The mathematical basis for the forthcoming Angular Liquid Bridge investigation on board Mir is described. Our mathematical work is based on the classical Young-Laplace-Gauss formulation for an equilibrium free surface of liquid partly filling a container or otherwise in contact with solid support surfaces. The anticipated liquid behavior used in the apparatus design is also illustrated

Concus, Paul

Finn, Robert

English

We describe here recent mathematical results that form the basis of our forthcoming space experiment, developed jointly with Mark Weislogel of NASA Lewis Research Center, which is scheduled for the Glovebox on the Mir 23 / NASA 4 Mission in December 1996. The mathematical basis for the Angular Liquid Bridge is described. The anticipated liquid behavior used in the apparatus is illustrated

Concus, P.

Finn, R.

UNT Digital Library

LBNL-38841 UC-405 1 3 1 Lawrence Berkeley Laboratory e UNIVERSITY OF CALIFORNIA Physics Division Mathematics Department Presented at the Third Microgravity Fluid Physics Conference, Cleveland, OH, June 13-15,1996, and to be published in the Proceedings Equilibrium Fluid Interface Behavior Under Low- and Zero-Gravity Conditions. 11 P. Concus and R. Finn May 1996 Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098 DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or The Regents of the University of California. Ernest Orlando Lawrence Berkeley National Laboratory is an equal opportunity employer. LBNL-38841 EQUILIBRIUM FLUID INTERFACE BEHAVIOR UNDER LOW- AND ZERO-GRAVITY CONDITIONS. 11* Paul Concus Lawrence Berkeley Laboratory and Department of Mathematics University of California Berkeley, CA 94720 Robert Finn Department of Mathematics Stanford University Stanford, California 94305 May 1996 "This work was supported in part by the National Aeronautics and Space Administration under Grant NCC3-329, by the National Science Foundation under Grants DMS-9400778 and DMS-9401167, and by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, Department of Energy, under Contract Number DEAC03-76SF00098. ~ISTRlBUnON OF MIS DOCUMENT IS UNLIMITED EQUILIBRIUM FLUID INTEWACE BEHAVIOR UNDER LOW- AND ZERO-GRAVITY CONDITIONS. I1 Paul Concus Lawrence Berkeley Laboratory and Department of Mathematics University of California Berkeley, CA 94720 Robert Finn Department of Mathematics Stanford University Stanford, CA 94305 ABSTRACT The mathematical basis for the forthcoming Angular Liquid Bridge investigation on board Mir is described. The anticipated liquid behavior used in the apparatus design is illustrated. INTRODUCTION We describe here recent mathematical results that form the basis of our forthcoming space experiment, developed jointly with Mark Weislogel of NASA Lewis Research Center, which is scheduled for the Glovebox on the Mir 23 / NASA 4 Mission in December, 1996. Our mathematical work is based on the classical Young-Laplace-Gauss formulation for an equilibrium free surface of liquid partly filling a container or otherwise in contact with solid support surfaces. In this formulation, when gravity is absent or can be neglected, which is the situation we consider here, the mechanical energy E of the system is given by E = a(S - s* cosy). (1) The interfacial liquid-vapor surface tension parameter a and the relative adhesion coefficient cos y of the liquid with the container walls are assumed to depend only on the material properties, which are taken here to be homogeneous (the same value of cosy on all parts of the container, as will be the case for the experiment). S and S* are, respectively, the areas of the liquid-vapor free surface and of the solid-liquid interface. Equilibrium configurations are those providing stationary values of the energy functional E subject to the condition of fixed liquid volume [l]. The equilibrium liquid-vapor free surfaces so determined are surfaces of constant mean curvature meeting the bounding walls with contact angle y. We consider here values of the contact angle 0 < y < 7r. Of particular interest in our mathematical studies are situations in which small changes in contact angle or geometry can result in large changes, possibly discontinuous, of the equilibrium fluid configuration. Impetus for the present experiment arise largely from recent doctoral dissertations of two students associated with our study, John McCuan [2] and Lianmin Zhou [3], from whose contrasting results striking inferences can be drawn. 