Capillary Surfaces in Circular Cylinders

Thomas I. Vogel

Abstract. In the absence of gravity, a capillary surface within a circularcylinder will have constant mean curvature and will make a constantcontact angle with the cylinder. Two types of surfaces with these prop-erties are Delaunay surfaces and cylinders (i.e., the free surface of theliquid is another circular cylinder). Stability and energy minimality ofthese capillary surfaces are investigated.

Mathematics Subject Classication (2010). Primary 76B45; Secondary53A10.

1. Introduction

The purpose of this paper is to investigate stability and energy minimality ofcapillary surfaces in circular cylinders in the absence of gravity. Although thiscapillary problem is not as well studied as others, it is of physical signicance.For example, it relates to bubbles which form in pipes in spacecraft, and ithas been studied in the biomathematics of pulmonary passages (see [1], [6],and [7]). A standard reference for general capillary surfaces is [3].

In the absence of gravity, capillary surfaces are determined by the prob-lem of minimizing surface energy (free surface area plus a constant timeswetted surface area) subject to a volume constraint. To be specic, supposethat Ω corresponds to a drop of liquid contacting a xed surface Γ. Then theboundary of Ω consists of two parts: the wetted region on Γ, which we willcall Σ∗, and the free surface Σ. The capillary problem consists of minimizing|Σ|−C|Σ∗|, C ∈ [−1, 1], subject to holding the volume of Ω constant. Here Cis a constant depending on the materials involved. The rst order necessaryconditions for a minimum are that the mean curvature of Σ be constant, andthat the contact angle γ be constantly arccos(C) along the curve σ of contactbetween Σ and Γ. (The angle γ is measured between the normals to Σ and Γ.)We will call a surface satisfying these rst order conditions a stationary cap-illary surface. For convenience, we will refer to Σ as the liquid-air interface,although it could easily be the interface between two immiscible uids.

2 T. I. Vogel

There are two obvious surfaces of constant mean curvature which haveconstant contact angle with a circular cylinder: spherical caps and circularcylinders with axes parallel to axis of the original xed cylinder. Perhaps notquite as obviously, there is also a family of toroidal drops whose free surfacesare Delaunay surfaces. In addition to these surfaces, there are non-symmetricsolutions, which were investigated numerically in [1]. The bifurcation anal-ysis of a similar problem in [12] suggests that there are further stationarysurfaces bifurcating from the unstable cylinders of liquid as well, so I makeno claim that the surfaces mentioned are all of the capillary surfaces in cir-cular cylinders. The types of surfaces which will be examined in this paperare Delaunay surfaces and cylindrical free surfaces.

We will dene a capillary surface to be stable if it satises the secondorder necessary conditions to be a local energy minimum. As in [9], (see also[13]), the quadratic form relating to stability is

M(ψ,ψ) =

¨Σ

|∇ψ|2 − |S|2ψ2 dΣ +

˛σ

ρψ2 dσ, (1.1)

where σ = ∂Σ. Here |S|2 is the square of the norm of the second fundamentalform of Σ. (In terms of mean curvature H and Gaussian curvature K, |S|2may be written as 2(2H2−K), and in terms of the principal curvatures, |S|2may be written as k2

1 + k22.) The coecient ρ is given by

ρ = κΣ cot γ − κΓ csc γ, (1.2)

where κΣ is the curvature of the curve Σ ∩ Π and κΓ is the curvature ofΓ∩Π, if Π is a plane normal to the contact curve ∂Σ. The planar curvaturesare signed; refer to [9], Figure 2, for an example where both are negative. Acapillary surface will be stable if

M(ψ,ψ) ≥ 0 (1.3)

for all suciently regular ψ for which˜

Σψ dΣ = 0, and will be unstable if

this is not true.Again following [9], dene the dierential operator L by

L(ψ) = −∆ψ − |S|2ψ (1.4)

where ∆ is the Laplace-Beltrami operator on Σ. The eigenvalue problemwhich arises is given by

