Stability of Delaunay Surface Solutions to

Capillary Problems

Thomas I. Vogel

Texas A&M University, College Station, TX, 77843tvogel@math.tamu.edu

Abstract. The stability of rotationally symmetric solutions to capillaryproblems is examined, with applications to three specic problems.

Keywords: Delaunay surface, constant mean curvature, capillarity

1 Introduction

Capillary surfaces are all around us. If you've blown a soap bubble, you've con-ducted a capillary experiment. If you've measured a liquid for a recipe, you'vehad to take the meniscus into account while using a measuring cup. Capillaryeects are particularly pronounced in the absence of gravity, which is why NASAfunds research into capillarity. It's important to know where uids are going inyour space craft.

This paper concerns rotationally symmetric capillary surfaces and their sta-bility. The reader of this paper will nd few formally stated theorems and proofs.My intent, rather, is to tie together a number of papers I've written concern-ing capillary stability (see bibliography) by bringing out the common theme ofstability of Delaunay surfaces. Throughout this paper we will consider capillarysurfaces only in the absence of gravity.

If two immiscible uids, such as air and water, are in static equilibrium, thefree surface between them is called a capillary surface. Examples are the meniscusin a capillary tube, or the surface of a drop of water resting on a tabletop. Theshape of a capillary surface arises from minimizing a certain energy functional.To be specic, assume that Γ is a xed solid surface, and Ω is a blob of liquidadhering to it. Then the shape of the free surface Σ of Ω is determined byminimizing the functional

E(Ω) = |Σ| − c|Σ1|,

subject to holding the volume of Ω xed, where Σ1 is the wetted region on Γ ,and c is a material constant. (In the case that gravity is present, an additionalpotential energy term is present in the energy functional.) The rst order nec-essary condition for Ω to be a minimum of E , subject to the volume constraint,is that the mean curvature H of Σ is constant and that the contact angle (theangle between the normals to Σ and Γ along the curve of contact) is constantly

2 T. I. Vogel

arccos(c). (See [7].) We will call such a surface stationary. There are many sta-tionary capillary surfaces which are not stable. For example, the free surface ofan innite circular cylinder of liquid has constant mean curvature, but wouldnever been seen in real life (even in the absence of gravity). It would break updue to the well-known Rayleigh instability. It is important, therefore, to con-cern ourselves with stability of capillary surfaces (dened in Section 3), and notto just consider stationary surfaces. Of course, even those stationary solutionswhich are not energy minima are of mathematical interest.

We will be concerned with three related problems in this paper. The rst isthat of a liquid bridge between parallel planes ([8], [16], [17]). No such bridge isa strict energy minimum, since translations parallel to the planes leave surfaceenergy and volume unchanged. However, we may seek energy minima modulotranslations, i.e., seek a bridge such that any nearby bridge with the same surfaceenergy and containing the same volume must simply be a translation of theoriginal bridge. The second problem is that of a liquid bridge between two solidballs ([11], [12]). The third problem is that of a toroidal drop inside of a xedcircular cylinder ([10]). A common feature of these three problems is that theyhave stationary solutions which are rotationally symmetric. It turns out that theproblem of the bridge between planes also has deeper stability implications forother rotationally symmetric capillary surfaces. This is used in the discussion ofthe other two problems mentioned, and a generalization is discussed in Section9.

Of the three problems, probably the problem which is physically most im-portant is the second, that of a bridge between solid balls. This problem has tobe considered when a bed of particles, e.g., sand, is wetted by a liquid. Morepapers have appeared concerning bridges between parallel planes, but that isprobably because it is mathematically more accessible. From a physical pointof view, the bridge between planes is often considered as an approximation tothe bridge between solid balls. The third problem, a toroidal drop in a circularcylinder, has received much less attention. However, it is also of some physicalinterest. One use is in understanding how well pipes and tubes function in theabsence of gravity. This problem also arises in mathematical modeling of lungfunction. See [10] for more references on applications.

