On the critical bubble volume at the rise velocity jump discontinuity in viscoelastic liquids

J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

On the critical bubble volume at the rise velocity jump discontinuity in viscoelastic liquids C. Pilz, G. Brenn ∗ Institute of Fluid Mechanics and Heat Transfer, Graz University of Technology, Inffeldgasse 25/F, 8010 Graz, Austria Received 9 October 2006; received in revised form 22 May 2007; accepted 23 May 2007

Abstract Bubbles rising in viscoelastic liquids may exhibit a jump discontinuity of the rise velocity as a critical bubble volume is exceeded. We carried out detailed experiments to investigate the occurrence of this discontinuity with single air bubble rising in various polymer solutions without influence of surfactants. The polymer solutions were characterized thoroughly by means of shear and elongational rheometry, as well as tensiometry. The experiments showed that a jump discontinuity can exist only if the non-dimensional group λE (g3 ρ1 /σ)1/4 , found by dimensional analysis, exceeds a critical value. A universal correlation of non-dimensional numbers for the non-dimensional critical bubble volume at the jump discontinuity was found. The non-dimensional numbers represent the relevant rheological and dynamic liquid properties. This is the first time that the prediction of the critical bubble volume as well as the potential of the solution to exhibit a bubble rise velocity discontinuity becomes possible based on liquid properties only. In the correlation found, the relaxation time of the polymer solutions in elongational flow of the viscoelastic liquid was found to play a key role. © 2007 Elsevier B.V. All rights reserved. Keywords: Viscoelastic liquids; Bubble rise velocity; Jump discontinuity; Elongational flow; Shear rheometry

1. Introduction For applications in biotechnology, bio-process engineering, and others it is of interest to know the rise behaviour, and therefore the residence time, of bubbles in viscoelastic liquids. Since the pioneer paper by Astarita and Apuzzo, it has been known that single bubbles rising in quiescent viscoelastic liquids may exhibit a rise velocity jump discontinuity, once their volume exceeds a critical value [1]. This discontinuity brings about a sudden increase of the steady rise velocity, which may raise the velocity by up to an order of magnitude. The abrupt change of the bubble rise velocity goes along with a change in the bubble shape from a convex to a “teardrop” shaped surface. The two authors of [1] suggested, that the sudden increase of the bubble rise velocity within a small range of volume change represents a change in the boundary conditions at the bubble surface form rigid to free, which is equivalent to the transition from the Stokes to the Hadamard–Rybczinsky regime in a Newtonian fluid. Calderbank et al. [2] investigated the shape, motion, and mass transfer of single carbon dioxide bubbles in an aqueous polyethylene ∗

Corresponding author. Tel.: +43 316 873 7341; fax: +43 316 873 7356. E-mail address: [email protected] (G. Brenn).

0377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2007.05.015

oxide solution. Comparison of the measured mass transfer coefficient with theoretical models for rigid spheres as well as for circulating spheres in creeping flow provided by Levich [3] in the respective regions before and after the discontinuity confirmed the hypothesis of Astarita and Apuzzo. Experiments with glass spheres moving in viscoelastic liquids by Leal et al. [4] showed no discontinuity in the velocity–volume plot and therefore also confirmed the suggestion of Astarita and Apuzzo. Furthermore an investigation of the contribution of shear-thinning effects on the rise velocity after the discontinuity by a numerical analysis of the creeping flow equations, neglecting elastic effects, indicated that only a fraction of the experimentally observed velocity jump is due to the shear dependence of the viscosity. Acharya et al. [5] suggested that polymer molecules might act as surfactants, generating the surface stresses which oppose the circulatory motion within the gas bubble, and that the partial cleansing of the surface responsible for the rapid velocity change is likely to occur much more abruptly in the case of a viscoelastic liquid than for Newtonian fluids. They compared the available data with a criterion based on a Bond number of the order of one: Bo =

ρl gR2c ≈ 1, σ

(1)

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

Nomenclature Latin symbols A ensemble-averaged dyadic of the end-to-end distance vector of the polymer chain b effective bond length of a macromolecule Bo Bond number c polymer mass concentration Ca, Ca capillary numbers as defined in [7,15], respectively dc critical sphere diameter of a bubble df liquid filament diameter df,0 initial filament diameter diameter of a polymer rod drod De Deborah number E¨oc critical E¨otv¨os number g gravitational acceleration G storage modulus G loss modulus K1 time constant of the √ Carreaumodel capillary length = σ/ρl g lcap L length of a single polymer rod m exponent in the relaxation time/mass fraction relation m1 “power-law exponent” of the Carreau model M molecular mass Ma Marangoni number Mo Morton number Ms molecular mass of a polymer segment N number of polymers per unit volume NA Avogadro constant R radius of volume-equivalent sphere RC position vector of the centroid radius of gyration Rg R2 coefficient of determination t time  √  tconv convective time scale = dc /g ux , uy , uz velocity components UT terminal bubble rise velocity v excluded volume parameter V bubble volume Vc critical bubble volume w polymer mass fraction x, y, z Cartesian coordinates X, Y, Z global Cartesian coordinates Greek symbols ␣i , βi fit parameters γ˙ shear rate ε˙ strain rate η shear viscosity η0 zero shear viscosity η0,a zero shear viscosity of aqueous solutions shear viscosity at high shear rates η∞ ηE transient elongational viscosity ηE,t steady terminal elongational viscosity

125

shear viscosity of the solvent intrinsic viscosity intrinsic viscosity of aqueous solutions relaxation time number of polymers per unit volume dimensionless group (=λE (g3 ρ1 /σ)1/4 ) threshold value of Π, necessary for the occurrence of a jump ρl density of the liquid phase ρs density of the solvent σ surface tension σs surface tension of the solvent , ϕ, x cylindrical coordinates (local coordinates)  i , xi discrete points of the bubble meridian curve ω angular frequency

ηs [η] [η]a λE ν Π Π min

(critical bubble radius Rc , gravitational acceleration g, liquid density ρ1 , and surface tension σ against the gas in the bubble). This is an inappropriate criterion for predicting the discontinuity since even for Newtonian fluids Bo can be unity and yet no discontinuity has ever been observed in Newtonian fluids. Zana and Leal [6] investigated the dynamics and dissolution of air and carbon dioxide bubbles in viscoelastic liquids and observed a discontinuous increase of the rise velocity with the bubble volume only in case of non-dissolving (constant-volume) air bubbles. For the dissolving (varying-volume) carbon dioxide bubbles the transition from rigid to free surface conditions is found to be smooth. Furthermore the authors presented and discussed two qualitative models (the film model and the surfactant model) in terms of the observed difference in the transition between constant-volume and shrinking bubbles. Liu et al. [7] studied the cusp at the rear pole of bubbles rising in viscoelastic liquids. They observed cusp formation (“teardrop - like” bubble shape) and the associated increase of rise velocity near a critical capillary number: Ca =

η0 UT ≈ 1, σ

(2)

(zero shear viscosity η0 and terminal bubble rise velocity UT ). The capillary number expresses the balance between viscous forces and surface tension forces; elastic forces are not included. The authors of [7] showed that data previously published by several authors also appear to correlate with a critical capillary number of order one. They noted further that the presence of a cusp is not a sufficient condition for the appearance of a discontinuity, which was confirmed, for instance, in the papers of De Kee et al. [8,9], Rodrigue and Blanchet [10], and also in the present work, where despite a clearly visible cusp at the rear stagnation point in some cases no jump appeared. Extensive investigations on the rise velocity jump discontinuity of bubbles in polymer solutions can be found in [10–15], which represent the recent developments on this phenomenon. The latter papers also include investigations on the effect of surfaceactive agents in the liquid phase. Rodrigue et al. [12] suggested

