7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The Rise Velocity Jump Discontinuity
of Bubbles in Viscoelastic Liquids
Gtnter Brenn and Christian Pilz
Graz University of Technology, Institute of Fluid Mechanics and Heat Transfer
Inffeldgasse 25/F, Graz, A8010, Austria
brenn@fluidmech.tugraz.ac.at
Keywords: Bubble rise velocity, viscoelastic liquids, jump discontinuity
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
bubbles 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 nondimensional group H, found by dimensional analysis, exceeds an
empirically found critical value. A universal correlation of nondimensional numbers for the nondimensional
critical bubble volume at the jump discontinuity was found. The nondimensional numbers represent the relevant
theological 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. Further, a correlation for the rise velocities of the bubbles is presented.
Nomenclature
Archimedes number ()
sphere radius of a bubble (m)
Bond number ()
capillary number ()
polymer concentration (g/cm3)
Deborah number ()
sphere diameter of a bubble,
filament diameter (m)
EOtvos number ()
Froude number ()
gravitational acceleration (m/s2)
constant in Carreau model (s)
length scale (m)
molecular mass (kg/kmol)
Marangoni number ()
Morton number ()
exponent in the relaxation time / mass fraction
relation ()
exponent in Carreau model ()
position vector (mm)
coefficient of determination ()
mass fraction ratio ()
t
U
V
w
x, Y, Z
X,Y,Z
Greek lett
a, 3
[r]
0
A
M
time (s)
bubble rise velocity (m/s)
bubble volume (mm3)
polymer mass fraction ()
global Cartesian coordinates (mm)
Cartesian coordinates (m)
ers
fit parameters ()
shear rate (1/s)
strain rate (1/s)
shear viscosity (Ns/m2)
intrinsic viscosity (cm3/g)
time scale (s)
relaxation time (s)
H dimensionless group (=
p
a
T
W7 p, X
g3/4 AE1P/4/1/4)
density (kg/m3)
surface tension (J/m2)
transport time (s)
local cylindrical coordinates (m, rad, m)
Subscripts
0 1st Newtonian plateau
initial
0, a 1st Newtonian plateau of aqueous solution
a aqueous solution
C centroid coordinates
c critical
cap capillary
conv convective
E elongational
f filament
1 liquid phase
M model
min minimum
red reduced
SC spherical cap
T terminal
oc 2nd Newtonian plateau
Introduction
For applications in biotechnology, bioprocess engineer
ing, and others it is of interest to know the rise be
haviour, 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 ris
ing in quiescent viscoelastic liquids may exhibit a rise
velocity jump discontinuity, once their volume exceeds
a critical value Astarita & Apuzzo (1965). This dis
continuity 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. As
tarita & Apuzzo (1965) suggested, that the sudden in
crease of the bubble rise velocity within a small range
of volume change represents a change in the bound
ary conditions at the bubble surface form rigid to free,
which is equivalent to the transition from the Stokes to
the HadamardRybczinsky regime in a Newtonian fluid.
Calderbank et al. (1970) investigated the shape, motion,
and mass transfer of single carbon dioxide bubbles in
an aqueous polyethylene oxide solution. Comparison
of the measured mass transfer coefficient with theoret
ical models for rigid spheres as well as for circulating
spheres in creeping flow provided by Levich (1962) in
the respective regions before and after the discontinuity
confirmed the lip'olh',sis of Astarita and Apuzzo. Ex
periments with glass spheres moving in viscoelastic liq
uids by Leal et al. (1971) showed no discontinuity in the
velocityvolume plot and therefore also confirmed the
suggestion of Astarita and Apuzzo. Furthermore an in
vestigation of the contribution of shearthinning effects
on the rise velocity after the discontinuity by a numer
ical analysis of the creeping flow equations, neglecting
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
elastic effects, indicated that only a fraction of the ex
perimentally observed velocity jump is due to the shear
dependence of the viscosity. Acharya et al. (1977) sug
gested that polymer molecules might act as surfactants,
generating the surface stresses which oppose the circu
latory motion within the gas bubble, and that the partial
cleansing of the surface responsible for the rapid veloc
ity change is likely to occur far 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 P= ., Z 1 (1)
(critical bubble radius ac, gravitational acceleration g,
liquid density pi, and surface tension a against the
gas in the bubble). This is an inappropriate criterion
for predicting the discontinuity, since even for Newto
nian fluids Bo can be unity and yet no discontinuity
has ever been observed in Newtonian fluids. Zana &
Leal (1978) investigated the dynamics and dissolution
of air and carbon dioxide bubbles in viscoelastic liq
uids and observed a discontinuous increase of the rise
velocity with the bubble volume only in case of non
dissolving (constantvolume) air bubbles. For the dis
solving (varyingvolume) 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 constantvolume and shrinking
bubbles. Liu et al. (1995) studied the cusp at the rear
pole of bubbles rising in viscoelastic liquids. They ob
served cusp formation ("teardrop like" bubble shape)
and the associated increase of the rise velocity near a
critical capillary number
Ca 1 (2)
The capillary number expresses the balance between vis
cous forces and surface tension forces; elastic forces
are not included. Liu and coworkers Liu et al. (1995)
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. (1986), De Kee et al. (1990)
and Rodrigue & Blanchet (2002), and also in the present
work, where despite a clearly visible cusp at the rear
stagnation point in some cases no jump appeared. Ex
tensive investigations on the rise velocity jump discon
tinuity of bubbles in polymer solutions can be found in
Rodrigue et al. (1996), Rodrigue et al. (1998), Rodrigue
& De Kee (1999), Rodrigue & De Kee (2000), Rodrigue
& Blanchet (2002), Rodrigue & Blanchet (2004), which
represent the recent developments on this phenomenon.