2 Delaunay Surfaces

The only rotationally symmetric surfaces of constant mean curvature are theDelaunay surfaces. (An elementary introduction to these surfaces is given in[5].) Delaunay surfaces are generated by rolling a conic section along a line,tracing the path of a focus, and revolving the resulting curve around the line.For an ellipse, the curve traced by the focus is called an undulary, and theresulting surface is an unduloid. Similarly, the focus of a rolling parabola tracesa catenary, and the resulting surface is the well-known catenoid. The focus of ahyperbola traces a nodary, with the resulting surface being a nodoid.

Delaunay Surfaces 3

It will be convenient to consider a parameterization of Delaunay surfaces asfollows. We assume that the axis of revolution is the x axis, that the prole curvegoes through the point (0, 1) with angle of inclination 0 at that point, and thatthe mean curvature is a parameter A. Parameterizing the prole curve by arclength s leads to the system of ordinary dierential equations

dx

ds= cosφ (1)

dy

ds= sinφ (2)

dφ

ds= cosφ

y + 2A, (3)

for coordinates x(s) and y(s), and inclination angle φ(s), with initial conditions

x(0) = 0y(0) = 1φ(0) = 0.

Fig. 1. Sample proles (x(s;A), y(s;A))

The behavior of the solutions is as follows.

For A ∈ (−∞,−1) ∪ (0,∞), the solution to the system is a nodoid. For A ∈

(−1,− 1

2

)∪(− 1

2 , 0), the solution is an unduloid.

For A = −1, the solution is a circular arc. For A = − 1

2 , the solution is a horizontal line. For A = 0, the solution is a catenoid.

4 T. I. Vogel

If the solution is an unduloid, sinφ oscillates between ±(2A+ 1). For the othersurfaces, the range of φ is obvious. Refer to [12] for more details. We will refer tosolutions to (1)(3) as x(s;A), y(s;A), and φ(s;A), to emphasize the dependenceon the parameter A. In [12], elementary formulas are given for y(s;A) and forcosine and sine of φ(s;A), and from these x(s;A) is given as an integral. Figure 1gives several proles in this family. The uppermost curve is given by A = 0.2. AsA decreases from that value down to A = −1.2, the other curves are generated.

The point of considering Delaunay surfaces parameterized by A in this wayis that this approach can be used to give solutions to certain symmetric capillaryproblems by the method of cutting and scaling. This was rst used to constructbridges between solid balls in [12]; however, it works easily for constructingbridges between parallel places and toroidal drops in circular cylinders as well.

As an example, suppose we wish to numerically study bridges between paral-lel planes. Suppose that the separation between the planes is one unit, and thatwe are interested in bridges which make a contact angle of π3 with both planes.For such a bridge, the inclination angles of the prole at the points of contactwill be −π6 at the left endpoint and π

6 at the right endpoint. We may nd allbridges which make the correct contact angles by taking A ∈

[− 1

4 ,∞), where

this interval comes from the above remarks on the dependence of solutions onA. Take A0 in that interval. Find an s0 < 0 for which φ(s0, A0) = −π6 , and ans1 > 0 for which φ(s1;A0) =

π6 . The curve (x(s;A0), y(s;A0)), s0 ≤ s ≤ s1 will

have the correct inclination angles at the endpoints, but the surface must bescaled by a factor of 1

x(s1;A0)−x(s0;A0)to form a bridge between planes separated

by one unit.Notice that in the procedure to construct a bridge between parallel planes,

if the reference curve is an unduloid, then there are innitely many choicesavailable for s0 and s1. This reects the fact that, after the contact angles arexed, there are innitely many rotationally symmetric stationary bridges makingthose contact angles with the planes. However, almost all of these are likely tobe unstable it is known ([17]) that for equal contact angles, any prole withan interior inection is unstable.

Constructing toroidal drops in circular cylinders by the cutting and scalingprocedure is also straightforward. Suppose that the contact angle γ is given.Take an A0 > − 1

2 , and let s0 < 0 be such that φ(s0;A0) = −γ, and φ(s;A0) < 0on (s0, 0). Set s1 = −s0. Then the curve (x(s;A0), y(s;A0)), s0 < s < s1 hasthe correct inclination angles at the ends, but must be scaled by a factor of

1y(s0;A0)

to be the prole of a free surface of a toroidal drop in a cylinder of

radius 1. Details may be found in [10]. The cutting and scaling procedure usedto construct bridges between solid balls is a bit more involved, and the reader isdirected to [12].

