysed for solutions up to a concentration of 0.20 wt.% PAA. For 0.40 and 0.80 wt.% PAA solutions the bubble velocity (and, therefore, the liquid film velocity) is dependent on the fluid history and a study of a larger number of bubbles with the same conditions should be made. The axial liquid velocity in the annular film at a fixed radial position is plotted as a function of the axial distance to the bubble nose in Figs. 7 9, for 0.01, 0.10 and 0.20 wt.% PAA solutions, respectively. As seen from these figures, for the 0.01 wt.% PAA solution the liquid film velocity does not stabilise, increasing continuously until the end of the bubble (z = 9.6D). For 0.10 and 0.20 wt.% PAA solutions, the liquid film velocity stabilises around z = 8D and z = 6D, respectively, in spite of the large scattering of the velocity data. For 0.10 and 0.20 wt.% PAA solutions, the average velocity profiles were determined using more than 600 velocity profiles from the region where the film thickness and the liquid film velocity are stabilised. These average profiles are represented in Fig. 10. A mass balance in a frame of reference moving with the bubble was done to compare the liquid flow rates far ahead of the bubble, Q A (Eq. (7)), and in the stabilised liquid film, Q B (Eq. (8)) Q A = Q B = R 0 R R δ (V exp (r) U b )2πr dr (7) (V z (r) U b )2πr dr (8) Fig. 16. Average velocity field (fixed frame of reference) in the wake of Taylor bubbles rising in a 0.01 wt.% PAA solution. where V exp is the axial velocity of the liquid ahead of the bubble nose due to the bubble expansion during the rise. In Table 5, these average flow rates are presented for the two solutions and also the corresponding relative differences in percentage. As seen from this table, the relative differences are around 10%. This is an acceptable value when taking into account the considerable scattering of the velocity data and the acquisition of the data in only one plane, which is assumed to be representative of the flow around the entire Taylor bubble. The positive values might also indicate that a very small misalignment of the column was responsible for a slight asymmetry in the flow. To illustrate the influence of fluid history on the stabilised liquid film, two average velocity profiles around bubbles with the same length rising in a 0.40 wt.% PAA solution are represented in Fig. 11. This figure shows that due to different fluid histories two bubbles of the same length have different velocities, which entails different velocity profiles in the liquid film; the bubble with the higher velocity presents higher liquid velocities in the film.

9 24 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 17. Instantaneous flow fields in the wake of a Taylor bubble rising in a 0.10 wt.% PAA solution; fixed frame of reference.

10 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 18. Instantaneous streamlines in the wake of a Taylor bubble rising in a 0.10 wt.% PAA solution; frame of reference moving with the bubble. In the 0.80 wt.% PAA solution the bubble velocity is even more dependent on the fluid history, and different values of the film thickness were observed for each side of the bubble image (paper plane), as shown in Fig. 12. This asymmetry has no preferential side, indicating that its occurrence should be related to the fluid history since a non-homogeneous viscosity field could be formed after the passage of each bubble Flow in the Taylor bubble wake The main differences from the flow in CMC solutions [23] appear in the wake region. In the less concentrated, 0.01 wt.% PAA solution, the bubble trailing edge oscillates three dimensionally, creating a wake flow pattern that is not axisymmetric. Since PIV measurements were only made in the central vertical plane of the column and shadowgraphy only gives a two-dimensional projection of the bubble shape, it was impossible to capture all the flow features from instantaneous images. Even so, and despite the very low weight percentage of polymer, the turbulence level is substantially lower than those observed in water and the higher CMC concentrations [23]. In Fig. 13, two consecutive flow fields in the wake of a Taylor bubble rising in a 0.01 wt.% PAA solution are represented. The ordinate variable, z, is the axial distance to the bubble trailing edge. In Fig. 13(a), a central region with high upward velocity (approximately four times the bubble velocity) is observed. Near the wall, the liquid coming from the film around the bubble keeps (approximately) the same thickness and high downward velocity magnitude for a long distance. Due to the long wake length, it is not possible to obtain the complete wake flow field in the same image with a reasonable resolution. In Fig. 13(b) the flow field in the subsequent image (4.11 Hz of acquisition frequency) is represented. Only after z = 2.0D does the expansion of the liquid coming from the falling film occur, asymmetrically, due to the three-dimensional flow. The upward velocity magnitude in the centre of the wake is four times higher than the bubble velocity, indicating that the fluid is in recirculation. This recirculation is clearly seen when the instantaneous streamlines are represented in a frame of reference moving with the bubble, as shown in Fig. 14. Within the long recirculation region, smaller recirculations are seen in the outer area of the wake. The instantaneous axial liquid velocity at the core of the column was taken from several consecutive images and is represented in Fig. 15. This figure shows that for up to around z = 2.5D, the central liquid is moving upwards with a velocity higher than the bubble velocity, meaning this fluid is being transported upwards. Only at z = 3.0D does the liquid velocity in the central region start to move downwards, due to the expansion of the liquid film. The oscillations observed in the axial liquid velocity after z = 5.0D are caused by a train of small bubbles formed during the bubble injection, rising up below the Taylor bubble. The average flow field in the wakes of Taylor bubbles rising in a 0.01 wt.% PAA solution is represented in Fig. 16, obtained by

