Capillary Surface Interfaces, Volume 46, Number 7

Capillary Surface

Interfaces

Robert Finn

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nyone who has seen or felt a raindrop,
or who has written with a pen, observed
a spiderweb, dined by candlelight, or in-
teracted in any of myriad other ways
with the surrounding world, has en-

countered capillarity phenomena. Most such oc-
currences are so familiar as to escape special no-
tice; others, such as the rise of liquid in a narrow
tube, have dramatic impact and became scientific
challenges. Recorded observations of liquid rise in
thin tubes can be traced at least to medieval times;
the phenomenon initially defied explanation and
came to be described by the Latin word

capillus

meaning hair.

It became clearly understood during recent cen-

turies that many phenomena share a unifying fea-
ture of being something that happens whenever
two materials are situated adjacent to each other
and do not mix. We will use the term

capillary sur-

face

to describe the free interface that occurs when

one of the materials is a liquid and the other a liq-
uid or gas. In physical configurations such as the
capillary tube, interfaces occur also between these
materials and rigid solids; these latter interfaces
yield in many cases the dominant influence for de-
termining the configuration.

In this article we describe a number of such

phenomena, notably some that were discovered
very recently as formal consequences of the highly
nonlinear governing equations. These discoveries
include discontinuous dependence on the bound-
ary data, symmetry breaking, failure of existence

under physical conditions, and failure of unique-
ness under conditions for which solutions exist.
The predicted behavior is in some cases in strik-
ing variance with predictions that come from lin-
earizations and formal expansions, sufficiently so
that it led to initial doubts as to the physical va-
lidity of the theory. In part for that reason, exper-
iments were devised to determine what actually oc-
curs. Some of the experiments required
microgravity conditions and were conducted on
NASA Space Shuttle flights and in the Russian Mir
Space Station. In what follows we outline the his-
tory of the problems and describe some of the
current theory and relevant experimental results.

The original attempts to explain liquid rise in a

capillary tube were based on the notion that the
portion of the tube above the liquid was exerting
a pull on the liquid surface. That, however, cannot
be what is happening, as one sees simply by ob-
serving that the surface fails to recede (or to change
in any way) if the tube is cut off just above the in-
terface. Further, changing the thickness of the
walls has no effect on the surface, thus suggest-
ing that the forces giving rise to the phenomenon
can be significant only at extremely small dis-
tances (more precise analysis indicates a large por-
tion of these forces to be at most molecular in
range). Thus, for a vertical tube the net attractive
forces between liquid and wall must by symmetry
be horizontal. It is this horizontal attraction that
causes the vertical rise. Molecules being pulled to-
ward the walls force other molecules aside in all
directions, resulting in a spread along the walls that
is only partly compensated by gravity. Liquid is
forced upward along the walls, and cohesive forces

Robert Finn is professor emeritus of mathematics at
Stanford University. His e-mail address is

finn@gauss.

stanford.edu

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carry the remaining liquid column with it. The low-
ered hydrostatic pressure at the top of the column
is compensated by the curvature of the surface, in
essentially the same manner as occurs with a soap
bubble. This basic observation as to the nature of
the acting forces appears for the first time in the
writings of John Leslie in 1802.

An early attempt to explain capillarity phe-

nomena was made by Aristotle, who wrote circa 350
B.C. that

A broad flat body, even of heavy mater-

ial, will float on a water surface, but a long thin one
such as a needle will always sink

. Any reader with

access to a needle and a glass of water will have
little difficulty refuting the statement. On the other
hand, Leonardo da Vinci wrote in 1490 on the me-
chanics of formation of liquid drops, using ideas
very similar to current thinking. But in the ab-
sence of the calculus, the theory could not be made
quantitative, and there was no convincing way to
test it against experiments.

The achievements of the modern theory de-

pend essentially on mathematical methods, and
specifically on the calculus, on the calculus of vari-
ations, and on differential geometry. When one
looks back on how that came about, one is struck
by the irony that the initial mathematical insights
were introduced by Thomas Young, a medical
physician and natural philosopher who made no
secret of his contempt for mathematics (and more
specifically for particular mathematicians). But it
was Young who in 1805 first introduced the math-
ematical concept of

mean curvature H

of a surface

and who showed its importance for capillarity by
relating it to the pressure change across the sur-
face:

∆

= 2

σ H

, with

equal to surface tension.

Young also reasoned that if the liquid rests on a
support surface

, then the fluid surface

meets

in an angle

(contact angle)

that depends only

on the materials and not on the gravity field, the
shape of the surface, or the shape or thickness of

; see Figure 1.

Young derived with these concepts and from the

laws of hydrostatics the first correct approxima-
tion for the rise height at the center of a circular
capillary tube of small radius

immersed vertically

in a large liquid bath:

(1)

≈

2 cos

κa

ρg

;

here

is the density change across the free sur-

face,

the magnitude of gravitational accelera-

tion. See Figure 2.

Young’s chief competitor in these developments

was Laplace, who relied heavily on mathematics.
Laplace derived a formal mathematical expression

(2)

≡

div

T u,

T u

≡

1 +

for the mean curvature

of a surface

(

x, y

) ; he

was led to

Theorem 1.

The height

(

x, y

)

of a capillary sur-

face interface lying over a domain

Ω

in a vertical

gravity field satisfies the differential equation

(3)

div

T u

κu

λ.

Here

is a constant to be determined by phys-

ical conditions (such as fluid volume) and bound-
ary conditions;

is positive when the denser fluid

lies below the interface; in the contrary case, the
sign of

reverses. For the problem described

above, considered by Young,

= 0 and (3) becomes

(4)

div

T u

κu.

