Nonlinear dynamics and breakup of free-surface flows
Jens Eggers
Universita
¨
t Gesamthochschule Essen, Fachbereich Physik, 45117 Essen, Germany
Surface-tension-driven flows and, in particular, their tendency to decay spontaneously into drops have
long fascinated naturalists, the earliest systematic experiments dating back to the beginning of the 19th
century. Linear stability theory governs the onset of breakup and was developed by Rayleigh, Plateau,
and Maxwell. However, only recently has attention turned to the nonlinear behavior in the vicinity of
the singular point where a drop separates. The increased attention is due to a number of recent and
increasingly refined experiments, as well as to a host of technological applications, ranging from
printing to mixing and fiber spinning. The description of drop separation becomes possible because jet
motion turns out to be effectively governed by one-dimensional equations, which still contain most of
the richness of the original dynamics. In addition, an attraction for physicists lies in the fact that the
separation singularity is governed by universal scaling laws, which constitute an asymptotic solution of
the Navier-Stokes equation before and after breakup. The Navier-Stokes equation is thus continued
uniquely through the singularity. At high viscosities, a series of noise-driven instabilities has been
observed, which are a nested superposition of singularities of the same universal form. At low
viscosities, there is rich scaling behavior in addition to aesthetically pleasing breakup patterns driven
by capillary waves. The author reviews the theoretical development of this field alongside recent
experimental work, and outlines unsolved problems. [S0034-6861(97)00303-6]
CONTENTS
I. Introduction 865
II. Experiments 869
A. Jet 869
B. Dripping faucet 872
C. Liquid bridge 873
III. Simulations 874
A. Inviscid, irrotational flow 875
B. Stokes flow 876
C. Navier-Stokes simulations 877
IV. Small Perturbations 878
A. Linear stability 878
B. Spatial instability 882
C. Higher-order perturbative analysis 883
V. One-Dimensional Approximations 885
A. Radial expansion method 886
B. Averaging method: Cosserat equations 888
C. Basic properties and simulations 889
D. Inviscid theory and conservation laws 892
VI. Similarity Solutions and Breakup 894
A. Local similarity form 894
B. Before breakup 895
C. Stability and the influence of noise 897
D. After breakup 900
VII. Away From Breakup 904
A. High viscosity—threads 904
B. Low viscosity—cones 907
C. Satellite drops 911
VIII. Related Problems 914
A. Two-fluid systems 914
1. Stationary shapes 915
2. Breakup 917
B. Electrically driven jets 919
C. Polymeric liquids 922
IX. Outlook 924
Acknowledgments 925
References 926
I. INTRODUCTION
The formation of drops is a phenomenon ubiquitous
in daily life, science, and technology. But although it is
plain that drops generically result from the motion of
free surfaces, it is not easy to predict the distribution of
their sizes or to observe the intricate dynamics involved;
see Fig. 1. Only the extremely short flash used by the
photographer, Harold Edgerton, clearly reveals the for-
mation of individual drops. Thus the subject has been
far from exhausted after more than 300 years of scien-
tific research, which in fact has gained considerable mo-
mentum only recently. On the one hand, the reason for
this interest lies in the tremendous technological impor-
tance of drop formation in mixing, spraying, and chemi-
cal processing, which leads to applications such as ink-jet
printing, fiber spinning, and silicon chip technology. On
the other hand, the modern theory of nonlinear phe-
nomena has created a new paradigm of self-similarity
and scaling, which opened a new perspective on this
classical problem.
The first mention of drop formation in the scientific
literature is in a book by Mariotte (1686) on the motion
of fluids. He notes that a stream of water flowing from a
hole in the bottom of a container decays into drops. Like
many authors after him, he assumes that gravity, or
other external forces, are responsible for the process. A
simple estimate shows, however, that uniform forces
cannot lead to drop formation and that another force,
which of course is surface tension, is responsible for the
eventual breakoff of drops. For reasons of mass conser-
vation, the rate at which the minimal cross section of a
fluid filament decreases is proportional to the cross sec-
tion itself, multiplied by an axial velocity gradient. As
long as this gradient is finite, as is to be expected when
only uniform forces are acting, the decrease of the mini-
mum thickness will be at most exponential, leading to
separation only in infinite time.
The basis for a more thorough understanding of drop
formation was laid by Savart (1833), who very carefully
investigated the decay of fluid jets. By illuminating the
jet with sheets of light, he observed tiny undulations
growing on a jet of water, as shown in Fig. 2. These
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undulations later grow large enough to break the jet.