The latter papers also include investigations on the ef
fect of surfaceactive agents in the liquid phase. Ro
drigue et al. (1998) suggested that, for an appropriate
jump criterion, elastic forces and shear thinning (vari
able viscosity) must be taken into account beside vis
cous forces, surface tension, and gravity forces appear
ing in the purely Newtonian case. Moreover, surface ten
sion gradient (Marangoni) forces can also be involved
when surfaceactive agents are present. In the paper
of Rodrigue et al. (1996) it is pointed out that poly
mer 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 gasliquid
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 in
terfacial and theological character. According to this hy
pothesis, Rodrigue et al. (1998) gave a physical inter
pretation 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 in
duce 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 & Blanchet (2002) it is shown
that the jump can be eliminated by using surfactant con
centrations 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 gasliquid in
terface. In the recent publication Rodrigue & Blanchet
(2004), the two authors presented the correlation
De
e = 0.0181 Ca5/3 (3)
Ma
as a demarcation line between experimental data below
and above the jump. The data were published in the liter
ature by Rodrigue's and another group around Leal and
Acrivos. In this relation, the viscous, elastic, surface ten
sion, 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. The authors suggest that Eq. (3) can be
used as a criterion for the jump discontinuity. Both the
bubble rise velocity UT and the volumeequivalent bub
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ble radius a occur in the characteristic numbers, since
all dimensionless groups appearing in the correlation in
clude an average shear rate j UT/a. From the cor
relation, 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.
Experimental Facility
The experiments of the present work were carried out
with the setup presented in Fig. 1. It consists of a test
column made of glass with an inner cross section of
120 x 120 mm2 within an aluminum frame and a plexi
glass bubblegenerating chamber attached to its bottom
in which the bubbles were produced, similar to the ex
perimental setup described in Liu et al. (1995). All alu
Light source
H / Microliter syringe
Hemispherical spoon
Capillary 00.09mm
Figure 1: Experimental setup.
minum 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. 1). Experiments were started when gas bubbles
in the fresh liquid had disappeared. In order to gener
ate 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
the pressure difference generated by the suction, air en
ters the chamber in discrete portions of about 13 mm3
through the stainless steel capillary with an inner diame
ter of 0.09 mm at the bottom of the chamber ("equal vol
ume" method Liu et al. (1995)). The reproducibility of
the single bubble formation under such quasistatic con
ditions with the given capillary diameter of 0.09 mm was
confirmed by means of measurements based on image
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
processing, which will be described below, and compar
ison with theoretical predictions by Mersmann for dem
ineralized water (see Brauer (1971) p. 278). For varying
the bubble volume, a well defined number of bubbles
with volumes ranging from 1mm3 to 3 mm3, depend
ing on the surface tension of the liquid, 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 De Kee et al. (1986), 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 re
sistance for the subsequent bubbles. The initial isotropic
configuration is recovered through molecular diffusion,
which requires a long period of time. Rodrigue et al.
(1996) reported injection periods as high as 300 s for
some polymer solutions. In an earlier paper, De Kee et
al. (1986) reported injection periods up to 600 s. How
ever, plots of the drag coefficient against the Reynolds
number including the injection period as a parameter
published by Carreau et al. (1974) show that injection
periods between 600 s and 3600 s have very little influ
ence on the drag curve, while a strong dependency on
injection periods between 4.7 s and 600 s is clearly visi
ble. Therefore, in the present experiments the time inter
vals 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 con
nected to a personal computer (PC) via FireWire. The
camera provides images at a max. framing rate of 30 Hz.
The camera position approximately 350mm above the
bottom of the column ensures that the bubble rises with
its terminal velocity, which was confirmed, as an exam
ple, by measuring the bubble rise velocity in an 0.8 wt.%
aqueous Praestol 2500 solution at different height posi
tions (250 mm, 300 mm, 350 mm and 400 mm above the
base plate). Results of these measurements confirm that
both the velocities and the detected critical bubble vol
ume are independent on the position of measurement in
the containment of our setup (see Fig. 2). This means
that in this region, the bubbles really move at their ter
minal rise velocity. We may also state that a slight in
crease 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 mo
tion (a height difference of 450 mm would result in a rel
ative isothermal change in the bubble volume of approx
imately 4 %, which is below the accuracy of our sizing
technique described in the following section).
V [mm3]
Figure 2: Terminal bubble rise velocity versus bubble
volume for an aqueous 0.8 wt.% P2500 so
lution measured at different height positions
above the column bottom.
