J. Math. Fluid Mech. 15 (2013), 397–413
c 2012 Springer Basel
1422-6928/13/020397-17
DOI 10.1007/s00021-012-0117-y
Journal of Mathematical
Fluid Mechanics
Liquid Bridges Between Balls: The Small Volume Instability
Thomas I. Vogel
Communicated by R. Finn
Abstract. Stability for a liquid bridge between two solid balls is studied by cutting and scaling pieces of a standardized
family of Delaunay surfaces. This theoretical framework is used to analyze the problem numerically.
Mathematics Subject Classification (2010).
Primary 76B45; Secondary 53A10.
1. Introduction
The physical problem we consider is that of a liquid bridge Σ between two fixed solid balls B1 and B2
(see Fig. 1). We will focus on the case that the solid balls have equal radius and are made of the same
material, thus the contact angles are equal (although some derivations are given which apply to the more
general case of unequal contact angles). To be more precise, let Ω be the region in space occupied by
the liquid, with Σ the free surface of the liquid, and Σ1 , Σ2 the wetted regions on the two balls. In the
absence of gravity or other external potentials, the shape of the bridge arises from minimizing the energy
E (Ω) = |Σ| − c |Σ1 | − c |Σ2 | ,
(1.1)
where |Σ| is the area of the free surface, |Σi | is the area of the wetted region on Bi , and c ∈ [−1, 1] is
a material constant. The minimization is subject to the constraint that the volume of Ω remains fixed.
Since the contact curves are free to move, the first order conditions from the minimization are that the
mean curvature of Σ must be constant, and that the angle between the normal to Σ and to Σi must be
constantly γ = arccos(c) (see [4]).
In [14], it was shown that the stability of a convex bridge between balls may be characterized simply.
If the bridge is part of a nodoid, it is unstable, and if it is part of a sphere or unduloid, it is stable (and
in fact a local energy minimum). The main purpose of this paper is to set up a framework to investigate
the instability which occurs as volume decreases from a convex bridge.
As in [14], the quadratic form relating to stability and energy minimality is
2
|∇φ| − |S|2 φ2 dΣ + ρφ2 dσ.
(1.2)
M(φ, φ) =
Σ
σ
2
Here |S| is the square of the norm of the second fundamental form of Σ. (In terms of mean curvature H
and Gaussian curvature K, |S|2 may be written as 2(2H 2 − K), and in terms of the principal curvatures,
|S|2 may be written as k12 + k22 .) The coefficient ρ is given by
ρ = κΣ cot γ − κΓ csc γ,
(1.3)
where κΣ is the curvature of the curve Σ ∩ Π and κΓ is the curvature of Γ ∩ Π, if Π is a plane normal to
the contact curve ∂Σ. The planar curvatures are signed; refer to [14], Fig. 2, for an example where both
are negative.
398
T. I. Vogel
JMFM
Σ
B
B2
1
Fig. 1. A liquid bridge between fixed, solid balls
Again following [14], define the differential operator L by
L(ψ) = −Δψ − |S|2 ψ
(1.4)
where Δ is the Laplace–Beltrami operator on Σ. The eigenvalue problem which arises is given by
L(ψ) = μψ
(1.5)
b(ψ) ≡ ψ1 + ρψ = 0
(1.6)
on Σ, with
on ∂Σ, where ψ1 is the outward normal derivative of ψ.
Enumerate the eigenvalues of (1.5), (1.6) as μ0 < μ1 ≤ μ2 ≤ · · ·. Conditions for a capillary surface to
be a local energy minimum are derived in [11] and expanded on in [13]. To summarize:
•
•
•
if 0 < μ0 , then the capillary surface is a local energy minimum.
if μ1 < 0, then the capillary surface is not a local energy minimum.
The case μ0 < 0 < μ1 is more complicated. If Σ ≡ Σ(0) can be embedded in a smoothly parametrized family Σ(t) of capillary surfaces, each with constant mean curvature H(t) (with respect to
the normal to Σ pointing out of the liquid, so that if the drop is a ball, the mean curvature is
Fig. 2. Graph of f (s, −0.7;
π
)
4
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
399
negative) and containing volume V (t), then if H (0)V (0) > 0 then Σ is a local energy minimum
and if H (0)V (0) < 0, Σ is not a local energy minimum.
We will not concern ourselves with the isolated cases of μ0 = 0, μ1 = 0, or μ0 < 0 < μ1 but H (0)V (0) = 0
in this paper.
2. Constructing Bridges Between Balls
A rotationally symmetric surface of constant mean curvature H may be found by solving the following
system of ordinary differential equations [9]:
dX
= cos Φ (s) ,
ds
dY
= sin Φ (s) ,
ds
dΦ
cos Φ (s)
=
+ 2H.
ds
Y (s)
(2.1)
(2.2)
(2.3)
The curve (X (s) , Y (s)) is the profile of a Delaunay surface, parametrized by arc length s, and Φ (s) is
the inclination angle of the profile. This is simple enough to do numerically for given initial conditions,
and in fact Y and Φ can be found explicitly and X can be written in terms of incomplete elliptic integrals
[2]. The difficulty is to generate a Delaunay surface which makes specified contact angles with two given
balls. One could attempt to do this using a shooting method. However, attempting to investigate behavior
near a bifurcation of a non-linear problem using a shooting method approach is numerically questionable.
Since we know that the profiles have constant mean curvature, a property preserved under scaling, an
alternative approach involving scaling suggests itself. We first derive a formula for shifting a solution of
(2.1)–(2.3) to form a bridge contacting two equal balls of radius r, with specified contact angle.
