International audienceWe study the pinch-off dynamics of droplets of yield stress and shear thinning fluids. To separate the two non-Newtonian effects, we use a yield stress material for which the yield stress can be tuned without changing the shear thinning behavior, and a shear thinning system (without a yield stress) for which the shear thinning can be controlled over a large range, without introducing too much elasticity into the system. We find that the pinch-off remains very similar to that of constant viscosity Newtonian liquids, and consequently thinning in shear flow does not imply a thinning in elongational flow

Amarouchene, Yacine

Aytouna, Mounir

Bonn, Daniel

Eggers, Jens

Moulinet, Sébastien

Paredes, Jose

Shahidzadeh-Bonn, Noushine

Wagner, Christian

English

Archive Ouverte en Sciences de l'Information et de la Communication

HAL Id: hal-00793062
https://hal.archives-ouvertes.fr/hal-00793062
Submitted on 28 Feb 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International
License
Drop Formation in Non-Newtonian Fluids
Mounir Aytouna, Jose Paredes, Noushine Shahidzadeh-Bonn, Sébastien
Moulinet, Christian Wagner, Yacine Amarouchene, Jens Eggers, Daniel Bonn
To cite this version:
Mounir Aytouna, Jose Paredes, Noushine Shahidzadeh-Bonn, Sébastien Moulinet, Christian Wagner,
et al.. Drop Formation in Non-Newtonian Fluids. Physical Review Letters, American Physical Society,
2013, 110 (3), pp.34501. ￿10.1103/PHYSREVLETT.110.034501￿. ￿hal-00793062￿
Drop Formation in Non-Newtonian Fluids
Mounir Aytouna,1 Jose Paredes,1 Noushine Shahidzadeh-Bonn,1 Se´bastien Moulinet,2 Christian Wagner,3
Yacine Amarouchene,4 Jens Eggers,5 and Daniel Bonn1,2
1Soft Matter Group, Van der Waals Zeeman Institute, IoP, Science Park 904, Amsterdam, Netherlands
2Laboratoire de Physique Statistique, ENS, 24 rue Lhomond, 75005 Paris, France
3Experimentalphysik, Universita¨t des Saarlandes, Postfach 151150, 66041 Saarbru¨cken, Germany
4
5
LOMA, Universite´ Bordeaux 1, 351 cours de la Libe´ration, 33405 Talence, France
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom 
We study the pinch off dynamics of droplets of yield stress and shear thinning fluids. To separate the
two non Newtonian effects, we use a yield stress material for which the yield stress can be tuned without
changing the shear thinning behavior, and a shear thinning system (without a yield stress) for which the
shear thinning can be controlled over a large range, without introducing too much elasticity into the
system. We find that the pinch off remains very similar to that of constant viscosity Newtonian liquids,
and consequently thinning in shear flow does not imply a thinning in elongational flow.
PACS numbers: 47.50. d, 47.55.D , 83.50.Jf
The process of drop formation is observed frequently in
everyday life, and its features are well understood for the
case of Newtonian fluids [1]. However, since many fluids
encountered in practice do not have a simple flow behavior,
recently the case of non-Newtonian fluids has attracted
much attention [2–9,9–14]. The main feature that has
been considered so far is the case of solutions of flexible
polymers, which are stretched in the elongational flow near
the pinch point, leading to a spectacular rise of the elonga-
tional viscosity. As a result, the formation of long-lived
fluid filaments provides a striking visual difference between
Newtonian and non-Newtonian breakup [2,15,16].
However, shear-thinning [17] and yield stress [18]
behaviors are at least as frequent features of non-
Newtonian flow as elasticity. How does the thinning in
shear flow affect the breakup dynamics, keeping in mind
that the shear rate encountered during breakup is small in
comparison to the rate of elongation [19]? In recent years,
it has become standard practice in the literature on drop
breakup to postulate the same rheological response for the
extensional flow near breakup [5,6,12] as for the thinning
behavior in shear flow. Other studies have included both a
yield stress and shear thinning, based on the same idea
[7,8]. The main conclusion of the theoretical analyses of
the breakup of shear-thinning fluids has therefore been that
breakup proceeds faster than in the Newtonian case,
increasingly so as the point of breakup is approached, since
the elongation rate strongly increases close to breakup.
In this Letter, we investigate the breakup dynamics for
