Content uploaded by Roux D
Author content
All content in this area was uploaded by Roux D on Aug 29, 2017
Content may be subject to copyright.
Influence of the nozzle shape on the breakup
behaviour of continuous ink jets
Maxime Rosello
Laboratoire Rh´
eologie et Proc´
ed´
es,
Univ. Grenoble Alpes, LRP,
F-38000 Grenoble, France
CNRS, LRP,
F-38000 Grenoble, France
Guillaume Maˆıtrejean
Laboratoire Rh´
eologie et Proc´
ed´
es,
Univ. Grenoble Alpes, LRP,
F-38000 Grenoble, France
CNRS, LRP,
F-38000 Grenoble, France
Denis Roux
Laboratoire Rh´
eologie et Proc´
ed´
es,
Univ. Grenoble Alpes, LRP,
F-38000 Grenoble, France
CNRS, LRP,
F-38000 Grenoble, France
Pascal Jay
Laboratoire Rh´
eologie et Proc´
ed´
es,
Univ. Grenoble Alpes, LRP,
F-38000 Grenoble, France
CNRS, LRP,
F-38000 Grenoble, France
Bruno Barbet
Markem-Imaje Industries,
ZA de l’Armailler
9, rue Gaspard Monge
26501 Bourg-L´
es-Valence
France
Jean Xing
Markem-Imaje Industries,
ZA de l’Armailler
9, rue Gaspard Monge
26501 Bourg-L´
es-Valence
France
ABSTRACT
In the present work, the influence of nozzle shape on microfluidic ink jet breakup is investi-
gated. First, an industrial ink used in continuous inkjet (CIJ) printing devices is selected. Ink
rheological properties are measured to ensure an apparent Newtonian behaviour and a constant
surface tension. Then, breakup lengths and shapes are observed on a wide range of disturbance
amplitude for four different nozzles. Later on, ink breakup behaviours are compared to the linear
theory. Finally, these results are discussed using numerical simulations to highlight the influ-
ence of the velocity profiles at the nozzle outlet. Using such computations, a simple approach is
derived to accurately predict the breakup length for industrial CIJ nozzles.
Nomenclature
Re Reynolds number
We Weber number
Oh Ohnesorgue number
Fr Froude number
ρDensity (kg.m−3)
UVelocity (m.s−1)
R,rJet radius (m)
ηDynamic viscosity (Pa.s)
σSurface tension (N.m−1)
gGravity acceleration (m.s−2)
ρDensity (kg.m−3)
LNozzle length (m)
Din,Dout Nozzle inlet and outlet diameters (m)
fdPiezoelectric actuator drive frequency (Hz)
˙
γShear rate (s−1)
tsFree surface life time (s)
∆PMean pressure difference between the ink tank and air (Pa)
xDimensionless wave number
kDimensional wave number (s−1)
λInstability wave length m
R0Undisturbed jet radius (m)
Rin,Rout Nozzle inlet and outlet radii (m)
LbJet breakup length (m)
zAxial position (m)
AVoltage disturbance amplitude (V)
∆vVelocity disturbance amplitude (m.s−1)
ε0Initial radius disturbance (m)
γInstability growth rate (s−1)
τcRayleigh time (s)
Ucap Capillary velocity (m.s−1)
lTank-nozzle junction length (m)
LRVelocity profile relaxation length (m)
LDi f f ,LConv Diffusion and convection correction terms (m)
εl,εcLinear theory and corrected breakup length errors
1 Introduction
Ink jet printing technology contributes to a wide range of industrial applications, from traditional
labelling to micro devices manufacture. Various instrumentation methods can be used to this end [1,2]
including the Continuous Ink Jet (CIJ) method. This process relies on the application of a periodical
disturbance to an ink jet, typically using a piezoelectric actuator upstream to the nozzle. When the wave
length of the radial disturbance is bigger than the jet circumference, the Rayleigh capillary instability
is triggered [3], resulting in the jet breakup and the generation of ink droplets. Several parameters can
be adjusted in order to control the breakup dynamics, including the fluid viscosity, the jet speed and the
disturbance frequency.
Jet breakup mechanisms have been widely studied. Eggers [4] did an excellent review of the recent
significant work found in the literature. For Newtonian fluids, the jet breakup is affected by the com-
petition between inertial, viscous, interfacial and gravitational forces. The influence of each force is
commonly assessed by the three dimensionless numbers, Reynolds, Weber, and Froude numbers, which
correspond respectively to the ratio of inertia to the viscosity, inertia to the surface tension, and inertia
to the gravity as defined below.
Re =ρUR
η,(1)
We =ρRU2
σ,(2)
Fr =U2
gR ,(3)
Here ρ[kg.m−3]is the fluid density, η[Pa.s]the fluid dynamic viscosity, R[m]the jet radius, U[m.s−1]the
mean velocity flow, σ[N.m−1]the surface tension between the fluid and surrounding air and g(m.s−2)
the gravity acceleration.
