Bubble velocities: further developments on the jump discontinuity

J. Non-Newtonian Fluid Mech., 79 (1998) 45 – 55

Bubble velocities: further developments on the jump discontinuity D. Rodrigue a,*, D. De Kee b, C.F. Chan Man Fong b a b

Department of Chemical Engineering, CERSIM, La6al Uni6ersity, Quebec G1K 7P4, Canada Department of Chemical Engineering, Tulane Uni6ersity, New Orleans, LA 70118 -5698, USA Received 6 August 1997; received in revised form 9 February 1998

Abstract The motion of a gas bubble in a non-Newtonian fluid has been further examined in order to determine the conditions for the possible existence of a discontinuity in the bubble velocity-bubble volume log – log plot. It has been proposed in the past that this phenomenon was the result of a sudden change in the hydrodynamics of the moving bubble, resulting in a transition from a Stroke to a Hadamard regime. Furthermore, this abrupt transition was only qualitatively attributed to the elasticity of the fluid. Using our data as well as those of Leal et al., we demonstrate here that the discontinuity results as a balance between elastic and Marangoni instabilities, providing another major difference between Newtonian and non-Newtonian hydrodynamics. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Bubble velocity; Jump discontinuity; Capillary; Deborah; Marangoni; Viscoelastic Mach numbers

1. Introduction In the past, several differences were observed between the hydrodynamics of polymeric and Newtonian liquids. These differences are represented by phenomena such as rod-climbing, hole-pressure error and Uebler effects, to name a few. More exhaustive lists are given in Bird et al. [1], Barnes et al. [2] and Carreau et al. [3]. A number of papers in the literature deal with the discontinuity observed in the bubble velocity versus bubble volume plot. It was assumed in the past that this phenomenon was the result of a sudden transition from a Strokes (rigid interface) to a Hadamard (free interface) regime. Astarita and Apuzzo [4], qualitatively attributed this transition to the elasticity of the fluid by referring to the rheological data of the fluid. Recent reviews on this subject are available in De Kee et al. [5] and in Rodrigue [6]. Rodrigue et al. [7] experimentally demonstrated that a * Corresponding author. Fax: +1 418 6565993. 0377-0257/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0377-0257(98)00072-X

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

46

surface active agent (polymer or surfactant molecules) as well as elastic forces must be simultaneously present, in order to modify the surface and generate a sudden jump in the velocities. In this paper, it is shown that the transition effectively represents a change in the hydrodynamic regimes and that viscoelasticity and surface tension are responsible for the sudden occurrence of this phenomenon. Using the experimental data of Leal et al. [8] on air bubbles rising in aqueous polyacrylamide (AP-30) solutions and those of Rodrigue et al. [7] on air bubbles rising in polyacrylamide (AP-273) in 50/50 wt.% aqueous glycerol solutions, we determine the role played by viscoelasticity and surface tension in order to establish a criterion for the occurrence of this jump and to shed more light on the elucidation of this phenomenon. Recently, a criterion was proposed (Rodrigue et al. [7]) to determine the volume at which a jump discontinuity may occur. It is now considered unwieldy since it was based on a simple data regression from experimental measurements. We now propose a simpler one, catching the influence of all forces involved in this particular flow.

2. Theory The motion of a spherical particle in Newtonian fluids has been extensively studied. Several correlation formulae are available to predict the drag coefficient (CD) as a function of the Reynolds number (Re) and they can be found in Clift et al. [9], Perry and Green [10] and more recently, in Maxworthy et al. [11]. In the case of gas bubbles, one can neglect both the density and the viscosity of the gas. By doing so, the drag coefficient can be written as CD =

4gD , 3U 2

(1])

where g is the gravitational constant, U is the terminal velocity of the particle and D is the particle diameter. In the case of a non-spherical bubble, the sphere equivalent diameter is generally used: D=

  6V p

1/3

,

(2)

where V is the bubble volume. Chhabra [12] reviewed the correlations available for bubbles and spheres moving in non-Newtonian fluids. For shear-thinning fluids, the power-law model of Eq. (3) is frequently used to represent the shear stress (t) as a function of the rate of deformation (y; ): t =m y; n

(3)

