Modes of Nonaxisymmetry in the Stability of Fixed Contact Line Liquid Bridges and Drops

Journal of Colloid and Interface Science 224, 28–46 (2000) doi:10.1006/jcis.1999.6667, available online at http://www.idealibrary.com on

Modes of Nonaxisymmetry in the Stability of Fixed Contact Line Liquid Bridges and Drops Brian J. Lowry Department of Chemical Engineering, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada Received July 27, 1998; accepted December 13, 1999

ˆ R. ˆ The volume of liquid bridges is uid bridge is then L = L/ 2 normalized as V = Vˆ /π Rˆ Lˆ so that V = 1 for any cylindrical bridge. A cylindrical liquid bridge becomes unstable to constant pressure disturbances at L = π (half wave) and to constant volume disturbances at L = 2π (full wave; (5–7)). The pressure jump across the interface, given by the Young–Laplace relation, ˆ where σ is the surface tension, is non-dimensionalized by σ/ R,

A method is presented for predicting the onset and stability character of nonaxisymmetric modes in liquid bridges and drops. The analysis applies to any fixed contact line axisymmetric interface in a steady force field. The onset and stability character of nonaxisymmetric equilibria in liquid bridges and drops is determined. Perturbation analysis is used to locate branches to nonaxisymmetry, and the configuration of the branches then gives the stability character. The number of unstable modes to both constant pressure and constant volume disturbances can be determined, so that changes in stability beyond the primary loss of stability may be examined. Although the first nonaxisymmetric mode tends to dominate higher order modes are significant for liquid bridges where length is less than radius and for drops at higher Bond numbers. At Bond numbers significantly greater than unity, the onset of the least unstable nonaxisymmetric modes tend to collapse between the fixed pressure and fixed volume axisymmetric modes of instability. For liquid bridges, two non-singular classes of nonaxisymmetric mode are distinguished: the predominant, classical shift mode; and a previously unreported tilt mode. The range over which the stability character of fixed contact line liquid bridges and drops is understood is significantly extended. °C 2000 Academic Press Key Words: liquid bridges; sessile drops; stability; perturbation; nonaxisymmetry; Bond number.

1p = κ, ¯

where κ¯ is the sum of the principal curvatures. For axisymmetric interfaces ((r, z) coordinates), these are the axial curvature, κ, and the radial curvature, z˙ /r , so that κ¯ = κ + z˙ /r . The arclength form of the generating equations for the shape of an axisymmetric interface is (cf. (2)) r¨ = −κ z˙ , z¨ = +κ r˙ ,

where dot superscripts denote derivatives by arclength, s, and where 1p = p(r, z) is a known function for a fixed force field. For example, p(r, z) = p0 − Bz for a uniform gravita2 tional field, where p0 = p(0, 0) and B = 1ρg Rˆ /σ is the Bond number (where g is gravitational acceleration and 1ρ is the change in density across the interface). For a fixed contact line drop, r (z = z 0 ) = 1 while for a fixed contact line liquid bridge r (z = z 0 ) = r (z = z 0 + L) = 1 when the bounds are equal (Fig. 1).

Methods for the determination of the stability of static axisymmetric liquid bridges and drops are well established. Recently, the computationally intensive determination of growth rates (e.g. (1)) has been to some extent superseded by analytical and variational methods (2, 3) which focus on the exact point of onset of instability instead. Slobozhanin et al. (4) have extended these methods to nonaxisymmetry by considering branches to nonaxisymmetry for fixed contact line liquid bridges. Their analysis forms the basis for this work.

1.2. Review The study of liquid bridge and drop stability has a long history dating back to before Plateau (5), who first observed fixed volume and fixed pressure instabilities of the cylindrical liquid bridge. Instabilities to nonaxisymmetric disturbances have also been known to occur since at least that time. In particular, in the absence of gravity, fixed contact line liquid bridges between equal ends become unstable to constant volume disturbances when the surface is perpendicular to the axis of symmetry at the contact line (2) (Fig. 2). The same holds for sessile drops under

1.1. Axisymmetric Formulation Drop and liquid bridge dimensions are non-dimensionalized ˆ where hats denote dimensional quantiby the bound radius, R, 3 ties. The volume of drops is normalized as V = Vˆ /πRˆ (so that V = 2/3 for a hemisphere). The dimensionless length of a liqC 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.

[2]

κ = 1p − z˙ /r,

1. INTRODUCTION

0021-9797/00 $35.00

[1]

28

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MODES OF NONAXISYMMETRY

comes the primary mode over the entire fixed volume stability envelope (11). There are several approaches available for stability analysis of liquid bridges and drops. The direct calculation of growth/decay rates for disturbances has been applied with some success, but is tedious numerically and restrictive in the types of disturbances considered. The computation of growth rates is well suited to systems where there is no fixed force field, such as viscous flows and electrostatic fields. However, the common problem of a liquid bridge or drop in a fixed force field (gravity, spin, etc.) is a simpler variational problem. This permits the use of bifurcation (4) or turning point conditions to locate changes in stability (3). Conjugate point conditions can also be used, but these depend strongly on the form of p(r, z) and to some extent on the nature of the instability (2). The method of Maddocks (12) is well suited to liquid bridges in fixed force fields, as demonstrated by Lowry and Steen (3). Stability changes to both constant pressure and constant volume disturbances always occur at turning points or singular (i.e., branch) points in ( p0 , V ) coordinates. The advantage to the method is that only equilibrium interface shapes need be computed, but disturbance growth rates are not determined. Both the turning point (Maddocks) and more specific variational methods are useful for examining stability changes beyond the primary instability. Numerical determination of stability for meshed surfaces is also possible via energy minimization (e.g., (1)). However, this approach suffers from discretization errors, and is best reserved for cases which are significantly nonaxisymmetric. Note that the turning point approach of Maddocks is also applicable to such interfaces. The importance of stability changes beyond the primary loss of stability has recently been made clear by the stabilization of cylindrical liquid bridges with L > 2π by acoustical means. The stability limit (L ≈ 8.6) and nature of the instability observed experimentally in the work of Marr-Lyon, Thiessen, and Marston (13), correspond roughly to the secondary constant volume instability (L ≈ 9). 1.3. Nonaxisymmetric Formulation FIG. 1. Coordinates and dimensions of (a) fixed contact line drop, (b) fixed contact line liquid bridge (external boundaries), and (c) fixed contact line liquid bridge (internal boundaries). The boundaries in (b) and (c) illustrate the variety of bounds required to accommodate some liquid bridge shapes physically, but the actual form of the bounds is irrelevant mathematically in this work.