Savart’s research showed that (i) breakup always occurs
independent of the direction of gravity, the type of fluid,
or the jet velocity and radius, and thus must be an in-
trinsic property of the fluid motion; (ii) the instability of
the jet originates from tiny perturbations applied to the
jet at the opening of the nozzle.
In spite of his fundamental insights, Savart did not
recognize surface tension, which had been discovered
some years earlier (de Laplace, 1805; Young, 1805) as
the source of the instability. This discovery was left to
(Plateau 1849), who showed that perturbations of long
wavelength reduce the surface area and are thus favored
by surface tension. On a level of quasistatic motion it
would thus be desirable to collect all the fluid into one
sphere, corresponding to the smallest surface area. Evi-
dently, as shown in Fig. 1, this does not happen. It was
Rayleigh (1879a,1879b) who noticed that surface tension
has to work against inertia, which opposes fluid motion
over long distances. By considering small sinusoidal per-
turbations on a fluid cylinder of radius r, Rayleigh found
that there is an optimal wavelength, l
R
'9r, at which
perturbations grow fastest, and which sets the typical
size of drops. Analyzing data Savart had obtained al-
most 50 years earlier, Rayleigh was able to confirm his
theory to within 3%.
Accordingly, the time scale t
0
on which perturbations
grow and eventually break the jet is given by a balance
of surface tension and inertia, and thus
t
0
5
S
r
3
r
g
D
1/2
, (1)
where
r
is the density and
g
the coefficient of surface
tension of the fluid. This tells us two important things:
Substituting the values for the physical properties of wa-
ter and r5 1 mm, one finds that t
0
is 4 ms, meaning that
the last stages of pinching happen very fast, far below
the time resolution of the eye. Secondly, as pinching
progresses and r gets smaller, the time scale becomes
shorter and pinching precipitates to form a drop in finite
time. At the pinch point, the radius of curvature goes to
zero, and the small amount of fluid left in the pinch
region is driven by increasingly strong forces. Thus the
velocity goes to infinity, and the separation of a drop
corresponds to a singularity of the equations of motion,
in which the velocity and gradients of the local radius
diverge.
Even in the case of an infinite-time singularity of the
equations of hydrodynamics, the physical event of
breaking may occur in finite time. That is, when the fluid
thread has become sufficiently thin, it may break owing
FIG. 1. A dolphin in the New England
Aquarium in Boston, Massachusetts; Edger-
ton (1977). © The Harold E. Edgerton 1992
Trust, courtesy of Palm Press, Inc.
866
Jens Eggers: Nonlinear dynamics and breakup of free-surface flows
Rev. Mod. Phys., Vol. 69, No. 3, July 1997
to microscopic effects that are outside the realm of hy-
drodynamics. The crucial distinction from the finite-time
singularity, which results from surface tension, is that
there is a chance to describe breaking in terms of con-
tinuum mechanics alone without resorting to micro-
scopic notions. The description of such singularities will
form a substantial part of this review.
For one hundred years after Rayleigh’s original work,
theoretical research focused on the extension of his re-
sults on linear stability. For example, Rayleigh (1892)
himself considered a highly viscous fluid, but the general
Navier-Stokes case was only treated in 1961 by Chan-
drasekhar. Tomotika (1935) took the surrounding fluid
into account; Keller, Rubinow, and Tu (1973) looked at
the growth of a progressive wave rather than a uniform
perturbation of a cylinder. To illustrate the power of
Rayleigh’s ideas even in completely different fields we
mention the application of linear stabilty to the breakup
of nuclei (Brosa, Grossmann, and Mu
¨
ller, 1990), in
which case the equivalent of a surface tension has to be
calculated from quantum mechanics. Another example
is the instability observed on pinched tubular vesicles
(Bar-Ziv and Moses, 1994), where entropic forces drive
the motion.
Meanwhile, experimental results had accumulated
that probed the dynamics of free surfaces beyond the
validity of linear theory. Early examples include Ray-
leigh’s photographs of jets (1891), Worthington’s study
of splashes (1908), and Edgerton, Hauser, and Tucker’s
(1937) photographic sequences of dripping faucets. Ex-
perimental techniques have also become available more
recently with sufficient resolution in space and time to
look at the immediate vicinity of the point of breakup.