126

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

that, for an appropriate jump criterion, elastic forces and shear thinning (variable viscosity) must be taken into account beside viscous forces, surface tension, and gravity forces appearing in the purely Newtonian case. Moreover, surface tension gradient (Marangoni) forces can also be involved when surface-active agents are present. In paper [11] by Rodrigue et al. it is pointed out that polymer molecules act as surfactants, since the presence of a polymer alters the surface tension of the solvent against a gas. They proposed further that the discontinuity is the result of an imbalance or an instability at the gas–liquid interface, and that the origin of the instability should be related to normal forces which, for certain conditions, may extract surfactant as well as polymer molecules from the bubble surface, leaving a zone of different interfacial and rheological character. According to this hypothesis, Rodrigue et al. [12] gave a physical interpretation of the jump as follows. At the rear stagnation point, where the local strains are large, causing strong curvature and local deformation, high normal stresses are developed. Polymer and/or surfactant molecules are stretched along the liquid streamlines and therefore induce a change in the fluid properties. The jump could be the result of the normal forces acting in the vicinity of the bubble, removing molecules from the bubble surface or causing a sudden change in the interfacial conditions. In the work of Rodrigue and Blanchet [10] it is shown that the jump can be eliminated by using surfactant concentrations above the critical association concentration (CAC). From their observations they concluded that the origin of the jump is most likely related to a change in interfacial conditions due to an imbalance in surface tension gradient and elastic forces at the gas–liquid interface. In the recent publication [15], Rodrigue and Blanchet presented the correlation De −5/3 , = 0.0181Ca Ma

(3)

as a demarcation line between experimental data before and after the jump. The data were published in the literature by Rodrigue’s group and a group around Leal and Acrivos. In this relation, the viscous, elastic, surface tension, and Marangoni forces acting on the rising bubble are represented by dimensionless numbers, namely the Deborah (De), the capillary (Ca ), and the Marangoni (Ma) numbers (we introduce the prime in Ca here to distinguish this capillary number from the one defined by Eq. (2) [7]). The authors suggest that Eq. (3) can be used as a criterion for the jump discontinuity. The terminal bubble rise velocity UT and the volume-equivalent bubble radius R define the representative shear rate γ˙ r = UT /R in the equations for the characteristic numbers, which read De = aγ˙ rb−n /2m, Ma = σ/mRγ˙ rn , and Ca = mγ˙ rn R/σ. The parameters a, b, m, and n come from the power-law models for the shear stress and the first normal stress difference in the fluids, τ = mγ˙ n and N1 = aγ˙ b , respectively. From the correlation (3), the critical bubble volume may be determined with the rise velocity known, and vice versa. It may be considered as a drawback of Eq. (3), however, that it does not allow for a determination either of the bubble velocity or the bubble volume at the jump independently, based on fluid properties only.

Flow visualization measurements by Funfschilling and Li [16] (PIV and birefringence) showed a very different flow field around bubbles rising in aqueous polyacrylamide solutions versus a Newtonian glycerol solution. For polyacrylamide solutions, the flow field around the bubble can be divided into three distinct zones: an upward flow in front of the bubble, similar to that in the Newtonian case; a downward flow in the central wake, as described earlier as a negative wake (see for example Hassager [17]); finally, a hollow cone of upward flow enclosing the region of negative wake. The birefringence visualization qualitatively revealed a butterfly-like spatial distribution of shear stresses around the bubble. Herrera-Velarde et al. [18] found that the flow configuration changes drastically below and above the critical bubble volume, and the flow situation described as a negative wake appears for bubble volumes greater than the critical one. They noted further that the size of the containment affects the magnitude of the jump, but not the critical bubble volume where the jump occurs. The present paper aims at developing a universal correlation for the bubble volume at the rise velocity jump discontinuity and a criterion for the potential of a given viscoelastic liquid to allow for the occurrence of this discontinuity. Our results will show that it is possible to represent these phenomena by properties of the liquid, provided that the relaxation behaviour against elongational deformation is included. The following section gives a short summary of the concentration regimes occurring in solutions of flexible or rigid rod-like polymers and briefly discusses the influence of the solvent quality as well as the theoretical background of self-thinning filaments of polymer solutions. In Section 3 we report about the materials used in the experiments. Section 4 describes the rheological and capillary characterization of the polymer solutions investigated. The experimental investigations are described in Section 5, results and discussion are given in Section 6. The conclusions are drawn in Section 7. 2. Theoretical background According to Doi and Edwards [19], polymer solutions in good solvents can roughly be divided into three concentration regimes: dilute, semidilute, and concentrated. A dilute solution is defined as one of sufficiently low concentration, where the polymer–polymer interaction has only a small effect, since the polymers are far apart from each other. For solutions of flexible polymers, any physical property can be expressed as a power series with respect to the polymer concentration c. With increasing concentration, an overlap of the polymer coils starts, which can be approximately estimated to set in when M 3 c ≥ c∗ ∼ . = NA 4π R3g

(4)

M denotes the molecular mass of the polymer, NA the Avogadro number, and Rg is the radius of gyration. It should be emphasised that for polymers of high molecular weight (M ≥ 106 kg/kmol) c* becomes quite small. Hence all polymer solutions used in

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

the present work that exhibited a velocity discontinuity can be attributed at least to the semidilute regime, where entanglement of the polymer chains can be assumed. Such a solution is known for large and strongly correlated spatial fluctuations in the segment density, the number of segments per unit volume, decreasing with increasing polymer concentration (see [19], p. 141). If the concentration becomes sufficiently large, the fluctuations become small and can be treated by mean field theory. The solution is then called concentrated and will approximately occur if the polymer concentration exceeds c** calculated according to v(Ms /NA ) , c∗∗ ∼ = b6

(5)

where v denotes the excluded volume parameter, Ms the molecular mass of a polymer segment, and b is the effective bond length. The distinction can be made similarly for rigid rod-like polymers, though the classification is done in terms of the number of polymers per unit volume ν=

c NA . M

(6)

The critical number of polymers per unit volume v1 indicating the transition from dilute to semidilute solutions can be expressed in terms of the length L of a single polymer rod: 1 ν1 ∼ = 3, L

(7)

since in semidilute solutions the rotation of each polymer is restricted by its neighbours. The dynamic properties are therefore completely changed since the polymers are not able to cross each other. The second characteristic concentration, ν2 , marks the polymer number concentration where excluded volume interactions become important: ν2 ∼ =

1 . drod L2

(8)

drod denotes the diameter of the polymer rod, since the polymer rods cannot be treated as mathematical lines without thickness (as done in Eq. (7)) if the number concentration exceeds ν2 . Solutions with number concentrations greater than ν2 are called concentrated. In contrast to solutions of flexible polymers, a fourth critical number concentration ν* exists, which marks the onset of anisotropic behaviour. Solutions with number concentrations above ν* are called liquid crystal solutions [19]. Despite the regime boundaries given by the above relations, the characteristic concentrations represent no sharp demarcation between the regimes, since the transition is fluent and can therefore only be used for a rough classification. Another important aspect beside the polymer concentration regime is the quality of the solvent which has a strong influence on the conformation occupied by the polymer chains due to the solvent effect on the size of the polymer dimension. In