gravity (8) (Fig. 3). In both cases the interface initially shifts off-axis initially (cf. (9,4)), losing its axis of symmetry. For liquid bridges between equal ends, instability to nonaxisymmetric disturbances occurs at all lengths for rotund liquid bridges but only at L < 0.8097 for slender liquid bridges. The stability envelope for the constant volume case is well-established (2, 10) (Fig. 4). When sufficient gravity (or analogously, bounds of unequal radius) is introduced, the nonaxisymmetric instability be-

Curvature and pressure. The onset of modes of instability to nonaxisymmetric disturbances coincide with branches to nonaxisymmetric equilibria. This had been illustrated for specific cases (4) and follows generally from the analysis of Lowry and Steen (3). These branches may be located by considering nonaxisymmetric perturbations to axisymmetric interfaces. Slobozhanin et al. (4) present the complete perturbation analysis for fixed contact line liquid bridges in the absence of an applied force field. Normal perturbations to the axisymmetric surface are considered, and branches to nonaxisymmetry occur where solutions to the perturbation problem exist. Here an outward normal perturbation to the surface, ²η(s, θ ), is considered so that (r, z) → (r + ²η˙z , z − ²η˙r ).

[3]

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BRIAN J. LOWRY

FIG. 2. Primary (m = 1) shift modes of nonaxisymmetry for B = 0 liquid bridges. A rotund liquid bridge with L = 2: (a) perturbed profiles with first-order perturbation only and (b) simulated images including second order perturbation, with uniformly increasing perturbation. A slender liquid bridge with L = 0.5: (c) perturbed profiles with first-order perturbation only.

The curvature of a surface subjected to an outward normal perturbation ²η(s, θ) is listed in Appendix A. The perturbed curvature is balanced against a fixed pressure field p(r, z) = p0 + p ∗ (r, z),

η = ηS (s) + O(²),

[6]

[4]

where in the work of Slobozhanin et al., p ∗ (r, z) ≡ 0. The base pressure jump p0 perturbs as p0 (²) = p0 (0) + ²δp0 + ² 2 δp1 + O(² 3 ),

a branch of axisymmetric equilibria. The axisymmetric solution has the form

[5]

where δp0 and δp1 are parameters which depend on the nature of the perturbation. Nonaxisymmetric modes. There is always an axisymmetric solution to the perturbation problem when the base state lies on

and includes the particular solution δp0 6= 0. There are also branches to axisymmetry where the above has multiple solutions. Branches to axisymmetry have been considered extensively elsewhere (3, 4, 14). Nonaxisymmetric solutions (at first order) become possible when solutions of the form η = ηA∗ (s, θ ) + O(²)

[7]

exist with the base pressure jump perturbed with δp0 = 0.

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MODES OF NONAXISYMMETRY

only appear at first order. The nonaxisymmetric solutions therefore take the form η = ηA (s) sin(mθ ) + [ηAS (s) + ηAA (s) cos(2mθ )]² + O(² 2 ), [9]

FIG. 3. Primary (m = 1) shift mode of nonaxisymmetry for a fixed contact line drop: perturbed profiles with first-order perturbation only and uniformly increasing perturbation.

Slobozhanin et al. determined the form of these nonaxisymmetric solutions to second order as (4) η = ηA (s) sin(mθ) + [ηAS (s) + ηAA (s) cos(2mθ ) + Q 2 ηA (s) sin(mθ)]¯² + O(¯² 2 ),

[8]

with the base pressure jump perturbed with δp0 = 0 but δp1 6= 0. The integer order of the nonaxisymmetric mode is m ≥ 1, and θ is polar angle. Their perturbation parameter ²¯ is defined in terms of the volume perturbation (singular at volume extrema), and Q 2 is a constant factor. In this work, ² is defined by the generally non-singular requirement that η˙ A (0) = 1, so that terms of the form ηA (s) sin(mθ )

where ² = (1 + Q 2 ²¯ )¯² + O(¯² 3 ). The form of the solution at higher order is preserved, though the dependence on ² is different. The definition of ² in terms of the boundary condition only fails for the special case η˙ A (0) = 0 (ηA (s) ≡ 0), where the perturbation is second order: η = O(²). The necessity for the second-order expansion of the nonaxisymmetric solution is that ηA (s) sin(mθ ) makes no contribution to either pressure (as δp0 = 0) or volume. Both the stability analysis of Slobozhanin and the related analysis presented in this work require variations in volume (and in this work, also in pressure). The classical nonaxisymmetric shift instabilities are surface perturbations with m = 1. In agreement with results presented in Myshkis et al. (2) and experimental observations (4, 9), the m = 1 mode is always the first to appear from stable axisymmetric liquid bridges and drops. System of equations. The system of equations used in this work to determine ηA (s), ηAS (s), and ηAA (s) is somewhat different than that presented by Slobozhanin et al. (4). A nonuniform pressure field, p ∗ (r, z) 6= 0 introduces several additional terms. Also, in this work, the form of solution in Eq. [8] has been assumed, as axisymmetric disturbances are considered mainly via direct computation of equilibria (cf. (3)). This permits simplification by removal of θ derivatives as well as all second derivatives (aside from linear operators). Finally, the singular definition of ² (based on volume perturbations) used by Slobozhanin et al. has been replaced by the simple non-singular requirement that η˙ A (0) = 1. As in Slobozhanin et al., the linear ODEs for ηA (s), ηAS (s), and ηAA (s) are L m (ηA ) = 0, L 0 (ηAS ) = −δp1 + NAS (ηA , ηA ),

[10]

L 2m (ηAA ) = NAA (ηA , ηA ), where here L j = κ2 +

z˙ 2 j 2 r˙ + p − + d/ds + d 2 /ds 2 n r2 r2 r

[11]

and the nonlinear terms NAS and NAA are listed in Appendix A. The normal derivative of pressure pn arises from the Taylor series for pressure in the normal direction: p ∗ (r + ²η˙z , z − ²η˙r ) = p ∗ (r, z) + pn ²η + + FIG. 4. Classical stability limits for a fixed contact line liquid bridge between ends of equal radius.