fluids with well-defined (and nonelastic) non-Newtonian
effects, which allow us to focus either on the shear thinning
or the yield stress behavior. We find that contrary to
previous theoretical analyses, breakup is described
completely by the equations for the breakup of simple
Newtonian fluids, controlled by either a viscous-capillary
or an inertial-capillary balance.
To study the dynamics of drop formation of the complex
fluids in air, we use a high-speed video camera (Phantom
V7 at 12 000 frames=s). The controlled release of the drops
is achieved using a syringe pump to set a low drop emission
rate and a syringe with a needle of diameter d ¼ 1:95 mm.
The rheological measurements for the same fluids, pre-
sented here were done by using a cone-plate geometry on a
Physica MCR 300 rheometer.
As a yield stress system with variable yield stress, we
make emulsions of castor oil droplets dispersed in water
stabilized by 1 wt% sodium dodecyl sulfate; water is the
continuous phase and the sodium dodecyl sulfate and the
castor oil were purchased from Sigma. The drop size
distribution in the emulsion was measured using confocal
microscopy; the average drop diameter is 3:14 m, with a
size polydispersity of 19%. Because of the small size of the
droplets compared to the minimum radius of the neck, we
do not anticipate any inhomogeneity during the thinning;
indeed in the experiments the filament radius remains
smooth. For the rheology of the emulsion, Fig. 1 shows
that the classical Herschel-Bulkley model  ¼ y þ k _n
provides a good description of the data. Varying the vol-
ume fraction, we can make the yield stress y vary over
several orders of magnitude, while the power-law exponent
n of the shear thinning remains remarkably similar and is
always around 0.5.
The evolution of the minimum neck radius (cf. Fig. 2)
was obtained from sequences of rapid camera images; each
experiment was repeated at least 5 times. The thinning of
the neck is usually fitted to a power-law dependence; here
we do the same and fit
Rmin ¼ Ajt0  tj; (1)
 1
where t0 is the breakup time and  is the power-law
exponent. For Newtonian fluids,  is either 23 or 1, depend-
ing on which forces are balanced. If the elongational
viscosity is rate dependent, the expectation is that  equals
the power-law exponent of the thinning:  ¼ n [5,7,8,12].
However, in our filament thinning experiments (cf. Fig. 2),
we find   0:67 for low yield stress emulsions, and unity
for very elastic emulsions (Fig. 3). Here the somewhat
gradual transition is the result of fitting over a limited
scaling range. This suggests that the emulsion behaves
very similarly to Newtonian fluids, where for low viscos-
ities one finds an exponent of 23 (inertia-capillary regime),
while for high viscosities a power of unity is observed
(capillary-inertial-viscous or capillary-viscous regime)
[1]. This picture is confirmed by the fact that at the cross-
over, a typical Reynolds number Re is of order unity, as
shown in the inset of Fig. 3.
In addition, the main visual difference between viscous
and inertial regimes is that for the former the breakup
exhibits up-down symmetry around the breakup point,
whereas the symmetry is broken if inertia is important.
Indeed, the asymmetry coefficient defined in [20], which
is plotted in Fig. 3, demonstrates a transition from asym-
metric to symmetric, as the power-law exponent changes
from 2
3 to 1.
Thus, the yield stress appears to be irrelevant and the
breakup exponent is unrelated to the rheology altogether.
To assess what the effective viscosity for the thinning is in
the experiment, we can use the fact that for Newtonian
fluids the rate of thinning in the viscous regime is given by
the capillary speed =, where  is the surface tension.
However, determining the latter is in fact a difficult prob-
lem, since most if not all of the usual methods for deter-
mining surface tensions of liquids fail when the material
has a yield stress.
We therefore developed a new method that makes use of
the breakup images. Once the drop is formed, the conical
shape of the interface (see the pictures in Fig. 4, inset)
relaxes under the influence of the surface tension. This
process stops as soon at the capillary pressure can no
FIG. 2. The minimum neck diameter Rmin of the emulsion vs
time, with low and high concentrations of castor oil (powers
0:67 0:02 for < 70% and 0:99 0:03 for > 72%).
FIG. 1. Yield stress system: shear stress vs shear rate for
