7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Experimental characterization of the whipping instability of charged jets inside
liquid baths.
G. Riboux*, A. G. Marfn* I. G. Loscertalest and A. Barrero*
Escuela Superior de Ingenieros, Universidad de Sevilla, Espaia
Escuela T6cnica Superior de Ingenieros Industriales, Universidad de MAlaga, Espaia
SPresent address: Physics of fluids group, University of Twente, Netherlands
griboux@us.es, a.gomezmarin@utwente.nl, loscertales@uma.es and abarrero@us.es
Keywords: Instability, ElectroHydroDynamic, Microjet
Abstract
The charged liquid microjet issued from a Taylor cone may develop a special type of nonsymmetric instability,
usually referred to in the literature as whipping mode. This instability usually manifests itself as a series of fast and
violent lashes of the charged jet, which makes difficult its characterization in the laboratory. Recently, we have found
that this instability may also develop when the host medium surrounding the Taylor cone and the jet is a dielectric
liquid instead of air. When the oscillations of the jet occur inside a dielectric liquid, their frequency and amplitude are
much lower than those taking place in air. Taking advantage of this fact, we have performed a detailed experimental
characterization of the whipping instability of a charged microjet within a dielectric liquid by recording the jet motion
with a high speed camera. Appropriate image processing yields the frequency and wavelength among other important
characteristics of the jet whipping as a function of the governing parameters of the experimental set up, the flow rate
and applied electric field. Alternatively, the results can be also written as a function of three dimensionless numbers:
the capillary and the electrical Bond numbers and the ratio between the electrical relaxation and the convective times.
Introduction
Micro and nanostructures of interest in several tech
nological fields can be generated from electrocapillary
liquid jets. Among the techniques capable of generat
ing these jets, electrospray and electrospinning are the
most popular ones for being able to generate particles in
the nanometric range, (Barrero and Loscertales (2007)).
These particles have a variety of applications in drug in
dustry, food additives, material engineering among oth
ers. In electrosprays, an electrified conical meniscus
(Taylor cone) is formed at the tip of a capillary tube and
a thin jet issuing from the vertex cone. The electrified
jet emitted from the cone vertex is inherently unstable
and it undergoes instability modes either due to capillary
instabilities leading to jet breakup or lateral instabilities
(whipping mode) owing to the action of the electric field.
To successfully obtain nanofibers from electrified liq
uid jets via electrospinning, the capillary instability,
which is responsible for the break up of the jet into
charged droplets, must be completely avoided. This is
accomplished, for example, by using polymeric solu
tions of sufficiently high electrical conductivity. In these
cases, the evaporation of the solvent induces polymer
ization and the jet becomes sufficiently viscoplastic to
support without breaking the intense stretching induced
by the electric field. On the other hand, the electrical
conductivity of the solution is responsible for the charge
density at the surface needed to shoot the whipping.
These and other properties of the solutions have been
named as the electrospinnability of a solution, i.e. the
capacity of making fibers by electrospinning. Unfortu
nately, most of the literature on electrospinning, which
produced hundreds of publications only in the year 2005,
deals with phenomenological studies testing the electro
spinnability and properties of different polymer solu
tions, but only some of them have developed theoret
ical models aiming to predict the behaviour of the jet
(Hohman et al. (2001a), Reneker et al. (2000), Yarin
et al. (2001)).
Recently, Barrero et al. (2004) operated an electro
spray in steady conejet mode inside a dielectric liquid
medium. Although electrosprays within a host liquid
medium are ruled by the same physical principles than
those operated in air, there are, however, some differ
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ences in their behaviours; the oscillations of the whip
ping instability inside a liquid are much less violent than
in air; The large inertia of the liquid bath makes the fre
quency of the jet oscillations several orders of magni
tude lower than those found in typical electrospinning
experiments (Shin et al. (2001); Hohman et al. (2001b);
Reneker et al. (2000)). Taking advantage of these prop
erties, we have carried out an experimental work of
the whipping instability of the jet of an electrospray of
glycerine operated within a liquid (hexane) bath. We
have used a setup entirely similar to that used in previ
ous works on electrosprays inside liquid media (Barrero
et al. (2004); Mrin et al. (2007)). The main aim of the
work is the characterization of the whipping instability,
its frequency, amplitude or wavelength in terms of the
governing parameters, flow rate and electrical field.