In most studies concerning capillary breakup description, the Ohnesorgue number,
Oh =√We
Re =η
√ρσR,(4)
is also used. This number describes the influence of viscosity over surface tension.
The earliest experimental study of capillary breakup process was performed by Savart [5] in 1833.
However, the first mathematical description of the instability growth is attributed to Lord Rayleigh [3].
He performed a linear analysis of the propagation of small radial disturbance on an infinite inviscid fluid
cylinder, assuming a negligible influence of both the surrounding gas and gravity. As a result, he pointed
out that the cylinder remains stable when the initial disturbance wave length is smaller than the cylinder
circumference. Otherwise, the disturbance grows and eventually breaks the jet up into droplets. Lord
Rayleigh also calculated the value of the optimal wave length for which the breakup is the fastest.
A few years after his first publication, Rayleigh studied the linear breakup of a viscous filament [6].
Later on, many improvements were made to Rayleigh’s linear theory. For instance, Chandrasekhar [7]
considered the first Navier-Stokes case and Tomotika [8] took into account the influence of surrounding
gas. In 1973, the static cylinder was replaced by a moving jet by Keller, Rubinow, and Tu [9]. In the
meantime, linear stability analysis was found unable to predict the jet breakup full behaviour. Indeed,
the generation of secondary droplets between main droplets, the so-called satellites, takes it origin in
the growth of non linear waves during the instability process. Non linear analysis were then performed
by several authors such as Yuen [10]. Eventually, a decade later, Pimbley and Lee described the satellite
generation mechanism using non linear spatial analysis successfully compared to experiments [11].
Most of the references cited above use the assumption of a constant axial velocity profile at the
nozzle exit. However, the development of a velocity boundary layer inside the nozzle [12] leads to
a non-uniform axial velocity distribution and a non-null radial velocity component. During jetting,
the zero shear-stress implied by the jet free-surface condition results in a velocity profile relaxation
which affects the instability growth. This relaxation process has been observed experimentally using
reconstructed velocimetry for undisturbed jets [13]. McCarthy and Molloy [14], Levannoni [15] and
Lopez [16] performed several experiments in order to assess the influence of the flow inside the nozzle
onto the jet dynamics and breakup. Leib and Goldstein [17, 18] chose to highlight the influence of the
nozzle shape through the influence of the jet velocity profile at the nozzle exit. They pointed out that
a parabolic velocity profile leads to a slower instability growth than plug flows. Eventually, Garcia et
al. [19] took into account nozzle contraction and calculated the breakup length of harmonically disturbed
capillary jets with an analytical method.
Nozzles used in CIJ devices have small aspect ratios L/Dout , with Lthe full nozzle length and Dout
the outlet diameter. Thus, the boundary layer is not fully developed in the fluid, resulting in a non-
parabolic velocity profile at the nozzle exit. To the author’s knowledge, there are no analytical breakup
behaviour descriptions in such cases. The development of such a description is the main goal of the
present work.
In the present study, the inks were diligently selected and characterized. As a matter of fact, in
this study, the use of an industrial ink instead of a model fluid was preferred so that the CIJ processes
conditions could be reproduced more precisely. However, the polymer composition of the fluid can
make the ink non-Newtonian given the high shear rates within the nozzle. It is critical to that an accurate
characterization of rheological properties of this ink is performed if one hopes to accurately predict the
velocity profile within the nozzle and downstream of the nozzle within the jetting fluid.
In chapter 4, the results from a series of experiments that were carried out in CIJ conditions for
several nozzle shapes and for a wide range of disturbance amplitude are presented. Breakup lengths and
shapes are discussed for both the linear and the non-linear regime. Of particular interest are phenomena
such as ‘inverted’ breakup at high disturbance amplitude. These results and their sensitivity to fluid
rheological parameters are highlighted and discussed in details. All breakup lengths are compared to
linear analysis description and discrepancies are investigated. Finally, series of numerical simulations
of the fluid flow inside the nozzle is presented and a quantitative correction of the analytical breakup
length prediction using the computational velocity profile is proposed.
2 Experimental setup
The jetting experimental device is depicted on figure 1. The axisymmetric fluid is jetted at high
speed (10m.s−1<U<20 m.s−1) through an axisymmetric micro–nozzle (Dout '50 µm). Four different
nozzle designs (Figure 2) with the same exit radius were tested (see Table 1 for exact dimension ratios).
A periodical pressure disturbance was applied to the jet using a ceramic piezoelectric actuator upstream
to the nozzle entry. The pressure disturbance was converted to a velocity disturbance inside the nozzle,
which propagated to the outgoing jet through the radius oscillations predominantly driven by the surface
tension.
The JetXpert c
imaging setup, similar to extensional rheometry measurement (ROJER) setup by
Keshavarz and al. [20], was used to image and track the breakup of the jet into droplets.. The jet was
strobed at given frequency synchronized with the drive frequency (10kHz <fd<100kH z) in order
to display a static image (1024 ×778 pixels with 1 px '1µm ). This method was very convenient as it
helped to avoid software and hardware challenges related to high-speed imaging. The visualization soft-
ware used was ImageXpert. The nozzle was mounted on a vertical axis translation stage controlled by a
stepper motor. The motor position was known with an accuracy of 1µm to ensure a better measurement
of the wave and breakup length.