Notable examples include the jet experiments of Rut-
land and Jameson (1970) as well as those of Goedde and
Yuen (1970) for water jets and of Kowalewski (1996) for
jets of high-viscosity fluids. A momentous paper by Per-
egrine, Shoker, and Symon (1990) not only helped to
crystallize some of the theoretical ideas, but also con-
tained the first high-resolution pictures of water falling
from a faucet. For higher viscosities, corresponding pic-
tures were taken by Shi, Brenner, and Nagel (1994).
By comparison, the development of computer codes
that would permit the calculation of free-surface flows
from first principles has been slow. Owing to the diffi-
culties involved in implementing both moving bound-
aries and surface tension, resolution has not been pos-
sible anywhere near the experimentally attainable limit,
even with present-day computers. An important excep-
tion is the highly damped case of the breakup of a vis-
cous fluid in another, which recently led to a detailed
comparison between experiment and numerical simula-
tion (Tjahjadi, Stone, and Ottino, 1992).
Only gradually did the theoretical tools evolve that
allowed for an analytical description of the nonlinear
dynamics close to breakup. The first was developed in
the theory of waves and often goes by the name of ‘‘lu-
brication theory’’ or ‘‘the shallow-water approximation’’
(Peregrine, 1972). It captures nonlinear effects in the
limit of small depths compared with a typical wave-
length. During the 1970s, lubrication approximations
were developed for the corresponding axisymmetric
problem, to study drop formation in ink-jet printers.
This is of particular relevance since a jet does not break
up uniformly, as predicted by linear theory, but rather
into main drops and much smaller ‘‘satellite’’ drops. The
satellite drops fundamentally limit the print quality at-
tainable with this technology, as drops of different sizes
are deflected differently by an electric field, which
should direct the stream of droplets to a given position
on the paper. Thus a fully nonlinear theory is needed to
understand and to control satellite formation. The first
dynamical equation, based on lubrication ideas, was in-
troduced by Lee (1974) for the inviscid case. His nonlin-
ear simulations indeed showed the formation of satellite
drops. But it took two decades until systematic approxi-
mations of the Navier-Stokes equation were found that
included viscosity (Bechtel, Forest, and Lin, 1992; Egg-
ers and Dupont, 1994).
FIG. 2. Perturbations growing on a jet of water (Savart, 1833).
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Rev. Mod. Phys., Vol. 69, No. 3, July 1997
Another important concept, which allows for the de-
scription of nonlinear effects, is that of self-similarity
(Barenblatt, 1996), which arises naturally in problems
that lack a typical length scale. In the case of a singular-
ity, the length scale of the solution will depend on time,
reaching arbitrarily small values in the process. Thus
self-similarity here means that the solution, observed at
different times, can be mapped onto itself by a rescaling
of the axes. In the context of flows with surface tension,
self-similarity was introduced by Keller and Miksis
(1983). Kadanoff and his collaborators (Constantin
et al., 1993; Bertozzi et al.., 1994) have looked at singu-
larities in a Hele-Shaw cell, which is the two-
dimensional analogue of the present problem, as a
simple model for singularity formation. They inge-
niously combined lubrication ideas and self-similarity to
arrive at a detailed description of the pinchoff of a
bubble of fluid.
In the wake of this success, Eggers (1993) and Eggers
and Dupont (1994) applied the same idea to the three-
dimensional case. As spelled out first by Peregrine et al.
(1990), the dynamics near breakup are independent of
the particular setup such as jet decay, a dripping faucet,
or even the complicated spraying shown in Fig. 1, but
rather are characteristic of the nonlinear properties of
the equations of motion. As the motion near a point of
breakup gets faster, only fluid very close to that point is
able to follow, making the breakup localized both in
space and time. Thus one expects the motion to become
independent of initial conditions, and the type of experi-
ment becomes irrelevant to the study of the singular mo-
tion. This brings about two crucial simplifications: (i) in
a local description around the point of breakup, the mo-
tion becomes ‘‘universal,’’ thus reducing the number of
relevant parameters. The only parameter upon which
the motion near the singularity still depends is the length
l
n
5
n
2
r
g
, (2)
which characterizes the internal properties of the fluid
(Peregrine et al., 1990; Eggers and Dupont, 1994); (ii) an
asymptotic analysis of the equations of motion reveals
that the motion close to the singularity is self-similar,
with the radius shrinking at a faster rate than the longi-
tudinal extension of the singularity. Near the pinch
point, almost cylindrical necks develop, making the mo-
tion effectively one-dimensional close to the singularity.