11 26 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 19. Average velocity field (fixed frame of reference) in the wake of Taylor bubbles rising in a 0.10 wt.% PAA solution. Fig. 20. Instantaneous axial velocity profile (fixed frame of reference) at r = 0 as a function of z /D behind Taylor bubbles rising in 0.10 and 0.20 wt.% PAA solutions. Fig. 21. Comparison of wake lengths in PAA, CMC [23] and Newtonian solutions [12].

12 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 22. Instantaneous flow fields in the wake of a Taylor bubble rising in a 0.40 wt.% PAA solution; fixed frame of reference. averaging 10 instantaneous flow fields. Here, the upward central region is clearly seen, surrounded by the falling liquid film. The next PAA solution studied in an increasing weight percentage of polymer was a 0.10 wt.% PAA solution. In this solution, a decrease in the frequency and amplitude of the oscillations of the bubble trailing edge was observed. Examples of consecutive images of the wake of a Taylor bubble rising in a 0.10 wt.% PAA solution are represented in Fig. 17. This figure shows that the flow pattern close to the bubble trailing edge is similar to that found in the 0.01 wt.% PAA solution. In the 0.10 wt.% PAA solution, however, a more uniform upward flow in the centre is noticed, with practically no radial velocity component. The liquid from the film around the bubble, maintains its thickness and velocity magnitude up to around 1.6D. The differences from the previous solution appear after this distance. An upward central flow is seen in a waving movement for several column diameters distance, which induces the downward displacement of the liquid in the outer regions by mass conservation. This behaviour is thought to be caused by high extensional stresses promoted by the bubble and wake displacement, which stretch the liquid. Near the wall, the falling liquid has low viscosity after being subject to high shear rates. In the CMC experiments, only a symmetric wake flow within a closed recirculation zone was observed. Fig. 18 represents the instantaneous streamlines in the wake of a Taylor bubble rising in a 0.10 wt.% PAA solution, in a frame of reference moving with the bubble. The central upward moving liquid is clearly seen, surrounded by recirculation regions formed due to the shear between the upward moving liquid and the downward moving liquid film. The average flow field represented in Fig. 19 was obtained from ten instantaneous flow fields, and shows the extent of the wake behind Taylor bubbles rising in a 0.10 wt.% PAA solution. In the 0.20 wt.% PAA solution, the tendency for trailing edge stabilisation with increasing polymer weight percentage is still observed. The wake flow pattern is very similar to that found in Fig. 23. Instantaneous axial velocity profile at r = 0 as a function of z /D behind a Taylor bubble rising in a 0.40 wt.% PAA solution.