L(ψ) = µψ (1.5)

on Σ, with

b(ψ) ≡ ψ1 + ρψ = 0 (1.6)

on σ, where ψ1 is the outward normal derivative of ψ.The relation between a capillary surface being stable and a capillary

surface being a local energy minimum over volume conserving perturbations iscomplex. The author has investigated this in several papers. Most relevant tothe present paper are Theorem 4.2 of [11] and Corollary 1 of [8]. To summarizethese, if the eigenvalue problem (1.5), (1.6) has two negative eigenvalues, then

Capillary Surfaces in Cylinders 3

Figure 1. Toroidal drop

the capillary surface is unstable and therefore not a local energy minimumin any sense. If the eigenvalue problem has precisely one negative eigenvalue,then we look at the eigenfunctions corresponding to the eigenvalue 0. If thesesimply correspond to energy neutral motions which take the xed surfaceinto itself (e.g., for the liquid bridge between parallel planes, these would betranslations which take the planes into themselves), then the zero eigenvaluesmay be ignored when one seeks energy minima modulo those motions. In thatcase, a condition on how H and V change through a parameterized family ofsolutions may be applied to determine energy minimality.

To be specic, suppose that the capillary surface of interest is embeddedin a family of stationary capillary surfaces Σ(ε) smoothly parameterized bysome ε, and that ε = ε0 gives the surface of interest. Then, in the case thatthere is precisely one negative eigenvalue, if H ′(ε0)V ′(ε0) > 0, then Σ(ε0)is stable and in fact a local energy minimum (modulo translations), and ifH ′(ε0)V ′(ε0) < 0, then Σ(ε0) is unstable. Here V (ε) is the volume containedby Σ(ε) and H(ε) is the mean curvature of Σ(ε) with respect to the normalpointing out of the liquid. For further details see [11] and [8].

2. Toroidal drops bounded by Delaunay surfaces

To make the problem specic, we will set the xed surface Γ to be the cylin-der x2 + z2 = 1, and consider capillary surfaces inside of this cylinder. Thecapillary surfaces in circular cylinders which we will consider in this sectionare those bounding regions which are topologically solid tori. (See Figure 1.)

4 T. I. Vogel

Figure 2. Liquid bridge between parallel planes

In this gure, the solid cylinder does not appear, although part of the cylin-der may be seen as a boundary of the drop.) We will see that the stability ofsuch a toroidal drop relates to the stability of a liquid bridge between parallelplanes bounded by the same Delaunay surface. In Figure 2, the same Delau-nay surface as in Figure 1 is now bounding a bridge between parallel planes.Although the liquid has ipped across the Delaunay surface, the surface re-mains as the Σ on which the integration forM(ψ,ψ) is performed, althoughthe contribution from the line integral will change. We will draw conclu-sions about stability of the toroidal drops from knowledge of the stability ofbridges between parallel planes and comparing the corresponding ρ values forthe two problems. Since we will be comparing dierent quadratic forms andeigenvalue problems, the notation will be that the subscript "o" (for "old")will correspond to the problem of the bridge between parallel planes, and thesubscript "n" (for "new") will correspond to a toroidal drop in a cylinder.For example, Lo is the operator L for the bridge between planes, and ρn isthe ρ value for the toroidal drop in a cylinder. The old and new notationparallels that used in [9], as does the approach which follows.

Lemma 2.1. Suppose that Σ is a section of a Delaunay surface which canbe considered as the boundary of a toroidal drop in a cylinder (as in Figure1) and also as the boundary of a liquid bridge between parallel planes (as inFigure 2). Suppose also that neither γo nor γn is zero. Then ρo, the value ofρ for the bridge, and ρn, the value of ρ for the toroidal drop, have oppositesigns or are both zero.

Capillary Surfaces in Cylinders 5

Proof. Consider equation (1.2). We have (κΓ)o and (κΓ)n equal to zero, sincethe intersection of the normal plane Π with the xed substrate in both casesconsists of lines, thus with curvature zero. We thus have only the rst termof (1.2) to consider. Both γo and γn will be in

(0, π2

)(they must be com-

plements), so that both cot γo and cot γn are positive. We have that whileΣo∩Π and Σn∩Π are the same curve, however, in computing κΣ, the normalpointing out of the drop is reversed, thus (κΣ)o = −(κΣ)n.