Using these ideas, a local solution of the Navier-
Stokes equation was found, which contained no free pa-
rameters (Eggers, 1993). To select a specific prediction
of this theory, the minimum radius of a fluid thread at a
given time Dt away from breakup found to be
h
min
5 0.03
g
rn
Dt. (3)
The surface profiles calculated from theory have been
compared quantitatively and confirmed by experiment
(Kowalewski, 1996). The columnar structure of the fluid
neck allows for a stability analysis of the flow close to
the breaking point, and is modeled closely on Rayleigh’s
analysis of a liquid cylinder (Brenner, Shi, and Nagel,
1994; see also Brenner, Lister, and Stone, 1996). As the
neck becomes sufficiently thin, it is prone to a finite-
amplitude instability, which may be driven by thermal
noise. This causes secondary necks to grow on the pri-
mary neck, which again have a self-similar form. The
corresponding complicated structure of nested singulari-
ties has also been observed experimentally (Shi, Bren-
ner, and Nagel, 1994).
At the same time stability analysis indicates that cy-
lindrical symmetry is not just a matter of convenience,
but rather a generic property near breakup. Rayleigh’s
analysis tells us that any azimuthal variation results in
only a relative increase in surface area and is thus unfa-
vorable. The universality and stability of the solution
near breakup therefore lead to answers of a much
greater generality than could be hoped for by investigat-
ing individual geometries and initial conditions. At the
same time, the singular motion is the natural starting
point for the calculation of nonlinear properties away
from breakup, which controls phenomena such as satel-
lite formation. Another advantage of universality is that
only one particular initial condition needs to be investi-
gated to construct a unique continuation of the Navier-
Stokes equation to times after the singularity (Eggers,
1995a). This establishes that breaking is described by
continuum mechanics alone, without resorting to a mi-
croscopic description, as long as observations are re-
stricted to macroscopic scales.
The scope of this review is limited mostly to the dy-
namics in the immediate vicinity of the point of breakup.
This is motivated by the expectation that pinching is uni-
versal under quite general circumstances, even if the
motion farther away from the singularity is more com-
plicated. In the nonasymptotic regime, our focus is on
the axisymmetric case of a jet with or without gravity.
This excludes many important examples of nonlinear
free-surface motion, such as drop oscillations, the dy-
namics of fluid sheets, and in particular the vast field of
surface waves.
We begin with an overview of the experimental basis
of the subject. Here and in the rest of this review, we
confine ourselves to cylindrical symmetry. In the case of
free surfaces, this is representative of the majority of
experimental work in the physics literature. But it ex-
cludes important effects like bending (Entov and Yarin,
1984; Yarin, 1993), branching (Lin and Webb, 1994), and
spraying (Yang, 1992) of jets. There is also substantial
work on splashes, i.e., the impact of drops on liquid
(Og
˜
uz and Prosperetti, 1990) or on solid surfaces (Yarin
and Weiss, 1995) surfaces. Mixing processes can also not
be expected to respect cylindrical symmetry. The outline
of experimental work in the second section is comple-
mented by a review of numerical work in the third sec-
tion. As indicated above, numerical simulations of the
full hydrodynamic equations are only slowly catching up
with the resolution possible in experiments. On the
other hand, important information on the velocity field
is not available experimentally. This and the superior
variability of simulations, for example, in the choice of
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Jens Eggers: Nonlinear dynamics and breakup of free-surface flows
Rev. Mod. Phys., Vol. 69, No. 3, July 1997
fluid parameters and of initial conditions, is bound to
make simulations an important source of information.
In the fourth section we give a detailed account of
linear stability theory, which is the classical approach to
the problem, but which remains an area of research to
the present day. Some nonlinear effects can be included
in perturbation theory, but the expansion quickly breaks
down near pinching.
The groundwork for the description of nonlinear ef-
fects is laid by the development of one-dimensional
models. We spell out two different approaches to the
problem and explain some of the properties of the re-
sulting models in Sec. V. In Sec. VI we study in detail
the universal self-similar solution leading up to breakup.
The solution can be continued uniquely to a new solu-
tion valid after breakup, which now consists of two
parts. The nonlinear stability theory of the asymptotic
solution explains the complicated structure seen in the
presence of noise.
Section VII explores the dynamics away from the
asymptotic regime, but where nonlinear effects are still
dominant. The area best understood is the case of highly
viscous jets, in which the pinching has not yet become
sufficiently fast for inertial effects to become important.
In the opposite limit of very low viscosity, the smoothing
effect of viscosity is missing, and gradients of the flow
field become large at a finite time away from breakup.