several volume fractions of castor oil; the shear thinning expo
nent is nearly constant, n  0:52 0:04 (a line of slope 12 has
been added at the right hand side).
FIG. 3 (color online). Left scale: power law exponent from fit
to the minimum neck radius (cf. Fig. 2) of the yield stress fluid
vs volume fraction (black circle). Right scale: asymmetry coef
ficient (open circles) In the inset, Reynolds number vs volume
fraction.
 2
longer overcome the yield stress of the material and so
from measuring the radius of curvature, as indicated in the
figure, the surface tension should follow from the force
balance =R ¼ y. We thus measure the curvature of the
interface a long time after breakup when it no longer
moves, and plot the curvature as a function of the yield
stress. We find roughly a straight line, the slope of which
should be the surface tension, as seen in Fig. 4.
We find a surface tension  ¼ 17 4 mN=m, close to
the expected value, since both the oil phase and the sur-
factant rich water phase have a surface tension  
20 mN=m. Then, knowing the surface tension, we can
obtain the effective viscosity from the filament thinning
rate in the viscous regime. For instance from the data
shown in Fig. 2, we conclude that the capillary velocity
is on the order of 0:025 m=s for the 74% sample, corre-
sponding to a viscosity of 0:032 Pa s. Comparing to the
rheology, this suggests that the value of the viscosity
deduced from the breakup experiment corresponds to a
typical shear viscosity at high shear rates.
Therefore, to study the relationship between pinch-off
and shear-thinning rheology in more detail and more
quantitatively, we will consider a simpler system. We use
aqueous xanthan solutions and solutions of xanthan in
glycerol-water mixtures (the glycerol was obtained from
Sigma) to be able to control the shear thinning. We use
xanthan since these solutions are known to be very
strongly shear thinning, and the shear thinning exponent
can be changed by changing the concentration, expressed
in ppm (weight parts per million). In addition, the xanthan
introduces very little elasticity into the system, which
is necessary to separate elastic from shear thinning
effects [21].
The data for the thinning of the neck is shown in Fig. 5,
which is again fitted by (1), the scaling exponent being
reported in Fig. 6. We find that the power remains constant
around 1, and does so both for aqueous xanthan solutions
and for xanthan in glycerol-water mixtures; again, each
experiment was done at least 5 times to ensure the repro-
ducibility of the results. The check with glycerol was done
to make sure inertia could not influence the results. From
the data, we show that again the rheology does not influ-
ence the breakup dynamics, although the shear-thinning
exponent varies over a large range.
This allows us to compare to a constitutive equation for
xanthan, which behaves like a suspension of rigid rods
[21]. The measured elongational viscosity, found from
the slope of Rmin in the viscous-capillary regime (taking
e ¼ 3 ¼ 0:213=j _Rminj [1]), is reported in Fig. 7. A
rigid rod will align in the direction of the extensional flow,
increasing the viscosity, since the elongational stress is
supported by the tension in the rods. Aligned rods are in
a stable equilibrium position, so even moderate rates of
FIG. 4. Inverse radius of curvature vs yield stress for different
volume fractions of castor oil. The inset shows the rounding of
the interface after drop detachment.
FIG. 5 (color online). Evolution of the diameter 2Rmin for
low and high concentrations of xanthan in water [1000 ppm
(slope dRmin=dt ¼ 0:0395 m=s;  ¼ 0:128 Pa s) and 3000 ppm
(dRmin=dt ¼ 0:0265 m=s;  ¼ 0:192 Pa s)]. Error bars are
smaller than the symbols.
FIG. 6. Left scale: power law exponent  from fit to Rmin
(cf. Fig. 5), vs concentration of xanthan in water as well as in
glycerol water mixtures. Right scale: power law exponent n
from the power law variation of the viscosity (Fig. 8) for
aqueous xanthan solutions.
 3
extension will lead to almost complete alignment, and the
elongational viscosity reaches the constant value [21]:
el ¼ 3s þ 6NkBT: (2)
Here s is the solvent viscosity (10
3 Pa s), N the
number density of the polymers in the suspension, T the
temperature, and  the relaxation time of the polymer
solutions, which we determine from shear measurements.