For a capillary surface in a cylindrical vertical

tube of homogeneous material and general hori-
zontal section

Ω

, the Young condition on the con-

tact angle yields the boundary condition

(5)

T u

= cos

∂

Ω

, with

the unit exterior normal on

Instead of a tube dipped into an infinite reser-

voir as considered by Young, one could imagine a
vertical tube closed at the bottom and partially
filled with a prescribed volume of liquid covering
the base. In general in this case

= 0, but addition

of a constant to

converts (3) to (4). From unique-

ness properties discussed below, it follows that

Figure 1. Fluid interface

, support surface

;

is the angle

between the two surface normals.

Figure 2. Capillary tube configuration.

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all cases, and independent of the volume of liquid,
the surfaces obtained are geometrically the same

This result holds also when

= 0 and a solution

exists; however, existence cannot in general then
be expected, as we shall observe below. To some
extent, these considerations extend to configura-
tions with

κ <

0, i.e., with the heavier fluid on top;

however, in general both existence and unique-
ness may fail when

κ <

is constant, we may normalize it to the

range 0

≤

. The range 0

≤

γ < π/

2 then in-

dicates capillary rise;

π/

< γ

≤

yields capillary

fall. It suffices to consider the former case, as the
other can be reduced to it. If

π/

2 , the only so-

lutions of (4), (5) with

κ >

0 are the surfaces

≡

= 0 , the only such surfaces are

≡

const.

κ <

0, nontrivial such solutions appear.

With the aid of (4) and (5), Laplace could improve

the approximation (1) to

(6)

≈

2 cos

κa

−

cos

−

2
3

−

sin

cos

Laplace did not prove this formula completely,
nor did he provide any error estimates, and in fact
it was disputed in later literature. Note that the
right side of (6) becomes negative if the nondi-
mensional “Bond number”

κa

8 , in which

case the estimate gives less information than does
(1). The formula is, however, asymptotically cor-
rect for small

; that was proved for the first time

almost two centuries later in 1980 by David Siegel,
who also provided explicit error bounds; a proof
yielding improved bounds was given by the pre-
sent author in 1984, and further improvements
were obtained by Siegel in 1989. The expression
(6) was extended by P. Concus (1968) to the entire
traverse 0

< r < a

. F. Brulois in 1981 provided the

full asymptotic expansion in powers of

; that re-

sult was obtained again independently by E. Mierse-
mann in 1994.

In 1830 Gauss used the

Principle of Virtual Work,

formulated by Johann Bernoulli in 1717, to unify
the achievements of Young and of Laplace, and he
obtained both the differential equation and the
boundary condition as consequences of the prin-
ciple. In the Gauss formulation the constant

(3) appears as a Lagrange parameter arising from
an eventual volume constraint.

Capillarity attracted the attention of many of the

leading mathematicians of the nineteenth and early
twentieth centuries, and some striking results were
obtained; however, the topic then suffered a hia-
tus till the latter part of the present century. A great
influence toward new discoveries was provided
by the “BV theory”, developed originally for min-
imal surfaces by E. de Giorgi and his co-workers.
In the context of this theory M. Emmer provided
in 1973 the first existence theorem for the capil-
lary tube of general section. For further references
to these developments, see [1, 2]. Other directions
were initiated by Almgren, Federer, Fleming, Si-
mons, and others and led to results of different
character; see, e.g., [10].

The Wedge Phenomenon

It is unlikely that anyone reading this article will
be unfamiliar with the name of Brook Taylor, as
the Taylor series figures prominently in every cal-
culus sequence. It is less widely known that Tay-
lor made capillarity experiments, almost one hun-
dred years prior to the work of Young and of
Laplace. He formed a vertical wedge of small angle
2

between two glass plates and observed that a

drop of water placed into the corner would rise up
into the wedge, forming contact lines on the plates
that tend upward in a manner “very near to the
common hyperbola”. Taylor had no theory to ex-
plain the phenomenon, but in the course of the en-
suing centuries a number of “proofs” of results
leading to or at least suggesting a hyperbolic rise
independent of opening angle appeared in the lit-
erature.

The actual behavior is quite different and varies

dramatically depending on the contact angle and
angle of opening.

There is a discontinuous transi-

tion in behavior at the crossing point

π/

In the range

γ < π/

2 , Taylor’s observation is

verified as a formal property that holds for any so-
lution. Specifically, we consider a surface

given

(

x, y

) over the intersection

Ω

of a wedge do-

main with a disk of radius

as in Figure 3. We ob-

tain:

Theorem 2.

Let

(

x, y

)

satisfy (4) with

κ >

Ω

and suppose

meets interior points of the bound-

ing wedge walls in an angle

such that

γ < π/

. Let

= sin

α/

cos

. Then in terms of

a polar coordinate system

(

r , θ

)

centered at the

vertex

, there holds

Ω

Figure 3. Wedge domain.

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(7)

≈

cos

−

sin

kκr

On the other hand, if

≥

π/

2 , then

(

x, y

)

is bounded, depending only on

and on the size

of the domain covered by

near

Theorem 3.