This makes the problem a hard one, and the understand-
ing of this regime is only in its early stages. However,
this subject is bound to remain an interesting and fre-
quently studied topic for the years to come, since low
viscosity fluids like water are the most common. From a
theoretical point of view, the understanding of the sin-
gularities of the Euler equation is one of the major un-
solved problems in hydrodynamics, and fluid pinching
serves as a particularly simple model system. To con-
clude Sec. VII, we describe some research on satellite
formation and present a numerical simulation of the sta-
tionary state of a decaying jet.
So far we have dealt only with free surfaces, with sur-
face tension being the only driving force. We relieve this
restriction in the final section, where we explore some
examples of related topics. First we look at two-fluid
systems, which are particularly important for the theory
of mixing and the hydrodynamics of emulsions. An
asymptotic theory for breakup in the presence of an
outer fluid has not yet been developed. Electric or mag-
netic fields represent another possible external driving
force. They force the fluid into sharp tips, where the
fields are strong, out of which tiny jets are ejected. This
allows for the production of very fine sprays. In chemical
processing, macromolecules are often present in solu-
tion. They result in non-Newtonian properties of the
fluid, to which we give a brief introduction in the context
of free-surface flows.
II. EXPERIMENTS
Historically, research on drop formation was moti-
vated mostly by engineering applications, hence the
three most common experimental setups, which are de-
scribed in more detail below: (1) Jets are produced when
a fluid leaves a nozzle at high speeds; (2) slow dripping
under gravity has been used for the measurement of sur-
face tension; and (3) liquid bridges are used to suspend
fluid in the absence of gravity. For a review on drop
formation in the context of engineering applications of
spraying, see Walzel (1988). Early work focused either
on the early stages of drop formation, characterized by
the growth of linear disturbances, or on the size and
number of the resulting drops. Either aspect of drop for-
mation is relatively easy to observe, but is highly depen-
dent on the experimental setup and on parameters like
nozzle diameter or jet speed.
Only slowly, as experimental techniques became
available for observing the actual evolution of the flow
during drop formation, did common features emerge
from the seemingly disparate results of individual ex-
periments. The last stages of the evolution are domi-
nated by the properties of the pinch singularity, which is
the same for all cases. This idea was first enunciated
clearly by Peregrine et al. (1990). The appearance of the
motion depends only on the scale of observation
l
obs
relative to the size of the internal length l
n
[see Eq. (2)]
of the fluid. If
l
obs
/l
n
is large, which is typical for flows
of low viscosity like water, the shapes near the pinch
point are cones attached to a spherical shell. After
breakup, as the fluid neck recoils, capillary waves are
excited. If on the other hand
l
obs
/l
n
is small, as for
fluids of high viscosity like glycerol, long and skinny
threads are observed, which rapidly contract into tiny
drops after breakup. At the highest viscosities, the
threads tend to break at random places. It is from this
universality that much of the physical interest of the sub-
ject of drop formation derives, and we try to emphasize
common features in discussing different kinds of experi-
ments.
A. Jet
By far the most widely used experimental setup in the
study of drop formation is that of a jet of fluid leaving an
orifice at high speeds. The earliest jet experiments were
performed with fluid being driven out of holes near the
bottom of a container (Bidone, 1823). The focus of the
early research was on the shape of jets produced by ori-
fices of different forms. It was Savart (1833) who dis-
tinctly noticed the inevitability of a decay into drops and
carefully investigated the laws governing it. By deliber-
ately disturbing the jet periodically at the nozzle, he pro-
duced disturbances on the surface of the jet with the
same frequency. Many other 19th-century researchers
repeated these experiments, notably Hagen (1849),
Magnus (1855), and Rayleigh (1879b,1882). Both Pla-
teau (1873) and Rayleigh were able to perform some
quantitative tests of their theories, but without photog-
raphy it was impossible to record the shapes of jets in
detail. Photographic methods were introduced by Ray-
leigh (1891), but these observations were only qualita-
tive in nature. The first quantitative experiments were
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Jens Eggers: Nonlinear dynamics and breakup of free-surface flows
Rev. Mod. Phys., Vol. 69, No. 3, July 1997
![FIG. 29. Two pictures of the thread in between drops for a highly viscous fluid, l n=2 mm [series GLY3 of Kowalewski, 1996], which is also the width of the frame. In (a), 2.8 ms away from the singularity, a Stokes similarity solution, represented by the solid line, was fitted by adjusting the axial scale. Part (b), 1.5 ms away from the singularity, contains no adjustable parameter.](/figures/fig-29-two-pictures-of-the-thread-in-between-drops-for-a-v59rmwan.webp)
