In the bubble-jet printing process, it has been observed that the drop that ultimately pinches off from the ink jet sometimes moves sideways rather than straight relative to the symmetry axis of the liquid jet.



We examined various mechanisms that might lead to the deflection of the ink drop. In particular, we focused on whether the liquid filament that connects the lead drop to the nozzle is capable of supporting lateral waves which might propagate from the nozzle toward the lead drop and break the symmetry at pinch-off

Barannyk, L.

Buckmire, R.

Cumberbatch, E.

Kas-Danouche, S.

Nadim, A.

Platte, R.

Savettaseranee, K.

Vellante, H.

English

Mathematics in Industry Information Service Eprints Archive

Symmetry Breaking in Jetting
17th Annual Workshop on Mathematical Problems in Industry
Rensselaer Polytechnic Institute, June 4–8, 2001
A. Nadim, Claremont Graduate University, ali.nadim@cgu.edu
E. Cumberbatch, Claremont Graduate University, ellis.cumberbatch@cgu.edu
L. Barannyk, New Jersey Institute of Technology
S. Kas-Danouche, New Jersey Institute of Technology
K. Savettaseranee, New Jersey Institute of Technology
H. Vellante, Rensselaer Polytechnic Institute
R. Buckmire, Occidental College
R. Platte, University of Delaware
Problem Presenter: John R. Andrews, Xerox Corporation, jandrews@crt.xerox.com
May 2002
Abstract
In the bubble-jet printing process, it has been observed that the drop that ultimately
pinches off from the ink jet sometimes moves sideways rather than straight relative to
the symmetry axis of the liquid jet. Our group examined various mechanisms that might
lead to the deflection of the ink drop. In particular, we focused on whether the liquid
filament that connects the lead drop to the nozzle is capable of supporting lateral waves
which might propagate from the nozzle toward the lead drop and break the symmetry
at pinch-off.
1 Introduction
We were introduced to the design of an ink-jet printer manufactured by Xerox [1]. The
print-head consists of an array of parallel cylinders, each containing ink. The cylinders
have square cross-section (approximately 30 µm in size), one end being connected to the
ink supply tank and the other end connected to a nozzle which tapers down to a small
circular hole (nozzles of square, triangular, elliptical and other cross-sections are also in
use). In the wall of the cylinder near the taper there is an electric heating element. A pulse
of current through this element causes an almost instantaneous vaporization of the ink in its
proximity. This gaseous vapor bubble forces the ink between it and the hole at the cylinder
end to be ejected through the hole. The ink vaporizes and the ink is ejected within a few
1
microseconds. The refilling process takes tens of microseconds. The maximum pressure
reached within the vapor bubble is about 50 atmospheres, occurring within a fraction of
a microsecond. The velocity of the ejected drop is anywhere in the range 1–40 m/s. The
volume of the main drop is about 10 picoliters [1].
Quality printing will be achieved if: (a) The ink which issues from the hole as a jet
forms a drop at a forward location, which then severs from the filament connecting it to the
cylinder, (b) the drop is propagated along the intended direction, i.e., the symmetry axis of
the cylinder, and (c) the ink left behind retracts back into the cylinder under the suction
and surface tension forces caused by the rapid collapse of the ink vapor bubble.
However the scenario set out in (a), (b) and (c) is not always obtained. Other scenarios
in evidence include: (i) The filament trailing from the lead drop adopts a wavy form, breaks
up into a number of smaller, satellite drops which scatter off the symmetry axis. (ii) The
lead drop propagates at an angle to the symmetry axis.
Xerox introduced the problem of identifying the mechanism(s) causing (ii). The mis-