(

x, y

)

satisfies (4) with

κ >

Ω

and if

meets the bounding wedge walls in an

angle

such that

≥

π/

, then

(8)

| ≤

/κδ

) +

throughout

Ω

It is important to observe that the boundary is

singular at

and the contact angle cannot be pre-

scribed there. Despite this singularity, neither
theorem requires any growth hypothesis on the
solution near

. Such a statement would not be

possible, for example, for harmonic functions
under classical conditions of prescribed values or
normal derivatives on the boundary. It is also note-
worthy that no hypothesis is introduced with re-
gard to behavior on

. The local behavior is com-

pletely controlled by the opening angle and by the
contact angle over the wedge segments near

and is an essential consequence of the nonlinear-
ity in the equation. (In another direction, it can be
shown that if

(

x, y

) satisfies (4) in all space, then

≡

0.)

If in an initial configuration there holds

γ > π/

2 and

is then continuously decreased

until equality is attained, (8) provides a uniform
bound on height up to and including the crossing
point. For any smaller

the estimate (7) prevails,

and the surface is unbounded at

This behavior was tested in the Stanford Uni-

versity medical school in a “kitchen sink” experi-
ment by T. Coburn, using two acrylic plastic plates
and distilled water. Figure 4 shows the result of a
change of about 2

◦

in the angle between the plates,

leading in the smaller

case to a measured rise

height over ten times the predicted maximum of
Theorem 2. That result was confirmed under con-
trolled laboratory conditions and for varying ma-
terials by M. Weislogel in 1992; it yields a contact
angle of water with acrylic plastic of 80

◦

. (See

the remarks in the next section.)

The relation (7) was extended by Miersemann

(1993) to a complete asymptotic expansion at

in powers of

κr

. It is remarkable that

the co-

efficients are completely characterized by

and

and are otherwise independent of the data and do-
main of definition for the solution

. Thus again the

behavior is strikingly different from that arising
in linear problems.

If gravity vanishes, the discontinuous depen-

dence becomes still more striking. In this case (3)
becomes

(9)

div

T u

= 2

so that the surface has constant mean curvature.
We find:

Theorem 4.

(

x, y

)

satisfies (9) in

Ω

and defines

a surface

that meets interior points of the wedge

walls in the constant angle

, then

≥

π/

and

(

x, y

)

is bounded at

Thus, the change is now from boundedness to

nonexistence of a solution.

It may at first seem strange that this physical

problem should fail to admit a solution; after all,
fluid placed into a container has to go someplace.
Figure 5 shows the result of an experiment con-
ducted by W. Masica in the 132-meter drop tower
facility at NASA Glenn Research Center, which pro-
vides about five seconds of free fall. Two cylindrical
containers of regular hexagonal section were
dropped, both with acrylic plastic walls but with
differing liquids, providing configurations on both
sides of the critical value. In this case, when

≥

π/

2, the exact solution is known as a lower

spherical cap, and this surface is observed in the
experiment. When

γ < π/

2 , the liquid fills out

the edges and climbs to the top of the container,
as indicated in Figure 6. Thus the physically ob-
served surface folds back over itself and cannot
be expressed as a graph over the prescribed do-
main. The physical surface thus exists as it has to,
but not in the form of a solution to (9), (5). It
should be noted that

for regular polygonal sections,

the change in character of the solution is discon-
tinuous

. That is clearly apparent, as (unique) so-

lutions exist as lower spherical caps throughout
the closed range

≥

π/

2 .

Figure 4. Water rise in wedge of acrylic plastic. (Left)

γ < π/

2 .

(Right)

≥

π/

2 .

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Existence and
Nonexistence; the
Canonical
Proboscis

The just-described con-
currence of experimen-
tal results with predic-
tion from the formal
theory provides a per-
suasive indication that
the Young-Laplace-
Gauss theory, beyond
its aesthetic appeal,
also correctly describes
physical reality. The
physical validity of that
theory has been ques-
tioned on the basis of
ambiguities that occur
in attempts to measure
contact angle. There are
experimental proce-
dures that give rise to
reasonably repeatable
measurements of what
may be described as an
“equilibrium angle”.
However, if one par-
tially fills a vertical cir-
cular cylinder with liq-
uid (symmetrically) and
then tries to move the
liquid upward by a pis-
ton at the bottom, the
contact line of liquid
with solid does not im-
mediately move, but in-
stead an increase in the
contact angle is ob-
served. The maximum
possible such angle be-
fore motion sets in is
known as the “advanc-
ing angle”. Corre-
spondingly, the small-
est such angle ob-
tainable before reverse
motion occurs is the
“receding angle”. The
difference between ad-
vancing and receding
angles is the “hysteresis
range”, which presum-
ably arises due to fric-
tional resistance to mo-
tion at the interface,
but could conceivably
also reflect inadequacy
of the theory. This
range can be very large;

for example, for the water and acrylic plastic in-
terface in the Coburn experiment described above,
it is over 20

◦

. Such apparent anomalies of mea-

surement have led to some questioning of “con-
tact angle” as a physical concept. The experimen-
tal confirmation of discontinuous dependence on
data at the critical opening in a wedge supports the
view that contact angle does have intrinsic physi-
cal meaning; in addition it suggests a new method
for measuring the angle in particular cases. One
need only place a drop of the liquid into a wedge
that is formed by vertical plates of the solid and
has initial opening sufficiently large that Theo-
rem 3 applies. The opening angle is then slowly de-
creased until the liquid jumps up in the corner to
a height above the bound given by (8). The crite-
ria of Theorems 2 and 3 then determine

For values of

close to

π/

2 , remarkably good

agreement has been obtained in this way with the
“equilibrium angle” measured under terrestial con-
ditions. On the other hand, for values of

to 0

◦

, the physical changes occur in the immedi-

ate neighborhood of

with an opening close to

, and measurements become subject to experi-

mental error. We thus seek domains in which the
discontinuous behavior is manifested over a larger
set. We can do so in the context of a general the-
ory of independent mathematical interest, apply-
ing to zero gravity configurations and determin-
ing criteria for existence and nonexistence of
solutions of (9), (5) in tubes of general piecewise
smooth section

Ω

Let

be a curve in

Ω

cutting off a subdomain

Ω

∗

⊂

Ω

and subarc

∗

⊂

∂

Ω

, as in Figure 7.