3 Stability

Suppose that we have a capillary surface Σ, not necessarily rotationally sym-metric. If Σ is perturbed by an innitesimal normal perturbation φN, then the

Delaunay Surfaces 5

second order change in E is the quadratic form

M(φ, φ) =

¨Σ

|∇φ|2 − |S|2φ2 dΣ +

˛σ

%φ2 dσ, (4)

where σ is the curve of contact of Σ and Γ . (Note that one has to introduce atangential component near ∂Σ to keep the contact curve on Γ . See [7] or [20] forfurther details.) Here |S|2 is the square of the norm of the second fundamentalform of Σ, and may be written as k21 + k22, where these are principal curvatures.

The constant % is given by

% = κΣ cot γ − κΓ csc γ. (5)

Here κΣ is the curvature of the curve Σ∩Π and κΓ is the curvature of Γ ∩Π, ifΠ is a plane normal to the contact curve ∂Σ. We will dene a stationary surfacefor which M(φ, φ) ≥ 0 for all φ with

˜ΣφdΣ = 0 to be stable. The condition

on the integral of φ is due to the volume constraint.One must be cautious in using the term stable. This word has a number of

dierent uses in mathematics (and related elds) whose meanings tend to blurtogether, and so it should be carefully dened when used. As we have denedstability in this paper, if a capillary surface is not stable, it is certainly nota constrained local energy minimum. However, a stable capillary surface neednot be an energy minimum. This is not surprising, since even for a function ofone variable, having the second derivative non-negative does not imply a localminimum. However, even the strengthened condition that M(φ, φ) > 0 for allnon-trivial φ with

˜ΣφdΣ = 0 is not enough to guarantee that Σ is a local

energy minimum. See [6], [18], and [14] for more detail on this important point.

4 Eigenvalue criteria

Integrating (4) by parts leads to an eigenvalue problem. Dene the dierentialoperator L by

L(ψ) = −∆ψ − |S|2ψ

where ∆ is the Laplace-Beltrami operator on Σ. The eigenvalue problem westudy is given by

L(ψ) = λψ (6)

on Σ, withb(ψ) ≡ ψ1 + %ψ = 0 (7)

on ∂Σ, where ψ1 is the outward derivative of ψ in the direction which is tangentto Σ and normal to ∂Σ. In [15] these operators are written out in coordinates.

The eigenvalues for this problem satisfy λ0 < λ1 ≤ λ2 ≤ · · ·. The conditionfor stability requires that M(φ, φ) be positive semi-denite on the space of allφ satisfying

˜ΣφdΣ = 0 (corresponding to innitesimally volume conserving

perturbations). If 0 ≤ λ0, this is certainly true. If λ1 < 0, this is false, and thesurface is unstable. The reason for instability in this case is that the subspace

6 T. I. Vogel

spanned by the two eigenfunctions φ0 and φ1 has dimension 2, and thereforehas a non-trivial intersection with the subspace of all φ for which

˜ΣφdΣ = 0

(which has co-dimension 1). We will not treat the case λ1 = 0.The most interesting case is when λ0 < 0 < λ1. In this case, in addition

to looking at the eigenvalues, there is another condition to check. If the freesurface may be embedded in a family smoothly parameterized by ε, and ifH ′(ε0)V

′(ε0) > 0, then the capillary surface is stable. If H ′(ε0)V′(ε0) < 0 then

the capillary surface is unstable. Here H(ε) is the mean curvature with respectto the normal pointing out of the liquid (negative for a spherical drop) and V (ε)is the volume of the liquid. We will refer to this as the dV/dH condition. This isan application of the general theory of stability of constrained problems: see [9].