Based on this model, we can represent the drag coefficient as a function of a generalized Reynolds number (Rep) and a correction function. For a rigid sphere, the drag coefficient is approximated by: CD =

24 X(n) Rep

(4)

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

47

where X(n) is a correction function dependent on the power-law index n, Rep is the generalized Reynolds number for the power-law model is defined as: Rep =

rU 2 − nD n m

(5)

and r is the density of the liquid. The correction factor has been approximated in the past using theoretical calculations and numerical methods (Chhabra et al. [13], Gu and Tanner [14]. Due to the scatter in the experimental data, no unique relation exists. Its value is unity for Newtonian fluids (n= 1) and increases for shear-thinning fluids (nB 1). The correction used for X(n) was developed by Tomita [15], corrected numerically by Wallick et al. [16] and represented graphically by Chhabra et al. [13] since it is the one describing more adequetly all data in this study. In the case of a gas bubble moving in a fluid free of contaminant, the drag coefficient can be represented by Eq. (6) for the creeping flow of an incompressible fluid around a spherical gas particle: CD =

16 Y(n), Rep

(6)

where Y(n) is the correlation developed by Rodrigue et al. [17] using a perturbation method up to the second order in (n–1)/2: Y(n) = (2)n − 1 (3)(n − 1)/2

1+ 7n−5n 2 . n(n+ 2)

(7)

In order to take the viscoelastic character of the fluids into account, the primary normal stress different (N1) was also modeled by a power-law relation [2]: N1 = a g; b

(8)

For viscoelastic fluids one also needs to introduce a dimensionless number such as a Deborah number (De) which will reflect the ratio of the elastic to the viscous stresses. Following Arigo and McKinley [18], as we define De as De=

N1(g; )= a b − n g; . 2t(g; ) 2m

(9)

We further note that the Reynolds and Deborah numbers represent the ratios of inertia and elastic forces to viscous forces, respectively.

3. Rheology of the fluids As mentioned earlier, power-law equations were used to model the rheological behaviour of the fluids, i.e. solutions of polyacrylamide (AP-273) in 50/50 wt.% aqueous glycerol [7] for concentrations ranging from 0.075 to 0.25 wt.% and for 0.5 and 1.0 wt.% aqueous AP-30 solutions [8]. The parameters a, b, m, and n and their range of validity for each solution are listed in Table 1. Fig. 1 represented typical rheological behaviour of the PAA solutions. It can

48

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

Table 1 Parameters determined for the power-law models PAA solution

m

n

a

b

Range of g;

U/R

(WT.%)

(Pa · sn)

(−)

(Pa · sb)

(−)

(s−1)

(s−1)

0.075a 0.10a 0.15a 0.20a 0.25a 0.5b 1.0b

0.161 0.563 0.751 0.853 1.079 0.990 3.22

0.700 0.523 0.513 0.536 0.499 0.453 0.393

0.166 1.05 1.59 1.88 2.52 1.40 8.14

0.834 0.790 0.817 0.880 0.856 0.686 0.598

0.1 – 100 0.1 – 400 0.1 – 400 0.1 – 400 0.1 – 400 0.5 – 1000 0.5 – 1000

10–71 1.0 – 51 1.0 – 50 0.9 – 38 0.6–31 3.0 – 49 0.5 – 34

a

Data from [7]; b data from [8].

be seen that for rates above 0.1 s − 1, the primary normal stress difference is greater than the shear stress, indicating dominant viscoelastic properties.

4. Results Fig. 2 illustrates representative velocity–volume data. For PAA concentrations equal to and below 0.10 wt%, no jump discontinuity was observed in solutions without surfactant. With the

Fig. 1. Rheological data for the 0.10 wt.% PAA solution [7]. ", N1; “,t; — , Eq. (3) with m =0.563 and n = 0.523; - - -, Eq. (8) with a =1.05 and b=0.790.