From the zero gravity equations (9) and (5) we ob-
tain, for the area

Ω

∗

and length

∗

(10)

Ω

∗

cos

T u ds.

The same procedure with

Ω

∗

Ω

yields

(11)

cos

Ω

In (10) we observe that

(12)

T u

| ≤

1 +

for any differentiable function

(

x, y

) . It follows

that whenever there exists a solution to the
problem, the functional

(

Ω

∗

;

)

≡ |

| −

∗

cos

+ 2

Ω

∗

satisfies

(13)

(

Ω

∗

;

)

for all strict nonnull subsets

Ω

∗

⊂

Ω

cut off by

smooth curves

, when

is determined by (11).

This inequality provides a necessary condition for
existence of a solution.

To make the condition sufficient, we appeal to

the properties of “Caccioppoli sets” in the BV

Figure 5. Liquids with different contact

angle in acrylic plastic cylinders of

hexagonal section during free fall.

(Top)

γ > π/

. (Bottom)

γ < π/

Figure 6. Sketch of side view at edge in

drop tower experiment,

γ < π/

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theory. Roughly speaking, these are subsets of

Ω

whose boundaries within

Ω

can be assigned a fi-

nite length, in a variational sense. Details of gen-
eral properties of these sets can be found in [1, 2];
reference [3] and references cited there give spe-
cific applications to capillarity. It was shown by
Giusti in 1976 that if we rephrase (13) as a prop-
erty of Caccioppoli sets, the condition also be-
comes sufficient; see also Theorem 7.10 of [3]. We
obtain:

Theorem 5.

A smooth solution

(

x, y

)

exists in

Ω

if and only if the functional

(

Ω

∗

;

)

is positive for

every Caccioppoli set

Ω

∗

⊂

Ω

, with

Ω

∗

not equal

∅

Ω

In order to use this result, one proves that if

Ω

is piecewise smooth, then

can be minimized

among Caccioppoli sets, and the minimizing sets
can be characterized geometrically. One obtains:

Theorem 6.

For

Ω

piecewise smooth, there exists

at least one minimizing set

Ω

for

Ω

. Any such

set is bounded within

Ω

by a finite number

nonintersecting subarcs of semicircles of radius

)

, each of which either meets

at a smooth

point with angle

measured within

Ω

or meets

at a corner point. The curvature vector of each
subarc is directed exterior to

Ω

Thus, the existence question is reduced to eval-

uation of particular configurations, which often can
be determined explicitly by the geometrical re-
quirements of prescribed radius and angle with

In general, there will be a number

≥

of sub-

arcs of semicircles satisfying those requirements.
We refer to these arcs as

extremals

. If

= 0, then

since

vanishes on the null set, we find

(

Ω

∗

;

)

0 , all

Ω

∗

⊂

Ω

with

Ω

∗

∅

Ω

, and

hence by Theorem 5 a smooth solution exists. That
happens, for example, in a rectangle when

≥

π/

(If

γ < π/

4, then by Theorem 4 there is

no solution for that problem.)

is finite and positive, then a finite number

of subdomains

Ω

∗

appears for examination. If one

or more of these sets yields

≤

0, then no solu-

tion of (9), (5) can exist in

Ω

. If the value zero but

no smaller value is achieved by

on the sets

Ω

∗

then there exists a set

Ω

among them such that

there will be a solution

(

x, y

) in the set

Ω

and

such that

→ ∞

Ω

is approached from within

Ω

. If we then set

≡ ∞

Ω

, we obtain a gen-

eralized solution in a sense introduced by M. Mi-
randa in connection with minimal surfaces in 1977.
This solution will have mean curvature determined
by (11).

If a domain

Ω

∗

exists with

(

Ω

∗

;

)

0 , then

that inequality holds also for each minimizer.
Again it can be shown [4] that there exists a solu-
tion

(

x, y

) in a subdomain such that

(

x, y

) tends

to infinity on a finite number of circular subarcs
of equal radius that intersect

in angle

. The

mean curvature

(and accordingly the radius of

the arcs) will, however, no longer be determined
by (11).

The case

∞

can occur, and it is this

case that provides the clue for obtaining domains
in which there is abrupt transition in height of so-
lutions over large sets at the critical angle sepa-
rating existence from nonexistence. Following joint
work by Bruce Fischer, by Tanya Leise, and by
Jonathan Marek with the author, we look for a
translational continuum of extremals, each of
which yields

= 0. We construct the domain

Ω

finding the curves that meet a given translational
family of circular arcs of common radius

in a

fixed angle

; these curves can be written ex-

plicitly, in the form

(14)

−

sin

−

cos

−

sin

cos

−

sin

We choose upper and lower branches of the

curves joining at a point on the

-axis as part of

the boundary

, and then complete the domain by

a circular “bubble” whose radius is adjusted so that
the arcs become extremals with

= 0, as indicated

in Figure 8. We refer to such a domain as a

canon-

ical proboscis

. It can be shown that when the con-

tact angle exceeds

, then a smooth solution of

(9), (5) exists for that domain; however, for

the surface becomes a generalized solution in the
sense of Miranda, infinite in the entire region cov-
ered by the extremals.