127

a good solvent, the chain will expand in order to increase its favourable interaction with the medium, which results in a positive excluded volume parameter v. In Θ solvents, however, the repulsive excluded volume effect balances the attractive forces between the segments. The excluded volume parameter v equals zero under Θ conditions [20]. Thus the solvent also affects the macroscopic rheological quantities of the polymer solutions. Quantities obtained from elongational rheometry have not been taken into account in investigations of the bubble rise velocity yet. Our hypothesis is that the elongational behavior of the polymer solution must play a significant role in the rise of bubbles, since the liquid flow field around the bubbles exhibits considerable elongational components. Since we assume that for the characterization of the dynamic bubble behavior the mechanism bringing about the elongational flow does not matter, our characterization of the polymer solution includes the relaxation time obtained from filament stretching experiments carried out by means of the elongational rheometer described in Stelter et al. [21] and shown in Fig. 1. The device enables one to measure the diameter decrease of a self-thinning liquid filament with time after inducing a step strain within a liquid bridge between two plates. In Stelter et al. [21], it was shown that the self-thinning of filaments of semi-dilute solutions of flexible polymers can be subdivided into two regimes. In the first regime, the filament diameter decreases exponentially with time, and in the second one it decreases linearly with time. In the first regime, the polymer solution exhibits viscoelastic behavior. The filament diameter decreases according to the law   t df = df,0 exp − , 3λE

(9)

where df is the filament diameter, df,0 its initial value (at time t = 0), and λE is the relaxation time. In the second regime complete extension of the polymers is reached and the polymer solution exhibits Newtonian-like behavior, which allows one to determine the steady terminal elongational viscosity ηE,t , from the relation df = df,0 −

σ t, ηE,t

where σ denotes the surface tension.

Fig. 1. Sketch of the elongational rheometer.

(10)

128

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

It should be pointed out that the rate of stretching within the filament 2 d df ∂uz =− , (11) ε˙ = ∂z df dt is constant during the self-thinning process in the first regime which can be shown by combining Eqs. (9) and (11): ε˙ =

2 . 3λE

(12)

Precisely the same results are obtained using a constitutive model for dilute and semidilute solutions of rigid rod-like polymers according to Doi and Edwards [19] as shown in Stelter et al. [22], despite the difference in the nature of the relaxation process. The relaxation time λE obtained from the self-thinning process of the filament can be related to the longest time constant of the Zimm spectrum obtained from shear rheology measurements (see Anna and McKinley [23]). It is therefore emphasized that, regardless of the polymer character, polymer concentration or solvent quality, the dependence of the filament diameter on time (Eq. (9)) does not change, and the relaxation time λE calculated from the measured evolution of df does not depend on any particular assumptions on the macromolecular model and polymer/polymer interactions [22]. This parameter λE will play an essential role in our further investigations below. 3. Materials used For our experiments we used Praestol 2500 (a linear polyacrylamide, degree of hydrolysis 3–4%) with a molecular mass of M ≈ 15–20 × 106 kg/kmol, and polyethylene oxide (PEO) with a molecular mass of M = 8 × 106 kg/kmol as nonionic polymers. The linear polyacrylamide Praestol 2540 (middle anionic,

degree of hydrolysis 40%, M ≈ 15–20 × 106 kg/kmol) was used as an anionic polymer. The Praestols were produced by Stockhausen Inc., Germany. The polyethylene oxide was provided by Sigma–Aldrich Chemical Company, Inc. All data about molecular masses and degrees of hydrolysis of the polymers were provided by these companies. All polymer solutions were prepared in pure de-ionized water, in solutions of glycerol in de-ionized water, or in ethylene glycol. The conductance of the de-ionized water provided by a reverse osmotic equipment was between 4 ␮S and 5 ␮S. Glycerol and ethylene glycol were provided by Carl Roth GmbH and Co. The data of all polymer solutions investigated are listed in Table 1 together with their material properties and the critical bubble volume Vc where the jump discontinuity (if existing) was observed. The material properties of the solvents used in the present experiments are shown in Table 2. According to the nomogram presented in Kulicke [24] (p. 210, Fig. 5.20) that allows a rough classification of polymer solutions in terms of molecular mass M and polymer concentration c, all investigated polymer solutions correspond to the semidilute or the transition regime from dilute to semidilute. 4. Rheometric and tensiometric characterization of the polymer solutions The polymer solutions were prepared using an anchor stirrer operating at speeds lower than 300 rpm to avoid mechanical degradation of the polymer chains. Concentrations that ensured the existence of a jump discontinuity of the bubble rise velocity ranged above 0.1% weight for aqueous solutions. Variations of the solvent (70:30 wt.% glycerol:water, ethylene glycol) showed, however, that the jump discontinuity can occur at by far lower concentrations of the same polymer (see Table 1). Before the experiments with rising bubbles, the solutions were

Table 1 List of all polymer solutions used for the investigations. The column with the figure numbers allows the data points in the diagrams to be identified with the various polymer concentrations. The compositions of mixed solvents are given in weight percent. The water was de-ionized. All experiments were carried out at room temperature (approximately 20 ◦ C) Polymer

Solvent

Water Praestol 2500 (PAM)

70:30 wt.% glycerol:water Ethylene glycol

w (wt.%)

ρ1 (kg/m3 )

0.1 0.3 0.8 1.0 0.025 0.1 0.05 0.2

998.8 999.4 1000.9 1001.6 1182.1 1183.1 1113.0 1113.1

Praestol 2540 (PAM)

Water

0.05 0.1 0.3

998.8 999.0 999.9

PEO (M = 8 × 106 g/mol)

Water

0.2 0.5 0.8

998.8 999.5 1000.0

Praestol 2500 (P2500) + Praestol 2540 (P2540)

Water

P2500: 0.075; P2540: 0.225 P2500: 0.15; P2540 0.15 P2500: 0.225; P2540 0.075

999.7 999.5 999.2

η0 (Ns/m2 )

σ (N/m)

λE (s)

Vc (mm3 )

0.012275 0.096511 1.5122 3.3929 0.047639 0.22159 0.031299 0.14183

0.07224 0.07315 0.07555 0.07709 0.06837 0.07275 0.05545 0.06125

0.050182 0.107551 0.207357 0.300125 0.170036 0.356929 0.095958 0.224624

– – 45.97957 48.2727 10.46728 11.07349 – 4.4055

1.0295 2.3576 9.4737

0.07651 0.07722 0.07707

0.114868 0.189046 0.465288

– 8.61165 19.48502

Fig. 11

0.071485 1.8029 7.0635

0.06359 0.06693 0.06672

0.094315 0.192229 0.421666

– 14.12959 21.80728

Fig. 13

6.391 4.1549 1.7111

0.07653 0.07642 0.07709

0.32832 0.233692 0.210188

16.66408 14.0174 40.82877

Fig. 15

Fig. 9

Fig. 17

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

129

Table 2 List of properties at room temperature (approximately 20 ◦ C) of all solvents used for the investigations. The last column lists the values of the intrinsic viscosity for solutions of Praestol 2500 (P2500) in the various solvents Solvent

ρs (kg/m3 )

ηs (Ns/m2 )

σ s (N/m)

[η] (P2500) (cm3 /g)

Water 70:30 wt.% glycerol:water Ethylene glycol

998.5 1181.3 1113.0

0.001 0.021 0.019

0.07181 0.07502 0.05292

3209.4 1350.7 315.06

thoroughly characterized by means of shear and elongational rheometry. The reason for looking at the elongational behaviour of the solutions was our hypothesis that the relaxation of stresses in elongational flow could play an important role in the physics of the rise velocity jump discontinuity. It was already noted in the work of Rodrigue and Blanchet [10], and in Rodrigue et al. [12], that care should be given to the possible effect of elongational viscosity of viscoelastic liquids for further investigations. As shown in the paper of Stelter et al. [21], the transient elongational viscosity ηE (t) is related to the relaxation time λE according to the relation