We must recall some facts about general rotationally symmetric capil-lary surfaces. A rotationally symmetric surface of constant mean curvaturewill be a Delaunay surface, which we can parameterize by s and θ, where sis arclength along the prole curve and θ is the angle with the x axis. If theboundary conditions are also rotationally symmetric, one can solve the eigen-value problem (1.5), (1.6) by separation of variables (see [13], also [11]). Theeigenvalues may be written as λjk. Symmetric eigenfunctions correspond toλj0. For k ≥ 1 there is a sequence of eigenvalues λjk, with each correspondingto a two dimensional eigenspace spanned by fjk(s) cos(kθ) and fjk(s) sin(kθ).Moreover, the eigenvalues satisfy

λ0k < λ1k < · · ·for each k, and

λ00 < λ01 < λ02 < · · · .For the problem of a liquid bridge between parallel planes, it is known thatλ01 = 0 (see [11]), since this corresponds to an energy neutral translationof the bridge. To keep the notation consistent, for a bridge between parallelplanes, we will refer to the eigenvalues as λojk, so that λo01 = 0. We will referto eigenvalues of a toroidal drop in a cylinder as λnjk.

Lemma 2.2. For any toroidal drop in a cylinder, λn10 = 0.

Proof. A translation parallel to the axis of the cylinder is energy neutral,hence corresponds to a zero eigenvalue. The eigenfunction is the dot prod-uct of innitesimal translation with the normal vector. This is rotationallysymmetric, hence in the k = 0 family. By the geometry of Delaunay surfaces,this dot product is zero exactly once. By standard Sturm-Liouville theory,this implies that this eigenvalue is the second smallest in this family, i.e.,j = 1.

Theorem 2.3. A toroidal drop in a cylinder whose prole is a graph withsecond derivative strictly positive must be unstable.

Proof. Call the free surface of the toroidal drop Σ. Considering this as bound-ing a drop in a cylinder, label the value of ρ for this surface ρn, with ρo beingthe value of ρ when Σ is considered as a bridge between parallel planes (asin Figures 1 and 2 respectively). Since the surface integrals in the quadraticformM are the same, we have

Mn(ψ,ψ)−Mo(ψ,ψ) =

˛σ

(ρn − ρo)ψ2 dσ (2.1)

6 T. I. Vogel

holding for any normal perturbation ψ ~N of Σ. When considered as a bridgebetween parallel planes, there is a volume conserving perturbation ψ forwhich Mo(ψ,ψ) = 0, namely the normal component of an innitesimaltranslation of the bridge in a direction parallel to the planes. To be spe-cic, dene ϕ on Σ to be the inclination of the prole curve at a point. ThenMo(cos(ϕ), cos(ϕ)) = 0 and

˜Σ

cos(ϕ) dΣ = 0 (see [11]). Note that the in-clination of the prole at each curve of contact must be π

2 − γo. Then from(2.1) it follows that

Mn (cos(φ), cos(φ)) = 4π(ρn − ρo) sin2 γ, (2.2)

since the line integral in (2.1) is the integral of the constant (ρn − ρo) sin2 γaround two circles of radius 1. Now, by the assumption of the theorem, theprole is concave up. Thus ρo > 0 since (κΣ)o > 0. From (2.2),

Mn (cos(ϕ), cos(ϕ)) < 0.

Since this perturbation is volume conserving, it follows that when Σ is con-sidered as the boundary of a toroidal drop in a cylinder, that drop is unsta-ble.

For the problem of a liquid bridge between parallel planes, the appear-ance of an inection in the prole corresponds to an increase in the numberof negative eigenvalues of Lo(ψ) = µψ on Σ, ρo(ψ) = 0 on ∂Σ, see [11]. Instark contrast, for the problem of a toroidal drop in a cylinder, the appear-ance of an inection in the prole corresponds to a decrease in the numberof negative eigenvalues. More precisely,

Theorem 2.4. Suppose that Σ is the boundary of a toroidal drop in a cylinder,and that the prole of Σ has an inection between the curves of contact withthe cylinder. Then λn00 < 0, λn10 = 0, and all other eigenvalues of the problemLn(ψ) = µψ on Σ, ρn(ψ) = 0 on ∂Σ are strictly positive.