direction of drops clearly is a cause of low-quality printing. Unfortunately there was little
reproducibility of the circumstances causing (ii); one cylinder would misdirect for a few
drops, then would re-establish the design behavior. No change in parameter values could
consistently cause misdirection.
A variety of mechanisms for symmetry breaking were suggested. These were divided
into two broad categories: those somehow associated with surface tension, and those not.
The latter category included possible action of an aerodynamic “lift” force on the drop that
would act if the drop were spinning after pinch-off, an asymmetric blockage of the nozzle
leading to the jet being deflected at the nozzle itself, or the non-uniformity of the velocity
profile in the fluid that reaches the tapered nozzle. Possible causes associated with surface
tension included the existence of lateral waves (supported by the tension) in the filament
that connects the drop to the nozzle initially, off-center pinch-off of the lead drop from the
filament, asymmetry in the retraction of the ink into the nozzle during the collapse of the
vapor bubble, and meniscus asymmetry prior to ejection due to wetting non-uniformities at
the rim of the nozzle. In our group effort, we focused on the question of lateral waves on the
filament and on whether asymmetry in the retraction of the ink into the nozzle could create
waves that propagate along the filament and affect the lead drop. The mathematical model
we proposed to analyze these effects is introduced below and the results of our analysis
are given in separate sections. Some of the existing literature on breakup of liquid jets
[2, 3, 4, 5, 6] is listed in the bibliography at the end of this report.
2
2 Mathematical Model
Consider an inviscid liquid jet moving primarily in the horizontal direction (denoted by
x), whose cross-sectional area is assumed to remain circular, with radius h(x, t), but whose
centerline is allowed to have small vertical deflections from the x-axis. Denote the position
of the centerline of the jet by y(x, t) and let the density of the liquid jet and its surface
tension be given by ρ and σ, respectively. The average horizontal velocity of the jet at
position x as a function of time t is denoted by u(x, t). For an incompressible liquid, the
differential form of the law of conservation of volume of the jet thus takes the form
∂h2
∂t
+
∂(uh2)
∂x
= 0 . (1)
The pressure, p(x, t), within the slender jet is related to the ambient pressure, po, by
the Young-Laplace equation
p = po + σ
[
1
h
−
∂2h
∂x2
]
, (2)
where the two terms within the brackets represent the principal radii of curvature of the jet
surface, one within the circular cross-section of radius h, and the other in the vertical plane
passing through the jet. The pressure is assumed to be uniform across the jet cross-section.
As such, the horizontal momentum equation for the jet is given by
Du
Dt
=
−1
ρ
∂p
∂x
=
σ
ρ
∂
∂x
[
∂2h
∂x2
−
1
h
]
. (3)
Here,
D
Dt
=
∂
∂t
+ u
∂
∂x
(4)
is the substantial time-derivative.
The small vertical deflections of the jet from the symmetry axis are governed by a
vertical momentum equation that is quite similar to the equation for a taut string under
tension. In the case of the jet, the equivalent tension is given by 2pihσ (the circumference
of the jet multiplied by the surface tension) and the vertical acceleration of the jet is given
by Dy/Dt. Applying Newton’s second law to a segment of the jet between x and x +∆x
yields:
D
Dt
[
ρpih2∆x
Dy
Dt
]
=
[
2pihσ
∂y
∂x
]∣∣∣∣
x+∆x
−
[
2pihσ
∂y
∂x
]∣∣∣∣
x
. (5)
Dividing by ∆x and taking the limit ∆x→ 0 then yields
D
Dt
[
h2
Dy
Dt
]
=
2σ
ρ
∂
∂x
[
h
∂y
∂x
]
. (6)
Eqs. (1), (3) and (6) constitute the complete set of partial differential equations that
describe the dynamics of the jet of ink, under the assumptions that the jet remains circular,
3