Experimental setup and image processing
The experimental setup consists of an open tank of
plexiglass with a 38 x 32 mm2 crosssection and 180
mm high, see figure la. The walls of the tank were
made of glass to allow visualization. The tank was filled
with hexane whose viscosity, density and relative elec
trical permittivity were respectively: p 0.3 mPa.s,
p, 660 kg.m3 and e/ec, 1.89, where e, is the
electrical permittivity of the hexane and o, is the vac
uum permittivity. A round electrodecollector of 23 mm
of diameter connected to electrical ground was located
at the bottom of the tank. A metallic needle of inner
radius a 115 pm immersed in the hexane was lo
cated at a fixed distance H = 27 mm above the col
lector. A conductive liquid, glycerine, was injected
through the needle at a flow rate Q; a Harvard PHD 4400
programmable syringe pump with a Hamilton Gastight
glass syringe of 32.6 mm diameter was used to control
the flow rate. The physical properties of the glycer
ine were: density p 1250 kg.m3, dynamic viscos
ity p 1280 mPa.s, electrical conductivity K 1.7
pS.m 1 and electrical permittivity e/c, 43. Surface
tension of the glycerinehexane pair was measured us
ing the pendant drop method; the measured value was
y 28.3 mN.m1. The metallic needle was connected
to a Bertran 205B10R high voltage power supply to
impose an electric field between the collector and the
needle exit (see Fig. la). The whipping motion of the
jet was captured with a Photron FASTCAM 1024 PCI
highspeed video camera. The motion of the charged jet
was recorded at an acquisition frequency equal to 2000
frames per second. We used a floodlight facing the high
speed video camera to obtain a good contrast of the jet
image. The dimensions in all pictures were 1024x512
pixels while the resolution was within a range of 8 16
pm. Figure lb presents an example of a digital image of
Needle Inner liquid
E (fpEK,Y)
2a1
h 7d
  
h !
SDroplets
A
External liquid
Collector
Z
High
 voltage
power
supply
(D)
Figure 1: (a) Sketch of the experimental setup. (b) An
example picture of the whipping instability
(Q 1.0 ml/h, 2.25 kV).
(a)
A .29.ms
At=29 ms
(b)
II mm
2 .t m  
Figure 2: Images of the whipping jet trajectories. (a)
Images of the lateral instability taken at two
times separated by one complete period; (b)
Wavelength measurements at a halfperiod
and (c) whipping envelope with 0 the maxi
mal angle of the droplet projection (Q 1.0
ml/h, D 2.25 kV).
the jet path.
To analyze the images of the jet motion, we have used
the image processing ImageJ software together with al
gorithms in house developed. An example of a digital jet
path, which appears quite neat, is presented in figure 2a.
This figure shows two digital images of the jet path. The
images were captured at two different times separated by
an interval of 29 miliseconds. Since the two images per
fectly superpose each other, one may conclude that the
whipping mode is periodic for this set of values of the
governing parameters (Q = 1.0 ml/h, D = 2.25 kV);
the period of the whipping oscillations being 29 milisec
onds. The measurement of the whipping period may be
estimated with an accuracy of 1 frames, which corre
sponds to 0.5 ms. Figure 2b shows two images of the
jet path separated by a lapse of time of a semiperiod
(14.5 ms); the superposition of the two digital images
permits the measurement of the wavelength Ai of the
whipping oscillation. It should make mention that as
a consequence of the electric field, the wavelength A, of
the instability slightly varies along the zaxis. Therefore,
we define a mean value A of the wavelength by averag
ing the values of the measured wavelength Ai of each jet
loop along the zaxis.