Proof. First note that the only Delaunay surface which can bound a toroidaldrop in a cylinder and have an inection interior to the prole is an unduloid.In fact, there must be two inections in this case; a single interior inectionis not possible. For this prole, we must also have the prole convex downat the ends of the prole. In contrast to Theorem 2.3, ρo < 0 < ρn. Thusin going from the old problem (of a bridge between parallel planes) to thenew problem of a drop in a cylinder, the corresponding eigenvalues of theproblems do not decrease (see [2]). From Lemma 2.2 we know that λn10 = 0and from [11] we know that

0 = λo10 ≤ λn10. (2.3)

It remains to verify that the inequality in (2.3) is strict. If λn01 = 0, then thecontradiction is found as at the end of section 4 of [11]. The minimum of theRayleigh quotient using ρn will be zero, and the minimizer for the Rayleighquotient using ρn will automatically be a minimizer for the Rayleigh quotientusing ρo. This will then be the eigenfunction for λ01 for the bridge betweenparallel planes, i.e., cosφ. From (2.2) we obtain a contradiction.

Capillary Surfaces in Cylinders 7

Figure 3. H vrs. V , inected prole, γ = π20

To show that a toroidal drop satisfying the hypotheses of Theorem 2.4is stable (modulo translations down the cylinder), what remains is to look atthe sign of V ′(ε)H ′(ε), as outlined in the introduction, for some parameter ε.A convenient way of parameterizing these toroidal drops is to cut and scalestandardized Delaunay curves as in section 2 of [10], where the parameter is A,the mean curvature of the standardized Delaunay surface before scaling. Thecurve is cut at points leading to the appropriate contact angle and scaled to tinside a cylinder of radius 1. Using this parameterization we can investigateinected toroidal drops numerically.

Figure 3 gives volume (vertical axis) and mean curvature (horizontalaxis) of the family of toroidal drops with contact angle γ = π

20 , and withtwo interior infection points in the prole. The right endpoint of this curvecorresponds to a drop with inection points very near the curve of contact,as in Figure 5. The left endpoint of this curve corresponds to a drop withinection points near the center, as in Figure 6. Figure 4 gives a similar plotfor contact angle π

6 .

Remark 2.5. The interval for the parameter A which will describe the family

of inected proles of toroidal drops with contact angle γ is(

sin γ−12 , 0

). The

standardized unduloid with inection at inclination angle equal to γ is givenby A = sin γ−1

2 . I will refer to this as the inection end of the family. As Aincreases to 0, the prole approaches two arcs of circles. I will refer to this asthe cap end of the family, since the surface approaches two spherical caps.Figure 5 is close to the inection end for γ = π

20 and Figure 6 is close to thecap end of that family.

Note 2.6. The fact that A = sin γ−12 gives a standardized Delaunay curve

whose inection point occurs with inclination angle equal to γ may be derivedby requiring dϕ

ds in equation (2.6) of [10] to be zero, and observing that y cosϕ+

8 T. I. Vogel

Figure 4. H vrs. V , inected prole, γ = π6

Figure 5. Prole with inections near contact, γ = π20

Ay2 is constantly 1 + A on the Delaunay curve. Requiring ϕ to be γ at theinection yields the system

cos γ + 2Ay =0 (2.4)

y cos γ =1 +A. (2.5)

Solving this for cos γ and y in terms of A gives cos γ = 2√−A−A2, from

which the desired value of A follows.

The product V ′(A)H ′(A) will be positive when dVdH > 0, thus Figure 3

indicates that, for contact angle γ = π20 , the toroidal drops corresponding to

the left part of the curve are stable. Thus, numerical work indicates that a

Capillary Surfaces in Cylinders 9

Figure 6. Prole with inections near center, γ = π20

toroidal drop with prole as in Figure 6 will be stable, and a toroidal dropwith prole as in Figure 5 will be unstable.

Other contact angles between 0 and π2 were investigated numerically,

and gures analogous to 3 and 4 were found for volume and mean curvaturefor the inected family. This suggests the following:

Conjecture 2.7. For every contact angle γ ∈(0, π2

), there exist toroidal drops

in cylinders which are stable (modulo translation parallel to the axis of thecylinder) which have two inections in their proles.