Propositions 2.1 and 2.3 are stated for two possibly different contact angles γ1 and γ2 . Certainly the
case γ1 = γ2 is the most interesting, and we will quickly specialize to that case. However, the proofs of
these two propositions are no easier in the special case of equal contact angles, so we present the more
general results.
Proposition 2.1. Suppose that a curve (X (s) , Y (s)) comes from a solution to system (2.1)–(2.3) for some
value of H. Suppose that r and R, with R > 2r > 0 are given, and that contact angles γ1 , γ2 ∈ [0, π] are
given. In addition, suppose that there are s∗1 , s∗2 , s∗1 < s∗2 , so that
Y (s∗1 ) sec (γ1 − Φ (s∗1 )) = Y (s∗2 ) sec (γ2 + Φ (s∗2 )) = r
and
X (s∗2 ) + Y (s∗2 ) tan (Φ (s∗2 ) + γ2 ) − X (s∗1 ) − Y (s∗1 ) tan (Φ (s∗1 ) − γ1 ) = R.
Then there is a C so that
2
1. the point (X (s∗1 ) − C, Y (s∗1 )) lies on B1 , the circle x + R2 + y 2 = r2 ,
2
2. the point (X (s∗2 ) − C, Y (s∗2 )) lies on B2 , the circle x − R2 + y 2 = r2 , and
3. the inclination angle of the vector −Y (s∗1) , X (s∗1) is γ1 less than that of the vector X (s∗1 )+ R2 ,Y (s∗1 )
and the inclination angle of the vector −Y (s∗2 ) , X (s∗2 ) is γ2 greater than that of X (s∗2 )− R2 , Y (s∗2 ) .
Here we use angle brackets when we want to think of ordered pairs as vectors rather than points.
Proof. Set β1 to be Φ (s∗1 ) +
π
2
− γ1 and β2 to be Φ (s∗2 ) +
Y
(s∗1 )
π
2
= r sin β1
and
Y (s∗2 ) = r sin β2
+ γ2 . Then we have
400
T. I. Vogel
JMFM
along with
X (s∗2 ) − X (s∗1 ) + Y (s∗1 ) cot β1 − Y (s∗2 ) cot β2 = R.
Thus,
X (s∗2 ) − X (s∗1 ) = R − r cos β1 + r cos β2 .
Set
R
,
2
and (X (s∗i ) − C, Y (s∗i )) will lie on Bi for i = 1, 2. Now, the angle that the normal to Bi
at (X (s∗i ) − C, Y (s∗i )) makes with the positive x axis is βi . The angle that the tangent to the
curve (X (s) − C, Y (s)) at the parameter value s∗i makes with the positive x axis is Φ (s∗i ). Since
−Y (s∗i ) , X (s∗i ) is a counterclockwise rotation of that tangent vector by π2 , the result follows.
C = X (s∗1 ) − r cos β1 +
The point of Propositions 2.1 and 2.3 which follows is to enable us to consider “standardized” Delaunay curves, and then cut, scale, and shift them to form bridges with appropriate contact angles with
given balls. Define x(s; A), y(s; A), and ϕ(s; A) to be solutions of the system
dx
= cos ϕ(s),
ds
dy
= sin ϕ(s),
ds
cos ϕ(s)
dϕ
=
+ 2A,
ds
y(s)
(2.4)
(2.5)
(2.6)
with initial conditions
x(0) = 0,
y(0) = 1,
ϕ(0) = 0.
Note 2.2. The solutions to (2.4)–(2.6) may be characterized as in Table 1. These are profiles of the
Delaunay surfaces, so “circular arc” corresponds to a sphere, and “horizontal line” corresponds to a
cylinder.
Proof. This follows easily from Lemma 2.2 of [14]. We have that c defined by
c ≡ y cos ϕ + Ay 2
(2.7)
is constant on Delaunay profiles, and that for Ac > 0 the profile is a nodary, and Ac < 0 the profile is an
undulary. But of course we can evaluate c at s = 0 as 1 + A, so if A(1 + A) > 0, the profile is a nodary
and if A(1 + A) < 0 the profile is an undulary. The special cases A = 0, A = − 12 and A = −1 are easily
dealt with. For A = − 12 , we consider a cylinder as a special case of an unduloid.
Table 1. Classification of solutions to (2.4)–(2.6)
A ∈ (−∞, −1)
A = −1
A ∈ −1, − 12
1
A = − 2 A ∈ − 21 , 0
A=0
A ∈ (0, ∞)
Nodary
Circular arc
Undulary
Horizontal line
Undulary
Catenary
Nodary
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Liquid Bridges Between Balls: The Small Volume Instability
401
Proposition 2.3. Suppose that r, R are given, with R > 2r, and that contact angles γ1 , γ2 are given.
Suppose that A is fixed. If s1 , s2 satisfy
y(s1 ; A) sec (ϕ(s1 ; A) − γ1 ) = y(s2 ; A) sec (ϕ(s2 ; A) + γ2 ) > 0
(2.8)
and
x(s2 ; A)
cos (ϕ(s2 ; A) + γ2 ) + sin (ϕ(s2 ; A) + γ2 )
y(s2 ; A)
x(s1 ; A)
R
cos (ϕ(s1 ; A) − γ1 ) − sin (ϕ(s1 ; A) − γ1 ) = .
−
y(s1 ; A)
r
(2.9)
Then (x(s; A), y(s; A)) may be scaled and s∗1 and s∗2 may be found so that the hypotheses of Proposition
2.1 are satisfied.