The envelope of the whipping jet paths can be also
obtained by overlapping the jet images captured at dif
ferent times. Figure 2c shows the overlapping of 300
images of the jet. This overlapping of the jet images
permits an easy digitalization of the envelope of the jet
paths and hence the definition of the envelope contour
as a function of the distance z to the needle. In figure
2c, we have plotted for comparison both the binary im
age of the envelope of the jet path (black region) and the
overlapping of the binary jet paths. Since the whipping
envelope is symmetrical, it is possible to obtain an av
eraged envelope meridional curve as a function of the
distance z to the needle from data of the binary images
and MATLAB algorithms. It should be noted that other
possible choices for the envelope detection, as the use
of video cameras with long time exposure shots yielded
less clean and neat results.
Dimensionless parameters: scaling
The experimental characterization of the whipping in
stability of an electrified liquid jet evolving within both
a gas atmosphere or a bath of a different liquid, is a com
plex problem due to the large number of physical param
eters that play an important role in the phenomenon. In
the general case of a microjet subject to a strong elec
trified field the relevant parameters are those associated
with the physical properties of the two fluids: viscos
ity p, density p, electrical permittivitty c and conduc
tivity K of each fluid and the surface tension of the
fluid couple. Also, the dependence on the geometrical
parameters characterizing each experimental set up: the
radius of the capillary a, capillary to collector distance
H, among others. Finally the control parameters: the
flow rate Q and the applied electrical voltage b.
Nonetheless, in our experimental case, the viscosity
and the electrical conductivity of the outer fluid (hex
ane) can be neglected due respectively to its low viscos
ity, compared to the viscosity of the glycerine, and its
dielectric character. Furthermore, an estimation of the
Reynolds number p Q/ (p a) leads to a value of the or
der of 10 3. In consequence, we neglected the inertia of
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
the inner liquid and thus, the behaviour of an electrical
viscous jet formulated in nondimensional form can be
governed by six independent dimensionless parameters.
Using the viscosity p of the liquid jet, surface tension
7, capillary radius a, and the electrical permittivity c,
of the outer fluid as dimensional scales of the problem,
we can define the following dimensionless parameters of
the problem: geometrical length ratio H/a, glycerine to
hexane electrical permittivity ratio 3=e/c, the dimen
sionless number S pea/p2 and
Ca _ ,
7ca2
S 2 T pKa
B 7 a e7
Observe that in our experiment, the dimensionless num
ber S 1.3 x 103 is extremely small indicating that
the relative importance in the dynamic of the outer flow
is rather small.
The three dimensionless numbers in equation (1),
which were already introduced in a numerical work by
Higuera (2006), are respectively: the capillary number,
the electrical Bond number and a dimensionless time de
fined by the ratio between the convection time ap/y and
the electrical relaxation time c/K. The capillary num
ber Ca compares the fluid velocity Q/a2 at the end of
the capillary to the viscocapillary velocity V, = y/p.
The electrical Bond number B compares the electrical to
capillary pressure. Finally, T compares the convection
time to the electrical relaxation time.
Here, we restricted ourselves to the study of the whip
ping jet instability as a function of the supply flow rate
Q and the applied electrical voltage b. Thus, we fixed
the geometrical parameter H/a=235 and electrical per
mittivity coefficient 3=23 while we varied the three di
mensionless numbers within the ranges Ca = 0.2 8.5,
B = 300 1870 and T 19 27. The variation of the
last dimensionless numbers was carried out by changing
the glycerine electrical conductivity which appears only
on the dimensionless group T.
Preliminary observations
Let us comment some preliminary observations before
to present the experimental results. Figure 3 shows the
whippingmap for the fluid pair, glycerinehexane. The
whippingmap yields the behavior of the jet as a func
tion of the control parameters Q and b; it permits to dis
tinguish the region where the jet develops an instability
of whipping type in the (Q, b) space, or equivalently
in the (Ca, B) space. This type of graphic was pre
viously introduced by Hohman et al. (2001b) and Shin
et al. (2001), who called it operating diagram, who con
sidered either an electrified jet of glycerine or a jet of a
mixture of PEOwater evolving in air. These diagrams
are more general than the ours since, in addition to the
whipping mode, they permit to distinguish other insta
bility modes of the jet such as varicose, bending and
steady state jet modes. Here, we have carried out four
sets of measurements whose results are collected in fig
ure 3:
(a) The first set of eight measurements was taken at
different values of the capillary number Ca while we
kept B 0; these set of measurements, labeled as set A,
are represented by square symbols and numbered from
1 to 8. In the range of capillary numbers considered, the
experimental set up produces glycerine drops with a uni
form radius of the order of 10 times the capillary radius;
the frequency of the droplet formation increasing with
the capillary number. A long and thin jet was formed
before the detachment of the droplet; the length of the
jet just at the breakup process increasing linearly with
the capillary number, Ca. At the jet breakup, one can
observed the formation of a neck just behind the droplet.