1 pnn ² 2 η2 2

1 pnnn ² 3 η3 + O(² 4 η4 ). 6

[12]

The boundary conditions at a fixed contact line are simply

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BRIAN J. LOWRY

ηA = 0, ηAS = 0, and ηAA = 0. The pressure perturbation δp1 is determined (following (4)) by the solvability condition Z F3 ηA ds = 0, [13] where the quantity F3 ηA contains a large number of terms and is listed in Appendix A. The system of equations (Eq. [10]) is readily solved for liquid bridges and other interfaces where r > 0 at all points. However, the tip of an axisymmetric drop occurs at r = 0 by definition. On the axis, the system of equations is singular, and a perturbation expansion about r = 0 is required. This expansion is detailed in Appendix B. The perturbation alters the volume of a liquid bridge by (cf. (4)) ¶ Z µ ²2 κ¯ 2 2r ηAS + ηA ds + O(² 3 η3 ). δVA = L 4

[14]

The volume of a drop is altered by the same quantity except that the radius R is substituted for length L. 2. DETERMINATION OF STABILITY

The stability of a fluid interface to constant pressure and constant volume disturbances can be determined via turning points and branch points in the equilibrium structure (3). The equilibria are plotted in the preferred coordinates ( p0 , V ) and the stability of all related equilibria follows from the stability of one state on the branch. The stability to each of constant pressure and constant volume disturbances is expressed by the number of unstable modes, which is equal to the number of negative eigenvalues. Branches with positive slope have equal numbers of unstable modes in both the pressure (k P negative eigenvalues) and volume (k V ) cases (12). Branches with negative slope have one more unstable mode in the fixed pressure case than in the fixed volume case (k P = k V + 1). All stability changes must occur either at turning points or at singular points in the preferred coordinates. Counterclockwise turning points result in a loss of stability (k P → k P + 1 at a pressure turning point, k V → k V + 1 at a volume turning point), while clockwise turning points result in an increase of stability (k P → k P − 1 or k V → k V − 1) (Fig. 5). 2.1. Nonaxisymmetric Branch Classification The theory of Maddocks strictly applies only to turning points, but in general branch points can be broken into pairs of turning points (3). In the case of nonaxisymmetric branches, the breaking of a branch point into turning points could be achieved by introducing a nonaxisymmetric force field. There are twelve possible nonsingular configurations for a nonaxisymmetric branch from a branch of axisymmetric equilibria (Fig. 6). These are reduced to six types by noting that rotation through 180◦ is irrelevant to the stability of the nonaxisymmetric equilibria. The type I branches originate from a branch of positive slope,

FIG. 5. Four possible turning points in preferred coordinates of volume versus base pressure, showing the changes in the numbers of negative eigenvalues for fixed pressure and fixed volume systems.

and type II branches originate from a branch of negative slope. The subscripts indicate whether the nonaxisymmetric branch is as stable as the more stable axisymmetric branch segment (S), or only as stable as the less stable axisymmetric branch segment (U ). The first subscript refers to stability to constant pressure disturbances, and the second to constant volume disturbances. The effect on axisymmetric stability of a branch to nonaxisymmetry follows a simple rule: when the nonaxisymmetric branch is to the left of the axisymmetric branch in preferred coordinates, the axisymmetric equilibria are more stable at lower volume; when the branch is to the right, axisymmetric equilibria are more stable at higher volume. 2.2. Branch Transitions As Bond number or liquid bridge length is changed, branches to nonaxisymmetry can change in type. The simplest type of transition is rotation of the branch, leading to transitions as illustrated in Fig. 7. Transitions which do not cross the axisymmetric branch are simple, as only the stability of the nonaxisymmetric equilibria is affected. However, transitions which cross the axisymmetric branch do alter the stability of the axisymmetric branch significantly. For this reason, transitions of this type only appear as pairs near turning points in a parameter such as Bond number or bridge length. The two nonaxisymmetric branches are manifestations of the same branch, both before and after that branch swings across the axisymmetric branch (Fig. 8a). When branch points do not occur in pairs, the nonaxisymmetric branch flips through 180◦ to leave the axisymmetric branch unaffected. At the point of the sudden rotation, the nonaxisymmetric branch is actually stationary in pressure and volume to second order. Such a transition is illustrated from type I SS to type IUU in Fig. 8b.

FIG. 6. Six classes of non-singular branch to nonaxisymmetry, showing the numbers of negative eigenvalues on each segment, (k P , k V ) (nonaxisymmetric branch in italics): (a) class I branches and (b) class II branches.

FIG. 7. Cycles of possible branch type transitions for class I and class II branches showing sectors about axisymmetric branch. The stability character of a nonaxisymmetric branch is determined solely by the sector in which it lies.

FIG. 8. Examples of (a) pair of nonaxisymmetric branches near limit point and (b) combined rotation and flip of nonaxisymmetric branch. 33

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BRIAN J. LOWRY

3. STABILITY RESULTS

The fixed contact line sessile drop and fixed contact line liquid bridge between equal bounds are simple systems which have been examined extensively. Their stability is considered here beyond the well established primary envelope of stability to constant volume disturbances (k V = 0). The motivation is to understand the nature and onset of nonaxisymmetric modes as they affect the stability of axisymmetric interfaces. However, the initial stability of nonaxisymmetric interfaces to constant pressure or constant volume disturbances (k P , k V ) is also presented. The extensions to stability analysis of these basic systems serve to illustrate the significance of higher modes of instability.

TABLE 2 Minimum Bond Numbers at Which the First Eight Nonaxisymmetric Modes First Appear (Bmin ) and Bond Numbers at Which They Cross the Volume Maximum Curve (Becoming More Likely to be Observed) m

Bmin

Bat Vmax

1 2 3 4 5 6 7 8

0 0.3180 1.312 3.005 5.395 8.479 12.26 16.72

0 0.3962 1.857 5.138 11.78 24.54 47.47 85.43

3.1. Fixed Contact Line Sessile Drops It has been known for some time that a fixed contact line drop under gravity becomes unstable to nonaxisymmetric constant volume disturbances when the interface is perpendicular to the axis of symmetry at the fixed contact line bound (˙r = ±1 at z = z 0 ; Fig. 1). That the instability coincides with r˙ = ±1 has been shown analytically (2) and is confirmed numerically in this work. In this shift instability (m = 1), the drop shifts horizontally off the axis of symmetry before falling downward on the bulging side (Fig. 3). Agreement with existing theory for the m = 1 limit is perfect as it is always found (numerically) to occur precisely at r˙ = 1. This stability limit lies between the constant pressure stability limit (at a pressure maximum) and the volume maximum (Fig. 9a). Note the presence of a previously unreported volume maximum in the (axisymmetric) constant pressure stability limit, at B = 1.565, where V = 1.429 (Fig. 9b). Drops with volumes below the pressure maximum are stable to both constant pressure and constant volume disturbances (k P = k V = 0). Beyond the pressure maximum, drops become unstable to constant pressure disturbances (k P = 1). The m = 1 limit adds an unstable mode for both types of disturbance so that beyond it, (k P , k V ) = (2, 1). The m = 1 limit is therefore the fixed volume stability limit for sessile drops at all Bond numbers. Slightly nonaxisymmetric m = 1 equilibria are always unstable in both the fixed pressure and fixed volume cases, but they are more stable in the fixed pressure case for B > 0.2678 (Table 1). At the volume maximum, a further unstable mode to constant volume disturbances appears. For small Bond number (B < 0.3962), the volume maximum is the next stability change beyond the m = 1 limit. However, for larger Bond number, m > 1 modes become more significant.