1 ANGULAR LIQUID BRIDGE In his work, McCuan found conditions under which an equilibrium tubular bridge in a wedge domain (Fig. 1) would be possible in zero gravity, and he gave the shape such a bridge might take. This work is a completely rigorous mathematical study, based on the classical formulation. Consider a wedge domain with opening angle 20, 0 < 20 < T. The results McCuan proved contain the following (if the contact angles on the two sides of the wedge are different, the following results hold if y on the left of the inequalities is their average): If y > n/2 + a, a bridge in the shape of a portion of a sphere making contact angle y with the walls exists. If 7 5 n/2 + a, no physically realizable bridge is possible. It has not yet been proved whether or not other shape bridges may be possible when y > n/2 + a, or whether the spherical bridges are stable (provide a local minimum for the energy). However, our numerical results and those of H. Mittelmann (private communication), obtained us- ing the Surface Evolver software package [4], indicate that the spherical bridges are stable, at least for the representative cases we considered. Also, no bridge shapes other than the sphere have been found numerically. Note that McCuan's results imply that a bridge is possible only for y > n/2. A spherical liquid bridge is shown in Fig. 4 for the case cy = 25O, y = 130". ' BRIDGE BETWEEN PARALLEL PLATES-DISCONTINUOUS BEHAVIOR The above results for liquid bridges in a wedge compare in a remarkable way with those for bridges between parallel plates (Fig. 2). This latter problem was studied initially from a rigorous mathematical point of view by Athanassenas [5] and by Vogel [6], and later using a more physical approach by Langbein [7]. (Note that in these papers, as is the case in [3] and here, the boundary conditions at the plates are prescribed contact angle, which arises from the variational condition for (1). For fixed end conditions, as considered in much of the materials science literature, the behavior of solutions is different.) In her doctoral dissertation, Zhou obtained definitive mathematical results that imply the following: For any value of the contact angle y and f o r  any liquid volume V greater than or  equal t o  a critical value Vi(y) ,  a unique stable liquid bridge exists between two parallel plates of given separa- tion. It is known that any equilibrium bridge must be rotationally symmetric [6], [8] and that its free surface is a Delaunay surface [3], [9], [lo]. For y > ~ / 2  and for a specific liquid volume V,(h) depending on the plate spacing h, the free surface is simply a portion of the surface of a sphere. For other values of the volume the Delaunay surface is different from a sphere. These results, when combined with the results for the wedge, imply that a bridge between parallel plates may change its configuration and position markedly when one of the plates is tilted, even by a small amount, or it even may cease to exist as a bridge altogether; a liquid bridge between parallel plates can behave discontinuously with respect t o  tilting of the plates. In stability studies such as [3], [6], and [7], limited to the parallel plate geometry, this liquid bridge instability with respect to plate tilt is not observed. As a specific example, consider a spherical bridge between parallel plates of spacing h. Suppose the top plate is tilted clockwise by an angle 2cr < 27 - T about a pivot line in the plate that is a distance 8 h tan Q from the symmetry axis of the bridge. Then the liquid remains an equilibrium bridge for the new tilted plate configuration, without any change in the radius of the sphere or in the bridge's position on the lower plate. However, a bridge with volume V different from V, 2 (and with the same contact angle) would change both position and shape in altering to a spherical bridge after the tilt, moving to the right for V < V, and to the left for V > V,. This is one of the phenomena we wish to study in our forthcoming experiment. OTHER CONFIGURATIONS When the conditions for a bridge in a wedge are not satisfied, liquid may assume a position as a blob in the shape of a portion of a sphere in contact with the edge, see Fig. 3. The condition for such a configuration to be possible is that Iy - n/21 5 a!. (Recall we consider here only the case 0 < 2a! < T.) Although the edge blobs have not been studied with the same mathematical completeness as have the bridges, they have been noted in [ll] and [12] and for some examples studied numerically. Our numerical computations indicate that, as for the angular bridges, the spherical edge blobs are stable, and as yet we have found no other edge blob shapes numerically. In our earlier work, which considers fluid behavior in the neighborhood of the vertex of a wedge, we have shown that if ct + y < n/2, then fluid cannot remain as a blob in the edge but must spread arbitrarily far along the edge [l], [lo]. See also [12] and the references there for a discussion of stability of liquid columns in a wedge. ANTICIPATED EXPERIMENT BEHAVIOR The liquid behavior one might expect in a physical experiment in space, based on the Laplace- Young-Gauss formulation, is summarized in Fig 4. This figure illustrates the information discussed above, based in part on mathematically rigorous results and, where these are not available, on computational evidence for particular cases. The numerical solutions