(b) The second set of measurements was taken at four
given values of the electrical Bond number B while the
value of the capillary number, Ca= 1, was kept constant.
These set of four measurements, labeled as B in figure
3, are represented by diamond symbols and numbered
from 1 to 4. We observed clearly the transition from a
dripping mode, image 3 corresponding to B = 260, to a
whipping mode, image 4 corresponding to B = 470.
Therefore, one may define a value of the Bond num
ber B B= in such us for values of the Bond number
B < Bi,,i the jet behaves in dripping mode while it
develops whipping instability when B > Bin. In this
set of measurements, we have observed that the diam
eter of the drops decreased linearly when the electrical
Bond number increased. Thus, for a given Ca, there ex
ists a value of the electric Bond number Bmin such as
for B > Bm,i, the jet bends and develops whipping in
stability.
(c) The third set of eight measurements was carried
out keeping the electrical Bond number at a given value,
B 915 while we varied the capillary number Ca in
the range 0.2 to 7.6. This set is labeled as C and the
experimental measurements are represented by triangu
lar symbols and numbered from 1 to 8. In this case, we
observed a transition from a jetting mode to a whipping
mode (transition between image 1 and 2). Thus, and
similarly to the previous case, there exists a capillary
number (Ca,in) separating a jetting mode region from
the one where the whipping instability develops. The
jet diameter and the whipping amplitude increase with
Ca. We have also observed a spatial destabilization of
the jet motion when the capillary number increases. For
Ca 1.5, see (image C5), the spatial behavior of the jet
appears more chaotic than for smaller values of Ca.
(d) Finally we have determined the transition between
the dripping mode to the whipping mode in the para
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
metric space (Ca, B) using a highspeed camera. The
experimental measurements of this set, labeled as D in
figure 3, represented by empty circles. In the range of
Ca = 0.3 6.2, the minimum electrical Bond number
increases linearly with the capillary number. Thus, for
the fluid pair, glycerinehexane, the drippingwhipping
transition boundary seems to evolve as Bi,, 15 Ca +
340. Regarding the jettingwhipping transition, we have
found that Cam,,, has constant value, close to 0.3, which
is completely independent of the range of values of B we
have explored.
In conclusion, the whipping instability appears only
when the couple of parameters, Ca or B, is respectively
larger than Cain, or B in. The transition from the
dripping mode to the whipping one occurs at a value of
the electrical Bond number Bin 15 Ca + 340 within
the explored capillary number range 0.3 < Ca < 6.2.
The transition from jetting to whipping appears at a
value of the capillary number Ca greater than Cai, ,
0.3; this value is constant and independent of the applied
Bond number B for the range of values of this parameter
that we have explored. Note that the whipping mode de
velops within the parametric region represented by the
white domain, figure 3 while no whipping instability de
velops into the region in gray color.
whippingmode characterization
This section is devoted to the experimental analysis of
the jet whipping. We have observed whipping motion in
all experimental cases since the capillary number Ca,
electrical Bond number B were larger than the minimum
values (Cain,, Bmi). Therefore, all the experiments
lay within the white whipping region in figure 3. We
have focused our experimental study on both the jet
frequency, wavelength or the whipping envelope as a
function of Ca and B.
Frequency and wavelength of the whipping instability
In figure 4, we have plotted the dimensionless whip
ping wavelength A/a, line 1, the dimensionless whip
ping frequency fa/V,, line 2, and the whipping phase
velocity f A/Vo, line 3 as a function of either the capil
lary number, see figures (a), or the electrical Bond num
ber in figures (b). Note that because the path of the
jet motion may be unstable, the wavelength A was only
measurable in the spatially reproducible part of the jet
trajectory.