Proof. Let k be the common value in (2.8). Define X, Y and Φ by
k
r
X (s) = x s ; A ,
k
r
k
r
Y (s) = y s ; A ,
k
r
k
Φ (s) = ϕ s ; A .
r
It’s straight-forward to check that X, Y and Φ satisfy (2.1)–(2.3) with H = Ak
r . Now, (2.9) may be written
as
1
1
R
x(s2 ; A) + sin (ϕ(s2 ; A) + γ2 ) − x(s1 ; A) − sin (ϕ(s1 ; A) − γ1 ) = ,
k
k
r
hence
r
r
r
r
x(s2 ; A) + k sin (ϕ(s2 ; A) + γ2 ) − x(s1 ; A) − k sin (ϕ(s1 ; A) − γ1 ) = R
k
k
k
k
which is
r
r
r
r
x(s2 ; A) + y(s2 ; A) tan (ϕ(s2 ; A) + γ2 ) − x(s1 ; A) − y(s1 ; A) tan (ϕ(s1 ; A) − γ1 ) = R.
k
k
k
k
This may be rewritten as
r r r r r r X
s2 + Y
s2 tan Φ
s2 + γ2 − X
s1 − Y
s1 tan Φ
s1 − γ1 = R, (2.10)
k
k
k
k
k
k
showing that X, Y , and Φ as defined above satisfy the hypotheses of Proposition 2.1, with s∗i = kr si . Note 2.4. The solution to system (2.4)–(2.6) has elementary expressions for y(s) and ϕ(s) which are often
useful. In fact, if A = 0,
2A + 1
(1 − cos (2As)),
(2.11)
y(s; A) = 1 +
2A2
and if A = 0,
y(s; 0) = s2 + 1.
For A = 0,
sin (ϕ(s; A)) = sgn (A) (2A + 1) sin (2As)
4A2
(2.12)
+ (4A + 2)(1 − cos (2As))
and
cos (ϕ(s; A)) = sgn (A) (2A + 1) cos(2As) − 1
4A2
+ (4A + 2)(1 − cos (2As))
,
(2.13)
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T. I. Vogel
JMFM
and, for A = 0,
sin φ(s; 0) = √
s
s2 + 1
and
cos φ(s; 0) = √
1
.
s2 + 1
(sgn (A) is the signum function.)
Proof. From (2.5) and (2.6) we have
d
(y sin ϕ) = sin2 ϕ + y cos ϕ
ds
= 1 + 2Ay cos ϕ,
and
d
(y cos ϕ) = sin ϕ cos ϕ − y sin ϕ
ds
= −2Ay sin ϕ.
cos (ϕ (s))
+ 2A
y(s)
cos (ϕ (s))
+ 2A
y(s)
Let a = y sin ϕ and b = y cos ϕ. Then we have the linear system
da
= 1 + 2Ab
ds
db
= −2Aa,
ds
with initial conditions a (0) = 0, b (0) = 1. For A = 0, the solution is
(2A + 1) sin (2As)
,
2A
(2A + 1) cos (2As) − 1
.
b (s) =
2A
Since sin ϕ = y , the expression for a yields
a(s) =
yy =
(2.14)
(2.15)
(2A + 1) sin (2As)
.
2A
Hence,
d 2 (2A + 1) sin (2As)
y =
,
ds
A
and therefore
2
y =−
2A + 1
2A2
cos (2As) + C.
Since y (0) = 1, we find (2.11), above. Now that we have an explicit formula for y(s; A), we may use
(2.14) and (2.15) to find the explicit formulas for sin (ϕ(s; A)) and cos (ϕ(s; A)).
The case A = 0 is simple (an arc length parametrization of a catenary) and the proof is omitted. Remark 2.5. Here are a few properties of x, y, and ϕ which we will need. For A = 0, y(s; A) and ϕ(s; A)
π
are periodic of period |A|
(this follows from the previous note). The functions x(s; A) and ϕ(s; A) are odd
functions of s and y(s; A) is an even function of s. Finally, x(s; A), y(s; A) and ϕ(s; A) are real analytic
functions of s and A on open sets in the (s, A) plane where y(s; A) is non-zero. (A good reference for
multi-variable real analytic functions is [7].)
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
403
Fig. 3. Bridge with A = −0.7, s = 1.659616906
Proof. To show the symmetry assertions, define x̃(s; A) to be −x(−s; A), ϕ̃(s; A) to be −ϕ(−s; A), and
ỹ(s; A) to be y(−s; A). It’s not hard to verify that x̃, ỹ, and ϕ̃ solve the system (2.4)–(2.6) with the same
initial conditions as before. By the theorem on uniqueness of solutions to ODE’s, the symmetry assertions
follow. The real analyticity assertion follows from well known results on ODE’s.
Proposition 2.3 naturally leads to an efficient numerical method of finding rotationally symmetric
bridges between equal balls, making equal contact angles (i.e., γ1 = γ2 ≡ γ) and which are symmetric
across the plane which is the perpendicular bisector of the line segment between the centers of the balls.
We will refer to this approach as “cutting and scaling”, and will also use that term in somewhat more
general cases. Take R to be the distance between the centers of the balls, and r to be the radii of the
balls. In seeking such bridges, we naturally seek solutions to (2.8) and (2.9) with s1 = −s2 . However,
when γ1 = γ2 , Remark 2.5 automatically gives us the equality in (2.8) (although the inequality in (2.8)
must still be verified). Thus we must only deal with solving (2.9). To solve that equation, using symmetry
properties from Remark 2.5, we seek s2 so that
x(s2 ; A)
R
cos (ϕ(s2 ; A) + γ) + sin (ϕ(s2 ; A) + γ) =
(2.16)
f (s2 , A; γ) ≡
y(s2 ; A)
2r
and then set s1 to be −s2 .