its radius varies “slowly” with respect to x (i.e., on a length scale much larger than its
radius) and its centerline has small vertical deflections from the horizontal axis. Starting
with this set of equations, we now examine several problems that provide insight into the
behavior of the jet and sideways pinch-off of the lead drop.
3 Lateral Waves on a Uniformly Stretching Filament
Consider the situation where the relatively massive lead drop has been given an impulsive
motion with constant velocity Uo. A thin cylindrical filament of initial length xo trails be-
hind and connects the drop to the nozzle (at x = 0), see Fig. 1. Without any perturbations,
the cylindrical filament would stretch to length
X(t) = xo + Uot , (7)
at time t, and the velocity along the filament would vary linearly with distance according
as
u(x, t) =
Uo x
X(t)
. (8)
Conservation of volume then dictates that
h2(t)X(t) = h2oxo , (9)
where ho is the initial radius of the cylindrical filament. This uniformly stretching and
thinning motion of the filament represents an exact solution to the conservation of volume
and x-momentum equations, (1) and (3).
Starting with this “base flow,” we now allow for small deflections of the centerline of
the filament,1 y(x, t), and obtain the following equation for the evolution of y:
(
∂
∂t
+ u
∂
∂x
)[
h2
(
∂
∂t
+ u
∂
∂x
)
y
]
=
2σ
ρ
∂
∂x
(
h
∂y
∂x
)
, (10)
in which u(x, t) and h(t) are given by Eqs. (8) and (9).
Starting with this equation we now ask whether a lateral disturbance of the center-
line initiated at the nozzle (x = 0) can possibly reach the lead drop that is at position
X(t). Thinking of the filament as a taut string with tension 2pihσ and with mass per
unit length pih2ρ, the wave-speed along the filament would be approximated by (2σ/hρ)1/2
which increases as the filament radius decreases. Since the velocity of the lead drop remains
constant, any wave generated at the nozzle should thus ultimately reach the drop, unless
the drop pinches off earlier.
1A lateral disturbance could be caused by a non-symmetric retraction of the filament as the ink is being
sucked into the nozzle.
4
Figure 1: Sketch of the uniformly stretching filament.
To analyze this issue further, we effect the change of variables from (x, t) to (ξ, s) where
s = t (11)
ξ = xxo/(xo + Uot) (12)
thereby transforming Eq. (10) to the form
1
X
∂
∂s
(
1
X
∂y
∂s
)
= c2o
x
3/2
o
X7/2
∂2y
∂ξ2
. (13)
In this equation
c2o =
2σ
ρho
, (14)
which is the square of the wave-speed corresponding to the initial radius of the filament.
The additional change of variable
τ = xos+
Uos
2
2
so that
∂
∂τ
=
1
X
∂
∂s
(15)
simplifies the last wave equation to the form
∂2y
∂τ2
= c2o
x
3/2
o
X7/2
∂2y
∂ξ2
, (16)
which is a standard wave equation with a time-dependent wave-speed.
5
The equation for the right-going characteristics is given by
dξ
dτ
= co
x
3/4
o
X7/4
, (17)
with ξ = 0 at τ = 0 the proper initial condition to find that characteristic which starts at
the nozzle. In terms of t, this is given by
dξ
dt
= co
x
3/4
o
X3/4
with x(0) = 0 . (18)
This is solved by
ξ(t)
xo
=
4co
Uo
[(
1 +
Uot
xo
)1/4
− 1
]
. (19)
This characteristic reaches the lead drop when the left-hand side of the last equation equals
unity. This occurs at time
t∗ =
xo
Uo
[(
Uo
4co
+ 1
)4
− 1
]
. (20)
Apparently, the lead characteristic starting at the nozzle always reaches the lead drop, since
the filament is getting thinner over time and the wave-speed of lateral waves on the filament
continually increases, whereas the velocity of the lead drop remains constant. However, if
the drop pinches off before any such disturbance reaches it, there will not be any symmetry
breaking due to such lateral disturbances originating from the nozzle. If the characteristic
time for pinch-off is denoted by tpo, the main conclusion is that the ratio t
∗/tpo determines
whether the lead drop would travel in a straight path or if it might be deflected due to