A mercury lamp as the light source, together with a
frosted glass plate, provided diffuse light for high con
trast between air bubble and liquid, and enabled the de
termination of the bubble volume as well as the bubble
rise velocity by means of image processing simultane
ously. A typical image of an air bubble rising in a vis
coelastic liquid of the present work is shown in Fig. 3.
The experiments were carried out at room temperature
Figure 3: Image of an air bubble rising in an aqueous
0.3 wt.% P2540 solution.
(w 20 C). Measurements of the liquid temperature re
vealed temperature fluctuations of max. 1 "C through
out 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 sur
factants 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 bub
ble 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 liq
uid throughout the experiments. Moreover we avoided
the use of detergents containing surfactants for cleaning.
#'A
j 400mm
* a 350mm (regular position)
300mm
A 250mm
Instead tap water and demineralized water were used for
cleaning and rinsing.
Bubble shape as well as bubble position relative to
the picture frame were extracted from each picture us
ing 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
A xx,
c
Figure 4: Determination of the volume and the rise ve
locity of the air bubble.
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 merid
ian 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) coordi
nate system 0o : (X, Y, Z) defined in Fig. 4, with the
X axis pointing upward, the Z axis pointing to the right
in the image plane, and the Y axis (not shown) point
ing towards the camera. Furthermore, the coordinates of
the bubble contour points were transformed into a local
(movable) coordinate system E : (w, p, x) with (w, x)
representing the principal axes of the bubble's merid
ian area computed via image processing tools of Matlab.
Since for the volume range of interest rotational symme
try around the direction of motion of the bubbles can be
assumed (see Fig. 3), and furthermore the bubbles rise
along a vertical path, the axis of symmetry (denoted x in
Fig. 4) points in the direction of the X axis of the global
coordinate system 0o which describes the direction of
motion. The volume of the air bubble is obtained from
an extraction and approximation of the meridian curve
& : F(w7, ) =0, (4)
in the local coordinate system E. The terminal bubble
rise velocity UT is calculated from the centroid coordi
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
nates Rc of two consecutive bubble images within the
fixed frame oE and the frame rate.
Materials Used
For our experiments we used Praestol 2500 (a linear
polyacrylamide, degree of hydrolysis 34 %) with a
molecular mass of M z 15 20 x 106 kg/kmol, and
polyethylene oxide (PEO) with a molecular mass of
M 8 x 106 kg/kmol as nonionic polymers. The lin
ear polyacrylamide Praestol 2540 (middle anionic, de
gree of hydrolysis 40 %, molecular mass M 15 
20 x 106 kg/kmol) was used as an anionic polymer.
The Praestols (PAMs) were produced by Stockhausen
Inc., Germany. The polyethylene oxide was provided by
SigmaAldrich Chemical Company, Inc. All data about
molecular masses and degrees of hydrolysis of the poly
mers were provided by these companies. All polymer
solutions were prepared in pure deionized water, in so
lutions of glycerol in deionized water, or in ethylene
glycol. The conductance of the deionized water pro
vided by a reverse osmotic equipment was between 4
and 5 pS. Glycerol and ethylene glycol were provided
by Carl Roth GmbH & Co. The data of all polymer so
lutions investigated are listed in Table 1 together with
their material properties and the critical bubble volume
V, where the jump discontinuity (if existing) was ob
served.
According to the nomogram presented in Kulicke (1986)
(p. 210, Fig. 5.20) which 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.
Liquid Characterization
The polymer solutions were prepared using an anchor
stirrer with a diameter of approximately 70mm operat
ing at speeds lower than 300rpm inside a glass beaker
(inner diameter z 160 mm) to avoid mechanical degra
dation of the polymer chains. Concentrations that en
sured the existence of a jump discontinuity of the bub
ble rise velocity ranged above 0.1 % weight for aqueous
solutions. Variations of the solvent (70:30wt.% glyc
erol:water, ethylene glycol) showed, however, that the
jump discontinuity can occur at by far lower concentra
tions of the same polymer (see Table 1). The solutions
were thoroughly characterized by means of shear and
elongational rheometry. The reason for looking at the
elongational behaviour of the solutions was our hypoth
esis that the relaxation of stresses in elongational flow
g 0 N Ot Cfl
I  n   n
m r, In co c m
rs S. %n. SiI Vi M
a a N
t 00V 0000000
n 0 0 0 0 0 0 0 0 0
v u (if ex isig tt
o aamm
3t 
Table 1: Material properties of all polymer solutions un
volumes (if existing).
t t
volumes (if existing).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
could play an important role in the physics of the rise
velocity jump discontinuity. It was already noted in the
work of Rodrigue & Blanchet (2002), and in Rodrigue
et al. (1998), that care should be given to the possible ef
fect of elongational viscosity of viscoelastic liquids for
further investigations. As shown in the paper of Stelter
et al. (2000), the transient elongational viscosity IE (t)
and the relaxation time AE in uniaxial elongational flow
are related according to
3ur
TIE( W)3 AE (5)
df(t)
where df is a filament diameter. The measurements of
the relaxation time AE were carried out by putting a
droplet with a volume of 10 pl via micropipette on the
lower, fixed plate of the elongational rheometer shown
in Fig. 5. The upper, movable plate is then brought into
solenoid
d orifice photo detector
laser diode movable plate
liquid filament 
 fixed plate *
Figure 5: Sketch of the elongational rheometer.