depicted in Fig. 4 were ob- tained using the Surface Evolver software package. The computations were carried out with initial approximations and transitions between configurations similar to those in which the experiment is designed to proceed, thereby enhancing appropriateness of the numerically based predictions on uniqueness and stability. The upper two rows of Fig. 4 depict the nonwetting case y > n/2: A liquid bridge between parallel plates is convex (part of a sphere for a specific fluid volume). Spherical tubular bridges and edge blobs exist for tilted plates, for the range of values indicated. Edge spread is not possible. For fixed y > n/2, transition from tubular bridges to edge blobs occurs as a! increases through the value 7 - n/2. For the wetting case y < n/2, a liquid bridge between parallel plates is concave. A tubular bridge between tilted plates is not possible, but the (spherical) edge blob and edge spread are. For fixed y < n/2, the transition from edge blob to unbounded edge spread occurs as a! decreases through the value n/2 - y. Computed edge blobs are shown (from different viewing perspectives) for the case a! = 25", y = 100' in the second row and for a! = 20°, y = 75' in the bottom row. The planned experiment will explore the transition between the configurations for a nonwetting and for a wetting fluid. As discussed above, when initially parallel plates are tilted, the fluid is predicted to behave discontinuously in general, the exception being the special case of a spherical bridge and a particular pivot line. The other transitions, horizontally across the second and fourth rows of Fig. 4 as a! changes value, are gradual, as can be demonstrated by the explicit spherical solutions. 3 CONCLUDING REMARXS We have described fluid behavior predicted mathematically and computationally for the forth- coming Angular Liquid Bridge investigation on board the Mir 23 / NASA 4 Mission. The predic- tions, which include discontinuous behavior, are based on the idealized classical Young-Laplace- Gauss formulation. In the experiment there will be an opportunity to check the predictions against physical behavior and to observe the effects of hysteresis and other phenomena not included in the classical formulation. ACKNOWLEDGMENTS We wish to thank Victor Brady for carrying out the numerical computations reported here and for preparing the graphical output shown in Fig. 4. We wish also to thank John McCuan for helpful conversations and to thank Hans Mittelmann for providing us with some of the results of his numerical experiments. This work was supported in part by the National Aeronautics and Space Administration under Grant NCC3-329, by the National Science Foundation under Grants DMS-9400778 and DMS-9401167, and by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, Department of Energy, under Contract Number DEAC03-76SF00098. 1 2 3 4 5 6 7 8 9 10. 11. 12. . .  :. , 1. 148-154. Finn, R.: Equilibrium Capillary Surfaces, Springer-Verlag, New York, 1986. Russian transla- tion (with Appendix by H.C. Wente), Mir Publishers, 1988. McCuan, J.: Symmetry via Spherical Reflection and Spanning Drops in a Wedge, doctoral dissertation, Dept. of Mathematics, Stanford Univ., Stanford, CA, 1995. Zhou, L.: The Stability Criterion for Liquid Bridges, doctoral dissertation, Dept. of Mathe- matics, Stanford Univ., Stanford, CA, 1995. Brakke, K.: Surface Evolver, software package and manual available via the Internet from http://www.geom.umn.edu/software/download/evolver.html . Athanassenas, M.: A variational problem for constant mean curvature surfaces with free bound- ary, J. reine angew. Math., 377 (1987), 97-107. Vogel, T. I.: Stability of a liquid drop trapped between two parallel planes, SIAM J. Appl. Math., 49 (1987), 516-525. Langbein, D.: Stability of liquid bridges between parallel plates, Microgravity Sci. Technol., 5 Wente, H. C.: The symmetry of sessile and pendant drops, Pacific J. Math., 88 (1980), 387-397. Delaunay, C. E.: Sur la surface de rkvolution dont la courbure moyenne est constante, J. Math. Pures Appl., 6 (1841), 309-315. Concus, P. and Finn, R.: On capillary free surfaces in the absence of gravity, Acta Math., 132 Mittelmann, H. D. and Hornung U.: Symmetric capillary surfaces in a cube, Report LBL- 31850, Lawrence Berkeley Lab., Univ. of Calif., Berkeley, CA, 1992. Langbein, D.: Liquid surfaces in polyhedral containers, Microgravity Sci. Technol., 8 (1995), (1992), 2-11. (1974), 177-198. 4 Figure 1. Tubular bridge in a wedge. Figure 2. Bridge between parallel plates. 2a Figure 3. Edge blob 5 Spherical bridge y - a > n / 2  NONWElTING LIQUIDS (y > n / 2) Bridge between parallel plates Edge spread not possible Edge blob y - a < n / 2  WETTING LIQUIDS (ye n / 2) Bridge between parallel plates Wedge bridge not possible Edge blob y + a > 7 c / 2  Figure 4. Fluid configurations. Upper two rows: nonwetting liquids; lower two rows: wetting liquids. . 6 