To illustrate this, suppose that B1 and B2 are unit balls, that their centers are six units apart, and
we seek bridges making contact angle π4 with the balls. Further, suppose that we seek a scaled section of
system (2.4)–(2.6) with A = −0.7.
Determine numerical solutions to that system with that value of A, and consider the plot of
f (s, −0.7; π4 ), in Fig. 2. There are at least three numerical solutions to f (s, −0.7; π4 ) = 3 : s = 1.659616906,
2.738663658, and 3.934538440. By using the scaling arguments in Proposition 2.3, each of these roots leads
to a liquid bridge with the appropriate contact angles. The bridges corresponding to the first two roots
are illustrated in Figs. 3 and 4. Using this method for differing values of A, we will investigate bifurcation
in families of bridges making contact angle γ with fixed balls in Sect. 3.
There are rotationally symmetric bridges which are not symmetric across the plane x = 0, and these
may also be found without resorting to a shooting method. The approach is the following. Fix A and
solve the system (2.4)–(2.6). Then, numerically solve the system (2.8), (2.9), seeking s1 , s2 , which don’t
sum to zero. In practice, it is convenient to find s∗ and s2 so that s∗ = s2 , but
y(s∗ ) sec (φ(s∗ ) + γ) = y(s2 ) sec (φ(s2 ) + γ)
(2.17)
and
x(s∗ )
x(s2 )
R
cos (φ(s∗ ) + γ) + sin (φ(s∗ ) + γ) +
cos (φ(s2 ) + γ) + sin (φ(s2 ) + γ) = ,
(2.18)
y(s∗ )
y(s2 )
r
and then, using symmetry results from Remark 2.5, set s1 = −s∗ . It is helpful to think of it in this fashion
because solutions to (2.17) are simply two s coordinates of intersections of the graph of y(s) sec (φ(s) + γ)
with a horizontal line. Figure 5 shows the profile of such a bridge, again with A = −0.7 and contact angles
π
4 . All of the bridges in Figs. 3, 4, 5 are obtained by shifting and scaling the same unduloid (the solution
to (2.4)–(2.6), with A = −0.7).
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T. I. Vogel
JMFM
Fig. 4. Bridge with A = −0.7, s = 2.738663658
Fig. 5. Bridge with A = −0.7, not symmetric across x = 0
In most cases that we use cutting and scaling, we will compute f (s, A; γ) numerically. However, there
are a few cases in which it is straight-forward to determine f analytically.
Proposition 2.6. f (s, −1; γ) = sin γ sec s.
Proof. It’s easy to verify that the solution to system (2.4)–(2.6) in the case A = −1 is x(s) = sin s, y(s) =
cos s and φ(s) = −s. Substituting these into (2.16) gives the result after a little manipulation.
Proposition 2.7. f (s, − 12 ; γ) = s cos γ + sin γ.
Proof. In this case, x = s, y = 1, ϕ(s) = 0, from which the result follows.
Proposition 2.8.
√
ln s + s2 + 1
√
f (s, 0; γ) =
cos (arctan s + γ) + sin (arctan s + γ) .
s2 + 1
Proof. For A = 0 in (2.4)–(2.6), observe that
dy
dϕ
dϕ
ds
= y tan ϕ, which separates to give y = sec φ. Then
= y, so that x = ln (sec ϕ + tan ϕ). Finally,
= cos2 ϕ, which separates to give ϕ = arctan s. Plug
√
√
2
this into x and y to get x = ln s + s + 1 and (as before) y = s2 + 1. Substituting in, the result
follows.
dx
dϕ
3. Bifurcation in the A, s Plane
Instability often arises from an eigenvalue crossing zero, and these will typically be signaled by a bifurcation in the V, H plane. Pitchfork bifurcations in the A, s plane will correspond to pitchforks in the V, H
plane, leading us to study bifurcations in the A, s plane. For completeness, we also characterize fold-over
bifurcations in the A, s plane.
Define a function g by
g(s, A; γ) = y(s, A) sec (φ(s, A) + γ) ,
(3.1)
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
405
and take f (s, A; γ) as defined in 2.16. Then the system given in 2.17 and 2.18 may be written as
g(s∗ , A; γ) − g(s2 , A; γ) = 0
(3.2)
R
(3.3)
f (s∗ , A; γ) + f (s2 , A; γ) = .
r
For fixed γ, we wish to investigate values of s∗ , s2 , A for which there is a bifurcation of (3.2), (3.3).
Because of the simple form of (3.2), (3.3), elementary observations concerning solution curves in the
A, s plane follow. We are interested in the behavior of the family of bridges which are symmetric across
the plane x = 0. For these, we have s∗ = s2 in (3.2), (3.3), although bifurcations which are not symmetric
may occur from this family. As noted above, the symmetric family is determined by s and A solving
R
(3.4)
f (s, A; γ) = .
2r
As long as f1 (s, A; γ) = 0, the implicit function theorem implies that s is determined as a function of A.
When this fails, there may be a “fold-over” bifurcation. In fact,
R
, f1 (ŝ, Â; γ) = 0, f2 (ŝ, Â; γ) = 0, and f11 (ŝ, Â; γ) = 0, then (ŝ, Â) is a
Proposition 3.1. If f (ŝ, Â; γ) = 2r
fold-over bifurcation for (3.4) in the sense that for A on one side of  there are no solutions to (3.4)
near the point (ŝ, Â), and on the other side of  there are two solutions to (3.4).
Proof. This is clear when one considers A as a function of s.