As seen in Fig. 7, measured and theoretical values for the
extensional viscosity are in reasonable agreement, so we
have obtained a quantitative picture of the relation between
shear and elongation viscosity in a strongly shear thinning
system.
The predicted shear viscosity of the same rigid dumbbell
model, on the other hand, shows a behavior that is opposite
to the extensional behavior in every aspect. Increasing
alignment in the shear direction leads to a decreasing
viscosity with shear rate, as particles become better ori-
ented in the direction of shear, offering less and less resist-
ance to the flow. However, the aligned state is unstable in
shear flow, so the rod flips continuously between two
equilibria, the transient time decreasing gradually with _,
leading to the power-law dependence [21]:
ð _Þ ¼ s þ 0:678NkBT2=3 _1=3: (3)
In Fig. 8 we fit (3) to the high-shear end of our shear
measurements, treating  as an adjustable parameter,
used in (2) to calculate the shear viscosity.
In conclusion, we have studied the breakup of non-
Newtonian fluids for the cases of shear thinning and yield
stress materials; we conclude that both the emulsion and
the polymer solution behave very similarly to Newtonian
fluids: neither the yield stress nor the shear thinning come
into play. Both the shear and the elongational viscosity are
modeled quite well by the rigid dumbbell model for high
shear rates. During drop formation, the elongational
viscosity is always saturated at its high deformation rate
value.
Our physical picture does not take into account the
possible formation of complex structures created by highly
interacting particles. Measurements of a liquid crystalline
fluid in the smectic phase have found an extensional rheol-
ogy in line with shear measurements [10]. However, other
liquid crystalline phases showed a strong discrepancy
between shear rheology and breakup measurements [11].
A correspondence between shear and elongation rheology
was also reported in the case of even more complex (and
less well characterized) fluids like ketchup, shaving foam,
and hair gel [22,23]. We have to conclude that there may be
liquids forming complex networks, whose rheology is
beyond our current understanding.
[1] J. Eggers and E. Villermaux, Rep. Prog. Phys. 71, 036601
(2008).
[2] Y. Amarouchene, D. Bonn, J. Meunier, and H. Kellay,
Phys. Rev. Lett. 86, 3558 (2001).
[3] C. Wagner, Y. Amarouchene, D. Bonn, and J. Eggers,
Phys. Rev. Lett. 95, 164504 (2005).
[4] C. Clasen, J. Eggers, M.A. Fontelos, J. Li, and G.H.
McKinley, J. Fluid Mech. 556, 283 (2006).
[5] M. Renardy and Y. Renardy, J. Non Newtonian Fluid
Mech. 122, 303 (2004).
[6] P. Doshi and O.A. Basaran, Phys. Fluids 16, 585 (2004).
[7] N. J. Balmforth, N. Dubash, and A. C. Slim, J. Non
Newtonian Fluid Mech. 165, 1139 (2010).
[8] N. J. Balmforth, N. Dubash, and A. C. Slim, J. Non
Newtonian Fluid Mech. 165, 1147 (2010).
[9] D. Porter, J. R. Savage, I. Cohen, P. Spicer, and M.
Caggioni, Phys. Rev. E 85, 041701 (2012).
[10] J. R. Savage, M. Caggioni, P. T. Spicer, and I. Cohen, Soft
Matter 6, 892 (2010).
[11] A. A. Verhoeff and H.N.W. Lekkerkerker, New J. Phys.
14, 023010 (2012).
[12] O. E. Yildrim and O.A. Basaran, J. Non Newtonian Fluid
Mech. 136, 17 (2006).
FIG. 7. Elongational viscosity calculated from j _Rminj (solid
circle) and from (2) (open circle) vs xanthan concentration.
FIG. 8. Viscosity vs shear rate, for several concentrations
of aqueous xanthan; straight lines are fits of (3) to the high
shear end.
 4
[13] O. E. Yildirim and O.A. Basaran, Chem. Eng. Sci. 56, 211
(2001).
[14] P. Doshi, R. Suryo, O. E. Yildirim, G. H. McKinley, and
O.A. Basaran, J. Non Newtonian Fluid Mech. 113, 1
(2003).
[15] R. Sattler, C. Wagner, and J. Eggers, Phys. Rev. Lett. 100,
164502 (2008).
[16] R. Sattler, S. Gier, J. Eggers, and C. Wagner, Phys. Fluids
24, 023101 (2012).
[17] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics
of Polymeric Liquids (Wiley, New York, 1987), Vols. 1
and 2.
[18] P. Moller, A. Fall, V. Chikkadi, D. Derks, and D. Bonn,
Phil. Trans. R. Soc. A 367, 5139 (2009).
[19] J. Eggers and T. F. Dupont, J. Fluid Mech. 262, 205
(1994).
[20] A. Rothert, R. Richter, and I. Rehberg, Phys. Rev. Lett. 87,
084501 (2001).
[21] D. Bonn and J. Meunier, Phys. Rev. Lett. 79, 2662
(1997).
[22] P. Coussot and F. Gaulard, Phys. Rev. E 72, 031409
(2005).
[23] F.M. Huisman, S. R. Friedman, and P. Taborek, Soft
Matter 8, 6767 (2012).
 5


