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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).

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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)