What is more interesting, and more useful in the study of stability, is when non-symmetric solutions
to (3.2), (3.3) bifurcate from solutions to (3.4). We may isolate solutions for which s∗ = s2 by defining a
new function G(s∗ , s2 , A; γ) by
∗
g(s ,A;γ)−g(s2 ,A;γ)
, if s∗ = s2 ;
∗
s∗ −s2
(3.5)
G(s , s2 , A, ; γ) =
g1 (s∗ , A; γ),
if s∗ = s2 .
The point is that for s∗ = s2 , we can replace the system (3.2), (3.3) by
G(s∗ , s2 , A; γ) = 0
(3.6)
R
(3.7)
f (s∗ , A; γ) + f (s2 , A; γ) = .
r
Note that in the special case of A = − 12 , which corresponds to a cylinder, (see Table 1) G is identically
zero. Solutions to (3.6), (3.7) for which A = − 21 (which we will see in Sect. 5) do not lead to physically
interesting bifurcations, since they simply correspond to translations of the cylinder.
Since we will be using the implicit function theorem, we are naturally interested in the differentiability
of G(s∗ , s2 , A; γ), and later on we will need the existence of second derivatives as well. It is straightforward, although not elegant, to use the fact that g(s, A; γ) is real analytic where it is defined (which follows
from Remark 2.5) to obtain these derivatives.
Lemma 3.2. Suppose that, for fixed γ, g(s, A; γ) is real analytic on α < s < β, A1 < A < A2 . Then
G(s∗ , s2 , A; γ) is real analytic on α < s∗ < β, α < s2 < β, A1 < A < A2 .
Proof. Take a point (t∗ , t2 , A∗ ) in the described set. We must show that G(s∗ , s2 , A) equals a power series
in powers of (s∗ − t∗ ), (s2 − t2 ) and (A − A∗ ) on a disk centered at (t∗ , t2 , A∗ ) with positive radius. If
t∗ = t2 this is easy: g(s∗ , A) equals a power series in powers of (s∗ − t∗ ) and (A − A∗ ) on a disk centered at
(t∗ , A∗ ), similarly expand g(s2 , A) in a power series centered at (t2 , A∗ ) with equality on a disk of positive
1
around (t∗ , t2 ) with equality on a disk of positive radius. Plugging all
radius, and similarly expand s∗ −s
2
of these power series into the definition of G will result in a power series which equals G on a ball whose
radius is at least the minimum of the radii of the three disks.
The more interesting case is t∗ = t2 . We shall simply call the common value t∗ . By looking at a slightly
∞
i
j
smaller radius, if necessary, we can assume that g(s, A) equals a power series i,j=0 bij (s − t∗ ) (A − A∗ )
406
T. I. Vogel
JMFM
which converges absolutely and uniformly on a disk of radius r > 0 centered at (t∗ , A∗ ). Using this power
series,
∞
∞
∗
∗ i
∗ j
∗ i
∗ j
i,j=0 bij (s − t ) (A − A ) −
i,j=0 bij (s2 − t ) (A − A )
∗
(3.8)
G (s , s2 , A) =
s∗ − s2
∞
i−1
i−2
bij [(s∗ − t∗ )
+ (s∗ − t∗ )
(s2 − t∗ )
= b10 +
i,j=1
+ · · · + (s2 − t∗ ) ] (A − A∗ ) ,
√
for all (s∗ , s2 , A) in a ball of radius 2r centered at (t∗ , t∗ , A∗ ), at least for s∗ = s2 . However, on the set
∗
∗
s∗ = s2 , the power series in (3.8) simplifies
√ to the Taylor series for ∗g1 (s , A) centered at (t , A). Thus
equality holds on the entire ball of radius 2r, concluding the case t = t2 .
i−1
j
Proposition 3.3. If
•
•
•
•
R
f (ŝ, Â; γ) = 2r
,
g1 (ŝ, Â; γ) = 0,
f1 (ŝ,
Â; γ)= 0, f2 ŝ, Â; γ g11 ŝ, Â; γ − f1 ŝ, Â; γ g12 ŝ, Â; γ = 0, and
• g11 ŝ, Â; γ = 0,
then there is a pitchfork bifurcation at ŝ, ŝ, Â , in the sense that there is one curve of solutions to
(3.2), (3.3) through ŝ, ŝ, Â along which s∗ = s2 , and a second curve of solutions to (3.2), (3.3) through
ŝ, ŝ, Â for which s∗ = s2 for points near but not equal to ŝ, ŝ, Â .
∗
Proof. The condition that
f1 (ŝ,
Â; γ) = 0 guarantees that there is a curve of solutions satisfying s = s2
which passes through ŝ, ŝ, Â , as noted above. The condition from the implicit function theorem which
ensures that the system (3.6), (3.7) has a solution with s2 and A defined implicitly as differentiable
∞
is met,
s2
(s1 ) and A(s
1 ) will be
functions of s1 is that 2f2 G2 − f1 G3 = 0. In fact, once
this condition
C
(see [6]). When put in terms of f and g, this is f2 ŝ, Â; γ g11 ŝ, Â; γ − f1 ŝ, Â; γ g12 ŝ, Â; γ = 0.
Finally, we must check
that the
two curves are different, i.e., that there is not a curve of solutions to (3.6),
(3.7) going through ŝ, ŝ, Â which happens to also satisfy s∗ = s2 . This could only hold if g1 (s, A; γ)
were zero in a neighborhood of ŝ, ŝ, Â , which is excluded by the condition on g11 .