Drop Formation in Non-Newtonian Fluids

We study the pinch-off dynamics of droplets of yield stress and shear thinning fluids. To separate the two non-Newtonian effects, we use a yield stress material for which the yield stress can be tuned without changing the shear thinning behavior, and a shear thinning system (without a yield stress) for which the shear thinning can be controlled over a large range, without introducing too much elasticity into the system. We find that the pinch-off remains very similar to that of constant viscosity Newtonian liquids, and consequently thinning in shear flow does not imply a thinning in elongational flow

AYTOUNA, Mounir

PAREDES, Jose

SHAHIDZADEH-BONN, Noushine

MOULINET, Sébastien

WAGNER, Christian

AMAROUCHENE, Yacine

EGGERS, Jens

BONN, Daniel

Oskar Bordeaux

Physical Review Letters

Hal-Diderot

We study the pinch-off dynamics of droplets of yield stress and shear thinning fluids. To separate the two non-Newtonian effects, we use a yield stress material for which the yield stress can be tuned without changing the shear thinning behavior, and a shear thinning system (without a yield stress) for which the shear thinning can be controlled over a large range, without introducing too much elasticity into the system. We find that the pinch-off remains very similar to that of constant viscosity Newtonian liquids, and consequently thinning in shear flow does not imply a thinning in elongational flow.</p

Drop Formation in Non-Newtonian Fluids

Abstract

Similar works

Full text

Available Versions

Archive Ouverte en Sciences de l'Information et de la Communication

Oskar Bordeaux

Hal-Diderot

Oskar Bordeaux

Explore Bristol Research

Crossref

Search4Dev

NARCIS