We give a brief overview of the physical significance of singularities in
fluid mechanics

Eggers, Jens

English

arXiv

The equations of hydrodynamics are nonlinear partial differential equations, so they include the possibility of forming singularities in finite time. This means that hydrodynamic fields become infinite or at least non-smooth at points, lines, or even fractal objects. This mathematical possibility is the price one has to pa

Jens Eggers

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Hydrodynamic Singularities
Jens Eggers
Universit¨ at Gesamthochschule Essen, Fachbereich Physik,
45117 Essen, Germany
1 Introduction
The equations of hydrodynamics are nonlinear partial diﬀerential equations, so
they include the possibility of forming singularities in ﬁnite time. This means
that hydrodynamic ﬁelds become inﬁnite or at least non-smooth at points, lines,
or even fractal objects. This mathematical possibility is the price one has to pay
for the enormous conceptual simpliﬁcation a continuum theory brings for the
description of a many-particle system. Near singularities, the microscopic struc-
ture re-emerges, as the ﬂow changes over arbitrarily small distances. Eventually,
the singularity is cut oﬀ by some microscopic length scale such as the distance
between molecules.
The most fundamental question is whether the microscopic structure be-
comes relevant for features of the ﬂow much larger than the microscopic ones.
If this is the case, the continuum description is no longer self-consistent, but
has to be supplemented by microscopic information. There are some well-known
cases where singularities are artifacts of neglecting diﬀusive eﬀects like viscos-
ity in the hydrodynamic equations, and there is no more smoothing on small
scales. Examples are the singularities widely believed to be produced by the
three-dimensional Euler equation (Majda (1991)) or those of Hele-Shaw ﬂows at
zero surface tension (Dai, Kadanoﬀ, and Zhou (1991)). There does not seem to
be a direct correspondence between the intricate structure of spatial singular-
ities produced by these equations and real ﬂows. For a realistic description of
experiments, some tiny amount of viscosity or surface tension needs to be added.
The reason hydrodynamic singularities have nevertheless attracted consider-
able attention in recent years is the observation that certain singularities have
direct physical signiﬁcance and are not a consequence of inadequacies of the
equations. In particular, free-surface ﬂows exhibit a rich variety of experimentally
observable singularities, which are responsible for phenomena like the breakup
of jets, coalescence of drops, and bubble entrainment.
2 Physical singularities
We attempt to divide these physical singularities into two categories.2 Jens Eggers
2.1 Dynamical singularities
These are singularities which are conﬁned to a point in time. Usually, they are
associated with topological transitions like the breakup of a piece of ﬂuid into two
pieces or the joining of two pieces into one. For example, the breakup of a viscous
jet of ﬂuid is driven by surface tension, which tries to reduce the surface area by
diminishing the radius of the jet (Eggers (1997a)). Inertial forces constrain the
motion to become more and more localized, since smaller and smaller amounts
of ﬂuid have to move. This causes the jet to break at a point in ﬁnite time.
Only the smoothing eﬀect of viscosity prevents inﬁnite gradients from occurring
before the local radius goes to zero.
As the local radius of a ﬂuid thread becomes smaller and smaller during
pinching, it inevitably reaches a microscopic scale ℓmicro where the equations
cease to be valid. For thread diameters between 10 and 100 nm, short-ranged
van der Waals forces come into play, and for even smaller diameters the con-
cept of a sharp interface will certainly loose its meaning. Moreover, a stability
analysis (Brenner, Shi, and Nagel (1994)) shows that the pinching thread is very
sensitive to thermal ﬂuctuations. This makes even threads of micron size unsta-
ble, and leads to a new structure of nested singularities, driven by microscopic
ﬂuctuations. After the thread has dissolved, new surfaces form on either side,
whose rapid retraction is again governed by the Navier-Stokes equation, but with
a new topology. It would appear as if the continuation to the new Navier-Stokes