Equilibrium fluid interface behavior under low- and zero-gravity conditions. II

NASA Technical Reports Server

EQUILIBRIUM FLUID INTERFACE BEHAVIOR
UNDER LOW- AND ZERO-GRAVITY CONDITIONS. II
Paul Concus
Lawrence Berkeley Laboratory and Department of Mathematics
University of California
Berkeley, CA 94720
Robert Finn
Department of Mathematics
Stanford University
Stanford, CA 94305
I •
ABSTRACT
The mathematical basis for the forthcoming Angular Liquid Bridge investigation on board Mir
is described. The anticipated liquid behavior used in the apparatus design is illustrated.
INTRODUCTION
We describe here recent mathematical results that form the basis of our forthcoming space
experiment, developed jointly with Mark Weislogel of NASA Lewis Research Center, which is
scheduled for the Glovebox on the Mir 23/NASA 4 Mission in December, 1996. Our mathematical
work is based on the classical Young-Laplace-Gauss formulation for an equilibrium free surface
of liquid partly filling a container or otherwise in contact with solid support surfaces. In this
formulation, when gravity is absent or can be neglected, which is the situation we consider here,
the mechanical energy E of the system is given by
E = a(S - S" cos _,). (1)
The interracial liquid-vapor surface tension parameter a and the relative adhesion coefficient cos
of the liquid with the container walls are assumed to depend only on the material properties, which
are taken here to be homogeneous (the same value of cos 7 on all parts of the container, as will be
the case for the experiment). S and S* are, respectively, the areas of the liquid-vapor free surface
and of the solid-liquid interface.
Equilibrium configurations are those providing stationary values of the energy functional E
subject to the condition of fixed liquid volume [1]. The equilibrium liquid-vapor free surfaces
so determined are surfaces of constant mean curvature meeting the bounding walls with contact
angle V. We consider here values of the contact angle 0 < "7 < lr. Of particular interest in
our mathematical studies are situations in which small changes in contact angle or geometry can
result in large changes, possibly discontinuous, of the equilibrium fluid configuration. Impetus for
the present experiment arises largely from recent doctoral dissertations of two students associated
with our study, John McCuan [2] and Lianmin Zhou [3], from whose contrasting results striking
inferences can be drawn.
649
https://ntrs.nasa.gov/search.jsp?R=19970000459 2020-06-18T00:06:13+00:00Z
ANGULAR LIQUID BRIDGE
In his work, McCuan found conditions under which an equilibrium tubular bridge in a wedge
domain (Fig. 1) would be possible in zero gravity, and he gave the shape such a bridge might take.
This work is a completely rigorous mathematical study, based on the classical formulation.
Consider a wedge domain with opening angle 2c_, 0 < 2a < 7r. The results McCuan proved
contain the following (if the contact angles on the two sides of the wedge are different, the following
results hold if '7 on the left of the inequalities is their average):
I]'7 > 7r/2 + el, a bridge in the shape of a portion of a sphere making contact angle "7 with the
walls exists.
If'7 _- r/2 + a, no physically realizable bridge is possible.
It has not yet been proved whether or not other shape bridges may be possible when
'7 > lr/2 + a, or whether the spherical bridges axe stable (provide a local minimum for the energy).
However, our numerical results and those of H. Mittelmann (private communication), obtained us-
ing the Surface Evolver software package [4], indicate that the spherical bridges are stable, at least
for the representative cases we considered. Also, no bridge shapes other than the sphere have been
found numerically. Note that McCuan's results imply that a bridge is possible only for "7 > 7r/2.
A spherical liquid bridge is shown in Fig. 4 for the case c_ = 25 °, '7 = 130 °.
BRIDGE BETWEEN PARALLEL PLATES - DISCONTINUOUS BEHAVIOR
The above results for liquid bridges in a wedge compare in a remarkable way with those for
bridges between parallel plates (Fig. 2). This latter problem was studied initially from a rigorous
mathematical point of view by Athanassenas [5] and by Vogel [6], and later using a more physical
approach by Langbein [7]. (Note that in these papers, as is the case in [3] and here, the boundary
conditions at the plates are prescribed contact angle, which arises from the variational condition for