Note 1. In what follows we will often be comparing the same surface Σ, but withdierent xed boundaries Γ , and dierent contact angles. In fact, this is a majortheme of this paper. As an example, the free surface of the liquid bridge in Figure2 and the free surface of the toroidal drop in Figure 4 are the same surface: apiece of an unduloid. In switching between xed surfaces but keeping the samefree surface, the surface integral in (4) is the same in both cases: the only changein this quadratic form would be in the line integral. An important point to realizeis that if the value of % increases, then the corresponding eigenvalues betweenthe two problems also increase (or at least, do not decrease). If the value of %decreases, then the corresponding eigenvalues between the two problems alsodecrease (or at least, do not increase). This is one of the consequences of theextremum properties of eigenvalues discussed in [4].

5 Separation of variables

In Sections 3 and 4 we dealt with general capillary surfaces. We will now spe-cialize to the Delaunay surfaces used in the cutandscale procedure of Section2. Take the capillary surface to be parametrized as

r (s, θ) = 〈kx (s) , ky (s) cos θ, ky (s) sin θ〉 ,

where x, y, and θ are from the standardized Delaunay curves of Section 2, withdependence on A suppressed, and k is a constant of scaling. Then (see [12]), fora function ψ(s, θ) dened on Σ, the operator L may be expressed as

L(ψ) = −(

1k2

(ψss +

1y2ψθθ +

sinϕy ψs

))− 2k2

(2A2 + cosϕ

y

(cosϕy + 2A

))ψ.

(8)

The boundary conditions for the eigenvalue problem (6), (7) in coordinates willbe

b(ψ) = ±1

kψs + %ψ = 0,

with + at the right endpoint and − at the left.

Delaunay Surfaces 7

We now separate variables. Set ψ to be P (s)Q (θ). Then (8) becomes

P ′′Q+1

y2PQ′′ +

sinϕ

yP ′Q+

(4A2 +

4A cosϕ

y+

2 cos2 ϕ

y2

)PQ = −k2λPQ,

which is

y2P ′′

P+Q′′

Q+ y sinϕ

P ′

P+

(4A2y2 + 4Ay cosϕ+ 2 cos2 ϕ

)= −k2λy2,

or

y2P ′′

P+ y sinϕ

P ′

P+(4A2y2 + 4Ay cosϕ+ 2 cos2 ϕ

)+ k2λy2 = −Q

′′

Q= m2,

where the separation constant is m2 with m an integer (since Q must be a linearcombination of sinmθ and cosmθ). We obtain

yP ′′ + sinϕP ′ +

(4A2y + 4A cosϕ+

2 cos2 ϕ−m2

y

)P = −λk2yP.

This is

d

ds(y (s)P ′ (s)) +

(λk2y (s) + 4A2y + 4A cosϕ+

2 cos2 ϕ−m2

y

)P = 0, (9)

written in the form of equation (1) in [2], Chapter 10. For the boundary condi-tions, Q will cancel, so we obtain

±P ′ (si) + %P (si) = 0. (10)

The eigenvalues may be then found numerically by a Prüfer substitution (see[2]).

For xed m, (9) and (10) form a standard Sturm-Liouville problem, so it isnatural to label the eigenvalues as λjm. As in [20], we have

λ0m < λ1m < λ2m < · · ·

and

λ00 < λ01 < λ02 < · · · ,

and eigenfunctions corresponding tom = 0 are radially symmetric. For k ≥ 1, theeigenvalue λjk corresponds to a two-dimensional eigenspace spanned by functionsof the form fjk(s) cos(kθ) and fjk(s) sin(kθ). This labeling of eigenvalues is aslight abuse of the notation used in Section 4, since we were previously usingonly a single subscript for the eigenvalues λ. The smallest eigenvalue (which welabeled λ0 in section 4) is denitely λ00. The second smallest eigenvalue, labeledλ1 in section 4, is one of λ01 or λ10, depending on %.

8 T. I. Vogel

Fig. 2. Liquid bridge between parallel planes

6 Bridges between parallel planes

The problem of a liquid bridge between parallel planes in the absence of gravity(see Figure 2) has been addressed in a number of papers, e.g., [1], [8], [16],[17]. If the contact angles with the planes are constant (possibly two dierentconstants), then the argument proving symmetry in [21] may be used to showthat the free surface must be rotationally symmetric, hence a Delaunay surface.Since translations parallel to the planes are energy neutral, it follows that aninnitesimal translation is an eigenfunction with eigenvalue equal to zero. It'snot dicult to show that this is the eigenvalue λ01. Thus for the liquid bridgebetween parallel planes the lowest eigenvalue λ00 must be negative. The bridge isis unstable if either λ10 is negative, or the dV/dH condition for stability in section4 is violated. It is useful to think of the eigenvalues arranged as in Table 1. The

Table 1. Arrangement of eigenvalues for the bridge between planes

......