contact with the droplet, so that both plates are wetted by
the liquid. Then the movable plate is quickly pulled up
wards by a solenoid in order to produce a selfthinning,
cylindrical liquid filament of constant length on a time
scale much shorter than the relaxation time of the liq
uid. Since the filament is very thin, the motion is domi
nated by capillary forces (inertial and gravity forces are
negligible). 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 rela
tionship between the reduction of the light power and the
filament diameter is linear and obtained by a calibration
procedure. A detailed description of the measuring tech
nique is given in Stelter et al. (2000), where it is shown
that the filament thinning leads to an elongational flow
with a constant straining rate i. There is no possibility
to vary the straining rate arbitrarily, since the relaxation
time AE is a material property of the liquid tested. There
fore the Deborah number De = AE = 2/3 is constant
for all measurements, since the liquid is allowed to select
its own time scale. The decrease of the filament diameter
for viscoelastic liquids can be approximated by
df df, exp ( (6)
3 AE/
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
as follows from a theoretical consideration in Stelter et
al. (2000). The exponent of the function obtained by a
data fit determines the relaxation time. Typical results
of the elongational characterization are shown in Fig.
6. The reproducibility is given by a peak standard de
tO'
10
100
102
102
103
102 101
w [wt.%]
t [1/s]
* P2540 Figure 7: Flow curves for aqueous P2500 solutions of
 aE w8 various polymer mass fractions w.
. 0. o w 88
10 10' produced by Carreau (see Barnes et al. (1989), p. 18)
100 10' produced by Carreau (see Barnes et at (1989), p. 18)
Figure: Relaxation time versus
aqueous PAM solutions.
mass fraction for
aviation 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 scaling law
AE c Y m (7)
Comparison of the exponent m with the corresponding
measurements reported in Stelter et al. (2000) using the
same polymer solutions shows good agreement with the
present results and confirms the reliability of the mea
suring technique.
Flow curves as well as storage and loss moduli of the
various solutions were measured with the rotational vis
cosimeter Paar Physica UDS 200 in configuration MS
KP 25, which denotes a coneandplate device accord
ing to DIN 53018. For solutions exhibiting viscosities
lower than 0.02 Ns/m2, 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 Fig. 7 as an example. Similar re
sults are obtained for the other liquids used in the present
work. The flow curves (shear viscosity 9 vs. shear rate
7 see Fig. 7) exhibit shear thinning behaviour increas
ing with polymer concentration. For small shear rates,
a constant value, the so called zero shear viscosity 0o,
is reached, as expected for polymer solutions. The flow
curves can be approximated by the empirical model in
n no 1 m / (
(8)
o0 1 [1+(Ki )2] m/2
using the software provided by Paar Physica. The val
ues of the zero shear viscosity (= 1st Newtonian plateau)
were obtained from the data fit procedure.
Besides the theological properties, the surface tension a
against the gaseous phase and the density of the liquids,
pi, play an important role in bubble dynamics. The sur
face tension was measured by means of the dropvolume
tensiometer Lauda TVT 2. The density was measured
with the oscillating Utube density meter DMA 45 by
Anton Paar with an accuracy of 0.1 kg/m3. The values
of the surface tension presented in Fig. 8 are determined
with a standard deviation less than 1 %.
0.2 0.4 0.6
w [%]
0.8 1 1.2
Figure 8: Surface tension versus polymer mass fraction.
Since different solvents were used for preparing Praestol
2500 solutions, measurements of the intrinsic viscos
* *
U .
P2500 in water
P2540 in water
PEO in water
A P2500 in glycerolwater
v P2500 in ethylene glycol
ity [r] were carried out by means of an Ubbelohde vis
cosimeter at a controlled temperature of 20 C in order
to obtain a quantity representing the quality of the sol
vent. The measured intrinsic viscosities depicted in Fig.
9 show a clearly visible dependency on the quality of the
solvents.
2500 r
0 0.2 0.4 0.6
c [g/ml]
0.8 1 1.2
x 103
Figure 9: Intrinsic viscosities of P2500 solutions pre
pared with various solvents.
Results and Discussion
Figures 10 14 depict data of the terminal rise velocity
UT versus bubble volume V and bubble shapes for aque
ous solutions of various polymer concentrations of three
different polymers. For solutions of the nonionic poly
acrylamide Praestol 2500 (Fig. 10) a clearly identifiable
critical bubble volume exists, where the rise velocity in
creases 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 bub
ble volume. Moreover, Fig. 10 shows that a sufficient
100
10
101 102
v[mm3]
Figure 10: Terminal bubble rise velocity versus bubble
volume for aqueous P2500 solutions.