Nonaxisymmetric modes with m > 1 appear at limit points (in B) which always lie beyond the volume maximum. As suggested by Fig. 9b, c, the stability limits for the m > 1 modes are all similar in character. At sufficiently large Bond number, each m > 1 stability limit crosses the volume maximum and precedes it for all larger Bond numbers. These modes, unlike the m = 1 mode, do not occur when r˙ = ±1 at the fixed contact line bound. The m > 1 modes form envelopes of destabilization (followed by restabilization to that mode at the opposite edge). At sufficiently high Bond number, the destabilizing portion of each envelope precedes the volume maximum. The stabilizing portion recedes from the volume maximum on the other side, resulting in a broadening range of instability to each mode. The Bond numbers at which various modes appear and then cross the volume maximum curve are tabulated in Table 2. The most stable slightly nonaxisymmetric m > 1 equilibria undergo two changes in stability (Table 3). The first occurs shortly after the limit point in Bond number. The second is analogous to the change for m = 1 as the branch type changes from IIUU to II SU . Note that the crossing of the volume maximum has no effect on the stability of these equilibria. The most stable slightly nonaxisymmetric drops of mode m always occur at larger Bond number, where (k P , k V ) = (m, m). At very large Bond numbers, all primary stability limit curves are roughly coincident and parallel (Fig. 9c). The pressure and volume extremum curves border an increasing number of

TABLE 3 Selected Branch Types and Unstable Modes for m = 2 Branch Points for Drops

TABLE 1 Branch Types and Unstable Modes for m = 1 Branch Points for Drops B

Branch type

(k P , k V )

0 0.2678 ∞

IIUU IISU

(2, 1) (1, 1)

Bmin Vmax

B

Branch type

(k P , k V )

0.3180 0.3222 0.3962 0.5942 ∞

IUU IUS IIUU IISU

(3, 3) (3, 2) (3, 2) (2, 2)

MODES OF NONAXISYMMETRY

35

FIG. 9. Stability limits for fixed contact line drops: (a) overall stability limits, (b) detail showing onset of higher order nonaxisymmetric modes, and (c) detail showing collapse of limits at higher Bond number. Below the pressure maximum, (k P , k V ) = (0, 0); just above it, (k P , k V ) = (1, 0); above each subsequent line below the volume maximum, the number of negative eigenvalues increases as (k P , k V ) = (2, 1), (3, 2), (4, 3), . . .

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BRIAN J. LOWRY

bility limits for no force field (zero gravity, B = 0) including nonaxisymmetric modes beyond m = 1 are quite varied in type (Fig. 11a). The primary stability limits are classical (cf. Fig. 4) and others were considered in Lowry and Steen (3). However, with the addition of arbitrary nonaxisymmetric modes, all stability changes between the volume minimum and maximum can be included. The stability changes to axisymmetric modes take three forms: pressure and volume extrema and branch points. The upper branch point curve is a rotund analogue to the wellknown lower branch point curve and is the result of an underlying structure in liquid bridge equilibria (14). The nonaxisymmetric modes complete the picture.

FIG. 10. Drop on a hollow tube with B = 10, perturbed from m = 4 stability limit with ² = 1 (first-order perturbation only, simulated image).

nonaxisymmetric stability limits. As a shock to a drop system (e.g., a sudden increase in volume or a vertical acceleration) can result in the expression of higher modes, these limits are significant. In fact, many of these higher modes, albeit taken to an extreme where the perturbation expansion likely fails, show familiar characteristics (Fig. 10). 3.2. Fixed Contact Line Liquid Bridges The additional boundary condition for liquid bridges yields a more complex stability character than for drops. The sta-

Classes of nonaxisymmetry. The classical nonaxisymmetric modes of instability are the two m = 1 modes (rotund and slender limits) in which the bridge shifts off the axis of symmetry (Fig. 2). The boundary condition (˙r = ±1 at the contact lines) and the initial shift behavior are qualitatively identical to the fixed contact line drop instability described above. Analogous higher order modes occur for sufficiently short liquid bridges (Fig. 12). The presence of higher order modes for shorter bridges simply reflects the dominance of axial curvature over radial curvature for such bridges. Modes m > 1 ripple the bridge in the radial direction, strongly affecting radial curvature, but for short bridges this curvature is less significant and the changes can be accommodated. Longer bridges are characterized by relatively larger radial curvatures, so that higher modes cannot be realized.

FIG. 11. Complete stability limits for a fixed contact line liquid bridge between ends of equal radius (no force field): (a) types of stability limits including higher order nonaxisymmetric modes (dashed lines are transitions beyond volume extrema), and (b) numbers of negative eigenvalues over each stability region (unstable modes for either fixed pressure and fixed volume systems).

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MODES OF NONAXISYMMETRY

FIG. 12. Detail of stability limits for a fixed contact line liquid bridge between ends of equal radius (no force field): higher order nonaxisymmetric modes for (a) rotund liquid bridges and (b) slender liquid bridges (dashed lines are transitions beyond volume extrema).

A second type of nonaxisymmetry qualitatively different from the shift modes is found for m = 1. It lies past the upper branch point limit (Fig. 13) where bridges are so rotund that they cross what would be cylindrical bounds. Here an internal cylinder has been illustrated instead (cf. Fig. 1). In the “tilt” mode of instability the bridge tilts off of its axis of symmetry. This is unexpected behaviour, as the contact lines maintain the original axis. The tilt is quite persistent, with second-order behaviors aside from the tilt only becoming apparent for larger perturbations (Fig. 14). The m > 1 branches to nonaxisymmetry affect the stability of axisymmetric liquid bridges by forming envelopes of relative instability. By contrast, both the m = 1 shift and m = 1 tilt nonaxisymmetric branches appear as limits on stability rather than envelopes. The superposition of these distinct types of stability limit gives rise to a checkerboard pattern of stability regions for short, rotund liquid bridges (Fig. 12a). The stability of the slightly nonaxisymmetric liquid bridge equilibria can be determined via classification of branches. The m = 1 shift mode has been characterized by Slobozhanin et al. (4) for the fixed volume case. In the rotund case, there is a single change in stability with increasing length (Table 4a). Slightly nonaxisymmetric bridges (near the m = 1 rotund limit) with L > 1.0162 are stable to constant volume disturbances (k V = 0), as reported by Slobozhanin et al. The stability changes near the m = 1 slender limit are more complex (Table 4b). Slender nonaxisymmetric bridges are stable to constant volume disturbances only for 0.4483 < L < 0.8097. The change in stability

