Stability of Delaunay Surface Solutions to Capillary Problems

Transcription

1 Stability of Delaunay Surface Solutions to Capillary Problems Thomas I Vogel Texas A&M University, College Station, TX, tvogel@mathtamuedu Abstract The stability of rotationally symmetric solutions to capillary problems 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 conducted a capillary experiment If you've measured a liquid for a recipe, you've had to take the meniscus into account while using a measuring cup Capillary eects are particularly pronounced in the absence of gravity, which is why NASA funds research into capillarity It's important to know where uids are going in your space craft This paper concerns rotationally symmetric capillary surfaces and their stability 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 concerning capillary stability (see bibliography) by bringing out the common theme of stability of Delaunay surfaces Throughout this paper we will consider capillary surfaces only in the absence of gravity If two immiscible uids, such as air and water, are in static equilibrium, the free surface between them is called a capillary surface Examples are the meniscus in a capillary tube, or the surface of a drop of water resting on a tabletop The shape 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 liquid adhering to it Then the shape of the free surface Σ of Ω is determined by minimizing the functional E(Ω) = Σ c Σ 1, subject to holding the volume of Ω xed, where Σ 1 is the wetted region on Γ, and c is a material constant (In the case that gravity is present, an additional potential energy term is present in the energy functional) The rst order necessary 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 (the angle between the normals to Σ and Γ along the curve of contact) is constantly

2 2 T I Vogel arccos(c) (See [7]) We will call such a surface stationary There are many stationary capillary surfaces which are not stable For example, the free surface of an innite circular cylinder of liquid has constant mean curvature, but would never been seen in real life (even in the absence of gravity) It would break up due to the well-known Rayleigh instability It is important, therefore, to concern ourselves with stability of capillary surfaces (dened in Section 3), and not to just consider stationary surfaces Of course, even those stationary solutions which are not energy minima are of mathematical interest We will be concerned with three related problems in this paper The rst is that of a liquid bridge between parallel planes ([8], [16], [17]) No such bridge is a strict energy minimum, since translations parallel to the planes leave surface energy and volume unchanged However, we may seek energy minima modulo translations, ie, seek a bridge such that any nearby bridge with the same surface energy and containing the same volume must simply be a translation of the original bridge The second problem is that of a liquid bridge between two solid balls ([11], [12]) The third problem is that of a toroidal drop inside of a xed circular cylinder ([10]) A common feature of these three problems is that they have stationary solutions which are rotationally symmetric It turns out that the problem of the bridge between planes also has deeper stability implications for other rotationally symmetric capillary surfaces This is used in the discussion of the other two problems mentioned, and a generalization is discussed in Section 9 Of the three problems, probably the problem which is physically most important is the second, that of a bridge between solid balls This problem has to be considered when a bed of particles, eg, sand, is wetted by a liquid More papers have appeared concerning bridges between parallel planes, but that is probably because it is mathematically more accessible From a physical point of view, the bridge between planes is often considered as an approximation to the bridge between solid balls The third problem, a toroidal drop in a circular cylinder, has received much less attention However, it is also of some physical interest One use is in understanding how well pipes and tubes function in the absence of gravity This problem also arises in mathematical modeling of lung function See [10] for more references on applications 2 Delaunay Surfaces The only rotationally symmetric surfaces of constant mean curvature are the Delaunay 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 the resulting surface is an unduloid Similarly, the focus of a rolling parabola traces a catenary, and the resulting surface is the well-known catenoid The focus of a hyperbola traces a nodary, with the resulting surface being a nodoid