We have found that the wavelength A depends on the
dimensionless time ratio T and it can be expressed as
A ~ A, (T/T,) with A, 12 a 1.4 mm while the
frequency can be expressed as f f, (T/T,)1 with
f,o V,/3a 63 Hz; f, and A, are marked by
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
I   .
4..
.  _^  ". : n ) .
II
/ 1C I
i I
S DII
C1
                .     .................... ........... .. .
     ?
T ~       ............................ .......... . . . .: : : : : : i..... .. ........... :  .. .. .. .J 2 _
S3 h
fo a
ig Wi pping j t: ma Io Ih i c (
dashed lines and the two plain lines point out the limits
of the interval of 15' around these two values. Within
the range of 0.3 < Ca < 2.8, the dimensionless ra
tios (A/a)(To/T) and (fa/Vo,)(T/T) have a constant
value. For comparison, the value of the frequency re
ported by Shin et al. (2001) in their experiments with
a PEO (PolyEthylene Oxide)water mixture in air has
been also plotted and represented with white circle sym
bols in figure 4 (2a). In their experiment, these authors
fed a constant volumetric flow rate Q 0.2 ml/min
of a PEOwater mixture through a stainless steel cap
illary tube of inner diameter 1 mm. The electric field
in their experimental setup was E 1.11 kV/cm on a
distance of the order of 10 2 m. These experimental pa
rameters correspond to Ca = 0.4, T = 2.4 x 105 and
B ~ o(101 102). Resorting to long exposure times,
these authors experimentally recorded the whipping en
velope of this couple of fluids and fitted the experimen
tal envelope by using an exponential law for the whip
ping amplitude of the type A(z)/Ao ~ exp[f z/Uo];
U, and f 0.014 0.002 s1 being respectively the
average fluid velocity at the capillary end and the fre
quency. Note that the experimental frequency reported
by Shin et al. (2001) is in good agreement with ours re
sults, so that, the expression fo (T/To) 1 may be used
with confidence for the PEOwater mixture in spite of
the very large difference in the electrical conductivity of
the PEOwater mixture and the glycerine. Finally, ob
serve that for Ca > 2.8, the wavelength and the fre
quency of the whipping slightly increases and decreases
respectively with the capillary number. From this limit,
Ca > 2.8, we have observed that the the jet path is more
unstable in time and space, and corresponds practically
to a chaotic regime. In consequence of increasing and
decreasing, wavelength and the frequency of the whip
ping, as shown in figure 4 (3a), whatever the capillary
number, the phase velocity of the whipping jet defined
as the product of the dimensionless wavelength times
the frequency is constant and equal to fo Ao 4 Vo
with an error of l .'. as shown by the two horizontal
lines in the figure. Contrarily, the wavelength and fre
quency of the whipping jet strongly vary with the elec
trical Bond number and follow power laws of the type
B3/2 and B3 3/2 respectively, figure 4 (lb&2b). Note
that the whipping phase velocity f A is practically inde
pendent of the electrical Bond number within the error
band of 15%, figure 4 (3b).
In conclusion, the phase velocity of the jet whipping
only depends on the viscocapillary scale velocity Vo
and its value, equal to fo A, 4 V,, is 8.8 cm/s for the
glycerinehexane couple. The wavelength A can be ex
pressed as,
S= (B/B) 3/2(T/,) 1, (2)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
and the frequency f of the jet whipping can be expressed
as,
f = f(B/Bo)3/2(T/7o), (3)
with Bo 920, 1 23, A 12 a and fo = VO/3a.
These results confirm the destabilizing effect of the
electrical field on the jet trajectory as it has been typi
cally observed in electrospinning experiments (Hohman
et al. (2001b), Shin et al. (2001)), or in the experiments
carried out by Taylor (1969). In fact, an increase in the
applied voltage increases both the electrical field and
the charge density of the jet. The jet becomes more
unstable, increasing its whipping frequency, as it was
commented by Hohman et al. (2001b), and reducing
its characteristic wavelength. Interestingly enough, an
increase of the electrical field increases too the length
of the jet (see next section), and therefore the jet di
ameter must decrease since the flow rate is kept constant.