......

...

λ30 λ21 λ12 λ03

λ20 λ11 λ02

λ10 λ01 = 0

λ00

m = 0 m = 1 m = 2 m = 3 · · ·

inequalities mentioned in Section 5 imply that the eigenvalues strictly increaseas one moves up a column, and that the bottom elements of the columns strictly

Delaunay Surfaces 9

increase as one moves right. From this, the statement about stability dependingon the sign of λ10 and of dV/dH is perhaps easier to see.

For equal contact angles, the appearance of an inection in the prole signalsthat λ10 is negative, and hence that the bridge is unstable. To describe stabilitybehavior, we look at cases depending on the contact angle. (The following threeparagraphs summarize the appendix to [8].)

For γ > π2 , the limiting prole as volume tends to innity is an arc of a circle,

corresponding to a stable bridge. As volume decreases, we pass through a familyof nodoids, until we reach another circular arc, which is a section of a sphere. Asvolume continues to decrease, the bridge remains stable as we pass into a familyof unduloids. Finally, an inection appears on the boundary, signaling a secondeigenvalue crossing zero, and this is the minimum stable volume.

For γ = π2 , the only stable bridges are circular cylinders. As volume decreases,

instability occurs when the radius is 1π times the separation of the planes. This

is related to the Rayleigh instability of the cylinder.

For γ < π2 , the limiting prole for large volume is an arc of a circle. As volume

decreases, we pass though a family of nodoids until we reach a catenoid. Beyondthis, the proles are unduloids. For γ larger than a critical angle, about 31.14,the bridges are stable until the appearance of an inection on the boundaryindicating a second eigenvalue crossing zero causing instability. For γ less thanthat critical angle, bridges become unstable before the appearance of an inectionon the boundary, due to a violation of the dV/dH condition.

Since there are two ways that instability can arise in the problem of a bridgebetween planes with equal contact angles, it is natural to investigate whetherthe bridges become unstable in two dierent ways. A partial answer appears in[19]. Suppose that the planes are x = 0 and x = 1. If instability is due to aviolation of the dV/dH condition (i.e., for γ less than the critical angle of about31.14), then the bridge is unstable with respect to perturbations which aresymmetric across the plane x = 1

2 . If the bridge becomes unstable because of asecond eigenvalue crossing zero (i.e., for γ greater than the critical angle), thenthe bridge is stable with respect to perturbations symmetric across the planex = 1

2 . Roughly speaking, bridges with contact angle less than 31.14 breaksymmetrically (across x = 1

2 ) as volume decreases, whereas bridges with contactangle greater than that critical angle break asymmetrically across x = 1

2 .

The intuition behind the proof is this: instead of looking at a bridge betweenx = 0 and x = 1 with contact angles γ at both sides, consider a bridge betweenx = 0 and x = 1

2 , with contact angle γ with x = 0 and contact angle π2 with

x = 12 . Plotting V against H for the uninected family leads to the same curve

(except for some scaling) for both problems. However, in going from the γ − γbridge to the γ− π

2 bridge, the odd eigenvalues disappear. Thus an instability dueto a failure of the dV/dH condition for the γ − γ problem implies an instabilityfor the γ − π

2 problem, but an eigenvalue crossing zero for the γ − γ problemdoes not lead to an eigenvalue crossing zero for the γ − π

2 problem. For moredetails, see [19].

10 T. I. Vogel

For unequal contact angles, numerical experimentation suggests the follow-ing.

Conjecture 1. For unequal contact angles, the instability which occurs as volumedecreases is always due to a violation of the dV/dH condition, and never due toλ10 crossing zero.

One particularly important fact about bridges between parallel planes is thatif the bridge is convex (i.e., the prole curve is a function with negative secondderivative), then the bridge is stable ([17]). This is true for unequal as well asequal contact angles.