3 Delaunay Surfaces 3 It will be convenient to consider a parameterization of Delaunay surfaces as follows We assume that the axis of revolution is the x axis, that the prole curve goes through the point (0, 1) with angle of inclination 0 at that point, and that the mean curvature is a parameter A Parameterizing the prole curve by arc length s leads to the system of ordinary dierential equations dx = ds cos φ (1) dy = ds sin φ (2) dφ ds = cos φ y + 2A, (3) for coordinates x(s) and y(s), and inclination angle φ(s), with initial conditions x(0) = 0 y(0) = 1 φ(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, 2) 1 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 4 T I Vogel If the solution is an unduloid, sin φ oscillates between ±(2A + 1) For the other surfaces, the range of φ is obvious Refer to [12] for more details We will refer to solutions to (1)(3) as x(s; A), y(s; A), and φ(s; A), to emphasize the dependence on the parameter A In [12], elementary formulas are given for y(s; A) and for cosine and sine of φ(s; A), and from these x(s; A) is given as an integral Figure 1 gives several proles in this family The uppermost curve is given by A = 02 As A decreases from that value down to A = 12, the other curves are generated The point of considering Delaunay surfaces parameterized by A in this way is that this approach can be used to give solutions to certain symmetric capillary problems by the method of cutting and scaling This was rst used to construct bridges between solid balls in [12]; however, it works easily for constructing bridges between parallel places and toroidal drops in circular cylinders as well As an example, suppose we wish to numerically study bridges between parallel planes Suppose that the separation between the planes is one unit, and that we are interested in bridges which make a contact angle of π 3 with both planes For such a bridge, the inclination angles of the prole at the points of contact will be π 6 at the left endpoint and π 6 at the right endpoint We may nd all bridges which make the correct contact angles by taking A [ 1 4, ), where this interval comes from the above remarks on the dependence of solutions on A Take A 0 in that interval Find an s 0 < 0 for which φ(s 0, A 0 ) = π 6, and an s 1 > 0 for which φ(s 1 ; A 0 ) = π 6 The curve (x(s; A 0), y(s; A 0 )), s 0 s s 1 will have the correct inclination angles at the endpoints, but the surface must be 1 scaled by a factor of x(s 1;A 0) x(s 0;A 0) 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 choices available for s 0 and s 1 This reects the fact that, after the contact angles are xed, there are innitely many rotationally symmetric stationary bridges making those contact angles with the planes However, almost all of these are likely to be unstable it is known ([17]) that for equal contact angles, any prole with an interior inection is unstable Constructing toroidal drops in circular cylinders by the cutting and scaling procedure is also straightforward Suppose that the contact angle γ is given Take an A 0 > 1 2, and let s 0 < 0 be such that φ(s 0 ; A 0 ) = γ, and φ(s; A 0 ) < 0 on (s 0, 0) Set s 1 = s 0 Then the curve (x(s; A 0 ), y(s; A 0 )), s 0 < s < s 1 has the correct inclination angles at the ends, but must be scaled by a factor of 1 y(s 0;A 0) 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 used to construct bridges between solid balls is a bit more involved, and the reader is directed to [12] 3 Stability Suppose that we have a capillary surface Σ, not necessarily rotationally symmetric If Σ is perturbed by an innitesimal normal perturbation φn, then the

5 second order change in E is the quadratic form M(φ, φ) = φ 2 S 2 φ 2 dσ + Σ Delaunay Surfaces 5 σ ϱφ 2 dσ, (4) where σ is the curve of contact of Σ and Γ (Note that one has to introduce a tangential component near Σ to keep the contact curve on Γ See [7] or [20] for further details) Here S 2 is the square of the norm of the second fundamental form of Σ, and may be written as k k 2 2, where these are principal curvatures The constant ϱ is given by ϱ = κ Σ cot γ κ Γ csc γ (5) Here κ Σ is the curvature of the curve Σ Π and κ Γ is the curvature of Γ Π, if Π is a plane normal to the contact curve Σ We will dene a stationary surface for which M(φ, φ) 0 for all φ with φ dσ = 0 to be stable The condition Σ 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 blur together, and so it should be carefully dened when used As we have dened stability in this paper, if a capillary surface is not stable, it is certainly not a constrained local energy minimum However, a stable capillary surface need not be an energy minimum This is not surprising, since even for a function of one variable, having the second derivative non-negative does not imply a local minimum However, even the strengthened condition that M(φ, φ) > 0 for all non-trivial φ with φ dσ = 0 is not enough to guarantee that Σ is a local Σ 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 dierential operator L by L(ψ) = ψ S 2 ψ where is the Laplace-Beltrami operator on Σ The eigenvalue problem we study is given by L(ψ) = λψ (6) on Σ, with b(ψ) ψ 1 + ϱψ = 0 (7) on Σ, where ψ 1 is the outward derivative of ψ in the direction which is tangent to Σ and normal to Σ In [15] these operators are written out in coordinates The eigenvalues for this problem satisfy λ 0 < λ 1 λ 2 The condition for stability requires that M(φ, φ) be positive semi-denite on the space of all φ satisfying φ dσ = 0 (corresponding to innitesimally volume conserving Σ perturbations) If 0 λ 0, this is certainly true If λ 1 < 0, this is false, and the surface is unstable The reason for instability in this case is that the subspace