ηE (t) =

3σ λE , df (t)

(13)

where df is the filament diameter. The measurements of the relaxation time λE were carried out by putting a droplet with a volume of 10 ␮l via micropipette on the lower, fixed plate of the elongational rheometer shown in Fig. 1. The upper, movable plate is then brought into contact with the droplet, so that both plates are wetted by the liquid. Then the movable plate is quickly pulled upwards by a solenoid in order to produce a self-thinning, cylindrical liquid filament of constant length on a time scale much shorter than the relaxation time of the liquid. Since the filament is very thin, the motion is dominated by capillary forces (inertial and gravity forces are negligible). A detailed description of the measuring technique is given in Stelter et al. [21]. As shown in Eq. (12), the filament thinning leads to an elongational flow with a constant straining rate. There is no possibility to vary the straining rate arbitrarily, since the relaxation time is a material property of the liquid tested. Eq. (12) further ensures that all measurements are running under comparable conditions, since the Deborah number De = ε˙ λE = 2/3 is constant for all measurements since the liquid is allowed to select its own time scale. The filament diameter as a function of time can be measured by evaluating the power of light from the laser diode received by the photo detector, since the liquid filament is positioned in the laser beam. The relationship between the reduction of the light power and the filament diameter is linear and obtained by a calibration procedure. For determining the relaxation time, only the first regime of the diameter decrease is used and approximated by Eq. (9). The exponent of the function obtained by a data fit determines the relaxation time. Typical results of the elongational characterization are shown in Fig. 2. The reproducibility is given by a peak standard deviation of approximately ±5% for all relaxation times presented in the present work. The dependence of the relaxation time on the mass fraction w of the polymer can

be approximated using the following scaling law: λE ∝ wm .

(14)

Comparison of the exponent m with the corresponding measurements reported in Stelter et al. [21] using the same polymer solutions shows good agreement with the present results and confirms the reliability of the measuring technique. Flow curves as well as storage and loss moduli of the various solutions were measured with the rotational viscosimeter Paar Physica UDS 200 in configuration MS-KP 25, which denotes a cone-and-plate device according to DIN 53018. For solutions exhibiting viscosities lower than 0.02 Ns/m2 and the pure solvents, the flow curves were measured in configuration Z 1, which denotes a concentric cylinder double gap measuring system according to DIN 54453. The results of the shear experiments for Praestol 2500 solutions are shown in Figs. 3 and 4 as examples. Similar results are obtained for the other liquids used in the present work. The flow curves (shear viscosity η versus shear rate ˙ see Fig. 3) exhibit shear thinning behaviour increasing with γ, polymer concentration. For small shear rates, a constant value, the so-called zero shear viscosity η0 , is reached, as expected for polymer solutions. The flow curves can be approximated by the empirical model introduced by Carreau (see [25], p. 18): η − η∞ 1 = m /2 , η0 − η ∞ ˙ 2] 1 [1 + (K1 γ)

(15)

using the software provided by Paar Physica. The values of the zero shear viscosity presented in Table 1 were obtained from the data fit procedure. The shear rheometer also enables measurements of storage G and loss moduli G . Plotting storage and loss moduli against the angular frequency ω for various con-

Fig. 2. Relaxation time vs. mass fraction for aqueous PAM solutions.

130

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

Fig. 3. Flow curves for aqueous P2500 solutions of various concentrations.

centrations, as in Fig. 4(a) for aqueous Praestol 2500 solutions, showed clearly that for the higher concentrations a frequency range exists where the storage modulus exceeds the loss modulus. The solutions of Praestol 2500 in glycerol–water or ethylene glycol, however, exhibited a clearly visible dominance of the loss modulus G throughout the frequency range, as seen in Fig. 4(b).

Besides the rheological properties, the surface tension σ against the gaseous phase and the density of the liquids, ρ1 , play an important role in bubble dynamics. The surface tension was measured by means of the drop volume tensiometer Lauda TVT 2. The density was measured with the oscillating Utube density meter DMA 45 from Anton Paar with an accuracy of ±0.1 kg/m3 . The values of the surface tension presented in Table 1 are determined with a standard deviation less than ±1%. Since different solvents were used for preparing Praestol 2500 solutions, measurements of the intrinsic viscosity [␩] were carried out by means of an Ubbelohde viscosimeter at a controlled temperature of 20 ◦ C in order to obtain a quantity representing the quality of the solvent. The intrinsic viscosities of solutions with various solvents are shown in Table 2 together with material properties of the pure solvents as density ρs , (Newtonian) shear viscosity ηs , and surface tension σ s . 5. Experimental investigations 5.1. Apparatus and measuring techniques The experiments of the present work were carried out with the setup presented in Fig. 5. It consists of a test column made of glass with an inner cross section of 120 mm × 120 mm within an aluminum frame and a plexiglass bubble-generating chamber attached to its bottom in which the bubbles were produced, similar to the experimental setup described in Liu et al. [7]. All aluminum parts were anodized to avoid contamination of the liquid with ions and particles by corrosion. The whole setup was filled with the polymer solution to a liquid level of 450 mm measured from the base plate (see Fig. 5). Experiments were started when gas bubbles in the fresh liquid had disappeared. In order to generate a single bubble with a well defined volume, the ball valve was closed and liquid sucked out of the bubble-generating chamber using a microliter syringe. Due to

Fig. 4. Storage and loss moduli of P2500 solutions for various concentrations and solvents: (a) aqueous solutions and (b) solutions in glycerol–water (GW) and in ethylene glycol (EG).

Fig. 5. Experimental setup.

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

the pressure difference generated by the suction, air enters the chamber in discrete portions of about 2 mm3 through the stainless steel capillary with an inner diameter of 0.09 mm at the bottom of the chamber (“equal volume” method [7]). The reproducibility of the single bubble formation under such quasi-static conditions with the given capillary diameter of 0.09 mm was confirmed by means of measurements based on image processing, which will be described below, and comparison with theoretical predictions by Mersmann for demineralized water (see Brauer [26], p. 278). For varying the bubble volume, a well defined number of bubbles with volumes ranging from 1 mm3 to 3 mm3 , depending on the surface tension of the liquid against air, were produced and allowed to merge at the surface of a hemispherical spoon manufactured of plexiglass. Then the ball valve was opened and the spoon, now containing a single air bubble, was turned round to release the bubble. As noted in [8], a dependence of the rise velocity on the bubble injection period exists. This observation is attributed to the alignment of high molecular weight polymers due to the bubble motion, causing a lower resistance for the subsequent bubbles. The initial isotropic configuration is recovered through molecular diffusion, which requires a long period of time. Rodrigue et al. [11] reported injection periods as high as 300 s for some polymeric solutions. In an earlier paper, De Kee et al. [8] reported injection periods up to 600 s. However, plots of the drag coefficient against the Reynolds number including the injection period as a parameter published by Carreau et al. [27] show that injection periods between 600 s and 3600 s have very little influence on the drag curve, while a strong dependency on injection periods between 4.7 s and 600 s is clearly visible. Therefore, in the present experiments the time intervals between the formations of two bubbles were chosen about 10 min or longer in order to obtain rise velocities independent of the disturbances caused by the preceding bubble. The described technique ensures conditions of a single air bubble rising in a stagnant liquid. Pictures of the rising air bubble were taken with a CCD camera connected to a personal computer (PC) via FireWire. The camera provides images at a maximum framing rate of 30 Hz. The camera position approximately 350 mm above the bottom of the column ensures that the bubble rises with its terminal velocity, which was confirmed, as an example, by measuring the bubble rise velocity in an 0.8 wt.% aqueous Praestol 2500 solution at different height positions (250 mm, 300 mm, and 400 mm above the base plate). Results of these measurements are presented in Fig. 6. The data confirm that both the velocities and the detected critical bubble volume are independent on the position of measurement in the containment of our setup. This means that in this region, the bubbles really move at their terminal rise velocity. We may also state that a slight increase of the bubble volume caused by the decrease of the static pressure during rise of the bubble towards the liquid surface seems to have no influence on the rise motion (a height difference of 450 mm would result in a relative isothermal change in the bubble volume of approximately 4%, which is below the accuracy of our sizing technique described in the following section).