(1). For fixed end conditions, as considered in much of the materials science literature, the behavior
of solutions is different.) In her doctoral dissertation, Zhou obtained definitive mathematical results
that imply the following:
For any value of the contact angle 7 and for any liquid volume V greater than or equal to a
critical value Vo('7), a unique stable liquid bridge exists between two parallel plates of given separa-
tion.
It is known that any equilibrium bridge must be rotationally symmetric [6], [8] and that its
free surface is a Delaunay surface [3], [9], [10]. For '7 > 7r/2 and for a specific liquid volume Vs(h)
depending on the plate spacing h, the free surface is simply a portion of the surface of a sphere.
For other values of the volume the Delaunay surface is different from a sphere.
These results, when combined with the results for the wedge, imply that a bridge between
parallel plates may change its configuration and position markedly when one of the plates is tilted,
even by a small amount, or it even may cease to exist as a bridge altogether; a liquid bridge between
parallel plates can behave discontinuously with respect to tilting of the plates. In stability studies
such as [3], [6], and [7], limited to the parallel plate geometry, this liquid bridge instability with
respect to plate tilt is not observed.
As a specific example to illustrate the possibilities, consider the case 7 > 7r/2 and a bridge with
volume V_ between parallel plates of spacing h, so that the bridge is spherical. Suppose the top
plate is tilted clockwise by an angle 2c_ < 27 - n about a pivot line in the plate that is a distance
½h tan a from the symmetry axis of the bridge. Then this particular bridge remains an equilibrium
one for the new tilted plate configuration, without any change m the radius of the sphere or in the
650
bridge's position on the lower plate. However, a bridge with any volume V different from V_ (and
with the same contact angle) would change both position and shape discontinuously in altering to
a spherical bridge in conjunction with the tilt, shifting to the right for V < V_ or to the left for
V>Vs.
For -y _< _/2 an initial bridge would always behave discontinuously with respect to the tilt,
regardless of volume, as it cannot persist as a bridge. It has to be expected that the liquid will jump
to the edge of the plates in this case. If the tilted plates touch forming a wedge, then configurations
described in the following section may form. The above phenomena are ones we wish to study in
our forthcoming experiment.
OTHER CONFIGURATIONS
When the conditions for a bridge in a wedge are not satfsfied, liquid may assume a position
as a blob in the shape of a portion of a sphere in contact with the edge, see Fig. 3. The condition
for such a configuration to be possible is that h' - rr/2] _< c_. (Recall we consider here only the
case 0 < 2a < 7r.) Although the edge blobs have not been studied with the same mathematical
completeness as have the bridges, they have been noted in [11] and [12] and for some examples
studied numerically. Our numerical computations indicate that, as for the angular bridges, the
spherical edge blobs are stable, and as yet we have found no other edge blob shapes numerically.
In our earlier work, which considers fluid behavior in the neighborhood of the vertex of a
wedge, we have shown that if c_+ _/< 7r/2, then fluid cannot remain as a blob in the edge but must
spread arbitrarily far along the edge [1], [10]. See also [12] and the references there for a discussion
of stability of liquid colunms in a wedge.
ANTICIPATED EXPERIMENT BEHAVIOR
The liquid behavior one might expect in a physical experiment in space, based on the Laplace-
Young-Gauss formulation, is summarized in Fig 4. This figure illustrates the information discussed
above, based in part on mathematically rigorous results and, where these are not available, on
computational evidence for particular cases. The numerical solutions depicted in Fig. 4 were ob-
tained using the Surface Evolver software package. The computations were carried out with initial
approximations and transitions between configurations similar to those in which the experiment is
designed to proceed, thereby enhancing appropriateness of the numerically based predictions on
uniqueness and stability.
The upper two rows of Fig. 4 depict the nonwetting case 7 > 7r/2: A liquid bridge between