Note 3.4. The fold-over bifurcations observed in Proposition 3.1 have no particular relation to stability
of the bridge. The reason is that one expects bifurcations in the V, H plane when an eigenvalue of (1.5)
passes through zero, as in [9], but a fold-over bifurcation in the A, s plane does not translate to a fold-over
in the V, H plane. However, it is reasonable to expect that the pitchfork bifurcations given by Proposition 3.3 do relate to stability, since a pitchfork will remain a pitchfork no matter what the coordinate
system is used. Stability will be investigated numerically in Sect. 5, using, in part, Proposition 3.3.
4. Stability Computations
Typically, when an eigenvalue of (1.5) crosses zero, one expects a bifurcation to occur, as in the numerical
results of [9]. However, this does not automatically signal instability, since at a bifurcation one might have
either the second smallest eigenvalue dropping below zero (leading to instability) or the smallest eigenvalue rising above zero,possibly giving stability on both sides of a bifurcation (we will see this in Sect. 5,
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
407
π R
in the example of γ = 20
, r = 1.5). Therefore, it is important to compute eigenvalues of (1.5) directly
rather than attempting to infer stability from observing bifurcations. We will do this in coordinates, as
in [10].
Take the bridge 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 Sect. 2, with dependence on A suppressed.
The constant k is a scaling factor, and s is still an arc length parametrization. After some manipulation
along the lines of [10], one finds that the Laplacian on the surface in these coordinates is
1
sin ϕ
1
ψs .
(4.1)
ψ = 2 ψss + 2 ψθθ +
k
y
y
Using the formulas for mean and Gaussian curvature from [5], one finds that
2
|S| = 2 2H 2 − K
2
cos ϕ cos ϕ
2
+ 2A
,
= 2 2A +
k
y
y
so that
1
1
sin ϕ
ψs
L(ψ) = −
ψss + 2 ψθθ +
k2
y
y
2
cos ϕ cos ϕ
− 2 2A2 +
+ 2A
ψ
k
y
y
We now find the boundary conditions in coordinates. The normal derivative of ψ is the inner product
of the gradient of ψ with the unit vector in the tangent space of the surface which is normal to the
boundary. Referring to volume 4 of [8],
grad ψ =
n n
i=1 j=1
g ij
∂ψ ∂
∂xj ∂xi
∂
∂
+ k −2 y −2 ψθ ,
∂s
∂θ
∂
where g ij is the inverse of the metric tensor of the surface. A vector normal to the boundary curve is ∂s
.
1 ∂
This has length k (using the metric tensor, of course), so the unit normal is ± k ∂s . Therefore the normal
derivative is
1 2
1
0
k
±k
1
1 ψs k 2 ψ θ k 2 y 2
= ± ψs ,
0
k2 y2
0
k
= k −2 ψs
with “+” for the right endpoint and “−” for the left endpoint. Finally for ρ. Referring to [14], it is
1
1 dϕ
cot γ + csc γ
k
ds
r
1
1 cos ϕ
+ 2A cot γ + csc γ.
=
k
y
r
ρ=
Thus the eigenvalue problem, in coordinates, becomes
1
1
sin ϕ
cos ϕ cos ϕ
2
ψs − 2 2A2 +
+ 2A
ψ = λψ,
− 2 ψss + 2 ψθθ +
k
y
y
k
y
y
with boundary conditions
±
1
ψs +
k
1 cos ϕ
1
+ 2A cot γ + csc γ ψ = 0
k
y
r
on the end circles. The plus will apply to the right endpoint, the minus to the left.
(4.2)
(4.3)
(4.4)
(4.5)
408
T. I. Vogel
JMFM
We now separate variables. Set ψ to be P (s) Q (θ). Then (4.4) becomes
1
sin ϕ 4A cos ϕ 2 cos2 ϕ
P Q + 4A2 +
+
P Q + 2 P Q +
P Q = −k 2 λP Q,
y
y
y
y2
which is
Q
P 2 2
y 2 P +
+ y sin ϕ
+ 4A y + 4Ay cos ϕ + 2 cos2 ϕ = −k 2 λy 2 ,
P
Q
P
or
y 2 P P 2 2
Q
+ y sin ϕ
+ 4A y + 4Ay cos ϕ + 2 cos2 ϕ + k 2 λy 2 = −
= m2 ,
P
P
Q
where the separation constant is m2 , m an integer, since Q must be a linear combination of sin mθ and
cos mθ. We obtain
2 cos2 ϕ − m2
yP + sin ϕP + 4A2 y + 4A cos ϕ +
P = −λk 2 yP.
y
This is
2 cos2 ϕ − m2
d
2
2
(y (s) P (s)) + λk y (s) + 4A y + 4A cos ϕ +
P = 0,
ds
y
(4.6)
to get into the form of equation (1) in [1], Chapter 10. Now for the boundary conditions. Suppose that s
is in [s1 , s2 ] . Q will cancel, so the boundary conditions are
1 cos ϕ (si )
1
+ 2A cot γ + csc γ P (si ) = 0.
(4.7)
±P (si ) +
k
y (si )
r
Equation (4.7) may be interpreted as
±P (si ) + ρP (si ) = 0,
(4.8)
using (2.6) and the definition of ρ in (1.3). The eigenvalues may be then found numerically by a Prüfer
substitution (see [1]).
For fixed m, (4.6), (4.7) form a standard Sturm–Liouville problem, so it is natural to label the eigenvalues as λjm . As in [12], we have
λ0m < λ1m < λ2m < · · ·
and
λ00 < λ01 < λ02 < · · · ,
and eigenfunctions corresponding to m = 0 are radially symmetric. A crucial part of the argument in
[14] depended on the sign of λ01 in different circumstances. It will helpful to be able to make statements
about this sign in the case (unlike [14]) when the bridge is not convex.