131

Fig. 6. Terminal bubble rise velocity vs. bubble volume for an aqueous 0.8 wt.% P2500 solution measured at different height positions.

A mercury lamp as the light source, together with frosted glass, provided diffuse light for high contrast between air bubble and liquid, and enabled the determination of the bubble volume as well as the terminal bubble rise velocity by means of image processing simultaneously, as described in the next section. Typical images of air bubbles rising in viscoelastic liquids of the present work are shown in Fig. 7. The experiments were carried out at room temperature (≈20 ◦ C). Measurements of the liquid temperature revealed temperature fluctuations of maximum ±1.0 ◦ C throughout an experiment, which ensured a sufficiently constant temperature at which all liquid properties were measured in the characterization experiments. We emphasize that, in order to achieve a high degree of reproducibility and reliability of the experiments without influence of surfactants on the bubble motion, every care was taken to avoid contamination of the liquids with foreign material which could be surface active. For doing this, the bubble column as well as the bubble generating chamber, including all parts in contact with the test liquids, were thoroughly cleaned before refilling with a new test liquid throughout the experiments. Moreover we avoided the use of detergents containing surfactants for cleaning. Instead tap water and demineralized water were used for cleaning and rinsing.

Fig. 7. Shapes of air bubbles in an aqueous 0.8 wt.% P2500 solution below (left, V = 45.8 mm3 ) and above (right, V = 51.6 mm3 ) the critical bubble volume.

132

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

5.2. Image processing Bubble shape as well as bubble position relative to the picture frame were extracted from each picture using various image processing tools provided by Matlab. First a reduction of the socalled “salt and pepper” noise was accomplished by a median filter option in order to preserve the contour of the bubble. The picture was then converted to a binary form using a global threshold value computed according to Otsu’s method provided by Matlab. Furthermore, the centroid C of the meridian area was determined, and the bubble contour points were filtered out by use of the Canny method of Matlab. The coordinates of the centroid and the bubble contour points  were transformed into the global (fixed) coordinate system 0 : (X, Y, Z) defined in Fig. 8, with the X axis pointing upward, the Z axis pointing to the right in the horizontal plane of the bubble column, and the Y axis (not shown) pointing towards the camera. Furthermore, the coordinates of the bubble contour points  were transformed into a local (movable) coordinate system : (, ϕ, x) with (, x) representing the principal axes of the bubble’s meridian area computed via image processing tools by Matlab. Since for the volume range of interest rotational symmetry around the direction of motion of the bubbles can be assumed (see Fig. 7), and furthermore the bubbles rise along a vertical path, the axis of symmetry (denoted x in Fig. 8) points in the direction of the X axis of the global coordinate system 0 which describes the direction of motion. The volume of the air bubble is obtained from an approximation of the meridian curve M : F (, x) = 0,

cent points ( i , xi ) and ( i+1 , xi+1 ) of the boundary curve. A partial volume Vi of the air bubble is then computed according to the equation:  π 2 2 (xi+1 − xi ). (17) i + i i+1 + i+1 Vi = 3 Those partial bubble volumes were then added up to obtain the entire volume. The bubble volume V was determined for a sequence of images of the same bubble according to the image processing described in this subsection with a peak standard deviation of ±5%. Finally, the terminal bubble rise velocity UT was obtained from the consecutive centroid coordinates RC in the X direction of one bubble in different images and the frame rate. 6. Results and discussion 6.1. Aqueous solutions Figs. 9–14 depict data of the terminal rise velocity UT versus bubble volume V and bubble shapes for aqueous solutions of various polymer concentrations of three different polymers. For solutions of the nonionic polyacrylamide Praestol 2500 (Fig. 9) a clearly identifiably critical bubble volume exists, where the rise velocity increases by a factor of 3.64 for w = 0.8 wt.%, and of 5.11 for w = 1.0 wt.% within a small variation of the bubble volume. Moreover, Fig. 9 points out that a sufficient polymer concentration is necessary for the occurrence of a jump discon-

(16)

which represents one half of the previously extracted boundary curve, by means of piecewise linear functions through two adja-

Fig. 9. Terminal bubble rise velocity vs. bubble volume for aqueous P2500 solutions.

Fig. 8. Coordinate systems used for determining the volume and the rise velocity of the air bubble on our images. The Y axis points towards the reader and is not shown in the sketch.

Fig. 10. Shapes of air bubbles with V = 11.1 mm3 (left) and V = 19 mm3 (right) in an aqueous 0.3 wt.% P2500 solution (without jump discontinuity).

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

Fig. 11. Terminal bubble rise velocity vs. bubble volume for aqueous P2540 solutions.

tinuity, since experiments carried out at lower mass fractions (w = 0.1 wt.% and w = 0.3 wt.%) did not reveal a critical bubble volume. Instead a continuous variation in the velocity–volume relation was observed, as one would expect for non-elastic shear thinning liquids (see [26], p. 303). The evolution of the bubble shape as a consequence of the discontinuous enhancement of the terminal rise velocity for the w = 0.8 wt.% and the w = 1.0 wt.% solutions is found to be similar to the observations reported in the literature (see for example [1] or [2]). Fig. 7 shows an air bubble just below (V = 45.8 mm3 ) and one above (V = 51.6 mm3 ) the critical volume of 45.98 mm3 for the aqueous w = 0.8 wt.% solution of P2500. The image of the subcritical air bubble has an entirely convex boundary curve, while the supercritical bubble shows a clearly visible, different shape of the boundary curve with a cusp at the rear stagnation point (similar to a teardrop). Furthermore, the curvature in the rear stagnation point of the subcritical bubble is finite. The bubbles in solutions without

133

velocity jump also undergo a transition in the bubble shape, as can be seen in Fig. 10. Therefore the appearance of a cusp at the rear stagnation point does not ensure the appearance of a velocity jump. This is consistent with the observations by Rodrigue and Blanchet [10]. The aqueous solutions of the anionic polyacrylamide Praestol 2540 exhibit slightly different transition behaviour, as shown in Fig. 11. For a mass fraction of w = 0.05 wt.%, no jump discontinuity was observed. The w = 0.1 wt.%, as well as the w = 0.3 wt.% solutions show a discontinuity in the velocity volume plot, though the transition does not appear as abrupt as in the case of the aqueous Praestol 2500 solutions. Rodrigue and Blanchet [10] obtained a similar transition behaviour by enhancing the concentration of a surfactant added to the polymer solution in order to eliminate the discontinuity (see Fig. 5 in [10]), which they did not identify as a jump any more. Nevertheless care should be taken in the comparison of the present results to the data provided in [10–15], since the presence of surfactants leads to additional (Marangoni) forces at the gas–liquid interface. Moreover, the surface tension of the polymer solutions investigated in [10–15] exhibiting a jump discontinuity is by far lower than the surface tension of the solvent, while the surface tension of the liquids used in the experiments of the present work (Table 1) does not differ much from the surface tension of the solvents (see Table 2). It can therefore be assumed that Marangoni forces play a minor role in the present investigations. It should be pointed out that the rapid velocity increase occurs at lower concentrations than for Praestol 2500 solutions, which corresponds to the rigid-like molecular behaviour of Praestol 2540 where higher order concentration effects set in at a significantly lower concentration than for the flexible molecules of Praestol 2500 ([19], p. 289). Differences regarding the bubble shape transition in comparison with the observations made for Praestol 2500 solutions are also clearly recognizable in Fig. 12. The transition from an “egg

Fig. 12. Shapes of air bubbles with volumes of 19.5 mm3 , 25.4 mm3 , 35.9 mm3 , 41.6 mm3 , 44 mm3 , and 55 mm3 (from top left to bottom right) in an aqueous 0.3 wt.% P2540 solution.