Note that uninected proles must lead to unstable toroidal drops byTheorem 2.3.

A partial result in the direction of Conjecture 2.7 may be proven bycomputing the volumes of the limiting congurations in the family of proleswith two inections.

Lemma 2.8. (Cap end volume) The limiting volume as A → 0− in the in-ected family of toroidal drops with contact angle γ is

V1(γ) = π

(4

3tan3 γ − 2 tan2 γ sec γ +

2

3sec3 γ

).

Proof. This is a routine calculation, simply subtracting the volumes of twospherical caps from the volume of the appropriate cylinder.

The limiting volume at the inected end is a bit more complicated. Itis based on the calculation in [4], section 4b, of the volume contained by anunduloid between successive inections. The calculation is somewhat tediousand perhaps peripheral to the purpose of this paper, so I will summarize theresult as the following lemma.

10 T. I. Vogel

Lemma 2.9. (Inected end volume) The limiting volume as A→ sin γ−12

+of

the inected toroidal drop in a cylinder with contact angle γ is

V2 =πh√−1− 1

A0

− πh3

4 (E (k)− k)3(−1− 1

A0

)3/2

−3k +

1

3k3 +

7 + k2

3E (k)− 4

3

(1− k2

)K (k)

.

(2.6)

Here A0 = sin γ−12 , k = sin γ,

K(k) =

ˆ π/2

0

dt√1− k2 sin2 t

is a complete elliptic integral of the rst kind, and

E(k) =

ˆ π/2

0

√1− k2 sin2 t dt

is a complete elliptic integral of the second kind.

We must also determine the limiting volumes at the two ends of thefamily of inected proles.

Lemma 2.10. The limiting mean curvature as A→ 0− of the toroidal drop ina cylinder is 0 for any contact angle in

(0, π2

). The limiting mean curvature

as A → sin γ−12

+in the family of inected toroidal drops with contact angle

γ is (1−sin γ)3/2

2√

1+sin γ.

Proof. To construct a torioidal drop from the standardized unduloid, one cutsthe undulary (the prole of the unduloid) at a point with inclination γ andscales the result out so that the point with inclination γ is one unit from theaxis of rotation. To take into account the change in sign of mean curvaturesince we are ipping the liquid to be on the outside of the unduloid, the meancurvature H of the drop corresponding to a parameter value A is

H = − A

yγ(A),

where yγ(A) is a y coordinate of a point on the standardized unduloid wherethe inclination is γ. There can be at most two dierent such y values. Sincewe are interested in toroidal drops with inections in their proles, we willchoose the larger of the two values.

As A tends to 0, the prole of the unduloid approaches a circulararc. The limit as A tends to 0 of yγ(A) is therefore cos γ from elementarytrigonometry. Thus as A tends to 0, H also tends to 0.

At the inected end, as A → sin γ−12

+, yγ(A) approaches the radius of

the inection on the undulary. Using formulas from [4], this is√−1− 1

A .

Substituting in what A approaches, the result is straight-forward.

Capillary Surfaces in Cylinders 11

Proposition 2.11. For any γ ∈(0, π2

)for which V1(γ) < V2(γ), there exist

stable toroidal drops with interior inection points.

Proof. As parameter values go from sin γ−12 (the inected end of the family)

to arbitrarily close to 0 (the cap end), the change in volume is V2 − V1, and

the change in mean curvature is (1−sin γ)3/2

2√

1+sin γ> 0. Thus when V2 > V1, there

will exist a parameter value for which dVdH > 0 by the Cauchy Mean Value

theorem (see [5], or almost any other textbook on advanced calculus).

Numerical experiments indicate the following: the equation V1(γ) =V2(γ) has one root γ0 ∈ (0, π), with γ0 ≈ 14.4. If γ > γ0, then V1(γ) < V2(γ).For these values of γ, Proposition 2.11 gives existence of inected toroidaldrops. Of course, numerical experiments also indicate that Conjecture 2.7 istrue, which is a stronger statement.