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

49

Fig. 2. Velocity–volume curves for two PAA solutions [7]. The circles and diamonds are for the 0.10 and 0.25 wt.% PAA solutions, respectively. The open and closed symbols correspond to 0 and 100 ppm added SDS, respectively.

addition of 100 ppm of SDS (sodium dodecyl sulfate), an anionic surfactant, a jump appears at a volume of  30 mL. This suggest that surfactants modify the conditions prevailing at the gas – liquid interface which is needed for a jump to occur. Furthermore, a jump occurs in the case of the 0.25 wt.% PAA solution without any added surfactant, suggesting that the PAA molecules could act as surface active agents and supporting the hypothesis of Acharya et al. [19] and Zana and Leal [20]. Haque et al. [21] obtained similar results with aqueous CMC solutions. In order to establish whether the jump is really a transition between two hydrodynamic regimes, it is useful to replot the data in a standard form, relating the drag coefficient to the generalized Reynolds number. Fig. 3 shows the data for the 0.10 wt.% solution of Fig. 2 replotted in this manner. The data can be divided into two regions. For the 0 ppm SDS solution, a smooth departure of the data from the upper line (X(n) =1.89) to the lower line (Y(n) =1.38) is seen to occur around Rep = 0.1. The positive deviation from the lower line at high Rep is believed to be caused by inertia, deformation and surface tension effects, which are known to increase drag coefficients. Fig. 2 also suggests that for the 100 ppm SDS solution, the transition is rather abrupt. Fig. 4 represents the data for the 0.25 wt.% PAA solution. As in the case of Fig. 3, all of the points prior to the jump follow the upper line (X(n) =1.90), the data after the jump (larger volume bubbles) follow the lower curve (Y(n) =1.40). Since this solution is more viscous, the jump occurs at lower Reynolds numbers and inertia effects are less important. By looking at Figs. 2 – 4, it is seen that when a discontinuity occurs, it effectively represents a sudden transition from an almost rigid interface to an almost free interface.

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

50

5. Discussion In order to establish a criterion for the appearance of a jump discontinuity, we wish to quantify the role played by the different forces affecting the flow. First, we define an average shear rate for characterizing the rheological properties of the fluids. This shear rate is defined by the ratio of the terminal velocity of the bubble to its spherical equivalent radius (R =D/2): g; = U/R.

(10)

It is worth noting that Table 1 justifies the use of the power-law models for N1 and t, i.e. the range for the average experimental shear rates is completely within the range of validity of the determined parameters. It has been postulated in the past, that viscoelasticity was responsible in some ways for this sudden change in regimes. One way to look at this is via the viscoelastic Mach number (M). This number represents the ratio of the characteristic flow velocity to the speed at which the shear waves propagate into the liquid. One expression for M is M = RepDe.

(11)

The influence of the viscoelastic Mach number has been investigated in the past for the flow around rigid spheres and cylinders. As reviewed recently by Becker et al. [22], the value of M is associated with the shift in the streamlines around a particle in a viscoelastic fluid, compared to a Newtonian one. It also represents the propagation of signals in the fluid. It is known that a change of character in the governing equations from an elliptic to a hyperbolic type occurs at M= 1. No direct correlation was found between the discontinuity and the viscoelastic Mach

Fig. 3. Drag coefficient (CD) as a function of the generalized Reynolds number (Rep) for the 0.10 wt.% PAA solution [7]. , 0 ppm SDS; “, 100 ppm SDS; —, Eq. (4) with X(n) = 1.89; - - -, Eq. (5) with Y(n) = 1.38.

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

51

Fig. 4. Drag coefficient (CD) as a function of the generalized Reynolds number (Rep) for the 0.25 wt.% PAA solution [7]. , 0 ppm SDS; “, 100 ppm SDS; —, Eq. (4) with X(n) = 1.90; - - -, Eq. (5) with Y(n) = 1.40.

number. However, in every case studied so far, it was found that the discontinuity occurs when M is order unity. In the past, the Bond criterion was used to evaluate the radius at which the discontinuity occurs. The Bond number, which represents gravity over surface tension forces, was set to unity at the transition. That is to say Bo =

rgR 2 = 1. s

(12)