134

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

Fig. 13. Terminal bubble rise velocity vs. bubble volume for aqueous PEO solutions.

shaped” convex bubble to a “teardrop-like” bubble with a cusp at the rear stagnation point occurs in a range of larger volumes than for bubbles in Praestol 2500 solutions. In case of Praestol 2540 solutions, the cusp appears in the transition regime (see Fig. 12, second row). The third family of polymer solutions investigated are aqueous solutions of polyethylene oxide (PEO), which can be regarded as non-ionic. The bubble rise behaviour for varying bubble volume depicted in Fig. 13 is therefore similar to the Praestol 2500 solutions, although the magnitude of the velocity jump is not as high as for Praestol 2500 solutions. This can be attributed to the higher shear viscosity of the PEO solutions at a given polymer concentration. Furthermore, the higher viscosity goes along with a higher elongational viscosity, which results in higher values of the relaxation time according to Eq. (13) (see Table 1). No discontinuity was observed for the PEO mass fraction of w = 0.2 wt.%. The bubble shape transition is similar to the one observed for aqueous Praestol 2500 solutions. Typical shapes of air bubbles in a PEO solution with w = 0.8 wt.% (Vc = 21.8 mm3 ) are shown in Fig. 14.

constant total polymer mass fraction of 0.3 wt.% and varying mixture ratio (see Table 1), similar to the experiments reported in Stelter et al. [22]. The mass fraction of 0.3% was chosen since the aqueous solution prepared with the flexible Praestol 2500 showed no velocity discontinuity at that polymer concentration, while the Praestol 2540 solution did. We therefore expected the jump to disappear with increasing mass fraction of Praestol 2500, which makes the polymer solution more “flexible” and reduces the relaxation time, as reported in the work of Stelter et al. [22]. Unfortunately it turned out that for Praestol 2500 contents greater than 0.225 wt.% one of the dissolved polymers precipitated, leading to opaque polymer solutions, which make image processing impossible. Therefore, our investigations with the polymer mixtures were impossible for solutions with Praestol 2500 mass fractions greater than 0.225 wt.%. The volume-rise velocity curves are plotted in Fig. 15. An increase of the Praestol 2500 mass fraction first shifts the velocity discontinuity to lower volumes. For larger mass fractions of P2500, however, where the solution can be regarded as flexible, the critical bubble volume increases again, which goes along with a decrease of the velocity increase factor. The abruptness of the velocity change increases with enhancement of the mass fraction of Praestol 2500, which corresponds to the observations made for flexible polymer solutions. The observations confirm our expectation that the interaction of rigid polymers with flexible ones may cause a velocity discontinuity due to the increase of the relaxation time in elongational flow. 6.3. Dimensional analysis and correlation for aqueous polymer solutions According to the observations made in the bubble rise experiments together with the characterization of the polymer solutions, we can set up the following list of parameters relevant for the critical bubble volume: dc , g, ρl , η0 , σ, λE ,

(18)

where dc denotes the critical bubble diameter at the jump, g the gravitational acceleration chosen as process variable, since the

6.2. Mixtures of flexible and rigid polyacrylamides In the next stage of our investigations we looked at the rise behaviour of air bubbles in aqueous polymer solutions prepared with mixtures of flexible and rigid rod-like polyacrylamides with

Fig. 14. Shapes of air bubbles in an aqueous 0.8 wt.% PEO solution just at (left, V = 21.8 mm3 ) and above (right, V = 22.4 mm3 ) the critical bubble volume.

Fig. 15. Terminal bubble rise velocity vs. bubble volume for aqueous solutions of P2500 and P2540 with the given mass fraction ratios wP2500 : wP2540 .

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

flow is driven by buoyancy, ρ1 the density of the liquid phase (the density of the gaseous phase is neglected), and η0 represents the zero shear viscosity of the liquid. The surface tension σ is considered as relevant, since we are dealing with a curved gas–liquid interface, and finally the elongational relaxation time λE comes in according to our hypothesis. Dimensional analysis according to the method described in [28] converted the relevant parameters into the following set of three non-dimensional numbers: E¨oc =

1/4

g3/4 λE ρl Π= σ 1/4

.

(19)

Among the other possibilities of combining the parameters, this set of dimensionless groups was found to represent the measurement data best, as will be seen later. It should be noted that the above definitions of the E¨otv¨os and Morton numbers differ slightly from those given in Clift et al. [29], where the two numbers are both defined with the density difference between continuous and dispersed phases rather than with the density of the continuous phase as in our above Eq. (19). In our investigations we worked with liquids having densities about 103 times higher than the density of the dispersed gaseous phase. Therefore, in the density difference we neglect the gaseous against the liquid density ρ1 . Therefore, apart from the difference in the length scale, our E¨otv¨os number appears equivalent to the Bond number according to Eq. (1). The parameter Π can be rearranged by introducing the critical bubble diameter dc to obtain Π=

λE g1/2 1/2

dc



dc2 gρl σ

1/4 .

(20)

In this representation of the non-dimensional group we recognize groups of parameters relevant √ to the rise of the bubbles: the convective time scale tconv = dc /g, which is the denominator √ in the fraction with λE , and the capillary length lcap = σ/ρl g, which appears in the fraction with the critical bubble diameter dc . The parameter Π may therefore be represented in the form Π≡

λE tconv



dc lcap

formulation of the function in Eq. (22) the ansatz Π = α1 E¨oαc 2 Moα3 + α4 ,

(23)

where the undetermined parameter α4 should denote a threshold value for the group Π, which must be exceeded for the velocity discontinuity to occur. Since we are interested in a critical bubble volume, i.e. a critical equivalent bubble diameter, we rearranged Eq. (23) before computing the yet undetermined parameters αi via data regression into the form E¨oc = β1 Moβ2 (Π − β4 )β3 .

gρl dc2 (critical E¨otv¨os number), σ

gη40 (Morton number), Mo = ρl σ 3

135

(24)

Note that the parameters αi are now expressed in terms of βi . Eq. (24) denotes a relation which allows the critical bubble volume to be calculated from material properties of the polymer solution and the gravitational acceleration exclusively. Other quantities, e.g. the velocity of bubble motion, do not appear in the equation. The best fitting parameters βˆ i describing the problem were determined using a nonlinear least squares method provided by Matlab, which resulted in the following equation for the critical E¨otv¨os number ˆ oc = 5.3688 (Π − 9.8938) E¨ . (25) Mo0.3282 The agreement of Eq. (25) with the measured data for the various liquids is shown in Fig. 16. The coefficient of determination R2 = 0.9906 indicates an excellent representation of the critical bubble volume in aqueous polymer solutions. The correlation Eq. (25) yields real numbers only in cases that the difference in round brackets is positive. Therefore, Π > 9.8938 is required to have a potential of the liquid for the jump to occur. This corresponds to our observation that Π ≤ 9.8938 only for liquids without jump discontinuity. The parameter Π may therefore be considered as an indicator for the existence of a jump discontinuity. Because of the previously mentioned absence of properties of the bubble motion, such as the terminal rise velocity, Eq. (25) may be attractive and useful for predicting the critical bubble volume in viscoelastic liquids. Nevertheless, the correlation is a purely empirical relation, and especially the parameter βˆ 4 representing the threshold value 0.9211

1/2 .