at the length maximum (L = 0.8097) is of the type illustrated in Fig. 8a. At L = 0.7223, the m = 1 slender limit crosses the volume minimum (Fig. 12b), but no change in stability occurs. This point is mistakenly identified in Slobozhanin et al. (4) as two separate changes in stability (perhaps due to the use of a volume-based perturbation). The change in stability at L = 0.6292 completes the envelope (which extends back to L = 0). Slightly nonaxisymmetric equilibria near the m = 1 tilt limit are considerably less stable than those near the m = 1 shift limits (Table 5). As L → 0, due to instabilities arising from the m > 1 shift modes, the slightly nonaxisymmetric equilibria near the tilt limit actually approach (k P , k V ) → (∞, ∞). However, from TABLE 4 Branch Types and Unstable Modes for m = 1 (a) Rotund and (b) Slender Branch Points for B = 0 Liquid Bridges L

Branch type

(k P , k V )

(a)

0 1.0162 ∞

IISU IISS

(1, 1) (1, 0)

(b)

0 0.4483 0.8097 0.7223 0.6292 0

IISU IISS IIUU IUS ISS

(1, 1) (1, 0) (2, 1) (2, 1) (1, 1)

L max Vmin

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BRIAN J. LOWRY

FIG. 13. Liquid bridges with L = 0.5 and B = 0 from (a) the cylinder through (c) the rotund limit (the m = 1 shift instability), (f ) the large volume axisymmetric branch point, (g) the m = 1 tilt instability, (m) the volume maximum, to (p) a second m = 1 shift instability which occurs past the volume maximum. Left side is optical simulation; right side shows bounds and interface as lines.

just prior to the crossover with the m = 2 rotund limit, the m = 1 tilt equilibria become maximally stable, with (k P , k V ) = (4, 3). Note that the crossing of the m = 2 rotund limit at L = 0.8257 does not affect the stability of m = 1 tilt equilibria (the converse is not true—see below). The upper portion of the m = 2 rotund envelope stabilizes the axisymmetric branch to exactly the same extent that the stability change in the m = 1 tilt branch decreases the stability of the nonaxisymmetric branch. The net result is no TABLE 5 Selected Branch Types and Unstable Modes for m = 1 Tilt Branch Points for B = 0 Liquid Bridges

m = 4 crossover m = 3 crossover m = 2 crossover Vmax p0,min

L

Branch type

(k P , k V )

0.1593 0.3018 0.8029 0.8071 0.8257 1.807 2.131 ∞

IIUU IIUU IISU IISS IIUU IUS IISS

(6, 5) (5, 4) (4, 4) (4, 3) (4, 3) (4, 3) (4, 3)

TABLE 6 Selected Branch Types and Unstable Modes for m = 2 (a) Rotund and (b) Slender Branch Points for B = 0 Liquid Bridges

(a)

L max

Branch point crossover Tilt (m = 1) crossover Vmax (b) L max

L

Branch type

(k P , k V )

0 0.3644 1.3036 1.3063 1.2907 1.2874 1.1198 0.8257 0.3689

IISU IISS IIUU IISS IISU IIUU IIUU IIUU

(2, 2) (2, 1) (3, 2) (2, 1) (2, 2) (3, 2) (4, 3) (5, 4)

0 0.2456 0.4111 0.4100 0.3916 0.3510 0

IISU IISS IIUU IISS IISU IIUU

(2, 2) (2, 1) (3, 2) (2, 1) (2, 2) (3, 2)

MODES OF NONAXISYMMETRY

39

FIG. 14. The m = 1 tilt mode of nonaxisymmetry for a liquid bridge with L = 0.5: (a) perturbed profiles with first-order perturbation only and (b) simulated images including second-order perturbation.

change in stability of the nonaxisymmetric m = 1 tilt equilibria near the branch point. The two changes appear to coincide exactly (from precise numerical results). The stability of slender and m > 1 rotund slightly nonaxisymmetric equilibria is very similar for a given m (Table 6). Within each subset (slender equilibria and rotund equilibria) the series of envelopes of increasing m are also of similar character.

To a point, the changes in branch type and number of unstable modes are identical for slender equilibria and rotund equilibria. However in both cases there are three closely spaced changes in stability along each envelope, and the ordering is slightly different near the length maximum (for rotund they bracket the maximum; for slender, they occur at and beyond the maximum). The similarity ends at L = 0 (slender case) and the branch point limit

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BRIAN J. LOWRY

FIG. 15. Stability limits for liquid bridges at high Bond number (dashed lines are m = 1 stability transitions beyond volume extrema to illustrate saddle bifurcation at B = 3.089): (a) B = 2.0 showing volume, pressure, and nonaxisymmetric stability bounds; (b) B = 3.0 and (c) B = 4.0 with m = 1 limits highlighted.

MODES OF NONAXISYMMETRY

41

FIG. 16. Schematics illustrating consistency of figure eight loop in preferred coordinates and impossibility of a consistent simple loop (without branches from it).

(rotund case). Beyond this point the m > 1 rotund slightly nonaxisymmetric equilibria become increasingly less stable. The maximal stability of slightly nonaxisymmetric m ≥ 1 liquid bridges is (k P , k V ) = (m, m − 1). 3.3. Gravity The introduction of gravity simplifies the liquid bridge stability limits by breaking the branch points into pairs of turning points (3, 14). This disconnects portions of the envelope beyond the branch point stability limits, so that features such as the m = 1 tilt instability and volume extrema (for B = 0) become less relevant (Fig. 15a). At higher Bond numbers, as for drops, the pressure and volume extrema collapse to similar limits, with nonaxisymmetric modes in between. The collapse for nonaxisymmetric stability limits begins with the formation of a closed loop for the m = 1 mode (Fig. 15b, c), as has been noted by Slobozhanin and Perales (11). The saddle bifurcation where the m = 1 loop becomes continuous precedes the Bond number at which the m = 1 loop becomes the sole primary fixed volume stability limit (B = 3.59423, as reported in (11)). The m = 1 limit becomes a loop at B = 3.089, but up to B = 3.59423 a short segment of the loop remains slightly beyond the volume extrema limit and is thus a secondary stability change in the fixed volume case. With further increase in Bond number, the m > 1 stability limits also collapse to form loops between the pressure and volume extrema limits. The fold in the volume maximum limit for short bridges is simply due to the coexistence of three volume extrema at large volume for a given length of liquid bridge (cf. Fig. 17a). This represents interaction between remnants of the B = 0 branch point curve (which becomes the upper volume limit for small Bond number) and remnants of the original B = 0 volume maximum (which exists chiefly as a secondary volume extremum at small Bond number). The fold never occurs particularly close to stable equilibria and is likely of little practical significance. Loops and Self-Consistency. Self-consistency is crucial in confirming that all possible branches have been detected. The checks below confirm the method for a few select cases, and a