As shown in De Kee et al. [5] and in Rodrigue [6], this criterion is inappropriate for predicting the discontinuity since even for Newtonian fluids, Bo can be unity and yet no discontinuity has even been observed in Newtonian fluids (with or without surfactants). At best, the Bond criterion can be used to estimate the transition point from the Strokes to the Hadamard regime as communicated in the original paper [23]. Liu et al. [24] studied the motion of a gas bubble in a viscoelastic liquid. They postulated that the appearance of the jump results from a transition in shape and the appearance of a cusp at the trailing edge of the bubble. Using a viscoelastic Mach number and a Capillary number based on the zero shear viscosity, they observed that the transition appears when these numbers approach unity and under these conditions, the rear end of the bubble is in the form of a cusp. By looking carefully at the experimental setup and conditions used, we are led to believe that the cusp formation may be caused, in part, by wall effects, since the dimensions of the tank were small compared to the bubble volumes (e.g. Fig. 10 of their paper shows a 6 cm3 bubble in a 1 in2 cross section tank). As the appearance of a cusp was not observed or mentioned in any of the previous reports on the jump discontinuity, future research should be focused on the bubble

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

52

shapes before and after critical volume. In order to investigate this hypothesis further and to compare with the results in Table 4 of Liu et al. [24], the Capillary number (Ca) is calculated for the solutions used here and the results are given in Table 2. The Capillary number is defined as h0U Ca = . (13) s It is shown that the critical Capillary number at the discontinuity is mostly near unity. However, the use of the zero shear viscosity in the definition of the Capillary number is questionable. The average shear rates at the discontinuity are clearly in the power-law region and this should introduce an error in the evaluation of the shear stresses. Secondly, even at a Capillary number around one, no discontinuities were observed in the case of zero surfactant for the 0.075 and 0.10 wt.% PAA solutions. This indicates that the Capillary number by itself cannot be used to determine the occurrence of a jump, however, it could indicate the region in the velocity – volume curve where it is most likely to occur. Rodrigue et al. [7] pointed out that the presence of surface active agents leads to a variation of surface forces and this is a possible cause for the discontinuity. It is necessary to introduce another parameter to include this effect. This can be done via the Marangoni number (Ma) which relates surface tension gradient stresses and viscous stresses in the following way: Table 2 Critical parameters before and after the jump PAA solution

SDS added

Volume

Velocity jump

Capillary

Mach

a

(wt.%)

(ppm)

(mL)

(cm s−1)

(−)

(−)

(−)

0.075a

30 300 50 100 200 300 0 30 300 0 30 300 0 7 30 100 300 0 0

50 65 25 30 30 38 25 30 35 25 30 30 25 25 30 35 35 101 103

11 – 16 12 – 19 1.7 – 4.9 1.6 – 5.1 1.8 – 5.6 3.0 – 6.6 1.2 – 2.3 1.1 – 3.1 1.3 – 4.1 0.87 – 2.2 0.68 – 2.5 0.68 – 2.2 0.58 – 1.6 0.42 – 1.6 0.42 – 2.0 0.40 – 1.9 0.47 – 1.9 2.23 – 11.2 0.50 – 2.10

0.86 – 1.27 1.14 – 1.77 0.55 – 1.56 0.53 – 1.72 0.68 – 2.17 1.40 – 3.04 0.49 – 0.96 0.54 – 1.54 0.99 – 3.07 0.38 – 0.97 0.32 – 1.18 0.46 – 1.47 0.36 – 1.0 0.27 – 1.0 0.30 – 1.4 0.28 – 1.4 0.42 – 1.6 1.00 – 5.0 1.33 – 5.6

3.93 – 5.51 4.38 – 6.34 0.68 – 1.89 0.62 – 1.77 0.69 – 1.92 1.12 – 2.21 0.47 – 0.85 0.43 – 1.12 0.51 – 1.42 0.34 – 0.80 0.28 – 0.90 0.28 – 0.80 0.22 – 0.58 0.17 – 0.57 0.16 – 0.63 0.19 – 0.67 0.21 – 0.66 0.56 – 2.34 0.11 – 0.39

0.87 – 1.93 0.72 – 1.75 0.39 – 2.53 0.26 – 2.17 0.21 – 1.73 0.47 – 1.91 1.10 – 3.71 0.25 – 1.73 0.22 – 1.78 0.67 – 3.92 0.21 – 2.39 0.10 – 0.86 0.33 – 2.28 0.18 – 2.11 0.10 – 1.41 0.12 – 1.58 0.12 – 1.13 0.37 – 5.71 0.83 – 7.28

0.10a

0.15a

0.20a

0.25a

0.5b 1.0b

a

Data from [7]; b data from [8].