Whipping envelope
In this part, we analyze the behavior of whipping en
velope as a function of the dimensionless parameters Ca
and B. Figure 2c shows an example of the whipping en
velope. The amplitude A of the envelope corresponds
to the maximum amplitude of the jet trajectory and the
maximum angle of the droplet trajectories. We defined
also the length h corresponding to the vertical distance
between the capillary end and the beginning of the whip
ping instability. We arbitrarily define the starting point
of the whipping instability (the value of h) as the jet
point in which the jet envelope reaches an amplitude
100 pm, which corresponds to 6 7 pixels in the im
ages. Results were obtained by taking different values
of the threshold; we have found that in all cases, the rel
ative differences in the value of h were smaller than 15' .
for values of threshold amplitude within a range between
50 200 pm. The second length hj was defined from
the point at which the jet bends and the whipping mode
starts, z = h, until the jet breakup into droplets (see fig
ure la). This length corresponding to the vertical length
of the whipping envelope. These characteristic lengths
of the whipping instability have been obtained from data
of the binary images and MATLAB algorithms.
In figure 5, we have plotted the evolution of the whip
ping envelope as a function of either the capillary num
ber, column (a) or the electrical Bond number, column
(b). The amplitude A and the axial distance z are nor
malized with the needle diameter a. As shown in figure
5 (la), two regimes may be identified: For Ca < 2.8,
the amplitude A and length h, of the envelope increase
with the capillary number while the envelope remains
unchanged for Ca > 2.8. Note that the downstream be
havior of the envelope also changes with the capillary
number until the envelope reaches an asymptotic
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Ca=0.5 10 3
I C = 1.0
I Ca = 1.4 '
S 10 1 /2
102 103 102
B
4 6 8 10 12 x102
I0 T=23
T= 27
100 ..
*,
10 
1 2 3 4 5
Ca
0.25
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
12 ,
F T= 231
T=27
10
8
4
2
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Ca
12
0 Ca=0.5
SC = 1.0
10 Ca = 1.4
8
B
Figure 4: (1), Wavelength A, (2), frequency f, and (3), phase velocity f A of the whipping jet as a function of either
(a) the capillary number or (b) electrical Bond number. In (a): B = 920 and To = 23; in (b): T = 23.
Circle white symbols in (2a) correspond to the results reported by Shin et al. (2001).
universal shape at Ca = 2.8 that remains unchanged
for values of the capillary number larger than 2.8. The
meridional shape of the different envelopes has been
compared with that obtained by Shin et al. (2001) using
a PEO (PolyEthylene Oxide)water mixture in air; the
amplitude of the latter envelope is represented by white
crosses meridional line in figure 5 (la). The dimension
less parameters of their experiment were Ca 0.4, T
2.4 x 105 and electrical Bond number B ~ o(101 102),
which are higher than the minimum dimensionless val
ues Camin and Bmi, for which the whipping instability
develops in the case of glycerinehexane fluid couple.
Note also that the dimensionless parameter T is greater
than T in the case of glycerinehexane fluid couple. The
envelope obtained by Shin et al. (2001) presents a dif
ferent downstream behavior than the glycerinehexane
whipping envelopes. In the former case, the jet begins
to bend at a smaller distance from the capillary, so that,
the envelope amplitude is near the capillary larger than
in the latter case; also the amplitude of the envelope of
the whipping jet is in their case two times larger than in
the ours. For Ca > 2, the envelope amplitude of the
2. 2
(a)
1.75
1,5
1.25
1
S0.75
0.5
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
1.