* P2500 water
* P2500 glycerol water
* P2500 ethylene glycol
[q] =3209.4 ml/g
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
polymer concentration is necessary for the occurrence
of a jump discontinuity, since experiments carried out at
lower mass fractions (w = 0.1 wt.% and w = 0.3 wt.%)
did not reveal a discontinuity. Instead, a continuous vari
ation in the velocityvolume relation was observed, as
one would expect for nonelastic shear thinning liquids
(see Brauer (1971) p. 303). The evolution of the bubble
shape as a consequence of the discontinuous enhance
ment of the terminal rise velocity for the w 0.8 wt.%
and w = 1.0 wt.% solutions is found to be similar to the
observations reported in the literature (for example As
tarita & Apuzzo (1965) and Calderbank et al. (1970)).
Figure 1 l(b) shows air bubbles below and one above the
(a)
(b)
Figure 11: Bubble shapes observed in aqueous P2500
solutions of different polymer mass fraction
(a) w 0.3 wt.% (no jump), (b) w
0.8 wt.% (jump occurs between the bubbles
left and right to the black line).
critical volume of 45.98 mm3 separated with a black line
for the aqueous 0.8 wt.% solution of P2500. The im
age of the subcritical air bubble has an entirely convex
boundary curve, while the supercritical bubble shows
a clearly visible, different surface shape with a cusp at
the rear stagnation point (similar to a "teardrop"). Fur
thermore, the curvature at the rear stagnation point of
the subcritical bubble is finite. The bubbles in solutions
without velocity jump also undergo a transition in the
bubble shape, as can be seen in Fig. 1 (a). Therefore the
appearance of a cusp at the rear stagnation point does not
ensure the appearance of a velocity jump. This is con
sistent with the observations by Rodrigue & Blanchet
(2002).
The aqueous solutions of the anionic polyacrylamide
Praestol 2540 exhibit slightly different transition be
haviour, as shown in Fig. 12. For a mass fraction of
0.05wt.%, no jump discontinuity was observed. The
0.1 wt.%, as well as the 0.3 wt.% solutions show a dis
continuity in the velocity volume plot, though the tran
sition does not appear as abrupt as in the case of the
aqueous Praestol 2500 solutions. Rodrigue & Blanchet
(2002) obtained a similar transition behaviour by in
[q] 1350.7 mUg
[] = 315.06 ml/g
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
U^
40, /0
..
S* w = 0.05%
w=0.1%
w =0.3%
V[mm3]
Figure 12: Terminal bubble rise velocity versus bubble
volume for aqueous P2540 solutions.
creasing the concentration of a surfactant added to the
polymer solution in order to eliminate the discontinuity
(see Fig. 5 in Rodrigue & Blanchet (2002)), 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 Rodrigue & Blanchet (2002), Ro
drigue et al. (1996), Rodrigue et al. (1998), Rodrigue
& De Kee (1999), Rodrigue & De Kee (2000) and Ro
drigue & Blanchet (2004), since the presence of surfac
tants leads to additional (Marangoni) forces at the gas
liquid interface. Moreover, the surface tension of the
polymer solutions investigated in those papers exhibit
ing a jump discontinuity is by far lower than the sur
face tension of the solvent, while the surface tension
of the liquids used in the experiments of the present
work does not differ much from the surface tension of
the solvents (see Fig. 8). It can therefore be assumed
that Marangoni forces play no role in the present inves
tigations. It should be pointed out that the rapid ve
locity 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 (Doi & Edwards (1986), p. 289).
The third family of polymer solutions investigated are
aqueous solutions of polyethylene oxide (PEO), which
can be regarded as nonionic. The bubble rise behaviour
for varying bubble volume depicted in Fig. 13 is there
fore 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 vis
cosity goes along with a higher elongational viscosity,
Figure 13: Terminal bubble rise velocity versus bubble
volume for aqueous PEO solutions.
which results in higher values of the relaxation time ac
cording to Eq. (5) (see Fig. 6). No discontinuity was
observed for the PEO mass fraction of 0.2 wt.%.
In the next stage of our investigations we looked at the
rise behaviour of air bubbles in aqueous polymer solu
tions prepared with mixtures of flexible and rigid rod
like polyacrylamides with constant total polymer mass
fraction of 0.3 wt.% and varying mixture ratio (see Ta
ble 1), similar to the experiments reported in Stelter et al.
(2002). The mass fraction of 0.3 wt.% 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 poly
mer solution more "flexible" and reduces the relaxation
time, as reported in the work of Stelter et al. (2002). Un
fortunately 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 in
vestigations with the polymer mixtures were impossible
for solutions with Praestol 2500 mass fractions greater
than 0.225 wt.%. The volumerise velocity curves are
plotted in Fig. 14. 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 crit
ical bubble volume increases again, which goes along
with a decrease of the velocity increase factor. The
abruptness of the velocity change is enhanced with in
creasing mass fraction of Praestol 2500, which corre
sponds to the observations made for flexible polymer
solutions. The observations confirm our expectation that
the interaction of rigid polymers with flexible ones may
10'
10
*
40
.m. w= 0.2%
S= 0.5%
Sw =0.8%
V[mm3]
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
E 102
o10
10
V [m3]
v [mm3]
Figure 14: Terminal bubble rise velocity versus bubble
volume for aqueous solutions of P2500 and
P2540 with the given mass fraction ratios
r = WP2500 : WP2540*
cause a velocity discontinuity due to the increase of the
relaxation time in elongational flow.