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

Ma =

Ds/R Ds/R = , t mg; n

53

(15)

where Ds is the difference between the surface tension of the pure solvent and the surface tension of the solution (solvent+ Polymer+surfactant) and g; is defined by Eq. (10). Rodrigue et al. [7] and Asnacios et al. [25] showed that polymer molecules can act as surfactants. By introducing this parameter, it was found that the jump discontinuity occurs when the Capillary number was of the same order of magnitude as the ratio of the Marangoni number to the Deborah number. We now propose a criterion which is dependant on a parameter a for the appearance of the discontinuity, where a=

Ca De . Ma

(16)

From the definition of a, we note that a will suffer a jump when there is a jump in velocity. It is observed that the discontinuity in velocity occurs as a tends to unity. Prior to the jump, aB 1 and after the jump, a \ 1. The parameter a represents the balance between the forces acting on the bubble, namely elastic, viscous, surface tension and surface tension gradients. Table 2 lists the critical values (before and after the jump) of the viscoelastic Mach number (M), the Capillary number (Ca) and for the proposed a criterion. The errors in one calculation of a can be due to experimental error associated mainly with surface tension measurements. These were measured using a Fisher Autotensiomat at the lowest measuring speed (1 mm min − 1) using the Du Nouy ring method. The error on s is 0.5 mN m − 1 [26]. Since a is inversely proportional to the product of s and Ds, great care should be used when measuring this parameter. Furthermore, the average rate of deformation is not exactly U/R and the bubble might not be exactly spherical. These will also introduce some error. As mentioned earlier, there is a jump discontinuity in a after the transition. We recall that a depends on both R and U. When measuring the velocity–volume curve experimentally, there is an increment in the radius of bubble tested and this can cause a relative increase in a. However, the major difference comes from the height of the velocity jump itself. As shown in the past [20], this height can be as high as 7. The parameter a depends on U 1 + b. For our data, b is around 0.8 and a is proportional to U 1.8. A difference in velocities of 3.5 (average in our case) will give a difference of 10 in a. Therefore, obtaining values of a near unity can be difficult, within experimental error. Nevertheless, from all of the results obtained so far, the jump may be physically interpreted as follows. At the rear stagnation point, the local strains are large and cause high curvature and local deformation. In this region, high normal stresses develop. Polymer and/or surfactant molecules are stretched and induce a change in the fluid properties. The jump could be the result of the action of normal forces in the vicinity of the bubble, removing molecules from the bubble surface, or causing a sudden change in interfacial conditions. Noh et al. [27] observed that the major contribution to the bubble deformation near the rear stagnation point is due to the uniaxially straining flow dominating there which causes the stretching of the molecules. Several papers studying the motion of gas bubbles in polyacrylamide particularly, did not observe a discontinuity in the velocity–volume curves and led to doubts concerning the existence of this observation. For example Carreau et al. [28] use 0.5 wt.% MG-700 wt.% glycerol/water mixture. No discontinuity was found in this case. It is believed that, if surface tension effects

54

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

were present, the volumes of the bubbles were too large to exhibit the phenomenon. Their smallest bubble volume was 100 mL and was probably just above the critical volume. De Kee et al. [5] showed that observed critical volumes examined so far were in the range 10–125 mL, depending on the polymer, the polymer concentration and the solvent.

6. Conclusions Experimental data regarding the abrupt discontinuity in the velocity–volume curves of the gas bubbles rising freely in viscoelastic fluids are reviewed. A new criterion is developed based on a ‘modified’ Capillary number, which takes into account the various characteristic forces in competition in this kind of flow. The transition from the Strokes to the Hadamard regime occurs at a Reynolds number near unity and this indicates that inertia could play a significant role. This might lead to a different type of governing equation when the viscoelastic Mach number (M) is of the order of unity. This results in a change in the way information travels through the fluid and drastically modifies the momentum and mass transfer at the gas–liquid interface, as both are coupled. The change in the type of flow greatly influences the magnitude of the discontinuity. As explained in Zana and Leal [20], the transition generates two fundamentally different kinds of dominating flow. Prior to the jump, the bubble interface is almost completely rigid and the flow is shear dominated. In contrast, following the jump, the interface is almost free and the flow is dominated by compressional and extensional stresses. This type of change should have a great influence on the rheology of the flow. Variables such as secondary normal stress differences and extensional viscosity should be investigated more carefully. No data on these is available in any of the previous studies. Nevertheless, experimental and theoretical contributions are needed now to gain more insight into this phenomenon. Finally, the negative wake behind a bubble first observed by Hassager [29], should be given some attention, as it is believed to be caused by extensional forces, which tend to stretch the rear of the particle as mentioned earlier.