problem should necessarily include the microscopic length ℓmicro at which the
thread broke. This is however not the case (Eggers (1997a)), as long as one is
looking at scales much larger than ℓmicro. Namely, the ﬁnal stages occur on very
small spatial and temporal scales, and do not aﬀect the ﬂow at a ﬁnite distance
away from breakup. Thus the outer part of the solution can be used as a bound-
ary condition for the new problem after breakup. A closer analysis reveals that
this is suﬃcient to determine the new solution completely. This means that the
dynamics very quickly “forget” the microscopic details of breaking, thus making
a consistent hydrodynamic theory of the topological transition possible.
2.2 Persistent singularities
The other important category of singularities are those which exist for a period of
time, being either stationary, or moving about in space like the classical example
of a shock wave. At ﬁnite viscosity, a shock wave is not a true singularity, but
maintains a ﬁnite width determined by the ratio of the viscosity and the shock
strength. However, the width δ of the shock wave is typically of the same order as
the mean free path of the gas it is moving in. What is important is thus the fact
that the solution remains consistent as δ goes to zero. Indeed, the dissipation
inside the shock remains ﬁnite in this limit, so on scales much larger than δ the
ﬂow ﬁeld is the same as if δ were zero.
In the realm of free-surface ﬂows, a beautiful example of a stationary singu-
larity has been discovered recently on the surface of the viscous ﬂow between twoHydrodynamic Singularities 3
counter-rotating cylinders (Joseph et al. (1991)). As seen in Fig.1, two counter-
rotating cylinders are submerged in a container ﬁlled with a very viscous ﬂuid.
The relative strength of viscous forces and surface tension is measured by the
capillary number
Ca =
η rc
γ
, (1)
Ω Ω
free surface h(z)
sheet  of  air
2rc
Fig.1. Sketch of the two-roller apparatus. At a critical capillary number, a pointed
cusp appears. At a second, much higher capillary number, a sheet emanates from the
cusp.
where γ is the coeﬃcient of surface tension, η the viscosity, and   and rc are the
cylinders’ rotation speed and radius, respectively. At a critical capillary number,
the surface appears to form a cusp, as indicated in Fig.1. Since the ﬂow remains
two-dimensional, this corresponds to a line of singularities of the surface. Assum-
ing that it is a true cusp, the original authors analyzed the ﬂow using a solution
due to Richardson (1968). This local solution leads to a logarithmic divergence
of the dissipation, and thus cannot be consistent with continuum theory or a ﬁ-
nite driving power. It was therefore suggested that the divergence is regularized
by some microscopic scale. Since the singularity is very weak, the dissipation at
the tip would still be small for realistic values of the microscopic length. How-
ever, this explanation for eliminating the logarithmic divergence would mean
that there is a dependence of the macroscopic ﬂow on a microscopic parameter.
Faced with this possibility, the problem was reanalyzed by Jeong and Moﬀatt
(1992), who solved the Stokes equation exactly, making the simplifying assump-
tion that the two rollers can be represented by a single dipole. The remarkable4 Jens Eggers
result of their calculation is that the radius of curvature R at the tip is in fact
ﬁnite, but exponentially small in the capillary number:
R = R0 exp[−32πCa]. (2)
For realistic values of the capillary number, this gives radii of curvature far below
any physical scale, but is still able to regularize the logarithmic singularity. Thus
one ﬁnds a ﬁnite value of the energy dissipation, making the macroscopic ﬂow
independent of the microscopic parameters of the ﬂuid.
For practical purposes, the theoretical value of R is far too small to be re-
alistic. Rather, it most likely is the gas above the ﬂuid which will set the value
of R, and this physical eﬀect has been neglected so far. Because of the no-slip
boundary condition, gas will be forced into the narrow channel formed by the
cusp. A simple calculation based on lubrication theory shows that for R = 0 this
will lead to a diverging pressure at the tip of the cusp. Thus the gas will force
the channel to widen to a ﬁnite radius, at which the gas pressure is comparable
to the pressure inside the ﬂuid.