6 6 T I Vogel spanned by the two eigenfunctions φ 0 and φ 1 has dimension 2, and therefore has a non-trivial intersection with the subspace of all φ for which φ dσ = 0 Σ (which has co-dimension 1) We will not treat the case λ 1 = 0 The most interesting case is when λ 0 < 0 < λ 1 In this case, in addition to looking at the eigenvalues, there is another condition to check If the free surface may be embedded in a family smoothly parameterized by ε, and if H (ε 0 )V (ε 0 ) > 0, then the capillary surface is stable If H (ε 0 )V (ε 0 ) < 0 then the capillary surface is unstable Here H(ε) is the mean curvature with respect to 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 is an 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 with dierent xed boundaries Γ, and dierent contact angles In fact, this is a major theme of this paper As an example, the free surface of the liquid bridge in Figure 2 and the free surface of the toroidal drop in Figure 4 are the same surface: a piece of an unduloid In switching between xed surfaces but keeping the same free surface, the surface integral in (4) is the same in both cases: the only change in this quadratic form would be in the line integral An important point to realize is that if the value of ϱ increases, then the corresponding eigenvalues between the two problems also increase (or at least, do not decrease) If the value of ϱ decreases, then the corresponding eigenvalues between the two problems also decrease (or at least, do not increase) This is one of the consequences of the extremum properties of eigenvalues discussed in [4] 5 Separation of variables In Sections 3 and 4 we dealt with general capillary surfaces We will now specialize to the Delaunay surfaces used in the cutandscale procedure of Section 2 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, with dependence on A suppressed, and k is a constant of scaling Then (see [12]), for a function ψ(s, θ) dened on Σ, the operator L may be expressed as ( )) L(ψ) = ( 1 k 2 ψ ss + 1 y ψ 2 θθ + sin ϕ y ψ s )) + 2A ψ ( 2 k 2A 2 + cos ϕ 2 y ( cos ϕ y The boundary conditions for the eigenvalue problem (6), (7) in coordinates will be b(ψ) = ± 1 k ψ s + ϱψ = 0, with + at the right endpoint and at the left (8)

7 Delaunay Surfaces 7 We now separate variables Set ψ to be P (s) Q (θ) Then (8) becomes P Q + 1 y 2 P Q + sin ϕ ( y P Q + 4A 2 4A cos ϕ ) cos2 ϕ y y 2 P Q = k 2 λp Q, which is or y 2 P P + Q Q + y sin ϕp P + ( 4A 2 y 2 + 4Ay cos ϕ + 2 cos 2 ϕ ) = k 2 λy 2, y 2 P + y sin ϕp P P + ( 4A 2 y 2 + 4Ay cos ϕ + 2 cos 2 ϕ ) + k 2 λy 2 = Q Q = m2, where the separation constant is m 2 with m an integer (since Q must be a linear combination of sin mθ and cos mθ) We obtain This is yp + sin ϕp + (4A 2 y + 4A cos ϕ + 2 cos2 ϕ m 2 ) P = λk 2 yp y d ds (y (s) P (s)) + (λk 2 y (s) + 4A 2 y + 4A cos ϕ + 2 cos2 ϕ m 2 y ) P = 0, (9) written in the form of equation (1) in [2], Chapter 10 For the boundary conditions, Q will cancel, so we obtain ±P (s i ) + ϱp (s i ) = 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 is natural to label the eigenvalues as λ jm As in [20], we have and λ 0m < λ 1m < λ 2m < λ 00 < λ 01 < λ 02 <, and eigenfunctions corresponding to m = 0 are radially symmetric For k 1, the eigenvalue λ jk corresponds to a two-dimensional eigenspace spanned by functions of the form f jk (s) cos(kθ) and f jk (s) sin(kθ) This labeling of eigenvalues is a slight abuse of the notation used in Section 4, since we were previously using only a single subscript for the eigenvalues λ The smallest eigenvalue (which we labeled λ 0 in section 4) is denitely λ 00 The second smallest eigenvalue, labeled λ 1 in section 4, is one of λ 01 or λ 10, depending on ϱ