Note 4.1. The result of Lemma 2.1 of [14] (which compare values of ρ for bridges between balls to values
of ρ for bridges between planes) was stated for convex bridges. However, it applies equally well to the
non-convex bridges we consider now, since the argument doesn’t depend on the curvature of the profiles
of the bridges. Therefore, in going from considering a bridge surface between parallel planes to considering the same bridge surface between balls [altering the appropriate constants in the quadratic form M
defined in (1.2) so that the bridge is still a stationary surface], the change in the value of ρ is the same
as the sign of
c ≡ y(s) cos ϕ(s) + Hy 2 (s),
which is constant on fixed Delaunay curves.
Lemma 4.2. For the standardized Delaunay curves from Sect. 2, if A ∈ (−1, ∞), then c > 0. (Note that
for the surface generated by one of these standardized curves, the mean curvature is A.)
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
Proof. The value of c at s = 0 is easily seen to be 1 + A.
409
Lemma 4.3. Suppose that on [s1 , s2 ] we have cos φ(s) = 0. Label the eigenvalues of the Sturm–Liouville
problem on [s1 , s2 ] for P (s) satisfying (4.6) with boundary conditions
1 cos φ(si )
+ 2A tan φ(si )P (si ) = 0
P (si ) +
(4.9)
k
yi (si )
as λij . Then λ01 = 0.
Proof. It is straight-forward, though tedious, to verify that P (s) = k1 cos φ(s) satisfies
d
2 cos2 ϕ − 12
2
(y (s) P (s)) + 4A y + 4A cos ϕ +
P = 0,
ds
y
(4.10)
and boundary conditions (4.9). Thus 0 is an eigenvalue for m = 1, with corresponding eigenfunction
cos φ(s). Since cos φ(s) is non-zero on [s1 , s2 ], standard results on Sturm–Liouville problems (see e.g., [1])
imply that 0 is the smallest eigenvalue for m = 1, as desired.
Note 4.4. Lemma 4.3 is not surprising when one realizes that the boundary conditions in (4.9) correspond
to a bridge between parallel planes, and k1 cos φ(s) corresponds to an energy neutral translation of the
bridge.
Theorem 4.5. If a rotationally symmetric bridge between balls is either a section of an unduloid, a cylinder, or a catenoid, or if it is a convex section of a nodoid, then λ01 > 0.
Proof. Let Σ be the bridge surface, and consider replacing the balls which Σ bridges by planes, changing the wetting energies so that Σ is still a stationary capillary surface. As in [14], we compare energy
functionals and stability of Σ as a bridge between balls and Σ as a bridge between planes. When Σ is
considered as a bridge between planes, Lemma 4.3 gives us that λ01 = 0. As in [14], Lemma 2.3 the
sign of the change of ρ in going from a bridge between planes to a bridge between balls is c. In all of
the cases considered in this theorem, A > −1, hence c > 0. Thus the value of ρ increases as we go
from a bridge between planes to a bridge between balls when the bridge surface is one of the surfaces
listed.
Using general formulas for eigenvalues of Sturm–Liouville problems, it is known (see [3,15]) that λ0m
is the minimum of the Rayleigh quotient
Dm (ψ(s))
,
H(ψ(s))
where
2
s2
m − 2 cos2 φ(s)
2
Dm (ψ(s)) = y(s) (ψ (s)) +
− 4A2 y(s) − 4A cos φ(s) ψ 2 (s) ds
y(s)
s1
+ ρy(s2 )ψ 2 (s2 ) + ρy(s1 )ψ 2 (s1 )
and
s2
H(ψ) =
k 2 y(s)ψ 2 (s) ds.
s1
Since the value of ρ has increased from the case of a bridge between planes to a bridge between balls,
and all other functions and constants remain the same, it follows immediately that λ01 ≥ λ01 = 0.
Now suppose that we have equality, i.e., that λ01 = 0, and let ψ(s) be the minimizer of the Rayleigh
quotient. Since λ01 is also zero, it follows that ψ(s) is also the minimizer of the Rayleigh quotient in the
case of a bridge between planes. However, the minimizer of the Rayleigh quotient must be a multiple of
the corresponding eigenfunction, in this case cos φ(s), by 4.3. On the bridge surfaces we are considering,
cos φ(s) is never zero, so that in particular, the Rayleigh quotient in the case of a bridge between planes
410
T. I. Vogel
JMFM
evaluated for ψ = cos φ(s) must be strictly less than the Rayleigh quotient for a bridge between balls.
This contradiction shows that λ01 > 0.
5. Numerical Illustrations
In this section, we present two numerical examples, illustrating how the small volume instability may
be found by the methods of the previous sections. The general outline is this: we start at a spherical
bridge making the correct contact angles, at A = −1. Increasing A from this value, we initially have
unduloidal bridges, which are stable as long as they remain convex [14]. As A continues to increase, an
instability will occur either from an eigenvalue of the problem (1.5), (1.6) changing sign from positive
to negative, or H V changing sign (as mentioned in the introduction). An eigenvalue crossing zero will
generically be associated with a bifurcation in the V, H plane. Thus we seek pitchfork bifurcations as in
Proposition 3.3 (since a pitchfork in the A, s plane will correspond to a pitchfork in the V, H plane), and
by plotting volume and mean curvature, look for fold-over bifurcations in the the V, H plane. However,
it is important to actually compute eigenvalues, since a bifurcation may also correspond to an eigenvalue
changing sign from negative to positive, which does not lead to an instability.