3. Cylindrical and Planar Free Surfaces

An obvious stationary capillary surface inside a xed cylinder Γ is a freesurface Σ which is also a cylinder contacting Γ with the prescribed contactangle γ. A planar free surface can be considered as a special case in thisfamily. Since the region occupied by liquid is unbounded, we must be carefulhow we interpret stability. As in [12], we say that a cylindrical free surfaceis stable if it is stable with respect to compact perturbations. Thus, we saythat the free surface is unstable if for some h > 0 there is a perturbation ofthe liquid which preserves the volume of the liquid between the planes y = 0and y = h for which the second variation of the energy of the region betweeny = 0 and y = h is negative. If no such h exists, then the cylindrical freesurface is stable.

We rst consider the planar case. Recall that we assume that the cylin-der is x2 + z2 = 1. For contact angle γ, we may consider the planar freesurface Σ to lie in the plane z = cos γ, and in fact to be parameterized asx = u, y = v, z = cos γ, with − sin γ < u < sin γ, and −∞ < v < ∞. Forh > 0, let Σh be the portion of Σ which lies between the planes y = 0 andy = h. We will see that for every contact angle γ the free plane is unstable:there exists an h suciently large so that there is a ϕ dened on Σh, equalto zero at y = 0 and y = h, with

˜ΣhϕdS = 0 andM (ϕ,ϕ) < 0.

Theorem 3.1. Stationary planar free surface solutions to the capillary probleminside xed cylinders are unstable.

Proof. We must address the eigenvalue problem (1.5), (1.6). From the pa-rameterization of Σh, L (ψ) = −ψuu−ψvv. For boundary conditions, we needρ = κΛ cot γ − κΣ csc γ. Of course, κΛ = 0, and κΣ is the sectional curvatureof circular cross section of the xed cylinder with respect to a normal point-ing into the uid, hence is +1. Thus, in coordinates, the eigenvalue problem

12 T. I. Vogel

is

−ψuu − ψvv = µψ,

ψu (sin γ, v)− 1

sin γψ (sin γ, v) = 0,

−ψu (− sin γ, v)− 1

sin γψ (− sin γ, v) = 0,

ψ (u, 0) = ψ (u, h) = 0.

We approach this using separation of variables. Set ψ (u, v) to be U (u)V (v),

thus −U′′

U −V ′′

V = µ. Since the boundary conditions imply that V will bea trigonometric function, we nd (omitting routine details) that V (v) =

C sin(kπh v)and U ′′

U = k2π2

h2 − µ. Since instability is related to the number ofnegative eigenvalues µ, we consider the ordinary dierential equation

U ′′

U= β2, (3.1)

with boundary conditions

U ′ (a)− 1

aU (a) = 0,

−U ′ (−a)− 1

aU (−a) = 0,

where for convenience we set sin γ to be a.We seek U (u) = Aeβu +Be−βu solving (3.1) . Since U ′ (u) = Aβeβu −

Bβe−βu, the boundary conditions become

Aβeβa −Bβe−βa − 1

a

(Aeβa +Be−βa

)= 0,

−(Aβe−βa −Bβeβa

)− 1

a

(Ae−βa +Beβa

)= 0,

which we consider as the following system in the unknowns A and B.

A

(βeβa − 1

aeβa)

+B

(−βe−βa − 1

ae−βa

)= 0, (3.2)

A

(−βe−βa − 1

ae−βa

)+B

(βeβa − 1

aeβa)

= 0. (3.3)

There will be non-trivial solutions A and B if and only if the determinant iszero. Thus the question is whether, given a ∈ (0, 1), is there a β such that(

βeβa − 1

aeβa)2

=

(βe−βa +

1

ae−βa

)2

?

Multiply both sides by a and let t = aβ to arrive at(tet − et

)2=(te−t + e−t

)2,

i.e.,

e4t =

(t+ 1

t− 1

)2

. (3.4)

Capillary Surfaces in Cylinders 13

By looking at limits of both sides as t → ∞ and t → 1+, clearly there is aroot. Numerically, equation (3.4) has a root at t0 ≈ 1.1997. Since β = t0

a , wend that for xed h and γ,

µ =k2π2

h2− t20

sin2 γ

is an eigenvalue. By setting h to be suciently large, we have an arbitrarynumber of negative eigenvalues. Since having two negative eigenvalues implyinstability, for large enough h, the rectangle Σh is unstable. Thus Σ is notstable with respect to compact perturbations.