8 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, eg, [1], [8], [16], [17] If the contact angles with the planes are constant (possibly two di erent constants), then the argument proving symmetry in [21] may be used to show that the free surface must be rotationally symmetric, hence a Delaunay surface Since translations parallel to the planes are energy neutral, it follows that an in nitesimal translation is an eigenfunction with eigenvalue equal to zero It's λ01 Thus for the liquid bridge λ00 must be negative The bridge is is unstable if either λ10 is negative, or the dv /dh condition for stability in section not di cult to show that this is the eigenvalue between parallel planes the lowest eigenvalue 4 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 increase as one moves up a column, and that the bottom elements of the columns strictly

9 Delaunay Surfaces 9 increase as one moves right From this, the statement about stability depending on the sign of λ 10 and of dv/dh is perhaps easier to see For equal contact angles, the appearance of an inection in the prole signals that λ 10 is negative, and hence that the bridge is unstable To describe stability behavior, we look at cases depending on the contact angle (The following three paragraphs summarize the appendix to [8]) For γ > π 2, 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 family of nodoids, until we reach another circular arc, which is a section of a sphere As volume continues to decrease, the bridge remains stable as we pass into a family of unduloids Finally, an inection appears on the boundary, signaling a second eigenvalue crossing zero, and this is the minimum stable volume For γ = π 2, the only stable bridges are circular cylinders As volume decreases, instability occurs when the radius is 1 π times the separation of the planes This is related to the Rayleigh instability of the cylinder For γ < π 2, 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 Beyond this, the proles are unduloids For γ larger than a critical angle, about 3114, the bridges are stable until the appearance of an inection on the boundary indicating a second eigenvalue crossing zero causing instability For γ less than that critical angle, bridges become unstable before the appearance of an inection on 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 bridge between planes with equal contact angles, it is natural to investigate whether the 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 a violation of the dv/dh condition (ie, for γ less than the critical angle of about 3114 ), then the bridge is unstable with respect to perturbations which are symmetric across the plane x = 1 2 If the bridge becomes unstable because of a second eigenvalue crossing zero (ie, for γ greater than the critical angle), then the bridge is stable with respect to perturbations symmetric across the plane x = 1 2 Roughly speaking, bridges with contact angle less than 3114 break symmetrically (across x = 1 2 ) as volume decreases, whereas bridges with contact angle greater than that critical angle break asymmetrically across x = 1 2 The intuition behind the proof is this: instead of looking at a bridge between x = 0 and x = 1 with contact angles γ at both sides, consider a bridge between x = 0 and x = 1 2, with contact angle γ with x = 0 and contact angle π 2 with x = 1 2 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 γ bridge to the γ π 2 bridge, the odd eigenvalues disappear Thus an instability due to a failure of the dv/dh condition for the γ γ problem implies an instability for the γ π 2 problem, but an eigenvalue crossing zero for the γ γ problem does not lead to an eigenvalue crossing zero for the γ π 2 problem For more details, see [19]

10 10 T I Vogel For unequal contact angles, numerical experimentation suggests the following Conjecture 1 For unequal contact angles, the instability which occurs as volume decreases is always due to a violation of the dv/dh condition, and never due to λ 10 crossing zero One particularly important fact about bridges between parallel planes is that if the bridge is convex (ie, the prole curve is a function with negative second derivative), then the bridge is stable ([17]) This is true for unequal as well as equal contact angles A dierence between bridges with equal contact angles and bridges with unequal contact angles relates to inections and stability For equal contact angles, an interior inection in the prole curve implies instability For unequal contact angles, however, there are proles of stable bridges which have inections As a 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 have an interior inection In doing some numerical experiments, W C Carter ([3]) observed an interesting fact concerning the minimum stable volume of a bridge between planes Allowing any choice of the contact angle, the stable liquid bridge with smallest volume that he observed had contact angles equal to π 2, and is the previously mentioned cylinder of radius 1 π times the separation of the planes Finn and Vogel ([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 of a liquid bridge between two solid spheres ([11], [12]) The symmetry argument outlined in Section 6 fails for a bridge between balls, and in fact there are other stationary solutions For some values of the parameters (radius of the balls, separation of the centers, contact angles, and volume of the bridge) there exist bridges whose free surfaces are also portions of spheres which are not symmetrically placed It is not known whether there are other solutions to this capillary problem besides these asymmetrically placed spheres and the Delaunay surfaces Using Note 1, we may draw conclusions about stability of bridges between solid balls from the theory of stability of bridges between planes Suppose that Σ is a piece of a Delaunay surface whose boundary consists of two circles in parallel planes, and suppose that the prole is convex We may consider Σ as a bridge between parallel planes, or, using dierent contact angles, a bridge between solid balls Lemma 23 in [11] shows that, if Σ is a section of a nodoid, then the value of ϱ decreases in going from considering Σ as a bridge between planes to a bridge between balls, and that if Σ is a section of an unduloid, the value of ϱ increases