A dierence between bridges with equal contact angles and bridges with un-equal contact angles relates to inections and stability. For equal contact angles,an interior inection in the prole curve implies instability. For unequal contactangles, however, there are proles of stable bridges which have inections. Asa simple example of the latter, consider contact angles which are complements,and not equal to π

2 . For suciently large volumes a stable bridge must exist.However, the prole of any bridge with contact angles as described must havean interior inection.

In doing some numerical experiments, W. C. Carter ([3]) observed an inter-esting fact concerning the minimum stable volume of a bridge between planes.Allowing any choice of the contact angle, the stable liquid bridge with smallestvolume that he observed had contact angles equal to π

2 , and is the previouslymentioned cylinder of radius 1

π times the separation of the planes. Finn andVogel ([8]) proved this observation for equal contact angles, and L. Zhou ([22])extended this to unequal contact angles.

For further discussion of the bridge between planes, see [8], [16], and [17].

7 Bridges between solid balls

Another capillary problem which has Delaunay surfaces as solutions is that ofa liquid bridge between two solid spheres ([11], [12]). The symmetry argumentoutlined in Section 6 fails for a bridge between balls, and in fact there are otherstationary solutions. For some values of the parameters (radius of the balls,separation of the centers, contact angles, and volume of the bridge) there existbridges whose free surfaces are also portions of spheres which are not symmetri-cally placed. It is not known whether there are other solutions to this capillaryproblem besides these asymmetrically placed spheres and the Delaunay surfaces.

Using Note 1, we may draw conclusions about stability of bridges betweensolid balls from the theory of stability of bridges between planes. Suppose that Σis a piece of a Delaunay surface whose boundary consists of two circles in parallelplanes, and suppose that the prole is convex. We may consider Σ as a bridgebetween parallel planes, or, using dierent contact angles, a bridge between solidballs. Lemma 2.3 in [11] shows that, if Σ is a section of a nodoid, then the valueof % decreases in going from considering Σ as a bridge between planes to a bridgebetween balls, and that if Σ is a section of an unduloid, the value of % increases

Delaunay Surfaces 11

in going from a bridge between planes to a bridge between balls. Intuitively,nodoidal bridges between balls are less stable than the corresponding surfacebridging between planes, and unduloidal bridges between balls are more stablethan the corresponding surface bridging between planes.

More precisely, one can show that the value of λ01 changes as in Table 2.Immediately, one can conclude that a convex bridge between balls which has a

Table 2. Arrangement of eigenvalues for a convex bridge between balls

......

......

...

λ30 λ21 λ12 λ03

λ20 λ11 λ02

λ10 λ01 < 0, nodoidλ01 > 0, unduloid

λ00

m = 0 m = 1 m = 2 m = 3 · · ·

nodoidal free surface must be unstable, since both λ00 and λ01 must be negative.

Comparing bridges between planes and bridges between balls also yields aconclusion about convex bridges between balls whose free surfaces are unduloids.If Σ is a convex bridge whose free surface is an unduloid, then the quadratic formin (4) increases when one goes from considering Σ as a bridge between planes toconsidering Σ as a bridge between balls. Since convex bridges between planes areknown to be stable (Section 6) it follows that convex unduloidal bridges betweenballs are also stable. Note that we can't simply cite Note 1 for this, since thestability criterion in Section 4 involves more than just the eigenvalues.

There are some numerical results on bridges between balls in [12], obtainedby cutting and scaling the standardized Delaunay surfaces in Section 2. Aninteresting phenomenon was observed: for certain values of the parameters in-volved, a (fold-over) bifurcation can occur without loss of stability. This occursif one starts with a stable bridge with a single negative eigenvalue and withH ′(A)V ′(A) > 0; then as A varies, the negative eigenvalue happens to cross zeroand becomes positive. As the eigenvalue crosses zero there is a bifurcation, butwith no negative eigenvalues, the bridge remains stable. A numerical exampleis given in Figure 3. The gure shows volume and mean curvature for a familyof symmetric bridges between two solid balls of radius 1 unit, whose centers areseparated by 3 units. The large-volume bridges corresponding to the right endof the curve are close to a symmetrically placed spherical bridge, and are stable.As volume decreases, note the rst fold-over in the curve, close to the point onthe curve at which volume equals 8. This fold-over bifurcation occurs when λ00crosses zero to become positive. As volume continues to decrease, a second fold-over occurs, as λ00 crosses zero again. The bridge is stable through all of this,

12 T. I. Vogel

only becoming unstable when a minimum volume is reached at about H = −0.8and the dV/dH condition fails.