(a) 4
a
2
1
1
2
3
4
2. 4
(a)
a1
C
ic
3. 1.2
(a)
S0.8
: 0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8
(z h)/hw
0 0.2 0.4 0.6 0.8
(z h)/h0
Figure 5: Amplitude of the envelope versus the axial distance as function of either (a) the capillary number or (b) the
electrical Bond number. (1) Digital meridional images of the whipping envelope. (2) Meridional curves of
the whipping envelope without the steady region and the droplet region. (3) Normalized whipping envelope.
glycerine jet at z > 40 a becomes larger than that of the
PEOwater mixture. Let us finally remark that the dif
ference between the case reported by Shin et al. (2001)
and the ours firstly lie in the fact that PEOwater mixture
is a nonnewtonian fluid and secondly that the dimen
sionless time ratio T was 104 times larger in the Shin
et al. (2001) case than in the ours. Finally, the differ
ence in the geometry of the electrical configuration (a
plateplate geometric configuration in the yours and a
capillaryplate configuration in the ours) may also play
an important role in the envelope behavior. The shape of
the whipping envelope depends on the electrical Bond
number, 5 (lb). Essentially, the length h, of whipping
envelope decreases due to the increase of the length h
with the electrical Bond number. For comparison, we
have plotted a meridional curve of an envelope (white
dashed line in figure 5 (lb)), which has been theoreti
cally obtained in the case of B = 540, see equation (4);
0 0
mmn,
0 10 20 30
(z h)/a
40
420
8=445
a=47
a=515
a=565
agb
a=645
Sa=675
a=700
=730
B=76
=790
B=820
S=855
B =915
_= 1055
40 50
a 400
a = 420
a = 445
a470
a 490
= 515
a 540
_=590
=
N 5
 =
=73
=79
5=l955
5=oa
Ca = 0.75
Ca 0.65
. C 1.1
Ca= 2.
C2 15
Ca 2
C. 2.8
,~~C 0 ,, *i
1 1.2
the agreement with the experimental envelope is quite
good.
For a better comparison of the different experimental
envelope curves, it proves convenient to plot the dimen
sionless amount (A rj)/a versus (z h)/a with rj the
jet radius at z h. As shown in figure 5 (2a), both the
amplitude of the envelope and the length of the whipping
increase with the capillary number whereas the whip
ping envelope are independent of the electrical Bond
number; therefore, the meridional curves of the enve
lope obtained at different values of B overlap. Also, the
dependence on the electrical Bond number of both the
length, hw, and the maximum amplitude of the whip
ping envelope, Aax, A(z = h), are rather small
compared to their dependence on the capillary number.
Note that contrarily to the behavior of the shape of the
envelope, the length h of the stable part of the jet de
pends on the electrical Bond number, see figure 5 (lb).
Observe also that the maximum angle 0 that forms the
droplet trajectories with the jet axis (see figure 2c for
the definition) depends on the capillary number but it
is independent on the electrical Bond number. For any
combination of values of both the capillary and electrical
Bond number, the maximum angle of droplet trajectories
was within the range of Omi 35 and 0max = 70.
The two extrema angles are plotted in figure 5 (2b).
Since whipping envelopes seem each other very
much, we have normalized them with the maximum am
plitude Amax while the abscise z h has been also nor
malized using the whipping length h Results, which
are plotted in figure 5 (3a) confirm the collapse of the
whipping envelope curves in an universal one. This self
similar behavior of the envelopes has been also observed
in experiments carried out as function of the electrical
Bond number at two other values of the capillary num
ber, Ca=1.0 and 1.4. Nonetheless, one should be aware
that this selfsimilar behavior is observed only for the
range of parameters 0.55 < Ca < 2.8, 400 < B < 1055
and T 23 27. We have calculated the mean curve
of the normalized amplitude of the envelopes for each
case: an example of the mean curves of the normal
ized envelopes is represented by dashed lines in figure
5 (3a&b); the two cases correspond respectively to the
mean curve for a given electrical Bond number and sev
eral values of the capillary number, 5 (3a) and, that in 5
(3b), to the mean curve for a given capillary number and
several values of the electrical Bond number. The mean
curves permit the comparison of the mean amplitude of
the envelopes for different values of Ca and B.