Since the relative size of the portion of such a

domain that is covered by extremals can be made
as large as desired, the canonical proboscides can
be used for determining small contact angles based
on rapid changes of fluid height over large sets (see
the initial paragraph of this section). The theory
was tested by Fred W. Leslie in the Space Shuttle
USML-2 using containers with two proboscides on

Σ∗

Ω∗

Ω

Figure 7. Existence criterion;

Ω

∗

is the portion of

Ω

cut off by an

arbitrary curve

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opposite sides of a “bubble” corresponding to con-
tact angles 30

◦

and 34

◦

. The fluid used had an equi-

librium contact angle measured on Earth as 32

◦

with a hysteresis range of 25

◦

. Effects of resis-

tance of the contact line to motion (see beginning
of this section) were observed in the experiment,
but after the astronaut tapped the apparatus the
successive configurations of Figure 9 (top row in
cover montage) appeared. In an analogous con-
tainer stored for some days, the fluid on the right
climbed over the top and descended on the other
side. Thus, although time is required (and per-
haps also thermal and mechanical fluctuations), the
resistance effects are overcome and a clearly de-
termined contact angle is evidenced.

Differing Contact Angles; the

Domains,

and Edge Blobs

We return to the discussion of wedges above and
ask what happens when two distinct contact an-
gles

, γ

are prescribed on the two sides of a

wedge. Again, new differences in possible behav-
ior appear. Setting

= cos

, one sees easily that

a necessary condition for existence of any surface
over a wedge domain

Ω

of opening 2

and with

normal vector continuous to

is that (

, B

) lies

in the closed ellipse

(15)

+ 2

cos 2

≤

sin

α.

See Figure 10. For data in the indicated domains

bounded between the square and the ellipse,

one can show the natural extension of Theorem 4
that

solutions of

(9), (5)

are precluded without re-

gard to growth hypotheses

For data lying in

the situation is very dif-

ferent. In this case solutions of (9), (5) have been
shown to exist under general conditions, but their
form must be very different from what occurs for
data in

, in which case spherical surfaces yield

particular solutions; that cannot happen in

, as

the normal is discontinuous for such data. The clas-
sical Scherk minimal surface provides a particular
example. But for data (

, B

) that are not on the

boundary of the square in Figure 10, the solution

(

x, y

) is bounded above and below. It has been

conjectured by J-T. Chen, E. Miersemann, and this
author that

(

x, y

) is itself discontinuous at

The conjecture was examined numerically, initially
in special cases by Concus and Finn (1994), and
later by Mittelmann and Zhu (1996) in a compre-
hensive survey of minimal surfaces achieving the
data (

−

) and (

−

B, B

) on adjacent sides of a

square. The calculations suggest that a disconti-
nuity does indeed appear in

, although at points

close to the boundary with

it was so small as to

create a numerical challenge to find it.

The local problem we have been discussing can

be realized either by a global surface in a capillary
tube, with prescribed contact angle along the walls,
or alternatively by a drop of liquid sitting in a
wedge with planar walls that meet in a line

and

extend to infinity. In the latter case, if data come
from

, a solution can be given explicitly for any

prescribed

H >

0, as the outer spherical surface

of a portion of a ball of radius 1

cut off by the

wedge. This is effectively the only possible such
surface. Specifically:

Theorem 7.

(

, B

)

lies interior to

and if

topologically a disk and locally a graph near its in-
tersections with

, over a plane

orthogonal to

then

is metrically spherical, and is uniquely de-

termined by its volume, up to rigid motion parallel
to

Theorem 7 can be viewed as an extension of the

H. Hopf theorem that characterizes genus zero
immersions of closed constant

surfaces as met-

ric spheres.

We now keep the volume of the drop constant

and allow the data to approach the boundary of

. For simplicity we restrict attention to the two

symmetry lines

joining the

regions,

and

−

joining the

regions. We find:

Figure 8. Proboscis domain with bubble showing extremals

Figure 10. Ellipse of regularity.

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OTICES OF THE

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Theorem 8.

As one increases

to move from

the ellipse

to the region in the parameter space

−

, the free surface of the drop changes in topo-

logical character from a disk to a cylinder. The
surface becomes a metric sphere with two caps re-
moved at the places where the drop contacts the two
plates. Figure 11a depicts a value of the parame-
ter shortly before the transition occurs. On ap-
proaching

within

on the same line, the radius

of the drop grows unboundedly, with the drop cov-
ering very thinly a long segment of

; see Figure

11b. As either

−

are entered from

on the

line

−

, the drop becomes a spherical cap

lying on a single plate and not contacting the other
plate; see Figure 11c.

As already mentioned, for data in

−

or in

no drop in the wedge can exist even locally as a
graph over a plane orthogonal to

. Theorem 8 sug-

gests that the same result should hold for

data. The statement is here, however, less clear-cut;
in fact, recent joint work of P. Concus, J. McCuan,
and the author [6] shows that capillary surfaces
with data in

can exist locally as graphs. The

matter may be related to the presence of vertices,
in a sense introduced in [5]. There a blob of liquid
topologically a ball was considered, resting on a
system of planes, each of which it meets in a pre-
scribed angle. Points at which intersection lines of
the planes meet the surface of the blob are called

vertices

. This concept extends also to configura-

tions in which the intersection point is not clearly
defined, as can occur for data in

. The follow-

ing results are proved, for a configuration with

vertices, and for which all data on intersecting
planes that it contacts lie in

Theorem 9.