Fig. 3. Volume and mean curvature for γ = π20, R = 3

.

This behavior, i.e., λ00 crossing zero to become positive, can't occur for abridge between parallel planes. In this case λ00 always remains negative sinceλ00 < λ01 = 0.

A case of particular physical and mathematical interest is a bridge betweentwo balls of equal radius which touch, with the same contact angle with both balls([13]). If the contact angle is less than π

2 , then existence of stable rotationallysymmetric bridges is shown for a large range of volumes. However, if contactangle is greater than or equal to π

2 , then no stable rotationally symmetric bridgesexist. Thus for a bridge between contacting balls, existence of a stable bridgedepends discontinuously on the contact angle.

8 Capillary surfaces in cylinders

Another example of a capillary problem with Delaunay surfaces as stationarysolutions occurs when the xed surface Γ is a circular cylinder and the regionΩ occupied by the liquid is interior to the cylinder (see section 2 of [10]). Theliquid drop is topologically a solid torus, as in Figure 4. In this gure, the xedcylindrical surface may be seen as part of the boundary of Ω.

For this problem, we can notice an energy-neutral translation, in a directionparallel to the axis of the cylinder. As for the problem of the liquid bridgebetween parallel planes, this leads to an eigenvalue equal to zero. However inthis case, the eigenfunction corresponding to this eigenvalue of zero is rotationally

Delaunay Surfaces 13

Fig. 4. Toroidal drop

symmetric (it is just the normal component of a vector directed along the axisof the cylinder). This eigenfunction equals zero once, at the midpoint of theinterval. Thus by standard Sturm-Liouville theory ([2]) this corresponds to λ10.Therefore λ10 = 0 for the toroidal drop in a circular cylinder. The eigenvaluesfor this problem may be arranged as in Table 3

Table 3. Arrangement of eigenvalues for the toroidal drop in a cylinder

......

......

...

λ30 λ21 λ12 λ03

λ20 λ11 λ02

λ10 = 0 λ01

λ00

m = 0 m = 1 m = 2 m = 3 . . .

We may again draw conclusions about this problem from facts about bridgesbetween parallel planes. The plan of attack is to consider a bridge between par-allel planes whose free boundary is the same Delaunay surface. (As an example,the free surface of the bridge in Figure 2 is the same Delaunay surface as thefree surface of the toroidal drop in Figure 4.) Call that Delaunay surface Σ. Fol-lowing the notation in [10], let %o be the value of % (as dened in (5)) when Σ isconsidered as a bridge between parallel planes, and let %n be the value of % whenΣ is considered as the free boundary of a toroidal drop in a circular cylinder.Here o stands for old, n for new. The reasoning behind the terminologyis that the bridge between planes is used as a starting point, and we use that to

14 T. I. Vogel

deduce results about a newer problem. Lemma 2.1 of [10] shows that %o and %neither have opposite signs or are both zero.

Note 2. A startling result which follows from Lemma 2.1 of [10] is that a toroidaldrop in a circular cylinder whose prole is a graph with second derivative strictlypositive must be unstable (Theorem 2.3 of [10]). The essential idea of the proof isconsider Σ both as as the free surface of a bridge between planes and as the freesurface of a toroidal drop. An innitesimal translation of the bridge parallel tothe planes is energy neutral. More precisely, dene φ on Σ to be the inclinationof the prole at each point. Then

Mo(cos(φ), cos(φ)) = 0,

with˜ΣcosφdΣ = 0. One can compute that %o > 0 in this set-up. Thus %n < 0,

and it follows (see [10] for details) that

Mn(cos(φ), cos(φ)) < 0.