Similarly, cylindrical free surfaces will be unstable.

Theorem 3.2. Stationary cylindrical free surface solutions inside xed cylin-ders are unstable.

Proof. Take the free cylinder to have radius r as in Figure 7. From the lawof cosines, the free cylinder can be parameterized as

~x =⟨r cosu−

√1 + r2 − 2r cos γ, v, r sinu

⟩, −β < u < β, −∞ < v <∞. Here we make the assumption that the regionoccupied by air is that interior to the free cylinder, and the liquid region isexterior to the free cylinder. Switching the role of "air" and "liquid" leadsto a completely equivalent problem, with contact angle γ being replaced byπ − γ and the signs of the sectional curvatures changing, so nothing is lostby neglecting that case. As in [12], using the parameterization one can verifythat

L(ψ) = − 1

r2ψuu − ψvv −

1

r2ψ. (3.5)

The relation between r, γ, and β can be found from Figure 7 is

tanβ =sin γ

r − cos γ,

using the law of sines.

The normal derivative of ψ at the boundary of the free surface is 1rψu at

u = β and − 1rψu at u = −β. Thus the boundary conditions for the eigenvalue

problem are

ψ =0, v = 0, h

1

rψu + ρψ =0, u = β,

−1

rψu + ρψ =0, u = −β.

There remains to determine ρ. We have

ρ = κΛ cot γ − κΣ csc γ =1

rcot γ − csc γ = −cotβ

r,

14 T. I. Vogel

Figure 7. Cross section of a free cylinder in a xed cylinder

therefore the boundary conditions to apply to (3.5) for the eigenvalue problembecome

ψ(u, 0) = 0 (3.6)

ψ(u, h) = 0 (3.7)

ψu(β, v)− ψ(β, v) cotβ = 0 (3.8)

ψu(−β, v) + ψ(−β, v) cotβ = 0. (3.9)

Since γ and r do not appear explicitly in the boundary conditions, we maydetermine stability by considering dierent cases for β.

We approach the eigenvalue problem L(ψ) = µψ, with boundary con-ditions (3.6)(3.9) by separation of variables as in the previous theorem. Setψ(u, v) = U(u)V (v), so that

− 1

r2U ′′V − UV ′′ =

(µ+

1

r2

)UV.

The boundary conditions suggest that V should be sin(kπh v), so we set

V ′′

V= − 1

r2

U ′′

U− µ− 1

r2= −k

2π2

h2.

We may therefore say

U ′′

U=r2k2π2

h2− µr2 − 1. (3.10)

We may use (3.10) to show instability for any value of β ∈ (0, π). Wedivide into cases.

• Case 1: β ∈(0, π2

). Consider the case that the right hand side of (3.10)

is positive, and call it q2. Then U(u) will be Aequ +Be−qu. Substitute

Capillary Surfaces in Cylinders 15

this into the separated boundary conditions arising from (3.8), (3.9) toobtain

Aeqβ (q − cotβ) +Be−qβ (−q − cotβ) = 0,

Ae−qβ (q + cotβ) +Beqβ (−q + cotβ) = 0,

which will have nontrivial solutions if and only if

e4qβ =

(q + cotβ

q − cotβ

)2

. (3.11)

In the case β ∈(0, π2

), (3.11) will have non-trivial roots. To see this,

note that as q → cotβ+, the right hand side of (3.11) will tend to +∞while the left hand side is bounded, whereas as q → +∞, the left handside tends to 1 while the right hand side tends to +∞. Looking at thederivatives of both sides with respect to q shows that for each β ∈

(0, π2

)there is in fact a unique positive root of (3.11), call it q0(β).

Thus for the case β ∈(0, π2

),

µ =k2π2

h2− 1

r2− q0(β)2

r2(3.12)

is an eigenvalue for L(ψ) = µψ, with boundary conditions (3.6)(3.9).Since r is xed, by choosing h suciently large, there are as many neg-ative eigenvalues as desired. In particular, taking h larger than πr√

1+q20,

both k = 0 and k = 1 will give negative values for µ. Therefore, as inTheorem 3.1, we have instability for the case of β ∈

(0, π2

).