Since it is evident that the conformation of the poly
mer molecules in the solvent affects macroscopic
theological properties of the solutions Stelter et al.
(2002), we were also interested in the influence of the
solvent on the bubble rise behaviour. We decided to
use a glycerolwater mixture (70:30 wt.%) and ethylene
glycol, both denoted as nonaqueous 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. (2002). While the glycerolwater
mixture can be regarded as a "good" solvent, ethylene
glycol is known as a "poor" one (see Kulicke (1986) and
Klein & Conrad (1980)). It can therefore be assumed
that the conformation of the polymer chains in ethylene
glycol is quite close to the state, where the polymers
behave like ideal chains Doi & Edwards (1986). The
bubble rise velocity is shown in Fig. 15 as a function
of the bubble volume for the nonaqueous solutions
of Praestol 2500. A velocity discontinuity exists in
both systems for solutions of sufficiently high polymer
concentration, 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 nonaqueous 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.
102
10i
v[mm3]
Figure 15: Terminal bubble rise velocity versus bubble
volume (a) for P2500 in 70:30wt.% glyc
erol:water, (b) for P2500 in ethylene glycol.
Dimensional Analysis and Data Regression
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 diameter:
dc
geometry
9 Pl, no, r, AE
process liquid properties
where dc denotes the critical bubble diameter at the
jump, g is the gravitational acceleration chosen as a pro
cess variable, since the flow is driven by buoyancy, pi is
the density of the liquid phase (the density of the gaseous
phase is neglected), and no represents the zero shear vis
cosity of the liquid. The surface tension a is considered
as relevant, since we are dealing with a curved gasliquid
interface. Finally the elongational relaxation time AE
comes in according to our I \l'pI l, cis,
Dimensional analysis according to the method described
in Zlokarnik (2005) converted the relevant parameters
103
102
10'
10o
o 0 oo W
S o0 r = 0.075% : 0.225%
o o o r= 0.15%: 0.15%
a r =0.225% : 0.075%
^Ba&^ '
(a)
4*0
S[ w= 0.025%
Sw=0.1%
(b)
A LAA A
*
A w=0.05%
A w =0.2%
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
into the following set of three nondimensional numbers:
E g p d"
(E
(7
... critical Eitvos number (9)
4
Mo = g0 ... Morton number
Pi (T 3
13/4 1E P/4
I (1/4
... new group
Among all possibilities of combining the parameters,
this set of dimensionless groups was found to represent
the measurement data best, as will be seen later. In order
to include data obtained from experiments carried out
with nonaqueaous solutions it was necessary to extend
the nondimensional groups with the ratio of the zero
shear viscosities and the ratio of the intrinsic viscosities,
,'u, ... ratio of zero shear viscosities
o/0
S ... ratio of intrinsic viscosities
[rI]a
each calculated with data of aqueous and nonaqueous
solutions at the same polymer mass fraction. It should
be noted that the above definitions of the Eitvis and
Morton numbers differ slightly from those given in Clift
et al. (1978), where the two numbers are both defined
with the density difference between continuous and dis
persed phases rather than with the density of the con
tinuous phase as in our above Eqs. (9) and (10). In
our investigations we worked with liquids of 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 pi. There
fore, apart from the difference in the length scale, our
Eitvis number appears equivalent to the Bond number
according to Eq. (1).
With the nondimensional groups of Eqs. (9) (13) we
are able to represent the critical state of the bubbles in
the form
ER Mo, Ho0,a [1]) 0. (14)
Together with the results from the rise velocity measure
ments, 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 AE can be re
lated to the polymer mass fraction (see Eq. (7) and Fig.
6), we choose for an explicit formulation of the function
in Eq. (14) the ansatz
E5, al Moa2 H (Ca [ri \ a0
(15)
where the undetermined parameter 04 should denote a
threshold value for the group H, which must be exceeded
for the velocity discontinuity to occur. Equation (15) de
notes 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
&i describing the problem were determined using a non
linear least squares method provided by Matlab, which
resulted in the following equation for the critical Eitvis
number
(H 9.9381)0.9087
eM 5.4801 Mo03268
X a) 1.2144 ]1.4389
T 0 [T[9]a)
(12) The agreement of Eq. (16) with the measured data for
the various liquids is shown in Fig. 16. The coefficient
1.5 2
E6c,M
Figure 16: Universal representation of the measured
data for the critical bubble volumes.
of determination R2 = 0.9923 indicates an excellent
representation of the critical bubble volume measured
with our test liquids. The correlation Eq. (16) yields
real numbers only in cases where the difference in round
brackets is positive. Therefore H > 9.9381 is required
to have a potential of the liquid for the jump to occur.