It is worth noting that the independence from microscopic parameters is by
no means self-evident. A famous counterexample is that of a moving contact
line, which occurs for example when a circular drop is allowed to spread on the
surface of a table (de Gennes (1985), Brenner and Bertozzi (1993)). Using kine-
matic arguments alone, one shows that there will be a logarithmic singularity
of the energy dissipation at the moving contact line. There is a vast literature
on this problem, dealing either with possible mechanisms for a microscopic cut-
oﬀ, or with a consistent mathematical description of the resulting macroscopic
dynamics. The important point to note is that continuum mechanics alone can-
not resolve the problem in a self-consistent fashion. It would predict that the
spreading of the drop is stalled, contradicting observation.
3 Scaling
The central assumption behind the description of singularities is that of locality.
Their spatial and temporal scale becomes arbitrarily small, so that the dynamics
should be removed from the large-scale features of the ﬂow. However, consistency
between the singular and the large scale dynamics has to be assured by matching
conditions between the inner and the outer problems.
A second, closely related assumption is that of self-similarity of the singular
ﬂow, which seems a natural concept for a class of problems which lack a typical
length-scale. In the case of time-dependent singularities it means that the inter-
face shapes at diﬀerent times can be mapped on one another by an appropriate
rescaling of the axes. For example, the surface proﬁle of the pinch singularity
when a ﬂuid thread breaks is (Eggers (1997a))
h(z,t) =
γ
η
|t0 − t|φ
￿
ρ1/2(z − z0)
(η|t0 − t|)1/2
￿
, (3)Hydrodynamic Singularities 5
where t0 and z0 are the temporal and spatial position where the ﬂuid breaks.
Remarkably, the scaling function φ is universal, independent of the type of ﬂuid
or of initial conditions. For the free surface cusp of Fig.1, the shape of the
interface has the scaling form
h(z) = h0 + R
1/2f
￿ z
R3/4
￿
, (4)
where (0,h0) is the position of the cusps’ tip.
Naturally, it is of particular interest to compute the scaling exponents. No
general understanding of what selects a particular set of exponents exists. Usu-
ally, local solutions like (3) or (4) are not exact solutions of the equations, but
only balance certain terms that are asymptotically dominant. In the case of
the pinch singularity these terms belong to surface tension, viscous, and inertial
forces. Knowing that, dimensional analysis alone leads to the correct power laws.
However, there are cases like the pinching of a very viscous thread (Papageorgiou
(1995)), where the exponents are ﬁxed to irrational values by other consistency
requirements.
z 0
rm
∆
R
h(z)
η
η / λ
Fig.2. Coalescence of two drops of radius R and R/δ. Asymptotically, the width w of
the bridge between the drops is much smaller than the minimum radius r.
What could possibly keep the local motion from behaving in a self-similar
fashion? Two examples for broken self-similarity come from the coalescence of
two drops. When two drops meet a point, surface tension will try to merge
them into one drop, so the minimum radius r of the bridge between the drops
will increase, (Fig.2). As long as the bridge is very small, inertial eﬀects can
be neglected and the problem is initially governed by the Stokes equation. For
geometrical reasons the width of the bridge w is much smaller than the minimum
radius r, so dimensionally
r(t) =
γ
η
t (5)6 Jens Eggers
is the only possible power law time dependence of the radius. However, a closer
analysis reveals (Eggers, Lister, and Stone (1997)) that (5) has to be corrected
by a logarithmic factor log(r/R), where R is the initial radius of a drop. The
reason for this change of the time dependence lies in the long-ranged character of
the Stokes equation, which couples very disparate length scales. Hence the width
of the bridge cannot be neglected, but enters as a logarithmic factor log(w/r) ∼
log(r/R).
Another mechanism for broken self-similarity is observed for the same prob-
lem, but for much smaller initial size of the drops. In that case the surface-
tension-driven motion will ﬁrst occur on the surface alone, rather than being
able to drive a ﬂow in the interior. The equation of motion for this surface dif-