11 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 surface bridging between planes, and unduloidal bridges between balls are more stable than the corresponding surface bridging between planes More precisely, one can show that the value of λ 01 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 λ 30 λ 21 λ 12 λ 03 λ 20 λ 11 λ 02 λ 10 λ 01 < 0, nodoid λ 01 > 0, unduloid λ 00 m = 0 m = 1 m = 2 m = 3 nodoidal free surface must be unstable, since both λ 00 and λ 01 must be negative Comparing bridges between planes and bridges between balls also yields a conclusion 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 form in (4) increases when one goes from considering Σ as a bridge between planes to considering Σ as a bridge between balls Since convex bridges between planes are known to be stable (Section 6) it follows that convex unduloidal bridges between balls are also stable Note that we can't simply cite Note 1 for this, since the stability criterion in Section 4 involves more than just the eigenvalues There are some numerical results on bridges between balls in [12], obtained by cutting and scaling the standardized Delaunay surfaces in Section 2 An interesting phenomenon was observed: for certain values of the parameters involved, a (fold-over) bifurcation can occur without loss of stability This occurs if one starts with a stable bridge with a single negative eigenvalue and with H (A)V (A) > 0; then as A varies, the negative eigenvalue happens to cross zero and becomes positive As the eigenvalue crosses zero there is a bifurcation, but with no negative eigenvalues, the bridge remains stable A numerical example is given in Figure 3 The gure shows volume and mean curvature for a family of symmetric bridges between two solid balls of radius 1 unit, whose centers are separated by 3 units The large-volume bridges corresponding to the right end of 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 on the curve at which volume equals 8 This fold-over bifurcation occurs when λ 00 crosses zero to become positive As volume continues to decrease, a second foldover occurs, as λ 00 crosses zero again The bridge is stable through all of this,

12 12 T I Vogel only becoming unstable when a minimum volume is reached at about H = 08 and the dv/dh condition fails Fig 3 Volume and mean curvature for γ = π 20, R = 3 This behavior, ie, λ 00 crossing zero to become positive, can't occur for a bridge between parallel planes In this case λ 00 always remains negative since λ 00 < λ 01 = 0 A case of particular physical and mathematical interest is a bridge between two 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 rotationally symmetric bridges is shown for a large range of volumes However, if contact angle is greater than or equal to π 2, then no stable rotationally symmetric bridges exist Thus for a bridge between contacting balls, existence of a stable bridge depends discontinuously on the contact angle 8 Capillary surfaces in cylinders Another example of a capillary problem with Delaunay surfaces as stationary solutions occurs when the xed surface Γ is a circular cylinder and the region Ω occupied by the liquid is interior to the cylinder (see section 2 of [10]) The liquid drop is topologically a solid torus, as in Figure 4 In this gure, the xed cylindrical surface may be seen as part of the boundary of Ω For this problem, we can notice an energy-neutral translation, in a direction parallel to the axis of the cylinder As for the problem of the liquid bridge between parallel planes, this leads to an eigenvalue equal to zero However in this case, the eigenfunction corresponding to this eigenvalue of zero is rotationally