This corresponds to our observation that H < 9.9381
only for liquids without jump discontinuity. The param
eter H may therefore be considered as an indicator for
the existence of a jump discontinuity. Because of the
previously mentioned absence of properties of the bub
ble motion, such as the terminal rise velocity, Eq. (16)
may be attractive and useful for predicting the critical
bubble volume in viscoelastic liquids.
Nevertheless, the correlation is a purely empirical rela
tion, and especially the parameter &4 representing the
threshold value Imin required for the occurrence of the
jump discontinuity needs further investigation to inter
pret Eq. (16) properly.
As already noted in the introduction of this paper, the
rise behaviour of air bubbles in viscoelastic liquids may
be of interest for applications in biotechnology and bio
process engineering. For example airlift fermenters re
quire a certain residence time of the gas bubbles in the
liquid phase. We therefore looked for a universal rep
resentation of the rise velocities of air bubbles with sub
critical volumes and also for the limit for large super
critical bubbles. Due to the complexity of the flow situa
tion we applied again Buckingham's n theorem together
with nonlinear data regression. The list of relevant pa
rameters now contains the sphere diameter d and the ter
minal rise velocity UT as the relevant bubble properties.
d UT, g pl, ro, a, AE
geometry process liquid properties
Pilz & Brenn (2007b) presented a correlation which re
sulted from an application of Buckingham's n theorem
to a parameter set similar to the one given here above,
but including the critical bubble diameter do. In order
to describe the subcritical bubble rise velocities includ
ing data of solutions prepared with nonaqueous sol
vents, the parameter set had to be extended with the ra
tio of the zero shear viscosities of the aqueous to non
aqueous solutions at the same polymer mass fractions,
the ratio of the intrinsic viscosities of the nonaqueous
to the aqueous solutions, and the ratio of the Morton
numbers of the nonaqueous to the aqueous solutions.
The resulting expression showed good agreement with
the subcritical experimental data (R2 = 11 *I' II. The
formula, however, appeared rather complicated. Since
a correlation for describing the rise velocities of sub
critical bubbles should also include a Froude number,
since the supercritical bubble rise velocities are repre
sented by Frsc 0.711. Therefore, Buckingham's n
theorem was applied again in order to describe the sub
critical bubble rise velocities with respect to the findings
for supercritical bubble rise.
Among all possibilities of grouping the parameters listed
above, the following set of nondimensional groups,
which can be achieved by arranging the relevant pa
rameters according to dimensional analysis according to
Zlokarnik (2005), turned out to represent the subcritical
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
o P2O60 in water w =0.8wt%
o P2OO inwaterw =1.Owt%
SP2540 inwater w =O.iwt.
o P240 inwaterw = 0.3wt.%
o PEO inwater w = 0.S6wt
o PEO in water w =0.8wt.%
P2600:P2540 in water w = O.OVwt% : 0 22wt.%
P2o00:P2540 in water w = 0.15wt% : 0.15wt%
P20OO:P2540 inwter w =0.225wt,% : O.O5wt.
o P2500 inQ0:30 wtb. glycerol:waterw= 0.025wt,%
o P200 in O0:30 wt.% glyoerol:wter w=O. iwt
EQP in ethylemn glycol w = 0. 2wt%
Model
....... S limit
10, f)e' = 0.9616
,o2 OR'=o.9
0
104 a... .... . ......
10
104 103 102 10" 100 10'
PrM
Figure 17: Nondimensional representation of the sub
critical bubble velocities in aqueous and
nonaqueous solutions of different polymers
(circles) by the correlation (19) and of the su
percritical velocities (squares) by the spheri
cal cap (SC) limit (20).
bubble rise behaviour best:
SUT
Fr=
ggd
9 pl d2
E6
gd3 Q2
Ar 
AE AE
,conv 1d/g
Froude number
EOtvos number
Archimedes number
ratio of time scales
With these nondimensional groups, again an ansatz was
chosen for the bubble Froude number Fr, which reads
E 31 Ar 2
Frm =6/tAr/ (18)
(1 + AE/con) (1 +4 E Ar)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The undetermined coefficients 3, were again obtained
from nonlinear data regression, yielding the following
correlation
E o0.5230 ArO.6975
FTM 05018 0 5578
(1 + AE/ conv)0.5018 (1 + 2.1859 E Ar)5578
(19)
As far as supercritical bubbles are concerned, As
tarita & Apuzzo (1965) showed that bubbles with vol
umes far greater than the critical one show a rise be
haviour, which can be described by a formula originally
proposed for sphericalcap bubbles, which is given in
Clift et al. (1978) (p. 206, equation (811)) and can be
rearranged into a Froude number of constant value:
UT
Frsc 0.711 (20)
The relation is based on the theoretical consideration of
the flow around a sphericalcap bubble given by Davies
& Taylor (1950), as remarked in Clift et al. (1978).
Experimental investigations revealed an opening angle
of approximately 50 for large sphericalcap bubbles,
which remains unaffected at variation of the bubble vol
ume. Comparison with the present experimental results
also showed good agreement with the spherical cap bub
ble for large bubble volumes.