3 Fluid characterization
The present study aims at highlighting the influence of the nozzle design on the capillary jet breakup.
In order to ensure high quality specification (quick drying, grip, heat / water resistance, etc), polymers
and resins are commonly added to ink composition. As a result, the high shear rate in the nozzle can
result in shear thinning effect (i.e. a viscosity decrease) due to molecular reorganisation. In addition,
potential molecular migration from the bulk to the interface might be experienced, resulting in dynamic
surface tension effects. Consequently, dimensionless groups like the Ohnesorgue number have a chance
to be flow dependant even though the nozzle radius was kept constant. Therefore, a diligent fluid
characterisation was performed in order better understand experimental flow regimes tested.
First, the dynamic viscosity was measured for both low (10s−1<˙
γ<1000s−1) and high (1000 s−1<
˙
γ<1,000,000s−1) shear rates with a rotational rheometer (ARG2 from TA-Instruments c
) and a capil-
lary rheometer (m-VROC from RheoSense c
) respectively. The combined results of these experiments
are depicted on Figure 3. Despite a slight decrease around 106s−1, the viscosity remains constant over
the whole shear rate range at a value of η=5.0±0.1mPa.s. In conclusion, the ink used in this study do
not demonstrate any significant shear thinning effects over the shear rate range considered.
The dynamic surface tension was also measured over a wide range of free surface life time (100µs ≤
ts≤1s). This characterization was performed with a Maximum Bubble Pressure method (MPT2 by
Lauda c
) and the surface tension was found to remain constant and equal to the solvent surface tension
of σ=22.8mN/m.
These experiments suggest that the Ohnesorgue number is not flow dependant and that the selected
industrial ink exhibits an Newtonian behaviour.
4 Breakup behaviour
4.1 Breakup length
A wide sweep of amplitude disturbances was performed for each nozzle. For each case, the wave
length, λ, was measured and the pressure difference, ∆P, was chosen to ensure a constant reduced wave
number of
x=kR0=2πR0
λ=0.6.(5)
Here R0is unperturbed jet radius. The Reynolds and Ohnesorgue number were constant and equal to
Re =110 and Oh =0.2. The Froude number was negligible with respect to Fr 1, so it is safe to
assume that gravity had no influence on the jet flow. The flow was thus laminar and the breakup driven
by a competition between inertial and capillary effects.
The jet breakup length, which is defined as the distance between the nozzle plate and the first jet
breakup, is a crucial parameter for inkjet printheads development. Indeed, this distance defines the
minimum space between the print head and the substrate. In order to avoid potential disturbances during
the jet flight, the breakup length is often minimized in industrial devices.
In figure 4, the value of the dimensionless breakup length Lb/Rout (with Rout =Dout /2 the nozzle exit
radius) is shown for all nozzles as a function of the disturbance amplitude applied to the piezoelectric
actuator. As predicted by linear and non-linear theories, the breakup length was found to decrease with
increasing amplitude. For the lowest amplitudes studied, the breakup length followed an exponential
decrease as the applied disturbance amplitude was increased. In this disturbance range, the jet breakup
is governed by linear growth of the imposed disturbance. This is why this regime is often defined as the
’linear regime’ [21]. In this regime, all nozzles were observed to present exponential decrease of the
breakup length with similar slopes, as predicted by Rayleigh’s theory [11]. Differences between nozzles
linear breakup lengths will be discussed in detail in section 5.2 using velocity profile computations.When
the disturbance amplitude was increased, the breakup length exponential slope was not observed any
more. In this regime, non linear effect have significant influence on the dynamics of the instability
development. As a result, in this regime, Rayleigh’s linear theory can no longer predict the breakup
length evolution of the jet as a function of the disturbance amplitude. From the ink studied here, the
transition from the linear to the non-linear regime was found to occur at an amplitude of approximately
20V.
In order to differentiate the effects of the different nozzle shapes, a series of zoomed in images of the
droplet breakup are presented in figure 4. The contrasts in the jet breakup resulting from the four differ-
ent upstream are significant. In particular, notice that nozzle N4 (filled triangles on figure 4) exhibited
the shortest breakup length in the linear regime. However, within the non-linear regime it displayed an
increase of the breakup length marked by the transition into an ‘inverted breakup’ phenomenon. This
particular phenomenon will be discussed in more details in section 4.2.
4.2 Breakup shape
The jet morphology close to the jet breakup is depicted on figures 5 for a constant wave number
x=0.6 and for the same range of disturbance amplitude as in section 4.1 (i.e. 2 Vto 60V).
In the linear perturbation regime (i.e. for disturbance amplitude lower than 20V), breakup dynam-
ics were found to be similar for all four nozzles and highlight four particular breakup shapes already
observed by Pimbley and Lee [11].