13 Delaunay Surfaces 13 Fig 4 Toroidal drop symmetric (it is just the normal component of a vector directed along the axis of the cylinder) This eigenfunction equals zero once, at the midpoint of the interval Thus by standard Sturm-Liouville theory ([2]) this corresponds to Therefore λ10 = 0 λ10 for the toroidal drop in a circular cylinder The eigenvalues for this problem may be arranged as in Table 3 Table 3 Arrangement of eigenvalues for the toroidal drop in a cylinder λ30 λ20 λ10 = 0 λ00 m=0 λ21 λ11 λ01 λ12 λ02 λ03 m = 1 m = 2 m = 3 We may again draw conclusions about this problem from facts about bridges between parallel planes The plan of attack is to consider a bridge between parallel 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 the Σ FolΣ is considered as a bridge between parallel planes, and let %n be the value of % when Σ is considered as the free boundary of a toroidal drop in a circular cylinder free surface of the toroidal drop in Figure 4) Call that Delaunay surface lowing the notation in [10], let %o be the value of % (as de ned in (5)) when Here o stands for old, n for new The reasoning behind the terminology is that the bridge between planes is used as a starting point, and we use that to

14 14 T I Vogel deduce results about a newer problem Lemma 21 of [10] shows that ϱ o and ϱ n either have opposite signs or are both zero Note 2 A startling result which follows from Lemma 21 of [10] is that a toroidal drop in a circular cylinder whose prole is a graph with second derivative strictly positive must be unstable (Theorem 23 of [10]) The essential idea of the proof is consider Σ both as as the free surface of a bridge between planes and as the free surface of a toroidal drop An innitesimal translation of the bridge parallel to the planes is energy neutral More precisely, dene φ on Σ to be the inclination of the prole at each point Then M o (cos(φ), cos(φ)) = 0, with Σ cos φ dσ = 0 One can compute that ϱ o > 0 in this set-up Thus ϱ n < 0, and it follows (see [10] for details) that M n (cos(φ), cos(φ)) < 0 Since this is an innitesimally volume conserving perturbation which reduces energy, this toroidal drop is unstable Note 3 Another startling result of Lemma 21 of [10] is that, if a toroidal drop in a cylinder has a prole with an inection, then there there is precisely one negative eigenvalue, λ 00 The eigenvalue λ 10 must be zero by previous remarks, and all the rest must be strictly positive (Theorem 24 of [10]) The basic idea of the proof is that in this case, ϱ o < 0 < ϱ n If we call the eigenvalues for the bridge (old) problem λ o ij and the eigenvalues for the toroidal drop in the cylinder (the new problem) λ n ij, increasing ϱ increases (or at least doesn't decrease) the corresponding eigenvalues Then λ o 10, which is negative for an inected bridge between planes, increases up to zero, and λ o 01, which is automatically zero for the bridge between planes, increases up to λ n 01 > 0 One can verify ([10]) that this is strictly positive by arriving at a contradiction if λ n 01 = 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 interior inection points in the prole The right endpoint corresponds to the unduloid with inection on the cylinder, and the left endpoint is the limiting case of the exterior of two spherical caps All of these proles have precisely one negative eigenvalue Those for which dv dh > 0 are stable; thus there is numerical evidence of stable toroidal drops in cylinders, which will have inections in their proles To contrast the toroidal drop with the liquid bridge between planes (with equal contact angles): a stable bridge between planes cannot have inections, but a stable 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 rotationally symmetric, then in determining stability one need consider only rotationally symmetric perturbations This is certainly false in general For example, nodoidal

15 Delaunay Surfaces 15 Fig 5 H vrs V, inected prole, γ = π 20 bridges between solid balls are unstable with respect to non-symmetric perturbations (Section 7) Similarly, toroidal drops in circular cylinders whose proles have their second derivatives bounded from zero are unstable with respect to non-symmetric perturbations (Section 8) However, it is natural to wonder under what circumstances we are safe in restricting consideration to symmetric perturbations in determining stability A partial answer comes from consideration of the bridge between parallel planes Lemma 1 Suppose that Σ is a rotationally symmetric piece of a Delaunay surface, parameterized by s 1 s s 2, 0 θ 2π as in Section 5 Suppose that Σ may be considered as a bridge between parallel planes, with contact angle γ o, and that Σ may also be considered as a capillary surface contacting a rotationally symmetric smooth surface Γ n, with contact angle γ n (As an example, the Delaunay surface in Figure 2 is the same surface as the free surface of the toroidal drop in Figure 4) Let ϱ o be the value of ϱ computed from (5) for Σ as a bridge between planes, and let ϱ n be the value of ϱ for Σ as a capillary surface contacting Γ n (As before, o stands for old and n stands for new) If ϱ n > ϱ o, then λ n 01 0 Proof This follows from Note 1 and the fact that λ o 01, the eigenvalue for the old problem of the bridge between planes, is automatically zero Proposition 1 Assume the set-up is as in Lemma 1 If ϱ n < ϱ o, then Σ is unstable as a capillary surface contacting Γ n with contact angle γ n Proof In the case ϱ n < ϱ o the argument follows that of Note 2: the energy neutral translation of a bridge between planes turns into an energy decreasing perturbation in the case of a capillary surface contacting Γ n, and we have instability