5.1. R = 4, r = 1, γ =
π
4
We
the case R = 4, r = 1, γ = π4 . Figure 6 superimposes the graphs of the level curves
firstπ consider
f s, A; 4 = 4 and g1 s, A; π4 = 0. (The closed curve is the level
for f , the other, thicker curve is
curve
π
the level curve for g1 .) The values of s and A which satisfy f s, A; 4 = 4 correspond to bridges which
are rotationally symmetric and symmetric across the plane x = 0, making contact angle π4 with unit
balls whose centers are 8 units apart. The point with A = −1 and s approximately 1.39 corresponds to a
sphere which is a limit of stability, as noted in [14]. For A less than −1, the resulting sections of nodoids
are unstable. As A increases from −1 and s increases from 1.39, the bridges initially are convex sections
of unduloids, and are stable (see [14]). Numerically, the first intersection of the level curves f = 4, g1 = 0
Fig. 6. Graph of f (s, A;
π
)
4
= 4, g1 (s, A) = 0
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
Fig. 7. Volume and mean curvature for R = 4, r = 1, and γ =
411
π
4
which this family encounters is at A = −.6961, s = 1.9995. We check that the other three conditions
of Proposition 3.3 hold, therefore a bifurcation occurs at this point. At A = −0.7, λ10 ≈ 0.016 and at
A = −.69, λ10 ≈ −0.03, so that the bifurcation is due to a second eigenvalue crossing zero and becoming
negative. Certainly bridges in this family are unstable beyond this bifurcation, i.e., as A increases through
−0.6961. To see if bifurcation actually signals the onset of instability, we must also see if H (A)V (A)
changes sign before the bifurcation. Figure 7 plots volume and mean curvature for A starting at −1 and
increasing to −0.67 (thus through the bifurcation). Mean curvature is the horizontal axis, and volume is
the vertical axis, with the largest volume occurring at A = −1. Since no change in sign of H (A)V (A) is
apparent, this indicates that
is precisely where the small volume instability occurs.
the bifurcation
Following the curve f s, A; π4 = 4 beyond this bifurcation, we reach a maximum A value in this
family. Continuing back up this curve, we approach the point A = −1, s = π2 . The behavior at
this point is interesting. As we approach this point from values of A larger than −1, we are going
through a series of unduloids whose necks narrow. The limiting hemisphere is tangent to the solid
balls at their closest points. However, if we approach this point with values of A less than −1, we
are going through a series of nodoids. To maintain the correct contact angles, these will almost engulf
the solid balls, and the limiting hemisphere in this case is tangent to the solid balls at their most distant
points.
5.2. R = 3, r = 1, γ =
π
20
π
The case of γ = 20
, R = 3, r = 1 is interesting. Figure 8 gives the mean curvature of the bridge as a
function of the parameter A. Note the local minimum at A ≈ −0.6. Since the same mean curvature gives
different bridges, this corresponds to a fold-over bifurcation in the V, H plane. This does not lead to
instability, however. What computation indicates to be occurring is that as A increases through about
−0.6, λ00 crosses zero to become positive. Thus for A between the two local extremes, all the eigenvalues
of (1.5), (1.6) are positive, and the bridge is a local energy minimum without any condition on the sign
dV
dV
. As A increases past the local maximum, λ0 again passes through zero, and the sign of dH
must
of dH
412
T. I. Vogel
Fig. 8. Graph of H(A) for γ =
JMFM
π
,R
20
Fig. 9. Volume and mean curvature for γ =
=3
π
,R
20
=3
again be examined to determine stability. Figure 9 gives volume and mean curvature for this family of
bridges. Starting with a large volume symmetrically placed sphere, the bridge remains stable as volume
decreases, even as it passes through the two fold-over bifurcations. The bridge remains stable until a
minimum volume is reached at H ≈ −0.86, occurring at A ≈ −0.16. A pitchfork bifurcation occurs a
little further on, at A ≈ −0.14.
Vol. 15 (2013)
Liquid Bridges Between Balls: The Small Volume Instability
413
References
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[2]
[3]
[4]
[5]
[6]
[7]
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Finn, R.: Equilibrium Capillary Surfaces. Springer, New York, NY (1986)
Gray, A.: Modern Differential Geometry of Curves and Surfaces. CRC Press, Boca Raton, FL (1993)
Krantz, S.G., Parks, H.R.: The Implicit Function Theorem. Birkhäuser, Boston, MA (2002)
Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions, 2nd edn. Birkhäuser, Boston, MA (2002)
Spivak, M.: A Comprehensive Introduction to Differential Geometry. 3rd edn. Publish or Perish, Inc., Houston,
TX (1999)
[9] Vogel, T.I.: Stability of a liquid drop trapped between two parallel planes II: general contact angles. SIAM J. Appl.
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[10] Vogel, T.I.: Stability and bifurcation of a surface of constant mean curvature in a wedge. Indiana Univ. Math.
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[11] Vogel, T.I.: Sufficient conditions for capillary surfaces to be energy minima. Pac. J. Math. 194(2), 469–489 (2000)
[12] Vogel, T.I.: Local energy minimality of capillary surfaces in the presence of symmetry. Pac. J. Math. 206(2),
487–509 (2002)
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B (Suppl), 862–867 (2005)
[14] Vogel, T.I.: Convex, rotationally symmetric liquid bridges between spheres. Pac. J. Math. 2, 367–377 (2006)
[15] Wente, H.C.: The stability of the axially symmetric pendent drop. Pac. J. Math. 88, 421–470 (1980)
Thomas I. Vogel
Department of Mathematics
Texas A&M University
College Station, TX 77843, USA
e-mail: tvogel@math.tamu.edu
(accepted: September 10, 2012; published online: December 24, 2012)