(21)

We will come back to the physical interpretation of this group in Section 6.5 below. With the three non-dimensional groups of Eq. (19) we are able to represent the rise behavior of the bubbles in the form Φ(E¨oc , Mo, Π) = 0.

(22)

Together with the results from the rise velocity measurements, which showed that the jump discontinuity occurs for a sufficiently high mass fraction of the polymer, and since it is evident that the relaxation time λE can be related to the polymer mass fraction (see Eq. (14) and Fig. 2), we choose for an explicit

Fig. 16. Dimensionless representation of the data points corresponding to the critical bubble volumes observed in aqueous solutions of different polymers.

136

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

Π min required for the occurrence of the jump discontinuity needs further investigation to interpret Eq. (25) properly. Before doing this, we generalize the correlation by adding investigations with polymer solutions in different solvents. 6.4. Analysis for varying solvents—experimental data and universal correlation Since it is evident that the conformation of the polymer molecules in the solvent affects the macroscopic rheological quantities [22], we were also interested in the influence of the solvent on the bubble rise behaviour. Moreover, the use of solvents other than water may lead to a verification of the correlation found. We decided to use a glycerol–water (70:30 wt.%) mixture and ethylene glycol, both denoted as non-aqueous solvents in the following discussion, to prepare solutions of the flexible Praestol 2500, since similar materials were used in the work of Stelter et al. [22]. While the glycerol–water mixture can be regarded as a “good” solvent, ethylene glycol is known as a “poor” one [24,30]. It can therefore be assumed that the conformation of the polymer chains in dilute solutions of ethylene glycol is quite close to the Θ state, where the polymers behave like ideal chains [19]. The bubble rise velocity is shown in Fig. 17 as a function of the bubble volume for the non-aqueous solutions of Praestol 2500. A velocity discontinuity exists in both systems

Fig. 17. Terminal bubble rise velocity vs. bubble volume: (a) for P2500 in 70:30 wt.% glycerol:water and (b) for P2500 in ethylene glycol.

for solutions of sufficiently high polymer concentrations, but the corresponding concentrations are lower than for aqueous solutions of the same polymer, which clearly indicates the influence of the solvent. The viscosities of the investigated non-aqueous solutions are lower, which results in higher bubble rise velocities, but the magnitude of the velocity jumps is not as high as for aqueous solutions, although clearly visible. The evolution of the bubble shape is similar to the one observed for aqueous Praestol 2500 solutions. For the non-aqueous solutions the viscoelastic group Π found in the dimensional analysis still fulfilled the jump criterion established for the aqueous solutions: in cases where the threshold value Π min found for the non-aqueous solutions was exceeded, a jump discontinuity occurred, and vice versa. This fact seems to confirm our hypothesis that the relaxation time obtained from the elongational rheometer is a relevant quantity for the description of the occurrence of this jump phenomenon, and that the non-dimensional group Π is relevant for estimating the potential for a jump to occur. In order to make the data points of the non-aqueous solutions collapse on the curve described by Eq. (25), it was necessary to extend the model by two ratios of quantities representing the solvent: 0.9087 

(Π − 9.9381) E¨ oc = 5.4801 Mo0.3268

η0,a η0

1.2144 

[η] [η]a

1.4389 . (26)

The first additional factor on the right of Eq. (26) represents the ratio of the zero shear viscosities η0,a of the aqueous and η0 of the non-aqueous solutions at the same polymer mass fractions. The second factor denotes the ratio of the intrinsic viscosities of the non-aqueous to the aqueous solutions. The number factor and the exponents occurring in Eq. (26) were again determined by means of the nonlinear regression method provided by Matlab and differ only slightly from the values in Eq. (25). Fig. 18 shows the data points corresponding to the critical bubble volumes measured in the present work. The coefficient of determination R2 = 0.9923 indicates again excellent agreement of the calculated critical E¨otv¨os numbers, E¨ oc , with the mea-

Fig. 18. Dimensionless representation of the data points corresponding to the critical bubble volumes observed in aqueous and non-aqueous solutions of different polymers.

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

137

sured E¨oc although the correlation is purely heuristic for the moment. 6.5. Physical interpretation of the threshold criterion In the present section we give a physical interpretation of the significance of the threshold criterion Π > Π min for a viscoelastic liquid to have a potential to exhibit a bubble rise velocity discontinuity. For this purpose, we will interpret the group Π following the derivations which led to Eq. (21) above. When looking at the time and length scales occurring in Eq. (21), we have the following process in mind. The rising motion of the bubble through stagnant liquid may be regarded as equivalent to a steady flow of the polymer solution around the bubble. As the liquid approaches the bubble, it experiences viscous and elastic stresses built up in the stagnation zone around the upper part of the bubble. These stresses are partly elongational in nature due to velocity gradients ∂ux /∂x, and represent mechanical energy stored in the dissolved polymer molecules. In the subsequent flow down the upper hemisphere of the bubble, these stresses tend to relax. This postulated behaviour is consistent with results of numerical simulations presented in the paper of Noh et al. [31], the work by Harlen [32], and very recently in the work of M´alaga and Rallison [33]. All three papers present plots of tr A (where the tensor A represents the ensemble-averaged dyadic of the end-to-end distance vector of the polymer chain), or of components of the tensor, versus the position on the bubble surface. In the results of [31] and [33], the quantity tr A exhibits a minimum on the lower part of the bubble contour. This indicates that, at that location, the polymer chains exhibit a nearly unstretched conformation with almost no normal stress. According to results of Noh et al. [31], the polymer relaxation is characterized by a transformation from a state stretched in the direction of the polar angle to one stretched in the radial direction of a spherical coordinate system. This change of conformation is associated with forces, which are felt by the bubble and may lead to a partial compensation of the drag force coming from the polymer stretching during the deformation in the upper stagnation zone of the bubble. A qualitative plot of the tensor A as given in [33], represented in analogy to the representation in [32], where the main axes lengths of ellipses are proportional to the eigenvalues of A, is given in Fig. 19. While the above described polymer relaxation process takes place on the time scale λE , a given parcel of relaxing polymer molecules√is transported downstream on the convective time scale tconv = dc /g. The ratio of these two time scales, which appears in the group Π, decides at which part of the bubble the stresses may have relaxed to an extent that they may be felt by the bubble, so that the force may act as a resistance or as a push upward. For fast relaxation, with small λE , this process may hinder the bubble rise, while for slow relaxation, with large λE at sufficiently high polymer concentration, the relaxation may lead to forces pushing the bubble upwards. In the ratio of the length scales dc and lcap , also involved in the group , the capillary length represents the necessity to have a fluid–fluid interface for the discontinuity to occur, which does not exist for rigid spheres. This ratio stands to the power of 1/2. Together with

Fig. 19. Qualitative plot of the tensor A representing the configuration of the polymer. In analogy to the work by Harlen [32], the tensor A is represented by the ellipse |A−1 ·x|2 = constant.