1. As seen in Figure 6a, the fluid thread between drops were first found to break downstream and then
upstream. The time interval between these breakups is called the ‘satellite interaction time’. During
this interval, momentum is transferred from the upfront droplet to the satellite, which slows it down,
resulting in a rear-merging satellite.
2. When the disturbance amplitude was increased, the ‘satellite interaction time’ was found to decrease
until the back and front break occur at the same time, as can be sees in Figure 6b. In such cases, the
satellite has the same velocity as the main drops and does not merge. This particular condition is
called ‘infinite satellite condition’.
3. As the disturbance amplitude was further increased, the back break was found to occur before the
front break (Figure 6c ). As a result, the satellite was found to accelerate during the ‘satellite inter-
action time’ and forward-merged with the leading droplet.
4. Finally, at the highest amplitude disturbances studied in the linear range, the front break up no longer
occurred, and the satellite vanished, as presented in figure 6d. The absence of satellite and the low
breakup length obtained in this disturbance amplitude range makes this disturbance amplitude the
most suitable for industrial printing.
In the non linear regime (i,e, higher than 20V), particular breakup shapes can be observed (see
figures 5). While big drops with small tails were observed for nozzles N1, N2 and N3, nozzle N4
exhibited an ‘inverted breakup’. As pointed out by Kalaadji et al. [22], the inverted breakup phenomenon
is accompanied by an increase of the breakup length, which is confirmed by the data in figure 4. More
precisely, we observed that the onset of breakup length increase matches exactly with the onset of the
inverted breakup satellite dynamics.
To the author’s knowledge, there are currently no theory able to explain the outbreak of the inverted
breakup phenomenon. In her thesis, McIlroy [23] made the assumption of a coupling between aero-
dynamic instability and velocity profile relaxation that would result in such breakup. In other words,
‘inverted breakup’ would occur when the jet velocity profile is not fully relaxed as a uniform plug flow
profile during the breakup. We will discuss this assumption in section 5.2.
5 Discussions
In this section, experimental results are compared with both analytical and numerical ones. A partic-
ular attention is paid to the linear regime behaviour as the literature provides more quantitative elements
of comparison for this range of disturbance.
5.1 Instability growth
The temporal linear theory predicts an exponential growth in time and a periodical evolution in space
for the capillary instability. Thus, a solution for the jet radius evolution (6) can be found [24] :
r(z,t) = R0+ε0cos(kz)eγt,(6)
with R0the unperturbed jet radius, ε0the initial radius disturbance, kthe wave number and γthe insta-
bility growth rate. Starting from this assumption, Rayleigh [6] found a dispersion relation (7) between
x=kR0and τcγ, respectively the dimensionless wave number and growth rate. τc=qρR3
0/σis the
characteristic time scale for the capillary instability, also called Rayleigh time.
τcγ=r1
2(x2−x4) + 9
4Oh2x4−3
2Ohx2,(7)
According to the spatial analysis of Keller et al. [9], Rayleigh’s temporal analysis gives the right
predictions for UUcap with Ucap =pσ/2ρRbeing the capillary velocity. This assumption is equiva-
lent to OhRe √2/2 . In the experiments performed here, OhRe '30, which supports the comparison
with temporal analysis dispersion relation.
The analytical dimensionless growth rate has been calculated using equation (7) and has been found
equal to γ=0.248. The experimental growth rate has also been calculated from the slopes of the breakup
length versus the disturbance amplitude for each nozzle (see figure 4.1). Results are summed up in Table
2. The experimental results are excellent agreement with the analytical predictions.
Pimbley and Lee’s (8) [11] gave the following relation between the breakup and the velocity distur-
bance ∆v:
LA
b=U tb=−U
γln π∆v
2λγ ,(8)
where tbis the breakup time. This calculation was derived from a one-dimensional temporal anal-
ysis [25] considering a periodical axial velocity as the inlet condition. In order to compare analytical
prediction to experiments, one must establish the link between the disturbance amplitude in volts and the
disturbance amplitude in m/s. To that end, the voltage disturbance amplitude was converted into mean
pressure disturbance amplitude in the flow before the nozzle. This conversion has been set industrially
for the piezoelectric actuator used in the present work and several inks and nozzles. In order to deter-
mine the correlation between between the pressure disturbance and the velocity disturbance, numerical
simulations were performed using OpenFoam. To simplify the simulations, only the flow through tank
upstream of the nozzle and the nozzle itself have been computed so that a single phase numerical simu-
lation could be performed using an axisymetrical geometries. A periodical Dirichlet boundary condition
was applied at the tank inlet and the variation of the mean axial velocity at the nozzle exit was obtained.
Within the linear disturbance regime, the conversion between pressure, voltage and velocity disturbance
amplitudes turned out to be all linear.