13 28 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 24. Instantaneous asymmetric flow fields in the wake of Taylor bubbles rising in a 0.40 wt.% PAA solution; fixed frame of reference. the 0.10 wt.% PAA solution, but with a shorter wake length, and so the flow field will not be represented. The axial liquid velocity in the centre of the column is represented in Fig. 20 for both 0.10 and 0.20 wt.% solutions. The axial velocity magnitude increases up to around z = 2.0D and z = 0.5D for the 0.10 and 0.20 wt.% PAA solutions, respectively. From these points downwards, the velocity slowly decreases, reaching the bubble velocity magnitude at around z = 5D and z = 2D. Afterwards, there is still an upward liquid movement but always with a lower velocity than the bubble, indicating that the liquid in the centre of the column is being stretched. This stretched flow pattern has not been described before, so far as the authors are aware, and should be object of deeper study to be fully understood. To stress the differences between the wake flow patterns in the PAA, CMC and Newtonian solutions, the wake length is represented in Fig. 21, as a function of the dimensionless number (gd 3 ) 1/2 /ν, where ν = η/ρ and η is the viscosity in the liquid film for a shear rate given by the ratio between the mean velocity in the falling film and the film thickness. This figure clearly shows that for the CMC solutions, the behaviour is not very different from that found in the Newtonian solutions, while in the PAA solutions, the wake lengths are much higher due to the mentioned extensional viscoelastic effects. Although in the 0.20 wt.% PAA solution there is already some influence of fluid history on the bubble velocity, in the 0.40 and 0.80 wt.% PAA solutions this effect becomes critical. The fluid history not only affects the bubble velocity but also the flow pattern in the wake. Although the interval time between consecutive bubbles was approximately the same and always much higher than the calculated relaxation time, it was not enough for the recovery of the stagnant liquid viscosity. For this reason, the following results are presented only to emphasise the importance, and to describe effects, of the fluid history on the wake flow patterns. In Fig. 22, instantaneous flow fields in the wake of a Taylor bubble rising in a 0.40 wt.% PAA solution are represented. This example shows a case where the flow around the bubble is approximately symmetric. In Fig. 22(a) the presence of a negative wake is clearly observed, with a wake similar to those found in the solutions of higher CMC polymer concentration [23]. The liquid film expands just after the bubble trailing edge, giving origin to a recirculation region with a downward flow in the centre of the column and upwards flow away from the centre. In Fig. 22(b), the flow in the lower region of the wake is represented with a different vector scale to highlight the existence of secondary recirculation regions. These recirculations are rotating in alternate directions (similar to gears where a set of toothed wheels rotate in opposite directions), but with decreasing velocity magnitude. More detailed explanations on the formation of negative wakes behind Taylor bubbles rising in shear-thinning viscoelastic fluids can be found in [24] and behind spheres settling in shear-thinning viscoelastic fluids in [37].

14 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 25. Flow field in the wake of Taylor bubbles rising in a 0.80 wt.% PAA solution; fixed frame of reference. Four consecutive recirculation regions are clearly seen in Fig. 23, where the axial liquid velocity in the axis of the column is represented as a function of the distance to the bubble trailing edge. This flow pattern was not commonly seen among the studied bubbles. Due to different fluid histories the bubble might travel through a non-homogeneous viscosity field, which induces an asymmetric flow in the annular liquid film and an asymmetric flow in the wake. Two examples are represented in Fig. 24. As mentioned earlier, in the 0.80 wt.% PAA solution the fluid history has even more influence on the bubble velocity and wake flow pattern. The flow field in the annular liquid film around the bubble was always asymmetric, as presented in Fig. 12. Fig. 25 shows two examples of the asymmetric flow in the wake of Taylor bubbles rising in a 0.80 wt.% PAA solution. The asymmetric bubble trailing edge causes a radial movement of the liquid in a preferential direction which induces a recirculation in the fluid below, as shown in Fig. 25(c) and (d) with a different vector scale. In order to compare the flow in the wake of Taylor bubbles rising in different PAA solutions, Fig. 26 shows the axial velocity profiles along the radial position at z = 0.4D, for the solutions studied. The reducing velocity magnitude as the concentration of polymer increases, and the appearance of

15 30 R.G. Sousa et al. / J. Non-Newtonian Fluid Mech. 135 (2006) Fig. 26. Axial velocity profile at z = 0.4D in the wake of Taylor bubbles rising in PAA solutions. the negative wake in the 0.40 wt.% PAA solution is clearly seen. 4. Conclusions This work studied the flow around Taylor bubbles rising in different concentrations of polyacrylamide solution. For all the PAA solutions studied the bubble shape is similar, as is the flow around the bubble nose. Accurate fitting correlations for the bubble shape are presented. In concentrations higher than 0.01 wt.% a narrow and stabilised liquid film flowing around the bubble was observed. For 0.1 and 0.2 wt.% PAA solutions, below long regions of liquid in recirculation there is liquid being stretched in the central region of the column. To the best of our knowledge this is a flow pattern not previously described in the literature. The fluid history has significant importance in solutions with polymer (PAA) concentrations above 0.20 wt.%. 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