wide variety of checks strongly imply the absence of undetected branches. Unless such branches were to occur in pairs at pressure extrema they could not exist without singularities. However, a proper proof that all branches to nonaxisymmetry have been found is beyond the scope of this work. The self-consistency of the method for determining stability is most easily tested on equilibrium branches which form loops. All stability changes around a loop must sum to no change in stability. The theory of Maddocks precludes branchless loops which are open in preferred coordinates, as these are never selfconsistent. However, figure eight loops are permissible and do in fact occur, both with and without branches to nonaxisymmetry (Fig. 16). Note that the apparent crossing point in the figure eight is not a point of coincidence: the loop is open in some alternate coordinate system. An interesting and somewhat varied sequence of loops occurs for moderate length liquid bridges as Bond number increases. For example, with L = 8, the stable liquid bridges exist on a loop for B > 0.0365 [14]. However, for B > 0.05410 there are no stable bridges on the loop, and beyond B ≈ 0.0567 the loop ceases to exist. The loop is not an ideal figure eight at B = 0.04, but the stability is self-consistent (Fig. 17a). Two m = 1 branches to nonaxisymmetry are evident on the lower segment of the loop. These have no net effect on consistency but destabilize the entire upper segment of the loop. The stable equilibria (constant volume) lie between the leftmost m = 1 branch and the volume minimum. The loop is more ideal at B = 0.052, and the stability on the lower segment is identical to B = 0.04 (Fig. 17b). However, the rightmost m = 1 branch crosses the pressure maximum at B = 0.05343 and undergoes an interesting transition (Fig. 17c). In order to maintain self-consistency around the loop, when the m = 1 branch crosses the pressure maximum the angle of the branch must change (from decreasing volume to increasing volume). This change is singular as the axisymmetric second order part of the perturbation is a solution to L 0 (ηAS (s)) = NAS (ηA , ηA ) (6=0 in general),

[15]

while the equation for the particular solution, L 0 (ηS (s)) = −δp0 ,

42

BRIAN J. LOWRY

FIG. 17. Equilibrium loops for liquid bridges with L = 8 and various Bond numbers showing numbers of unstable modes (fixed pressure and fixed volume) and branches to nonaxisymmetry: (a) B = 0.04, (b) B = 0.052, (c) B = 0.05385, (d) B = 0.054, (e) B = 0.055. Inset schematics for b–d illustrate changes in stability at lower right ends of loops.

MODES OF NONAXISYMMETRY

43

FIG. 17—Continued

becomes L 0 (ηS (s)) = 0

[16]

at pressure extrema. Therefore, the solution for ηAS (s) is indeterminate by an arbitrary multiple of ηS (s) when a branch point coincides with a pressure maximum. Numerically, this is evidenced by a singularity in the magnitude of δVA , the volume change partly due to ηAS (s). The volume change δVA → ±∞ as the branch approaches the pressure maximum from either side. The lower (rightmost) m = 1 branch crosses the volume minimum at B = 0.053861, and the stable region is then bounded on both sides by m = 1 nonaxisymmetric instabilities (Fig. 17d). The two branches meet at the limit point B = 0.05410 so that for B > 0.05410 there are no stable equilibria (Fig. 17e). Shortly beyond this point the loop itself ceases to exist (cf. (14)). There are several variations on the figure eight loop, some involving pairs of branch points and disconnection instabilities. For shorter liquid bridges the equilibrium branches generally form a semiinfinite double helix rather than loops, both with and without gravity (14), making self-consistency more difficult to verify, but with a very wide variety of lengths, force fields, and to some extent boundary conditions, the numbers of negative eigenvalues have never been found to be inconsistent. 4. ACOUSTIC STABILIZATION

The higher order stability changes described here all lie outside of the constant volume stability envelope. However, ultra-

sonic acoustics have recently been used (coupled with an imaging control system) to stabilize portions of the lower (rightmost) (k P , k V ) = (2, 1) region to constant volume disturbances (13). This was achieved for near-cylindrical bridges by suppressing axial asymmetry (asymmetry along the z axis) of the fixed volume mode which appears at the lower axisymmetric branch point stability limit. An analogous axial asymmetry appears at the upper branch point limit as well. The existing acoustic method could likely be extended to remove both of these limits. Additionally, it is conceivable that a similar method could be used to suppress the nonaxisymmetric shift mode, which could be readily observed by a two-camera imaging system. Combined suppression of (axisymmetric) axial asymmetric modes and the m = 1 nonaxisymmetric shift modes would result in the stability envelope for constant volume disturbances shown in Fig. 18. The rotund bridge volumes attainable with such stabilization would be roughly four times greater for L > 3 than without stabilization. The existing acoustic work has shown experimental evidence of otherwise unobservable axisymmetric liquid bridges and instabilities. The instabilities take the form of axially symmetric wavenumber 3 undulations which occur at the volume minimum (3). This is in contrast to the classical wavenumber 2 instabilities for the cylinder at L = 2π (which are axially asymmetric and which occur at a branch point). Removal of (axisymmetric) axial asymmetric modes and the m = 1 nonaxisymmetric shift modes could potentially permit direct experimental observation of the m = 2 shift and m = 1 tilt modes of nonaxisymmetry.

44

BRIAN J. LOWRY

FIG. 18. Potential expanded range of fixed volume stability where axial symmetry and axisymmetry (to m = 1 disturbances) are enforced. Dashed lines are stability limits which vanish when axial asymmetry and m = 1 nonaxisymmetry are not permitted.