In figure 6a, we have plotted for comparison the dif
ferent mean curves of the envelopes in a loglog repre
sentation. In the range of (z h)/h, between 10 2 and
10 1, the mean envelope behaves linearly with the nor
malized axial distance and closely follows a law of the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
(a)1io
+3/2
10
SlO +1 f(B) at Ca = 0.5
 f(s) at Ca = 1.0
f(B)atCa = 1.4
f(ca) at T= 23
f(C atTr=27
1010 10' 100
(z h)/h,
(b) 20
B= 420
5  B=540
 B=675
10
5
10
15
20
0 5 10 15 20 25 30 35 40 45
(z h)/a
Figure 6: (a) Comparison of the normalized mean
curves of the whipping envelopes from differ
ent experiments. (b) Comparison of the whip
ping jet images (straight line ) and the cor
responding theoretical formula (dashed line
) for different values of the electrical Bond
number (Ca = 0.5, T 23).
type [(z h)/h,] for different electrical Bond and cap
illary numbers. Observe that this region (z h)/h,
O[10 2 10 1] is very small compared to the total
length scale of the envelope; it is approximately of the
order of 10 % of the total length of the envelope. Within
the range of the abscisa 0.1 < (z h)/h, < 0.4, which
corresponds to the 30 % of the total length of the enve
lope, the different curves collapse in just one universal
one that follows a power law of the type [(z h)/h,]3/2
provided that the capillary and the electrical Bond num
bers are within the range of parameters Ca z 0.5 2.8
and B 400 1055. Let us, finally, remark that this
result is quite different from the experimental observa
tion reported by Shin et al. (2001) for an electrified jet
of a mixture of PEO (PolyEthylene Oxide) and water,
for which they found a normalized whipping amplitude
A(z)/Ao following the exponential law exp[w z/Uo].
Using the experimental results properly, we have
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
modeled the amplitude of the whipping oscillation as,
A(z,t) L(z h
cos 2 + 27(f t + p) (4)
A /
where the exponents a 3/2 and ( z 1 take into ac
count the downstream evolution of the wavelength; c and
L are characteristic lengths and p is a phase amplitude.
The mean value of c and L were 0.19 mm and 0.78 mm
respectively. A comparison between several examples of
the whipping jet images and the corresponding theoreti
cal formula are plotted in figure 6b. The theoretical and
experimental curves match quite well. Of course, the
theoretical envelope obtained by overlapping the theo
retical curves on one period (1/f) is also in good agree
ment with the experimental whipping envelopes (white
dashed line in figure 5 (2a&b)). These results show the
pertinence of expression (4) for modeling the whipping
instability in the range of parameters Ca z 0.5 2.8
and B z 400 1055.
Conclusion
We have presented new experimental results about the
lateral jet instability in an electrified jet of glycerine dis
persed in a bath of hexane. With the help of a high
speed video camera and after specific image processing,
we have obtained measurements of the different charac
teristic scales, frequency, wavelength and shapes of the
whipping instability as a function of the two governing
parameters: the electrical Bond number B and the capil
lary number Ca. We have observed the important effect
of the electrical Bond number on the frequency and the
wavelength, which prones the instability by increasing
f and decreasing A. Furthermore, the wavelength A and
frequency f appears independent of the capillary num
ber and can be expressed as Ao (B/Bo) 3/2(T/T) 1
and fo (B/Bo)32(T/T); where Bo 880, To = 23,
Ao = 12 a and fo = Vo/3a. In consequence, whatever
the capillary number and electrical Bond number, the
phase velocity of the whipping jet defined as the product
of the dimensionless wavelength and frequency is con
stant and equal to fo A 4 Vo with an error of 15'.
as shown by the experimental results. That is, the phase
velocity of the jet whipping depends only on the visco
capillary scale velocity Vo and its numerical value for the
glycerinehexane couple is 8.8 cm/s. The envelope of
the jet whipping has been also obtained from the exper
iments. The detected whipping envelope showed a self
similarity behavior after appropriate normalization. The
amplitudes of the different mean whipping envelopes
normalized with the maximum amplitude evolved down
stream towards a sole curve showing a 3/2 power law of
the normalize distance. The obtained results shown il
lustrate clearly the general behaviour of these lateral in
stabilities and can be useful to understand the complex
underlying dynamics. Finally, experiments with other
couple of liquids should be carried out to confirm the dif
ferent experimental behavior observed, in particular, the
dependency of the frequency and wavelength or whip
ping jet behaviour as function of the dimensionless pa
rameters.
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