The comparison between analytical prediction for Lband experiments are shown on figure 7, where
the linear theory results are represented by the filled thin line. As predicted by Leib and Goldstein [17],
the breakup length calculated at any given amplitude for a plug velocity profile (temporal analysis)
was found to be smaller than the breakup length measured with any of the four different nozzles. As a
matter of fact, it is the relaxation of the non plug velocity profile after the nozzle exit that slows down the
instability growth and increases the breakup length. In the present work, nozzle N1 was found to deviate
the most from the linear theory for jet breakup length. Nozzle N1 also presents the shortest taper from
the tank to the nozzle and the longest straight cylindrical section. As a result, the development length
and the pressure loss due to wall friction are highest for nozzle N1, and the velocity profile of the jet
exiting nozzle N1 closest to a parabolic profile. Consequently, velocity profiles of the ink jets exiting
from nozzle N1 need more time to relax, resulting in a longer breakup length. In the following section,
a quantitative explanation of breakup length differences between nozzles will be discussed.
5.2 Velocity profile
Numerical simulations were performed using OpenFoam R
, and more specifically its two phase flow
solver InterFoam. This solver uses the Volume Of Fluid (VOF) method [26] to solve multiphase flow
problems. The flow inside the nozzle and the early stage of the jet were computed without disturbance
and a particular focus was given to the velocity profile relaxation after the nozzles exit. The geometry
was forced to be axisymmetric and meshes were structured. Each nozzles contained 94 cells in the axial
direction and 42 in the radial direction, the aspect ratio of all cells was set equal to 1. The timestep was
chosen automatically so that a mesh cell could not ”overflow”. To that end, the Courant number relating
the length of the time step to a function of the interval lengths of the grid and of the speed with which
information can travel in the physical space was introduced (9).
Co =kUk∆t
∆x<0.2 (9)
with ∆tthe timestep and ∆xthe size of a cell. Discretization schemes were chosen as Gauss linear
schemes for gradient and Laplacian operators. The divergence of the fluid volume fraction in the VOF
method was a Van Leer scheme.
The axial and radial velocity profiles at the exit of each nozzle are plotted in Figure 8 and 9 re-
spectively. As expected, the axial velocity profile was not constant, but was closer to a parabola due to
the shear flow inside the nozzle. The radial component of the velocity profile for each nozzle is much
smaller than the axial component (∀r∈[0,Rout ],Ur(r)/Uz(0)<2%) but is not zero.
As we can see on Figure 8, the velocity profiles are not Poiseuille’s profile. The development of
laminar pipe flows [12] scales as
Le
Dout
= [(0.619)1.6+ (0.0567Re)1.6]1/1.6.(10)
In the present study, Le=5.4L, which approves the non fully-developed velocity profile at the nozzle
exit.
Figure 8 provides some insight into the origin in the differences in breakup length in the linear
regime between the nozzle. The flatter the axial velocity profile , the faster it relaxes to a flat velocity
profile, and thus the shorter the breakup length. The trends predicted by the simulations in Figure 8 are
thus in excellent agreement with those in Figure 7. Nevertheless, nozzles N2 and N3 have the same
axial velocity profile even though jet breakup length exiting from them are different. Thus there seems
to be a correlation between radial velocity profile and breakup length.
The axial velocities along the jet are shown in Figure 10 both at the center (red) and at the surface
(green). These two velocities eventually converge, which depicts the velocity profile relaxation. Note
that all the nozzles exhibited a full relaxation at a distance LR'35Rout . Rupe [27] suggested that for
jet flows, the length required for a velocity profile to relax into a plug velocity profile is comparable
with the entry length for laminar flows in a pipe, i.e. Le=5.4L=12.7Rout . Here, this assumption is
supported.
In early stages of the jet flight, nozzle N4 was found to exhibit a velocity at the center of the jet much
lower than other nozzles. This could explain the high relative radial velocity component Ur/Uz(r=0)
observed on figure 9. Moreover, figure 10 highlights a velocity increase at the center of the jet for nozzle
N4 close to the nozzle (z<2Rout ).
Eventually, figure 10 depicts the same relaxation length for all nozzles. Furthermore, the ‘inverted
breakup’ observed for N4 occurs downstream the full velocity profile relaxation :
Lb'80Rout LR'35 Rout .(11)
Thus, inverted breakup occurs while the jet velocity profile is fully relaxed and we can not verify in
the present case McIlroy’s assumption [23], i.e. a coupling between aerodynamic forces and velocity
profile relaxation triggering. Further work needs to be done in order to investigate the role played by
radial velocity profile relaxation onto the onset of inverted breakup dynamics.
Although the axial velocity profile relaxation at the centerline and on the edges of the jet (Figure
10) does not provide any clue to quantify breakup length differences in linear regime, a more detailed
velocity distribution at the nozzle exit (Figures 8 and 9) might be use to that end.
The analytical solution LA
b(see eq. 8) gives the break up length assuming an inlet velocity plug flow.
However, as we have seen in industrial CIJ process, the nozzle outlet velocity is different from a plug
flow.