• Case 2: β = π2 . In this case, the boundary conditions for U(u) become

U ′(−β) = U ′(β) = 0, so that there is a separated solution when theright hand side of (3.10) equals zero. In this case, for every non-negative

integer k, µ = k2π2

h2 − 1 is an eigenvalue. As in the previous case, if h islarge enough, there will be more than one negative eigenvalue, and wehave instability in this case as well.• Case 3: β ∈

(π2 , π

). Now consider the case that the right hand side of

(3.10) is negative, and call it −w2. Then U(u) = A sinwu + B coswu,and the separated boundary conditions arising from (3.8), (3.9) become

A (w coswβ − cotβ sinwβ) +B (−w sinwβ − cotβ coswβ) = 0

A (w coswβ − cotβ sinwβ) +B (−w sinwβ − cotβ coswβ) = 0,

with non-trivial solutions if and only if either

w cotwβ = cotβ (3.13)

or

−w tanwβ = cotβ. (3.14)

The obvious root of w = 1 to (3.13) does not lead to negativeeigenvalues. Instead, look at (3.14). As w → 0+, −w tanwβ → 0. Asw → π

2β−, −w tanwβ → −∞. Since cotβ < 0 in this case, (3.14) will

16 T. I. Vogel

have a root in(

0, π2β

). But β > π

2 , so (3.14) will have a root in (0, 1),

call it w0(β).Thus for the case β ∈

(π2 , π

),

µ =π2k2

h2− 1

r2+w0(β)2

r2(3.15)

is an eigenvalue for L(ψ) = µψ, with boundary conditions (3.6)(3.9).Since w0 ∈ (0, 1), for h suciently large, (3.15) will yield as manynegative eigenvalues as desired, giving instability of the free cylinder inthis case as well.

References

[1] S. H. Collicot, W. G. Lindsey, and D. G. Frazer, Zero-gravity liquid-vaporinterfaces in circular cylinders, Physics of Fluids 18 087109, 2006.

[2] R. Courant and D. Hilbert, Methods of Mathematical Physics, volume 1. Inter-science Publishers, New York, NY, 1953.

[3] R. Finn, Equilibrium Capillary Surfaces, Spring-Verlag, New York, NY, 1986.

[4] R. Finn and T. I. Vogel, On the volume inmum for liquid bridges, Zeitschriftfuer Analysis und Ihre Anwendungen (1992), 323.

[5] P. Fitzpatrick, Advanced Calculus, second edition, Thomson Brooks/Cole, 2006.

[6] W. G. Lindsley, S. H. Collicot, G. N. Franz, B. Stolarik, W. McKinney, andD. G. Frazer, Asymmetric and Axisymmetric Constant Curvature Liquid-GasInterfaces in Pulmonary Airways, Ann. Biomed. Eng., vol. 33, no. 3, March 2005,365375.

[7] R. Lopez and J. Pyo, Capillary Surfaces of Constant Mean Curvature in a RightSolid Cylinder, Math. Nachr., 287, 1312-1319 (2014).

[8] T. I. Vogel, Comments on Radially Symmetric Liquid Bridges with InectedProles, Dynamics of Continuous, Discrete, and Impulsive Systems, SupplementVolume (2005), 862867.

[9] T. I. Vogel, Convex, rotationally symmetric liquid bridges between spheres,Pac. J. Math, (2) (2006), 367377.

[10] T. I. Vogel, Liquid bridges between balls: the small volume instability,J. Math. Fluid Mech., vol. 15, issue 2, 2013, 397413.

[11] T. I. Vogel. Local energy minimality of capillary surfaces in the presence ofsymmetry, Pac. J. Math., (2002), 487509

[12] T. I. Vogel, Stability and bifurcation of a surface of constant mean curvaturein a wedge, Indiana U. J. Math., vol. 41, no. 3 (1992), 625648.

[13] H. C. Wente, The stability of the axiallly symmetric pendent drop,Pac. J. Math, (1980), 411470.

Capillary Surfaces in Cylinders 17

Thomas I. VogelDepartment of MathematicsTexas A&M UniversityCollege Station, TX 77843e-mail: tvogel@math.tamu.edu