≤

, then either

= 0

and the blob

consists either of a spherical ball or a spherical cap
resting on a plane, or else

= 2

and the blob forms

a spherical drop in a wedge. If

= 3

, then the blob

covers the corner point of a trihedral angle; under

the supplementary orientation condition that the
mean curvature vector is directed away from the
vertex, the surface is spherical.

N >

3, the surface need not be spherical. The

surfaces described in the Appendix of [6] have
four vertices. A spherical drop in a wedge has two
vertices. We are led to:

Conjecture.

There exists no blob in a wedge, with

= 2

vertices and data in

Stability of Liquid Bridges I

A substantial literature has developed in recent
years on stability criteria for liquid bridges join-
ing two parallel plates separated by a distance

in the absence of gravity. Earlier papers considered
bridges joining prescribed circular rings, but more
recently the problem was studied with contact
angle conditions on the two plates. The initial con-
tributions to this latter problem, due to M.
Athanassenas and to T. I. Vogel, were for contact
angles

π/

2 for which a bridge exists as

a circular cylinder, and established that every sta-
ble configuration is a cylinder with volume

V ≥

/π

. A consequence is that instability occurs

at half the value of

that is indicated in classical

work of Plateau and of Rayleigh on stability of liq-
uid columns. The discrepancy arises because of the
use here of a contact angle condition, rather than
the Dirichlet condition employed by those authors.
It was later shown by Vogel and this author that
the same inequality holds regardless of contact an-
gles. The most complete results on stability of
such bridges are due to Lianmin Zhou in her 1996
Stanford University dissertation; Zhou established
in the general case of equal contact angles

on the

plates that regardless of

there is a unique sta-

ble bridge if the volume exceeds a critical

. In

1997 Zhou also obtained the striking result that
for unequal contact angles, the stability set in
terms of a defining parameter can be disconnected.

(a)

(b)

(c)

Figure 11. Spherical blobs in a wedge of opening

= 50

◦

for three choices of data near boundary of

. (a) Data in

near

−

; crossing the boundary into

−

yields continuous transition to a sphere with two caps removed.

(b) Data near

; transition to boundary yields spread extending to infinity along edge. Transition across

boundary not possible. (c) Data near

; transition across boundary yields fictitious drop realized physically as

drop on single plane.

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With regard to interpretation of these and also of
earlier results, see the editor’s note following her
1997 paper and the later clarifying papers by
Gro

e-Brauckmann and by Vogel. The stability cri-

teria just described should also be interpreted in
the context of the comments directly following:

Stability of Liquid Bridges II

Stable liquid bridges joining parallel plates are ro-
tationally symmetric but are in general not spher-
ical. The meridional sections (profiles) of these
surfaces were characterized by Delaunay in 1841
as the “roulades” of conic sections, and yield the
sphere as a limiting case. We obtain, however, the
result [6]:

Every nonspherical tubular bridge join-

ing parallel plates is unstable, in the sense that it
changes discontinuously with infinitesimal tilting of
either plate

. In the earlier stability results, tilting

of the plates was not contemplated.

From another point of view, there is the re-

markable 1997 result of McCuan:

Theorem 10.

If the data

(

, B

)

are not in

−

, then

there is no embedded tubular bridge joining inter-
secting plates.

It should be noted here that data in

−

are ex-

actly those for which tubular bridges that are met-
rically spheres are possible (cf. Theorem 8 above).
It is not known whether nonspherical embedded
bridges can occur in that case. H. C. Wente gave
an example of a nonspherical immersed bridge,
with contact angle

π/

2 on both plates (and thus

with data in

Exotic Containers; Symmetry Breaking

For solutions of (3), (5) in a bounded domain

Ω

with

given

≥

0, uniqueness for prescribed fluid vol-

ume and contact angle can be established under
much weaker conditions than are needed for lin-
ear equations. In fact,

arbitrary changes of the

data on any set of linear Hausdorff measure zero
on

∂

Ω

have no effect on the solution within

Ω

; this

behavior holds without growth condition on the so-
lutions considered

. Finn and Hwang showed if

κ >

(positive gravity) then

the stated result can be ex-

tended to unbounded domains of any form, with-
out conditions at infinity

. In the case of equations

(9), (5) (zero gravity), uniqueness can fail for un-
bounded domains.

A liquid blob resting on a planar support sur-

face is also uniquely determined by its volume
and contact angle, although the known proof pro-
ceeds along very different lines from the one for
the statement above. We may ask whether unique-
ness will persist during a continuous (convex) de-
formation of the plane to a tube. The negative an-
swer can be seen from Figure 12. If the tube on the
right is filled from the bottom until nearly the
maximum height of the conical portion with a liq-
uid making contact angle 45

◦

, then a horizontal sur-

face will result. But if it is filled to a very large
height, then a curved meniscus appears. Removal
of liquid can then lead to two distinct surfaces with
identical volumes and contact angles.

This line of thought can be carried further:
Capillary surfaces are characterized as station-

ary configurations for the functional consisting
of the sum of their mechanical (gravitational and
surface) energies. In any gravity field and for any

“exotic” containers can be constructed that yield

an entire continuum of stationary interfaces for fluid
configurations with the container as support sur-
face, all with the same volume and the same me-
chanical energy

. This can be done in a rotation-

ally symmetric container, so that all the surface
interfaces have the same rotational symmetry,
that no other symmetric capillary interfaces exist,
and in addition so that energy can be decreased
by a smooth asymmetric deformation of one of
the surfaces. Thus, none of the interfaces in the

Figure 12. For given volume, uniqueness holds in cylindrical tube for any section

Ω

; also drop on plane is

unique. But uniqueness fails in tube with curved bottom.