In order to account for the influence of the nozzle geometry which modifies the nozzle outlet ve-
locity, the analytically predicted break-up length LA
bcan be corrected by including the influence of the
effective (i.e. geometrically modified) inlet jet velocity profile. This influence is of two kinds:
1. the inlet jet velocity profile relaxes to a plug flow velocity profile form due to the viscous diffusion in
the jet. The velocity relaxation extends the breakup length as it requires an additional length (named
LDi f f
Rhereafter) for the jet to become a plug flow;
2. the radial velocity generated by the nozzle geometry carries the disturbance to the free surface of the
jet and thus contributes to the breakup. Conversely to the previous point, that phenomenon shortens
the final breakup length by reducing it by a length LConv
R.
For the first point, i.e. relaxation of the velocity profile to a plug flow by viscous effect, a simple
dimensional analysis gives the following LDi f f
Rlength:
LDi f f
R=ρ
ηZSkUz−UkdS (12)
where Sis the section of nozzle outlet and Uthe velocity norm. The term kUz−Ukcorresponds to the
difference of flow rate compared with an homogeneous flow field. The proposed solution is inspired
from the Prandtl problem solution of boundary layer [28]. The length LDi f f
Rin Eqn. (12) thus results
from the ratio between the diffusion time and the axial velocity component diverging from plug flow
velocity. It is worth pointing out that LDi f f
Rvanishes for plug flow inlet, i.e. when radial and axial
velocities are, respectively, null and constant.
As for the second term, LConv
R, it shortens the relaxation length by carrying momentum across the
jet. Furthermore one may assume that this term is not linear and, as the LDi f f
Rlength, it must vanish for
plug flow inlet velocity. With this assumption in mind,
LConv
R=C1
2
ρ
σZSkUrUzkdS (13)
with Ca constant representing the aerodynamical effects. One can see that LDi f f
Rand LConv
Rscale as,
respectively, Re ×Rand We ×R(with Rthe radius of the nozzle), the two dimensionless number char-
acterizing the Rayleigh-Plateau instability of a Newtonian jet.
Finally the analytical break-up length in the linear and moderate disturbance regime can be written
as
Lb=LA
b+LDi f f
R−LConv
R(14)
In Figure 11, the breakup length correction obtained for nozzles N1 to N4 using Eqn. (14) in the
linear regime (2V≤A(V)≤20V) is displayed. An excellent agreement between corrected predictions
(solid lines) and experimental dimensionless lengths (markers) is found when when Cis set to C=2π,
which supports the phenomenological approach retained in the present paper. Errors between analytical
prediction from linear theory for a flat velocity profile and experimental results were calculated and
were found to be between 25% ≤εl≤45% (see Table 3). The correction described by Eqn. (14) was
found to greatly improve the results, decreasing the relative error by a factor of nearly ten to εc≤5 %
(see Table 3). The breakup length correction (14) was particularly efficient for nozzle N1 with a relative
error between analytical and experimental results εc'1%, which is within the range of experimental
measurement error.
6 Conclusions and future work
In the present work, a Newtonian ink was jetted through four different nozzles and capillary breakup
lengths and shapes were observed. Satellite dynamics when the disturbance amplitude increases were
found to be in excellent agreement with linear theory. Then, with the help of numerical simulations, a
new correction method was introduced in order to predict breakup length differences between nozzles.
Although nozzle lengths and outlet radii were kept constant, topological changes in nozzle shapes
generated discrepancies in breakup dynamics. The flatter the velocity profile was at the nozzle exit, the
shorter the resulting breakup length. Conversely, a velocity profile close to Poiseuille profile resulted
in an increase of the breakup length due to the velocity profile relaxation. The present work aimed
at finding a method to address these differences both concerning the breakup length and the breakup
shapes.
Two characteristic lengths LDi f f and LConv were introduced with the goal of developing a correction
to the analytical breakup length. These lengths were calculated using a numerical approach to compute
velocity profiles. The simple correction suggestion developed within this paper was shown to be in
excellent agreement with experimental observations.
Further work is needed to include non-Newtonian behaviour to the current prediction method. Shear-
thinning as well as weakly elastic inks could eventually be used to cover the whole range of rheological
properties that inkjet processes address.
Finally, particular breakup shapes in the non linear regime are highlighted, although we could not
validate McIlroy’s assumption concerning ‘inverted breakup’ origins.
Acknowledgements
The authors thank Markem-Imaje c
, a Dover R
company, for their ongoing financial support, nozzle
manufacture and CIJ expertise. The authors would also like to thank Pr. Jonathan Rothstein for his
review and edits.
The Laboratoire Rh´
eologie et Proc´
ed´
es is part of the LabEx Tec 21 (Investissements d’Avenir -
grant agreement noANR-11-LABX-0030) and of the PolyNat Carnot Institut (Investissements d’Avenir
- grant agreement noANR-11-CARN-030-01).
References
[1] de Gans, B.-J., Duineveld, P. C., and Schubert, U. S., 2004. “Inkjet printing of polymers: state of
the art and future developments”. Advanced materials, 16(3), pp. 203–213.