5. CONCLUSIONS

The method applied here was extremely similar to that introduced by Slobozhanin et al. and verifies most of the results presented in that work within a reasonable margin of error (cf. (4)). In particular, the existence of stable (to constant volume disturbances) m = 1 slender and rotund slightly nonaxisymmetric liquid bridges was confirmed. Stable m = 1 slender slightly nonaxisymmetric liquid bridges exist for 0.4483 < L < 0.8097 and stable m = 1 rotund slightly nonaxisymmetric liquid bridges exist for all L > 1.0162. The extension of the analysis to gravitational force fields and also to drops did not result in the discovery of further stable slightly nonaxisymmetric interfaces. However, the stability character beyond the primary stability limits does show a number of interesting characteristics. Through consideration of stability to constant pressure disturbances as well as constant volume disturbances, the relationship between all stability limits for axisymmetric bridges becomes clearer. For drops, at higher Bond number the similarity between the axisymmetric fixed pressure (pressure maximum) and fixed volume (volume maximum) limits and the nonaxisymmetric limits is especially clear as the limits tend towards coincidence. Higher modes of nonaxisymmetry are significant for fixed contact line drops with large Bond number as well as for short fixed contact line liquid bridges. However, they appar-

ently never precede the m = 1 shift mode, which is the primary and classical mode of nonaxisymmetry. At higher Bond numbers, the axisymmetric fixed pressure stability limit and volume extrema tend to collapse around nonaxisymmetric stability limits for both drops and liquid bridges. The m = 1 tilt stability limit was discovered for liquid bridges in the absence of any force field. It appears that this type of instability is inherent to liquid bridges and not only characteristic of rotating liquid bridges. There are two basic classes of m = 1 nonaxisymmetry: the classical shift mode and the considerably less stable tilt mode. The maximal stability for slightly nonaxisymmetric m = 1 shift interfaces is (k P , k V ) = (1, 0) for fixed contact line liquid bridges between equal bounds and (k P , k V ) = (1, 1) for fixed contact line sessile drops. In contrast, the maximal stability for slightly nonaxisymmetric m = 1 tilt interfaces is (k P , k V ) = (4, 3) for fixed contact line liquid bridges between equal bounds (in the absence of a force field). The stability of slightly nonaxisymmetric m > 1 equilibria have also been examined. These equilibria are generally unstable, unlike the slightly nonaxisymmetric m = 1 shift liquid bridges characterized by Slobozhanin et al. (4). In fact, no m > 1 branches to nonaxisymmetry have been found originating from branches of stable axisymmetric equilibria. However, extensions to control methods (e.g. acoustic stabilization) could perhaps extend the envelope of fixed volume instability and render m > 1 equilibria or instabilities directly observable. Until that time, the m = 1 tilt and m > 1 instabilities will be observable only under strong force fields or shocked systems. They exist as further challenges to those who would stabilize drop or liquid bridges beyond the natural limits. APPENDIX A: CURVATURE PERTURBATION EXPRESSIONS

The sum of principal curvatures of the perturbed surface, ¯ θ )) may be determined as a perturbation series in ² κ¯ ² = κ(²η(s, and η(s, θ ) using a symbolic algebra package (Maple V was used here and the series was verified by hand as well as by comparison to Slobozhanin et al. (4), who first reported this perturbation expansion): ·µ ¶ ¸ z˙ z˙ 2 r˙ 2 2 κ¯ ² = κ + − κ + 2 η + ηs + ηss + ηθ θ /r ² r r r µ ¶ µ ¶ ·µ 3¶ z˙ z˙ 2˙r z˙ 1 3 2 κ− + κ + 3 η + ps + 2 ηηs + r r 2 r ·µ ¸ ¶ µ 2¶ ηθ z˙ ηθ θ 2 z˙ 4 2 4 × ηs − 2 + 2κηηss + 2 η 2 ² − κ + 4 η3 r r r r µ µ ¶ ¶ z˙ 2 r˙ z˙ −2κ 2 + 4κ + 2 + 3κ ps η2 ηs + r r r µ µ ¶ ¶ 2 z˙ 2 3 2 z˙ 1 2 z˙ 3˙z 2 η 2 κ − κ − 2 ηηs + − κ − κ + 2 η 2θ + 2 r 2r 2 r 2r r

MODES OF NONAXISYMMETRY

· ¸ 3 r˙ z˙ 2 z˙ 3 1 ηA η˙ A3 + −κ 3 − 2 2 κ + 3 − 2κ pn + pnn 8r r r 2 ¶ µ m2 z˙ 2 η (ηAS − 2ηAA ) × ηA2 (ηAS − ηAA /2) + 2 κ − r r A µ ¶¸ · z˙ r˙ 1 ps − κ− ηA [η˙ A (ηAS − ηAA /2) + 2 r r µ ¶ 1 z˙ κ− + ηA (η˙ AS − η˙ AA /2)] + 2 r

µ ¶ r˙ 3 z˙ 2 ηθ θ ηθ θ 3 ηs + 3κ 2 η2 ηss + 3 2 η2 2 − ηs2 ηss + 2 2r r r 2 2r µ ¶ ¸ 3ηθ θ r˙ η2 ηθ ηsθ 3 η2 1 ηss + 2 + ηs 2θ − 2ηs 2 [17] ² . − 2θ r 2 2r 2r r r

−

−

The subscript s indicates partial arclength derivatives (an overdot is used in this work for ordinary arclength derivatives), and the subscript θ indicates partial derivatives by polar angle. The arclength derivative of pressure may be expanded as ps = pr r˙ + pz z˙ and the normal derivatives of the pressure field may be expanded as

[18]

pnnn = prrr z˙ 3 − 3 prr z r˙ z˙ 2 + 3 pr zz r˙ 2 z˙ − prrr z˙ 3 . The second-order operator equations arising from equality between the curvature and the perturbed pressure field contain many nonlinear terms in ηA (s), which can be grouped as ·

1 z˙ 2 1 z˙ 3 1 NAS (ηA , ηA ) = − κ 3 − κ 2 + − κ pn + pnn 3 2 r 2r 4 · ¶¸ µ ¶ µ ¸ 3m 2 r˙ z˙ 1 z˙ + 2 κ− ηA2 − κ− − ps 4r r r r 2 µ ¶ z˙ 1 κ− η˙ A2 [19] × ηA η˙ A + 4 r · 1 3 z˙ 2 1 z˙ 3 1 κ +κ 2 − + κ pn − pnn NAA (ηA , ηA ) = 3 2 r 2r 4 · µ ¸ ¶¸ µ ¶ 5m 2 r˙ z˙ z˙ 1 − 2 κ− ηA2 + κ− − ps 4r r r r 2 µ ¶ z˙ 1 κ− η˙ A2 . [20] × ηA η˙ A − 4 r The solvability condition involves the nonlinear expression (in ηA (s), ηAS (s), and ηAA (s)): "

3 z˙ 2 8 r2

µ

z˙ r

¶2

3 3 3 − κ 2 pn + κ pnn − pnnn 8 8 48 # µ ¶ 7 z˙ 2 m 2 1 2 5 z˙ 1 m4 4 κ − κ + pn − 4 ηA + 2 − r 4 4 r 8 r2 16 8r · µ µ ¶ ¶ ¸ r˙ 3 2 z˙ 2 m2 3 k − 2 − 2 − κ ps ηA3 η˙ A + r 8 r 8r 8 · ¸ 3 2 3 z˙ 2 9 5m 2 2 2 pn + 2 ηA η˙ A + − κ − − 4 8 r2 16 8r

F3 ηA = −

κ−

× ηA η˙ A (η˙ AS − η˙ AA /2) − (κδp 1 ) ηA2 .