Figure 17 compares the measured bubble Froude
numbers Fr with the predicted values FrM. The cor
relation given in (19) differs from the one presented by
Pilz & Brenn (2007b). In contrast to that correlation,
the new correlation (19) includes no information about
the solvent and further includes an inelastic limit, i.e.
(AE/ conv) 0. The remaining nondimensional num
bers, i.e. Fr, E5, Ar, are typical parameters describing
the bubble rise in purely viscous fluids.
Characteristics of Sub and Supercritical
Bubbles
The jump discontinuity of the bubble rise velocity in vis
coelastic liquids has two aspects: 1.) It is a matter of
the liquid properties whether the jump discontinuity can
exist at all with a given liquid. The relevant liquid prop
erty is the quantity H g3/4 AE p/4/1/4, which must
exceed the threshold value of 9.9381 for the jump dis
continuity to exist. The threshold value was found em
pirically. 2.) For the state of motion of the bubble to be
supercritical in a liquid that has the potential to exhibit
the discontinuity, the bubble size (or volume) must ex
ceed the critical value given in nondimensional form by
correlation (16).
On aspect 1.) above we pointed out in Pilz & Brenn
(2007b) that the criterion H > 9.9381 represents the ra
tio of two accelerations, i.e. of two massspecific forces
1.4
1.2
1
0.8
S0.6
0.4
0.2
DI
+ LHS
o RHS

xZ
8 6 4 2 0 2
l(mm)
4 6
+ LHS
0 RHS
XE
8 6 4 2 0
l(mm)
2 4 6 8
Figure 18: Time scales for the flow of an aqueous 0.8
wt.% P2500 solution around a (top) subcriti
cal and (bottom) supercritical bubble. In the
supercritical case, the relaxation time is long
compared to the polymer transport time.
which must be large enough for the jump discontinu
ity to occur at a sufficiently large bubble size d d,.
About the aspect 2.), we looked at the Lagrangian trans
port of individual polymer macromolecules down the
bubble contour during the rising motion of the bubble.
In a frame of reference moving with the bubble, the
molecules approach the bubble from above its north pole
in a coiled equilibrium state, which is turned into an
elongated state upon approach of the stagnation point.
From that state, the molecule is transported down the
bubble contour and is subject to further elongational and
shearing stresses, which keeps an elongated state of the
molecule, as long as it is strong flow. The spatial varia
tion of the shearing and elongating flow field, however,
allows the molecule to relax due to Brownian motion and
to return to a coiled state before it is stretched again upon
arrival at the south pole of the bubble. The relaxation
process takes place on the time scale AE. The question
where at the bubble contour the coiled state is reached
again, and, therefore, where stresses set free due to the
molecular relaxation may be felt by the bubble, decides
about the hindering or pushing effect of the stresses on
the bubble motion. Short relaxation times relative to the
transport times of the molecules down the bubble con
tour cause a hindering effect, while long relaxation times
cause the stresses to push the bubble upward. The tran
sition from one state to the other may be quite sharp.
In the following we quantify the Lagrangian macro
molecule transport down the bubble contour in compar
ison with the relaxation times AE of the polymer solu
tions. For being able to do so, we need data on the ve
locity field in the liquid phase around the bubble con
tour. This velocity field was measured by PIV. We do
not present the details of these measurements here, but
give them elsewhere. With the liquid velocity u along
the bubble contour given, we are able to evaluate the in
tegral
T(S) = I d (21)
where T is the time elapsed for a liquid parcel to be
transported from the north pole of the bubble down to
the location I on the contour. Comparing the evolution
of this time scale along the bubble contour with the re
laxation time of the polymer solution, we find for sub
and supercritical states of bubbles in a 0.8 wt.% P2500
solution the results in Fig. 18. It is clearly seen that
the supercritical case is characterized by the long relax
ation time, which supports the above postulated model
mechanism. The same kind of results are found for the
P2540 soultions with jump discontinuity. This result
corresponds to the requirement that a time scale ratio
AE/conv AEUT/d must be large enough to allow for
supercritical motion.
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 theological and capillary
properties of the liquid only. In the development of
the correlation it was found that a thorough theological
and tensiometric characterization of the liquids is of
big importance. In the relevant liquid properties, the
relaxation behaviour of the liquids in straining flows
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
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 velocityvolume relation,
regardless of the polymer properties, and independent
of the quality of the solvents used for preparing the
solutions. 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. Furthermore, nondimensional representations of
the rise velocities for subcritical and far supercritical
bubbles were also developed.
Acknowledgements
Financial support of the present work from the Aus
trian Science Fund (FWF Fonds zur FOrderung der
wissenschaftlichen Forschung) under contract number
P17624N07 is gratefully acknowledged. The authors
wish to thank Professor V. Ribitsch at the Institute of
Chemistry of the University of Graz for making the
shear rheometers of his institute available for the char
acterization of our liquids, H. Katzer for her cooperation
during the shear experiments, and G. Kircher at the Insti
tute of Chemical Engineering and Environmental Tech
nology of Graz University of Technology for measuring
the surface tension of the liquids.
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