fusion was ﬁrst given by Mullins (1965). Based on simple scaling arguments,
Mullins (1965) gave
r(t) ∼ t
1/6 (6)
as the evolution of the radius, but this result could not be corroborated by
his own numerical simulations. To understand this failure, one needs to take a
closer look at the dynamics near the rising bridge (Eggers (1997b)). In contrast
to the viscous ﬂow problem, as the gap between the two spheres ﬁlls, a bulge of
material forms just above the minimum. Eventually, it grows enough for both
sides to touch, forming a void inside the material. Of course, at the point of
touching a new singularity occurs and the topology of the problem has changed.
The naive assumption of a single continuous evolution, underlying (6), is thus
incorrect. Self-similarity can at best exist in a discrete sense.
4 Birth of new structures
If one drives the two-roller apparatus of Fig.1 much harder than necessary for
the formation of a cusp, a second transition occurs, above which a thin sheet of
air emanates from the cusp, and is drawn continuously into the ﬂuid. This sheet
is stable in time, but undergoes a three-dimensional instability at its lower end,
where it decays into a curtain of tiny droplets. This provides a general mecha-
nism for the entrainment of bubbles into a ﬂow. The existence of the sheet was
pointed out by Moﬀatt (1994), and conﬁrmed in a series of qualitative exper-
iments (Eggers and Smock (1996)), using a silicone oil 10000 times as viscous
as water. At suﬃciently high driving, the air forced into the cusp experiences a
strong enough downward pull to form a stable sheet. A preliminary calculation
suggests that the thickness of the sheet scales like
δ ≈
￿
ηair
ηfluid
￿1/2
rc . (7)
This prediction is based on the assumption that there is a return ﬂow in the
sheet, which produces very high gradients. So when the sheet becomes very thin
the inner ﬂow is able to balance the high sheer stresses produced by the viscous
ﬂow. Both surface tension and gravity are not taken into account.Hydrodynamic Singularities 7
Fig.3. Tip-streaming in a Couette device, showing a drop of water in silicone oil 1000
times as viscous. To initiate tip-streaming, 200 ppm of surfactant has been added. The
image height is 0.5 mm. Photograph courtesy H. Leonhard.
Similar phenomena have been observed for a variety of other stationary singu-
larities. An example, known as “tip-streaming” (Taylor (1934),De Bruijn (1993)),
is shown in Fig.3. One sees a drop of water in a shear ﬂow of a very viscous
ﬂuid, which produces a cone-shaped singularity at both ends of the drop. Under
circumstances that are not well understood, a jet emanates from the tip, and
eventually decays into drops due to the Rayleigh capillary instability. This is the
precise analogue of the sheet in the two-roller apparatus, but the dimension of
the singularity and of the resulting structure are lowered by one. A second exam-
ple of a zero-dimensional singularity giving rise to a one-dimensional structure
is that of a dielectric drop in a strong electric ﬁeld, where a local cone-shaped
solution exists (Taylor (1964)). This “Taylor cone” is never stable, but either
oscillates between a rounded and a pointed state, or stabilizes itself by ejecting
a jet from its tip. Again, little is known about the conditions under which the jet
forms,but the striking similarities between diﬀerent systems suggest a unifying
explanation for the emergence of these structures.
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Majda A. J. (1991): Vorticity, Turbulence, and Acoustics in Fluid Flow.
SIAM Review 33, 349–388
Dai W., Kadanoﬀ L. P., Zhou S (1991): Interface dynamics and the motion of singu-
larities. Phys. Rev. A 43, 6672–6682
Eggers J. (1997a): Nonlinear dynamics and breakup of free-surface ﬂows.
Rev. Mod. Phys. 69, 865–9298 Jens Eggers
Brenner M. P., Shi X. D., Nagel S. R. (1994): Iterated Instabilities during Droplet
Fission. Phys. Rev. Lett. 73, 3391–3394
Joseph D. D., Nelson J., Renardy M., Renardy Y. (1991): Two-dimensional cusped
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manuscript in preparation
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Moﬀatt H. K. (1994): private communication
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unpublished experiments
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Hydrodynamic Singularities

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