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continuum can minimize the energy. Using a the-
orem due to Jean Taylor, we can prove that:

minimizer for energy will exist but is necessarily
asymmetric

Local minimizers for this “symmetry breaking”

phenomenon were obtained computationally by
Callahan, Concus, and Finn, and then sought ex-
perimentally, initially in drop tower experiments
conducted by M. Weislogel. In the lower left of the
cover montage are shown calculated meridian sec-
tions for the continuum of stationary symmetric
nonminimizing surfaces, corresponding to a con-
tact angle of 80

◦

and zero gravity. The second row

of the montage shows the planar initial interface
in the earth’s gravity field, and a “spoon” shaped
configuration that appeared near the end of free
fall in the 132-meter drop tower. The spoon sur-
face has the general form of the global minimizer
suggested by the calculations, which had also lo-
cated two local minimizers of larger energy (“potato
chip” and “lichen”). The time permitted by the
drop tower was, however, insufficient to deter-
mine whether the spoon configuration was in equi-
librium or whether it might continue to evolve to
some distinct further form.

In view of the ambiguity in the drop tower re-

sults, space experiments were later undertaken in
order to obtain a more extended low gravity envi-
ronment. These were performed initially by
Lawrence de Lucas on the NASA Space Shuttle
USML-1 and then by Shannon Lucid in the Mir
Space Station. Figure 13 (the lower right corner of
the cover montage) shows comparison of calcula-
tion with the latter experiment. The upper row
shows the calculated spoon and potato chip; the
lower row shows experimental observation. The ex-
periment thus confirms both the occurrence and
the qualitative structure of global and also of local
asymmetric minimizers, in a container for which
an entire continuum of symmetric stationary sur-
faces appears.

Re-entrant Corners; Radial Limits

It was shown by Korevaar in 1980 that if the open-
ing 2

of a wedge domain exceeds

, then even

, solutions

(

x, y

) of (3) can exist that are

discontinuous at the vertex

, in the sense that the

tangential limits along the two sides can differ. Lan-
caster and Siegel (1996) showed that under gen-
eral conditions radial limits at the vertex exist
from every direction. Further, if

(

x, y

) is discon-

tinuous, then there will exist “fans” of constant ra-
dial limit adjacent to each side. If 2

α < π

and if

(

, B

)

∈ E

, then the fans overlap, yielding conti-

nuity of

(

x, y

) up to

. Earlier work, initially for

the case

, due originally to L. Simon and later

extended by Tam, by Lieberman, and by Mierse-
mann, can be used to extend the result to the parts
of

∂

bordering

, yielding continuity of the

unit normal up to

, and shows that for data in-

terior to

(

x, y

) has first derivatives Hölder con-

tinuous to

. Note that no growth condition is im-

posed in any of this work.

If 2

α > π

then it can occur that a central fan of

opening

appears, in addition to the two fans at

the sides.

Pendent Drops

We consider what happens when

κ <

0 in (3).

That occurs when the heavier fluid lies above the
interface. Examples are a liquid drop hanging
from a “medicine dropper”, or a drop pendent
from a horizontal plate. The behavior of such so-
lutions is very different from what occurs in the
cases discussed above and, in general, instabili-
ties must be expected. This “pendent drop equa-
tion” was much studied toward the end of the last
century, and Kelvin calculated a remarkable par-
ticular solution of a parametric form of the equa-
tion; see Figure 14.

This surface is unstable; however its bottom

tip could be observed as a stable drop hanging from
a ceiling in a house with a leaking roof. The best
stability criteria were obtained in 1980 by Wente,
who showed that in the development of a drop by
increasing volume,

instability occurs after the ini-

tial appearance of an inflection in the profile, and
prior to appearance of a second inflection

. Wente

showed existence of stable drops with both “neck”

Figure 14. Pendent liquid drop (Thomson,
1886).

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and “bulge”. An example is the drop of colored
water in a bath of castor oil, shown in Figure 15.

Concus and Finn proved the existence of “Kelvin

drops” that are formal solutions of the paramet-
ric equation, and which have an arbitrarily large
number of bulges. In addition they proved existence
of a rotationally symmetric singular solution

(

)

≈

/κr

of (4); this solution was shown by

M.-F. Bidaut-Veron to extend to a strict solution for
all

r >

0. (If

≥

0 then any isolated singularity of

a solution of (3) is removable.) It was shown by Finn
that in the limit as the number of bulges increases
unboundedly, the Kelvin solutions tend, uniformly
in compacta, to such a singular solution. In a re-
markable work now in preparation, R. Nickolov
proves the uniqueness of that singular solution,
among all rotationally symmetric surfaces with a
nonremovable isolated singularity. The result holds
without growth hypotheses.

Gradient Bounds

In addition to the height bounds indicated in The-
orems 2, 3, and 4, gradient bounds can also be ob-
tained. Again, some of these have an inimical char-
acter, reflecting the particular nonlinearity of the
problems. We indicate two such results:

i) (Finn and Giusti, 1977).

There exists

= (0

5654064

. . .

)

and a decreasing function

(

)

with

(

) =

∞

and

(1) = 0

, such that

(

x, y

)

satisfies (9) with

≡

const. >

in a

disk

(0)

and

R > R

, then

(0)

(

)

. The

condition

R > R

is necessary. There is no solution

of (9) in

(0)

ii) (Finn and Lu, 1998).