the time scales appearing linear, this is typical of an accelerated motion, where a length scale may be proportional to the time squared. The whole group Π therefore may represent the ratio of two accelerations, an elasto-capillary one and a convective one. The ratio of the two must evidently exceed the threshold value of 9.9381 found empirically to enable the rise velocity jump discontinuity to occur, i.e., to enable a relaxation time appropriate to change the forces on the bubble as the bubble size varies. 7. Conclusions In the present work the dynamic behaviour of air bubbles rising in viscoelastic polymer solutions was investigated. Interest was focused on the rise velocity jump discontinuity, which may occur as a critical bubble volume is exceeded. A universal correlation was presented, which allows the critical bubble volume to be determined from relevant rheological and capillary properties of the liquid only. In the development of the correlation it was found that a thorough rheological and capillary characterization of the liquids is of big importance. In the relevant liquid properties, the relaxation behaviour of the liquids in straining flows was found to play a key role. It was further shown that the relaxation time in elongational flow is an appropriate measure for indicating the occurrence of a discontinuity in the rise velocity volume relation, regardless of the polymer properties, and independent of the quality of the solvents used for preparing the liquids. The correlation developed is interpreted as a threshold condition for a ratio of the time scale in the liquid flow around the bubbles and the relaxation time.

138

C. Pilz, G. Brenn / J. Non-Newtonian Fluid Mech. 145 (2007) 124–138

Acknowledgements Financial support of the present work from the Austrian Science Fund (FWF—Fonds zur F¨orderung der wissenschaftlichen Forschung) under contract number P17624-N07 is gratefully acknowledged. The authors wish to thank Professor V. Ribitsch at the Institute of Chemistry of the Karl-Franzens University of Graz for making his shear rheometers available for the characterization of our liquids, and H. Katzer for her cooperation during the shear experiments. We thank G. Kircher at the Institute of Chemical Engineering and Environmental Technology of Graz University of Technology for measuring the surface tension of the liquids. We also acknowledge valuable input from Dr. M. Schl¨uter at the Institute of Environmental Process Engineering of the University of Bremen in fruitful discussions about our work. References [1] G. Astarita, G. Apuzzo, Motion of gas bubbles in non-Newtonian liquids, AIChE J. 11 (1965) 815–820. [2] P.H. Calderbank, D.S.L. Johnson, J. Loudon, Mechanics and mass transfer of single bubbles in free rise through some Newtonian and non-Newtonian liquids, Chem. Eng. Sci. 25 (1970) 235–256. [3] V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. [4] L.G. Leal, J. Skoog, A. Acrivos, On the motion of gas bubbles in a viscoelastic liquid, Can. J. Chem. Eng. 49 (1971) 569–575. [5] A. Acharya, R.A. Mashelkar, J. Ulbrecht, Mechanics of bubble motion and deformation in non-Newtonian media, Chem. Eng. Sci. 32 (1977) 863–872. [6] E. Zana, L.G. Leal, The dynamics and dissolution of gas bubbles in a viscoelastic fluid, Int. J. Multiphase Flow 4 (1978) 237–262. [7] Y.J. Liu, T.Y. Liao, D.D. Joseph, A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid, J. Fluid Mech. 304 (1995) 321–342. [8] D. De Kee, P.J. Carreau, J. Mordarski, Bubble velocities and coalescence in viscoelastic liquids, Chem. Eng. Sci. 41 (1986) 2273–2283. [9] D. De Kee, R.P. Chhabra, A. Dajan, Motion and coalescence of gas bubbles in non-Newtonian polymer solutions, J. Non-Newtonian Fluid Mech. 37 (1990) 1–18. [10] D. Rodrigue, J.F. Blanchet, Surface remobilization of gas bubbles in polymer solutions containing surfactants, J. Colloid Interf. Sci. 256 (2002) 249–255. [11] D. Rodrigue, D. De Kee, C.F. Chan Man Fong, An experimental study of the effect of surfactants on the free rise velocity of gas bubbles, J. NonNewtonian Fluid Mech. 66 (1996) 213–232. [12] D. Rodrigue, D. De Kee, C.F. Chan Man Fong, Bubble velocities: further developments on the jump discontinuity, J. Non-Newtonian Fluid Mech. 79 (1998) 45–55.

[13] D. Rodrigue, D. De Kee, Bubble velocity jump discontinuity in polyacrylamide solutions: a photographic study, Rheol. Acta 38 (1999) 177– 182. [14] D. Rodrigue, D. De Kee, Recent developments in the bubble velocity jump discontinuity, in: Proceedings of the 13th International Congress on Rheology, vol. 2, British Society of Rheology, Glasgow, Cambridge, August 20–25, 2000, pp. 241–243. [15] D. Rodrigue, J.F. Blanchet, Recent developments on the velocity-volume bubble jump discontinuity, in: Proceedings of the 14th International Congress on Rheology, The Korean Society of Rheology, Seoul, August 22–27, 2004. [16] D. Funfschilling, H.Z. Li, Flow of non-Newtonian fluids around bubbles: PIV measurements and birefringence visualization, Chem. Eng. Sci. 56 (2001) 1137–1141. [17] O. Hassager, Negative wake behind bubbles in non-Newtonian liquids, Nature 279 (1979) 402–403. [18] J.R. Herrera-Velarde, R. Zenit, D. Chehata, B. Mena, The flow of non-Newtonian fluids around bubbles and its connection to the jump discontinuity, J. Non-Newtonian Fluid Mech. 111 (2003) 199– 209. [19] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. [20] M. Doi, Introduction to Polymer Physics, Clarendon Press, Oxford, 1996. [21] M. Stelter, G. Brenn, A.L. Yarin, R.P. Singh, F. Durst, Validation and application of a novel elongational device for polymer solutions, J. Rheol. 44 (2000) 595–616. [22] M. Stelter, G. Brenn, A.L. Yarin, R.P. Singh, F. Durst, Investigation of the elongational behavior of polymer solutions by means of an elongational rheometer, J. Rheol. 46 (2002) 507–527. [23] S.L. Anna, G.H. McKinley, Elasto-capillary thinning and breakup of model elastic liquids, J. Rheol. 45 (2001) 115–138. [24] W.M. Kulicke, Fließverhalten von Stoffen und Stoffgemischen, H¨uthig & Wepf Verlag, Heidelberg, 1986. [25] H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology, Rheology Series, vol. 3, Elsevier, Amsterdam, 1989. [26] H. Brauer, Grundlagen der Einphasen- und Mehrphasenstr¨omungen, Verlag Sauerl¨ander, Frankfurt am Main, 1971. [27] P.J. Carreau, M. Devic, M. Kapellas, Dynamique des bulles en milieu visco´elastique, Rheol. Acta 13 (1974) 477–489. [28] M. Zlokarnik, Dimensional analysis and scale-up in chemical engineering, Springer, Berlin, Heidelberg, 1991. [29] R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops and Particles, Academic Press, New York, 1978. [30] J. Klein, K.-D. Conrad, Characterisation of poly(acrylamide) in solution, Makromol. Chem. 181 (1980) 227–240. [31] D.S. Noh, I.S. Kang, L.G. Leal, Numerical solutions for the deformation of a bubble rising in dilute polymeric fluids, Phys. Fluids 5 (1993) 1315– 1332. [32] O.G. Harlen, The negative wake behind a sphere sedimenting through a viscoelastic liquid, J. Non-Newtonian Fluid Mech. 108 (2002) 411–430. [33] C. M´alaga, J.M. Rallison, A rising bubble in a polymer solution, J. NonNewtonian Fluid Mech. 141 (2007) 59–78.