lateral waves on the filament breaking the symmetry prior to pinch-off. (See the last part
of the Conclusions section for the orders of magnitude of some of these parameters.)
4 A Travelling Wave Description of the Receding Tail
We now consider whether the continuity and x-momentum equations, (1) and (3), which we
derived earlier can be used to actually describe the profile of the tail of the filament which is
receding into the nozzle. For this purpose, suppose that the centerline of the jet is straight
(y = 0) and that the nozzle is in the form of a long cylindrical tube of radius ho. Imagine
the jet to be receding in a steady fashion into the nozzle, with the jet interface making a
specified contact angle with the nozzle wall, see Fig. 2. Denote by V the velocity at which
the contact angles are receding along the tube. Upon scaling h and x with ho, and u with
V , in a frame of reference which is itself receding with speed V so that the jet appears to
be stationary, the dimensionless jet profile h(x) and horizontal velocity u(x) satisfy the pair
of ODEs
uu′ =
1
W
(h′′ − h−1)′ , (21)
(uh2)′ = 0 , (22)
6
Figure 2: Sketch of the receding jet of ink.
derived from Eqs. (1) and (3). These are to be solved subject to the conditions that at
x = 0 (which is the position where the contact lines touch the walls of the tube):
u(0) = 1 , h(0) = 1 , h′(0) = −α . (23)
In these equations, W = ρV 2ho/σ is the Weber number based on the velocity at which
the jet is receding, and α is related to the (receding) contact angle of the ink jet along
the nozzle. Integration of (22) yields uh2 = 1 using which u can be eliminated from the
equations to yield a single nonlinear second order ODE for h(x):
h′′(x) =
1
h
+
W
2h4
+ C , (24)
with
h(0) = 1 and h′(0) = −α . (25)
Here, C is a constant of integration. If we analyze this second-order ODE by considering
its two-dimensional phase plane (h, h′), we find that for negative values of C, the system
possesses a fixed point which turns out to be a center, with the trajectories forming closed
paths around that center. For positive values of C, there is no fixed point and all trajectories
are open in the phase plane. In either case, starting at x = 0, h(x) appears to decrease from
1 to reach a minimum value and either oscillate beyond that point (for the case when there is
a fixed point, see Fig. 3) or increase without bound (in the other case, see Fig. 4). That h(x)
reaches a minimum value turns out to be important in determining whether disturbances
generated at the contact lines can affect the pinch off of the lead drop connected to the jet.
We examine this issue in the next section.
0 2 4 6 8 10
0.6
0.8
1
1.2
1.4
1.6
x
h
h versus x, W=1, C=−1
Figure 3: h vs. x at W = 1, C = −1, α = pi/10.
0 0.05 0.1 0.15 0.2 0.25 0.3
0.98
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
x
h
h(x) versus x, W=1, C=1
Figure 4: h vs. x at W = 1, C = 1, α = pi/10.
8
5 Lateral Waves on the Receding Tail
Given the position dependent radius and velocity profiles of the jet, h(x) and u(x), from
the last section, we can examine the characteristics of lateral wave motion on the jet using
our vertical momentum equation (6). Substituting the steady solutions h(x) and u(x) into
Eq. (6), the result can be written as:
h2
{
∂2
∂t2
+
2
h2
∂2
∂t∂x
+
(
1
h4
−
2
Wh
)
∂2
∂x2
}
y = lower order derivatives . (26)
Since the operator on the left-hand side is of the generic form
L = a
∂2
∂t2
+ 2b
∂2
∂t∂x
+ c
∂2
∂x2
, (27)
its characteristics are the solutions to
dx
dt
= −b±
√
b2 − ac . (28)
For our problem, this reduces to:
dx
dt
=
−1
h2
±
√
2
Wh
. (29)
The minus sign on the right-hand side describes waves that move to the left, whereas the plus
sign might correspond to waves moving to the right, which are the ones we are interested
in. Therefore, depending upon whether the combination
−1
h2
+
√
2
Wh
(30)