[2] Hoath, S. D., 2016. Fundamentals of inkjet printing: The science of inkjet and droplets.
[3] Rayleigh, L., 1878. “On the instability of jets”. Proceedings of the London mathematical society,
1(1), pp. 4–13.
[4] Eggers, J., and Villermaux, E., 2008. “Physics of liquid jets”. Reports on progress in physics,
71(3), p. 036601.
[5] Savart, F., 1833. “M´
emoire sur la constitution des veines liquides lanc´
ees par des orifices circu-
laires en mince paroi”. Ann. Chim. Phys, 53(337), p. 1833.
[6] Rayleigh, L., 1892. “Xvi. on the instability of a cylinder of viscous liquid under capillary force”.
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(207),
pp. 145–154.
[7] Chandrasekhar, S., 2013. Hydrodynamic and hydromagnetic stability. Courier Corporation.
[8] Tomotika, S., 1935. “On the instability of a cylindrical thread of a viscous liquid surrounded by
another viscous fluid”. Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences, 150(870), pp. 322–337.
[9] Keller, J. B., Rubinow, S., and Tu, Y., 1973. “Spatial instability of a jet”. Physics of Fluids, 16,
pp. 2052–2055.
[10] Yuen, M.-C., 1968. “Non-linear capillary instability of a liquid jet”. Journal of Fluid Mechanics,
33(01), pp. 151–163.
[11] Pimbley, W., and Lee, H., 1977. “Satellite droplet formation in a liquid jet”. IBM Journal of
Research and Development, 21(1), pp. 21–30.
[12] Durst, F., Ray, S., ¨
Unsal, B., and Bayoumi, O., 2005. “The development lengths of laminar pipe
and channel flows”. Journal of fluids engineering, 127(6), pp. 1154–1160.
[13] Castrej´
on-Pita, J., Hoath, S., and Hutchings, I., 2012. “Velocity profiles in a cylindrical liquid jet
by reconstructed velocimetry”. ASME J. Fluids. Eng., 134(1), p. 011201.
[14] McCarthy, M., and Molloy, N., 1974. “Review of stability of liquid jets and the influence of nozzle
design”. The Chemical Engineering Journal, 7(1), pp. 1–20.
[15] Levanoni, M., 1977. “Study of fluid flow through scaled-up ink jet nozzles”. IBM Journal of
Research and Development, 21(1), pp. 56–68.
[16] Lopez, B., Soucemarianadin, A., and Attan´
e, P., 1999. “Break-up of continuous liquid jets: effect
of nozzle geometry”. Journal of Imaging Science and Technology, 43(2), pp. 145–152.
[17] Leib, S., and Goldstein, M., 1986. “The generation of capillary instabilities on a liquid jet”. Journal
of fluid mechanics, 168, pp. 479–500.
[18] Leib, S., and Goldstein, M., 1986. “Convective and absolute instability of a viscous liquid jet”.
Physics of Fluids (1958-1988), 29(4), pp. 952–954.
[19] Garc´
ıa, F., Gonz´
alez, H., Castrej´
on-Pita, J., and Castrej´
on-Pita, A., 2014. “The breakup length of
harmonically stimulated capillary jets”. Applied Physics Letters, 105(9), p. 094104.
[20] Keshavarz, B., Sharma, V., Houze, E. C., Koerner, M. R., Moore, J. R., Cotts, P. M., Threlfall-
Holmes, P., and McKinley, G. H., 2015. “Studying the effects of elongational properties on at-
omization of weakly viscoelastic solutions using rayleigh ohnesorge jetting extensional rheometry
(rojer)”. Journal of Non-Newtonian Fluid Mechanics, 222, pp. 171–189.
[21] Ashgriz, N., and Mashayek, F., 1995. “Temporal analysis of capillary jet breakup”. Journal of
Fluid Mechanics, 291, pp. 163–190.
[22] Kalaaji, A., Lopez, B., Attane, P., and Soucemarianadin, A., 2003. “Breakup length of forced
liquid jets”. Physics of Fluids (1994-present), 15(9), pp. 2469–2479.
[23] McIlroy, C., 2014. “Complex inkjets: particles, polymers and non-linear driving”. PhD thesis,
University of Leeds.
[24] Pimbley, W., 1976. “Drop formation from a liquid jet: a linear one-dimensional analysis considered
as a boundary value problem”. IBM Journal of Research and Development, 20(2), pp. 148–156.
[25] Lee, H., 1974. “Drop formation in a liquid jet”. IBM Journal of Research and Development, 18(4),
pp. 364–369.
[26] Hirt, C. W., and Nichols, B. D., 1981. “Volume of fluid (vof) method for the dynamics of free
boundaries”. Journal of computational physics, 39(1), pp. 201–225.
[27] Rupe, J. H., 1962. On the dynamic characteristics of free-liquid jets and a partial correlation with
orifice geometry. Jet Propulsion Laboratory, California Institute of Technology.