Acknowledgements The authors wish to acknowledge financial support from the Natural Science and Engineering Research Council of Canada (NSERC).

References [1] R.A. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, vol. 1, Fluid Mechanics, Wiley, New York, 1987. [2] H.A. Barnes, J.F. Hutton, K. Walters, An Introduction to Rheology, Rheology series 3, Elsevier, Amsterdam, 1989. [3] P.J. Carreau, D. De Kee, R.P. Chhabra, Polymer Rheology: Principles and Applications, Hanser, Munich, 1997.

D. Rodrigue et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 45–55

55

[4] G. Astarita, G. Apuzzo, AIChE J. 11 (1965) 815 – 820. [5] D.De Kee, R.P. Chhabra, D. Rodrigue, in: N.P., P.N. Cheremisinoff (Eds.), Handbook of Applied Polymer Processing Technology, Marcel Dekker, New York, 1996, pp. 87 – 123. [6] D. Rodrigue, Ph.d. thesis, University of Sherbrooke, 1996. [7] D. Rodrigue, D. De Kee, C.F. Chan Man Fong, J. Non-Newtonian Fluid Mech. 66 (1996) 213 – 232. [8] L.G. Leal, J. Skoog, A. Acrivos, Can. J. Chem. Eng. 49 (1971) 497 – 500. [9] R. Clift, J.R. Grace, M.E. Weber, Bubbles, Drops and Particles, Academic Press, New York, 1978. [10] R.H. Perry, D. Green (Eds.), Perry’s Chemical Engineers Handbook, 6th ed., McGraw-Hill, New York, 1984. [11] T. Maxworthy, C. Gnann, M. Ku¨rten, F. Durst, J. Fluid Mech. 321 (1996) 421 – 441. [12] R.P. Chhabra, Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Press, Boca Raton, FL, 1993. [13] R.P. Chhabra, C. Tiu, P.H.T. Uhlherr, Rheology 2 (1980) 9 – 16. [14] D. Gu, R.I. Tanner, J. Non-Newtonian Fluid Mech. 17 (1985) 1 – 13. [15] Y. Tomita, Bull. JSME 2 (1959) 469–474. [16] G.C. Wallick, J.G. Savins, D.R. Arterburn, Phys. Fluids 5 (1962) 367 – 369. [17] D. Rodrigue, C.F. Chan Man Fong, D. De Kee, 1997 (unpublished). [18] M.T. Arigo, G.H. McKinley, J. Rheol. 41 (1997) 103 – 128. [19] A. Acharya, R.A. Mashelkar, J. Ulbrecht, Chem. Eng. Sci. 32 (1977) 863 – 872. [20] E. Zana, L.G. Leal, Int. J. Multiph. Flow 4 (1978) 237 – 262. [21] M.W. Haque, K.D.P. Nigam, K. Viswanathan, J.B. Joshi, Chem. Eng. Commun. 73 (1988) 31 – 42. [22] L.E. Becker, G.H. McKinley, H.A Stone, J. Non-Newtonian Fluid Mech. 63 (1996) 210 – 233. [23] W.N. Bond, D.A. Newton, Phil. Mag. 5 (1928) 794 – 800. [24] Y.J. Liu, T.Y. Liao, D.D. Joseph, J. Fluid Mech. 304 (1995) 321 – 342. [25] A. Asnacios, D. Langevin, J.F. Arguillier, Macromolecules 29 (1996) 7412 – 7417. [26] D. Rodrigue, D. De Kee, C.F. Chan Man Fong, Can J. Chem. Eng. 75 (1997) 794 – 796. [27] D.S. Noh, I.S. Kang, L.G. Leal, Phys. Fluids A5 (1993) 1315 – 1332. [28] P.J. Carreau, M. Devic, M. Kapellas, Rheol. Acta 13 (1974) 477 – 489. [29] O. Hassager, Nature 279 (1979) 402–403.

.