is smaller or greater than zero, some waves may be able to propagate forward, toward the
lead drop. We thus find a critical value of h, given by
h∗ =
(
W
2
)1/3
, (31)
such that, for h < h∗ the characteristic speeds are negative while for h > h∗, they are
positive. This has an interesting implication. Namely, for a given Weber number W, if the
jet narrows to a thickness less than the critical value h∗ at some point after separating from
the nozzle, no waves generated at the nozzle itself can pass through that point. The smaller
the Weber number, the less likely it will be for the jet to become sufficiently thin to satisfy
this condition. For a large Weber number, however, this criterion is more easily satisfied.
Therefore, for stable symmetric operation of the ink jet printer, it is best to design for a
large Weber number.
9
6 Conclusions
The key results of our analysis are as follows: For the case when the lead drop moves
away from the nozzle at constant speed Uo, but remains connected to the nozzle through a
thinning cylindrical ink filament, in order to avoid asymmetric pinch-off, the pinch-off time
tpo must be designed to remain smaller than the value t
∗ given by
t∗ =
xo
Uo
[(
Uo
4co
+ 1
)4
− 1
]
, (32)
where xo is the initial length of the cylindrical filament and co =
√
2σ/ρho is the wave-speed
on the filament at the initial thickness ho. To make t
∗ large (compared to tpo), the ratio
Uo/co should be made as large as possible. The square of ratio is
(
Uo
co
)2
=
ρU2oho
2σ
, (33)
which is simply a Weber number based on the initial velocity of the lead drop.
At the same time, if another Weber number,
W =
ρV 2ho
σ
, (34)
defined based on the speed V at which the ink jet is receding into the nozzle and the nozzle
radius ho is made large enough, it may be possible to prevent any disturbance generated at
the nozzle to reach the lead drop. Both of these criteria suggest that designers of ink-jet
printers should aim for large Weber number operation. That is, the lead drop must be given
a sufficiently large impulsive velocity, and the ink and nozzle should be designed so that
the volume of ink that is not ejected in the form of a drop can recede quickly back into the
reservoir behind the nozzle.
Finally, note that using the characteristic values: ρ = 1000 kg/m3, σ = 0.07 N/m,
ho = 1.5 × 10
−5 m and Uo = 5 m/s, the initial velocity of lateral waves on the filament
would be approximately co ≈ 3 m/s, leading to co/Uo ≈ 0.6 and We = ρU
2
oho/σ ≈ 2.7. The
velocity of the lead drop itself is consistent with an estimate one could obtain by taking the
“impulse” delivered by the initial pressure (e.g., a pressure of 10 atmospheres acting over a
1 µs time interval, on an area given by pih2o), divided by the mass of the slug of ink that is
ejected, pih2oLρ, with L = 100 µm. This calculation yields a velocity of about 10 m/s which
is consistent with observations. We also estimated the Reynolds number of the translating
drop based on the viscosity of the surrounding air and found it to be about 10, suggesting
that in the aerodynamics inertia is important but not dominant. Our model, however, did
not include viscous effects.
10
References
[1] J.R. Andrews, “Symmetry-Breaking in Jetting,” Copies of presentation slides, MPI
Workshop, RPI, June 2001.
[2] N. Ashgriz & F. Mashayek, “Temporal Analysis of Capillary Jet Breakup,” J. Fluid
Mech., 291, 163–190 (1995).
[3] S.A. Berger, “Initial-Value Stability Analysis of a Liquid Jet,” SIAM J. Appl. Math.,
48, 973–991 (1988).
[4] J.N. Dewynne & P. Wilmott, “Slender Axisymmetric Fluid Jets,” Comput. Modelling,
18, 69–82 (1993).
[5] J. Eggers, “Nonlinear Dynamics and Breakup of Free-Surface Flows,” Rev. Modern
Phys., 69, 865–929 (1997).
[6] H.Q. Yang, “Asymmetric Instability of a Liquid Jet,” Phys. Fluids A, 4, 681–689
(1992).
11


Symmetry Breaking in Jetting

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