Suppose

(

)

with

(

)

≥

and

(

−∞

)

∞

)

. Then there exists

(

)

∞

such that if

(

x, y

)

satisfies (9) in

(0)

then

(0)

(

)

From the case

R < R

in (i) we see that the es-

timate of (ii) would be false if

is identically con-

stant. We note that the hypotheses of (ii) are sat-
isfied by the equation (4) for capillary surfaces in
a gravity field when

κ >

Past and Future

I have described only some of the many new re-
sults that have appeared in recent years; choices
had to be made as to what to cover, and they were
based largely on the simple criterion of familiar-
ity. Other directions of interest can be inferred from
items in the bibliography. Let me close with the
opening quotation from the 1851 Russian treatise
on capillarity by A. Yu Davidov:

The outstanding contributions made
by Poisson and by Laplace to the math-
ematical theory of capillary phenomena
have completely exhausted the subject
and brought it to such a level of per-
fection that there is hardly anything
more to be gained by its further inves-
tigation.

For the later French and German translations of the
book, Davidov changed the quotation to:

The outstanding contributions made
by Poisson, by Laplace, and by Gauss to
the mathematical theory of capillary
phenomena have brought the subject to
a high level of perfection.

The subject has advanced in significant ways

since the time of Davidov, but may nevertheless
still be in early stages. The material I have de-
scribed should indicate the kind of behavior to be
expected; some of it may serve as building blocks
toward creation of a cohesive and structured the-
ory.

Acknowlegments

The work described here was supported in part by
grants from the NASA Microgravity Research Di-
vision and from the NSF Division of Mathematical
Sciences. I am indebted to many colleagues and stu-
dents for numerous conversations, extending over
many years, that have deepened my knowledge and
understanding. I wish to thank the editors of the

Notices

for careful reading of the manuscript and

for perceptive comments from which the exposi-
tion has benefited greatly. The exposition has also
profited from comments by Paul Concus and by
Mark Weislogel.

—R. F.

Skeletal Bibliography

[1] E. G

IUSTI

Minimal Surfaces and Functions of Bounded

Variation

, Birkhäuser-Verlag, Basel, Boston, 1984.

[2] U. M

ASSARI

and M. M

IRANDA

Minimal Surfaces of

Codimension One

, North-Holland, Amsterdam, New

York, 1984.

Figure 15. Stable pendent drop exhibiting inflection on

the profile, also neck and bulge.

fea-finn.qxp 7/1/99 9:56 AM Page 780

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OTICES OF THE

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[3] R. F

INN

Equilibrium Capillary Surfaces

, Springer-

Verlag, New York, 1986; Russian translation (with ap-
pendix by H. C. Wente), Mir Publishers, 1988.

[4] R. F

INN

and R. N

EEL

, Singular solutions of the capil-

lary equation,

J. Reine Angew. Math.,

to appear

[5] R. F

INN

and J. M

UAN

, Vertex theorems for capillary

surfaces on support planes,

Math. Nachr.,

to ap-

pear.

[6] P. C

ONCUS

, R. F

INN

, and J. M

UAN

, Liquid bridges, edge

blobs, and Scherk-type capillary surfaces, Stanford
University preprint, 1999.

[7] K. L

ANCASTER

and D. S

IEGEL

, Existence and behavior of

the radial limits of a bounded capillary surface at a
corner,

Pacific J. Math.

176

(1996), 165–194; edito-

rial corrections in

Pacific J. Math.

179

(1997),

397–402.

[8] T. S

TÜRZENHECKER

, Existence of local minima for cap-

illarity problems,

J. Geom. Anal.

(1996), 135–150.

[9] A.D. M

YSHKIS

, V.G B

ABSKI

ı˘, N.D. K

OPACHEVSKI

ı˘, L.A.

LOBOZHANIN

, and A. D. T

YUPTSOV

Low-Gravity Fluid

Mechanics. Mathematical Theory of Capillary Phe-
nomena

, translated from the Russian by R. S. Wad-

hwa, Springer-Verlag, Berlin, New York, 1987.

[10] F. A

LMGREN

, The geometric calculus of variations and

modeling natural phenomena,

Statistical Thermo-

dynamics and Differential Geometry of Microstruc-
tural Materials (Minneapolis, MN, 1991)

, IMA Vol.

Math. Appl., vol. 51, Springer, New York, 1993, pp.
1–5.

[11] K. G

ROSSE

-B

RAUCKMANN

Complete Embedded Constant

Mean Curvature Surfaces

, Habilitationsschrift, Uni-

versität Bonn, 1998.

About the Cover

The top row exhibits “nearly discontinuous” de-
pendence on data for fluid in a cylindrical
“canonical proboscis” container in zero grav-
ity; the container section has two noses with
slightly different critical angles that encom-
pass the one for the fluid. Following successive
taps by the astronaut, the fluid remains
bounded in height below a predicted maximum
on one side, but rises well above that height on
the other side.

The second row shows symmetry breaking

for fluid in an “exotic container” designed for
zero gravity, shown just prior to and near the
end of free fall in a drop tower. The curves on
the lower left are calculated meridional sec-
tions of a continuum of symmetric equilibrium
surfaces bounding equal volumes in the con-
tainer; all these surfaces are unstable. The fluid
volume for the drop tower experiment is cho-
sen so that the initial section in the earth’s
gravity field can be identical to the horizontal
section of the calculated zero gravity family.

The lower right shows comparison of cal-

culation and experiment for surfaces yielding
asymmetrical critical points for mechanical en-
ergy in an exotic container during an experiment
on the Mir Space Station. The apparent global
minimum (“spoon”) and a local minimum of
larger energy (“potato chip”) are both observed
by the astronaut.

The cover montage was prepared by Gary

Nolan of the NASA Glenn Research Center.

—R. F.

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