[28] Petit, L., Hulin, J.-P., and Guyon, ´
E., 2012. Hydrodynamique physique 3e ´
edition (2012). EDP
sciences.
List of Figures
1 Experimental setup for drop generation and visualization . . . . . . . . . . . . . . . . 21
2 Nozzle design used for the present study. . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Dynamic viscosity of ink as a function of the shear rate . . . . . . . . . . . . . . . . . 22
4 Breakup length as a function of disturbance amplitude for various nozzle designs . . . 23
5 Breakup shapes obtained for different nozzles. . . . . . . . . . . . . . . . . . . . . . . 24
6 Satellite dynamics evolution observed in the linear regime. . . . . . . . . . . . . . . . 25
7 Comparison between experimental breakup length and analytical predictions in the lin-
ear regime. The analytical prediction is in filled thin line. . . . . . . . . . . . . . . . . 26
8 Axial velocity profiles at the nozzle exit obtained using numerical simulations. . . . . . 27
9 Radial velocity profiles at the nozzle exit obtained using numerical simulations. . . . . 28
10 Axial velocities at the center of the jet and near the interface along the jet obtained using
numericalsimulations. .................................. 29
11 Comparison between experimental results (markers) and breakup length predictions
(solidlines)fromeq14................................... 30
Nozzle
Periodic disturbance
Piezoelectric actuator
Strobe light
JetXpert c
Camera
Fig. 1: Experimental setup for drop generation and visualization
L
l
Din
Dout
Fig. 2: Nozzle design used for the present study.
2
3
4
5
6
7
8
10 100 1000 10000 100000 1e+06
Dynamic viscosity (mP a.s)
Shear rate (s−1)
ARG2 m-VROC
Fig. 3: Dynamic viscosity of ink as a function of the shear rate
50
100
150
200
250
300
350
5 10 20 30 40 50 60
Breakup length (Lb/Rout)
Disturbance amplitude (V)
N1
N2
N3
N4
Fig. 4: Breakup length as a function of disturbance amplitude for various nozzle designs
(a) N1
(b) N2
(c) N3
(d) N4
Fig. 5: Breakup shapes obtained for different nozzles.
(a) Front break resulting in rear-merging satellite. (Noz-
zle N1, A=2V).
(b) Infinite satellite condition. The front break and back
break occur at the same time (Nozzle N1, A=10V).
(c) Back break resulting in forward-merging satellite
(Nozzle N1, A=13V).
(d) Back break without any satellite generation (Nozzle
N1, A=20V).
Fig. 6: Satellite dynamics evolution observed in the linear regime.
150
200
250
300
350
2 3 4 5 6 7
0.1 0.2
Breakup length (Lb/Rout)
Disturbance amplitude (V)
Disturbance amplitude ∆v(m.s−1)
Linear theory
N1
N2
N3
N4
Fig. 7: Comparison between experimental breakup length and analytical predictions in the linear regime.
The analytical prediction is in filled thin line.
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.5 0 0.5 1
Uz/Uz(r= 0)
r/Rout
N1
N2
N3
N4
Plug flow
Fig. 8: Axial velocity profiles at the nozzle exit obtained using numerical simulations.
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
-1 -0.5 0 0.5 1
Ur/Uz(r= 0)
r/Rout
N1
N2
N3
N4
Plug flow
Fig. 9: Radial velocity profiles at the nozzle exit obtained using numerical simulations.
0.2
0.4
0.6
0.8
1
1.2
1.4
0 5 10 15 20 25 30 35 40 45 50
Axial velocities U/Umax
Axial position z/Rout
Bulk
Interface
N1
N2
N3
N4
Fig. 10: Axial velocities at the center of the jet and near the interface along the jet obtained using
numerical simulations.
50
100
150
200
250
300
350
1 2 4 6 8 10 20
0.1 0.2 0.3 0.4
Lb/Rb
A(V)
∆v(m.s−1)
LA
b
N1
N2
N3
N4
Fig. 11: Comparison between experimental results (markers) and breakup length predictions (solid
lines) from eq 14.
List of Tables
1 Summary of nozzles dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Experimental and analytical dimensionless growth rates γ................ 33
3 Error calculation of the breakup length prediction from the linear theory and for the
present correction. The error is calculated on the linear regime (2V≤A(V)≤20V) . . 34
Nozzles N1N2N3N4
l/L0.11 0.23 0.31 0.64
Din/Dout 1.19 1.40 1.36 1.40
Table 1: Summary of nozzles dimensions
γ
N1 0.241
N2 0.248
N3 0.241
N4 0.248
Temporal analysis 0.248
Table 2: Experimental and analytical dimensionless growth rates γ
Nozzles Linear theory error (%) Correction error (%)
N1 41.7 1.0
N2 39.7 3.4
N3 37.7 4.9
N4 27.8 3.1
Table 3: Error calculation of the breakup length prediction from the linear theory and for the present
correction. The error is calculated on the linear regime (2V≤A(V)≤20V)