Since this is an innitesimally volume conserving perturbation which reducesenergy, this toroidal drop is unstable.

Note 3. Another startling result of Lemma 2.1 of [10] is that, if a toroidal dropin a cylinder has a prole with an inection, then there there is precisely onenegative eigenvalue, λ00. The eigenvalue λ10 must be zero by previous remarks,and all the rest must be strictly positive (Theorem 2.4 of [10]). The basic ideaof the proof is that in this case, %o < 0 < %n. If we call the eigenvalues for thebridge (old) problem λoij and the eigenvalues for the toroidal drop in the cylinder(the new problem) λnij , increasing % increases (or at least doesn't decrease) thecorresponding eigenvalues. Then λo10, which is negative for an inected bridgebetween planes, increases up to zero, and λo01, which is automatically zero forthe bridge between planes, increases up to λn01 > 0. One can verify ([10]) thatthis is strictly positive by arriving at a contradiction if λn01 = 0.

Figure 5 gives volume (vertical axis) and mean curvature (horizontal axis)of the family of toroidal drops with contact angle π

20 , and with two interiorinection points in the prole. The right endpoint corresponds to the unduloidwith inection on the cylinder, and the left endpoint is the limiting case of theexterior of two spherical caps. All of these proles have precisely one negativeeigenvalue. Those for which dV

dH > 0 are stable; thus there is numerical evidenceof stable toroidal drops in cylinders, which will have inections in their proles.To contrast the toroidal drop with the liquid bridge between planes (with equalcontact angles): a stable bridge between planes cannot have inections, but astable toroidal drop must have inections.

9 When can one ignore non-symmetric perturbations?

A common misconception is that, if a problem and its solution are rotationallysymmetric, then in determining stability one need consider only rotationally sym-metric perturbations. This is certainly false in general. For example, nodoidal

Delaunay Surfaces 15

Fig. 5. H vrs. V , inected prole, γ = π20

bridges between solid balls are unstable with respect to non-symmetric pertur-bations (Section 7). Similarly, toroidal drops in circular cylinders whose proleshave their second derivatives bounded from zero are unstable with respect tonon-symmetric perturbations (Section 8). However, it is natural to wonder un-der what circumstances we are safe in restricting consideration to symmetricperturbations in determining stability. A partial answer comes from considera-tion of the bridge between parallel planes.

Lemma 1. Suppose that Σ is a rotationally symmetric piece of a Delaunay sur-face, parameterized by s1 ≤ s ≤ s2, 0 ≤ θ ≤ 2π as in Section 5. Suppose that Σmay be considered as a bridge between parallel planes, with contact angle γo, andthat Σ may also be considered as a capillary surface contacting a rotationallysymmetric smooth surface Γn, with contact angle γn. (As an example, the De-launay surface in Figure 2 is the same surface as the free surface of the toroidaldrop in Figure 4.) Let %o be the value of % computed from (5) for Σ as a bridgebetween planes, and let %n be the value of % for Σ as a capillary surface contact-ing Γn. (As before, o stands for old and n stands for new.) If %n > %o,then λn01 ≥ 0.

Proof. This follows from Note 1 and the fact that λo01, the eigenvalue for theold problem of the bridge between planes, is automatically zero.

Proposition 1. Assume the set-up is as in Lemma 1. If %n < %o, then Σ isunstable as a capillary surface contacting Γn with contact angle γn.

Proof. In the case %n < %o the argument follows that of Note 2: the energyneutral translation of a bridge between planes turns into an energy decreas-ing perturbation in the case of a capillary surface contacting Γn, and we haveinstability.

16 T. I. Vogel

Note 4. In the case %n > %o, we have λn01 ≥ 0 from the above Lemma. Since λ00

is the only negative eigenvalue, and this corresponds to a rotationally symmetriceigenfunction, it seems likely that rotationally symmetric perturbations are themost dangerous. One would expect that if Σ, as a capillary surface contactingΓn , is stable with respect to rotationally symmetric perturbations which arevolume conserving, then it is stable with respect to non-rotationally symmetric(volume-conserving) perturbations as well. I hope to give a formal proof to thisintuitively plausible statement in a later paper.

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