16 16 T I Vogel Note 4 In the case ϱ n > ϱ o, we have λ n 01 0 from the above Lemma Since λ 00 is the only negative eigenvalue, and this corresponds to a rotationally symmetric eigenfunction, it seems likely that rotationally symmetric perturbations are the most dangerous One would expect that if Σ, as a capillary surface contacting Γ n, is stable with respect to rotationally symmetric perturbations which are volume conserving, then it is stable with respect to non-rotationally symmetric (volume-conserving) perturbations as well I hope to give a formal proof to this intuitively plausible statement in a later paper References 1 Athanassenas, M, Variational Problem for Constant Mean Curvature Surfaces with Free Boundary, J Reine Angew Math, vol 377, (1987), Birkho, G, and Rota, J-C, Ordinary Dierential Equations, John Wiley and Sons, New York, NY, fourth edition, Carter, W C, The Forces and Behavior of Fluids Constrained by Solids, Acta Metall, vol 36, no 8, (1988), Courant, R, and Hilbert, D, Methods of Mathematical Physics, vol 1, Interscience Publishers, New York, NY, Eells, J, The Surfaces of Delaunay, Math Intelligencer, vol 9, no 1, (1987), Finn, R, Editorial comment on On Stability of a Catenoidal Liquid Bridge, by L Zhou, Pac J Math, vol 178, no 1, 197, Finn, R, Equilibrium Capillary Surfaces, Springer-Verlag, New York, Finn, R, and Vogel, T I, On the Volume Inmum for Liquid Bridges, Z Anal Anwend,, (1992), Maddocks, J H, Stability and Folds, Arch Rat Mech Anal, vol 99, (1987), Vogel, T I, Capillary Surfaces in Circular Cylinders, submitted to J Math Fluid Mech 11, Convex, Rotationally Symmetric Liquid Bridges Between Spheres, Pac J Math, vol 224, no 2 (2006), , Liquid Bridges Between Balls: the Small Volume Instability, J Math Fluid Mech, vol 15, issue 2, June, 2013, , Liquid Bridges Between Contacting Balls, J Math Fluid Mech, vol 16, no 4 (2014), , Local Energy Minimality of Capillary Surfaces in the Presence of Symmetry, Pac J Math, vol 206, no 2 (2002), , Stability and Bifurcation of a Surface of Constant Mean Curvature in a Wedge, Indiana U Math J, vol 41, no 3, (1992), , Stability of a Liquid Drop Trapped Between Two Parallel Planes SIAM J Appl Math, vol 47, (1987) , Stability of a Liquid Drop Trapped Between Two Parallel Planes II: General Contact Angles, SIAM J Appl Math, vol 49, (1989), , Sucient Conditions for Capillary Surfaces to be Energy Minima, Pac J Math, vol 194, no 2 (2000), , Types of Instability for the Trapped Drop Problem with Equal Contact Angles, Geometric Analysis and Computer Graphics, ed by P Concus, R Finn, and DA Homan, Springer-Verlag New York, 1991, pp

17 Delaunay Surfaces Wente, H C, The Stability of the Axially Symmetric Pendent Drop, Pac J Math, vol 88, no 2, (1980), Wente, H C, The Symmetry of Sessile and Pendent Drops, Pac J Math, vol 88, no 2, (1980), Zhou, L, On the Volume Inmum for Liquid Bridges, Z Anal Anwend, vol 12, no 4, (1993),