[21]

APPENDIX B: DROP TIP PERTURBATION EXPANSION

pn = pr z˙ − pz r˙ , pnn = prr z˙ 2 − 2 pr z r˙ z˙ + prr z˙ 2 ,

45

At the tip of a drop, where r = 0, quantities such as z˙ /r are difficult to treat numerically. The following perturbation expansions about a drop tip (here taken as s = 0) are therefore useful. The axisymmetric shape is easily derived by combining the first derivative for pressure ( p(s) = r˙ pr + z˙ pz ), the definition of axial curvature (Eq. [2]), and the definition of arclength (˙r 2 + z˙ 2 = 1). The shape of the drop is given by ¢ 1 2 3 p0 ¡ 3 p0 s + p0 − 12 p0 pz − 24 prr s 5 + O(s 7 ), 24 1920 [22] ¢ 4 1 1 ¡ 3 2 6 p − 3 p0 pz − 6 prr s + O(s ), z(s) = z 0 + p0 s + 4 192 0

r (s) = s −

where p0 = p(s = 0) and pz = ∂ p/∂z|s = 0 . The pressure and principle curvatures perturb as 1 p(s) = p0 + ( p0 pz + 2 prr )s 2 + O(s 4 ), 4 1 3 κ(s) = p0 + ( p0 pz + 2 prr )s 2 + O(s 4 ), 2 16 z˙ 1 1 (s) = p0 + ( p0 pz + 2 prr )s 2 + O(s 4 ). r 2 16

[23]

The nonaxisymmetric perturbations are somewhat more difficult, as they require solution of three second-order ODEs. However, they are nonsingular for all m ≥ 1. ηA (s)

¸ · (m 2 + m − 6) p02 + 12 pz m+2 m s CA + O(s m+4 ), = s + 48(m + 1) · ¸ ¢ 1¡ − p02 + 2 pz s 2 CAS ηAS (s) = 1 + 8 1 1 − δp1 s 2 + p0 s 2m CA2 + O(s 4 ), [24] 4 8 m ηAA (s) = s 2m CAA − p0 s 2m ln(s)CA2 + O(s 2m+2 ), 8

The constants CA , CAS , CAA , and δp1 are uniquely determined from the two matching conditions at the boundary of the drop

46

BRIAN J. LOWRY

(ηAS = ηAA = 0 at a fixed contact line), the definition of ² by the further condition η˙ A = 1 at the boundary, and the solvability condition. The condition ηA = 0 at the fixed contact line is used to locate branches to nonaxisymmetry and cannot be used as a constraint on CA . The solvability condition perturbs as Z

s

F3 ηA ds =

0

¡ 1 £ 4 p0 δp1 + 2 p03 − 8( p0 pz + 2 prr ) 2m + 1 ¢ ¤ − (m 2 − m)( p0 pz + pzz ) CAS CA2 s 2m+1

REFERENCES

+ O(s 2m+3 ). As this is an integral expression, it can be taken as zero with minimal numerical error, but the above series was used in the current work. Similarly, the volume perturbation near the tip is δVA = CAS s 2 + +

¢ ¤ 1 £¡ −2 p02 + 3 pz CAS − 3δp1 s 4 24

3 p0 CA2 2m+2 s + O(s 6 ), 8(m + 1)

where the author was a postdoctoral fellow. Professor Paul H. Steen has been a great source of encouragement and advice on this work. Philip Marston and David Thiessen’s work on acoustic stabilization presented this work with decidedly more practical potential. The work of Lev Slobozhanin and his collaborators was particularly useful in verifying numerical implementation of the rather complex equations arising in the perturbation expansion. This work was conducted mainly with support from the Natural Sciences and Engineering Council of Canada, but also startup funds provided by the University of New Brunswick. This support made possible numerical and imaging work as well as portions of the perturbation analysis.

[25]

which is generally not significant (relative to the rest of the drop) for small s. ACKNOWLEDGMENTS This work benefitted from numerous discussions with faculty in Chemical Engineering at the University of Alberta, particularly Professor K. Nandakumar,

1. Mittelmann, H. D., and Zhu, A., Microgravity Sci. Technol. 9(1), 22 (1996). 2. Myshkis, A. D., Babskii, V. G., Kopachevskii, N. D., Slobozhanin, L. A., and Tyuptsov, A. D., “Low-gravity fluid mechanics. Mathematical theory of capillary phenomena.” Springer-Verlag, New York, 1987. 3. Lowry, B. J., and Steen, P. H., Proc. R. Soc. London A 449, 411 (1995). 4. Slobozhanin, L. A., Iwan, J., Alexander, D., and Resnick, A. H., Phys. Fluids 9(7), 1893 (1997). 5. Plateau, J., Annu. Rep. Board Regents Smithsonian Inst. 207 (1863). 6. Plateau, J., Annu. Rep. Board Regents Smithsonian Inst. 285 (1864) (continued from Smithsonian Report for 1863). 7. Plateau, J., Annu. Rep. Board Regents Smithsonian Inst. 411 (1865) (continued from Smithsonian Report for 1864). 8. Michael, D. H., Annu. Rev. Fluid Mech. 13, 189 (1981). 9. Russo, M. J., and Steen, P. H., J. Colloid Interface Sci. 113, 154 (1986). 10. Gillette, R. D., and Dyson, D. C. Chem. Eng. J. 2, 44 (1971). 11. Slobozhanin, L. A., and Perales, J. M., Phys. Fluids 5(6), 1305 (1993). 12. Maddocks, J. H., Arch. Rational Mech. Anal. 99, 301 (1987). 13. Marr-Lyon, M. J., Thiessen, D. B., and Marston, P. L., J. Fluid Mech. 351, 345 (1997). 14. Lowry, B. J., Phys. Fluids, in press, 2000.