parallel plates is convex (part of a sphere for a specific fluid volume). Spherical tubular bridges
and edge blobs exist for tilted plates, for the range of values indicated. Edge spread is not possible.
For fixed _, > 7r/2, transition from tubular bridges to edge blobs occurs as ct increases through the
value _/- 7c/2.
For the wetting case 7 < 7r/2, a liquid bridge between parallel plates is concave. A tubular
bridge between tilted plates is not possible, but the (spherical) edge blob and edge spread are.
For fixed 3' < 7r/2, the transition from edge blob to unbounded edge spread occurs as cr decreases
through the value 7r/2 - 7. Computed edge blobs are shown (from different viewing perspectives)
for the case c_ = 25 ° , "_ = 100 ° in the second row and for ct = 20 °, 3' = 75° in the bottom row.
The planned experiment will explore the transition between the configurations for a nonwetting
and for a wetting fluid. As discussed above, when initially parallel plates are. tilted, the fluid is
predicted to 1)ehave discontinuously in general, the exception being the special case of a spherical
651
bridge and a particular pivot line. The other transitions, horizontally across the second and fourth
rows of Fig. 4 as c_ changes value, are gradual, as can be demonstrated by the explicit spherical
solutions.
CONCLUDING REMARKS
We have described fluid behavior predicted mathematically and computationally for the forth-
coming Angular Liquid Bridge investigation on board the Mir 23/NASA 4 Mission. The predictions,
which include discontinuous behavior, are based on the classical Young-Laplace-Gauss formulation.
In the experiment there will be an opportunity to check the predictions against physical behavior
and to observe the effects of hysteresis and other phenomena not included in the classical formula-
tion.
ACKNOWLEDGMENTS
We wish to thank Victor Brady for carrying out the numerical computations and for preparing
the graphical output shown in Fig. 4. We wish also to thank John McCuan for helpful conver-
sations and to thank Hans Mittelmann for providing us with some of the results of his numerical
experiments. This work was supported in part by the National Aeronautics and Space Adminis-
tration under Grant NCC3-329, by the National Science Foundation under Grants DMS-9400778
and DMS-9401167, and by the Applied Mathematical Sciences Subprogram of the Office of Energy
Research, Department of Energy, under Contract Number DE-AC03-76SF00098.
REFERENCES
1. Finn, R.: Equilibrium Capillary Surfaces, Springer-Verlag, New York, 1986. Russian transla-
tion (with Appendix by H.C. Wente), Mir Publishers, 1988.
2. McCuan, J.: Symmetry via Spherical Reflection and Spanning Drops in a Wedge, doctoral
dissertation, Dept. of Mathematics, Stanford Univ., Stanford, CA, 1995.
3. Zhou, L.: The Stability Criterion for Liquid Bridges, doctoral dissertation, Dept. of Mathe-
matics, Stanford Univ., Stanford, CA, 1995.
4. Brakke, K.: Surface Evolver, software package and manual available via the Internet from
http ://www.geom. umn.edu/software/download/evolver, html
5. Athanassenas, M.: A variational problem for constant mean curvature surfaces with free bound-
ary, J. reine angew. Math., 377 (1987), 97-107.
6. Vogel, T. I.: Stability of a liquid drop trapped between two parallel planes, SIAM J. Appl.
Math., 49 (1987), 516-525.
7. Langbein, D.: Stability of liquid bridges between parallel plates, Microgravity Sci. Technol., 5
(1992), 2-11.
8. Wente, H. C.: Ttle symmetry of sessile and pendant drops, Pacific J. Math., 88 (1980), 387 397.
9. Delaunay, C. E.: Sur la surface de r_volution dont la courbure moyenne est constante, J. Math.
Pures Apph, 6 (1841), 309--315.
10. Concus, P. and Finn, R.: On capillary free surfaces in the absence of gravity, Acta Math., 132
(1974), 177-198.
652
11. Mittelmann, H. D. and Hornung U.: Symmetric capillary surfaces in a cube, Report LBL-
31850, Lawrence Berkeley Lab., Univ. of Calif., Berkeley, CA, 1992.
12. Langbein, D.: Liquid surfaces in polyhedral containers, Microgravity Sci. Technol., 8 (1995),
148-154.
Figure 1. Tubular bridge in a wedge.
Figure 2. Bridge between parallel plates.
2c_
Figure 3. Edge blob.
653
NONWETTING LIQUIDS (7 > = / 2)
Bridge between parallel plates
Edge spread
not possible
Spherical bridge
y-(z>=/2
Edge blob
-/- c(<=/2
WETTING LIQUIDS (y < _ / 2)
Wedge bridge
not possible
Bridge between parallel plates
Edge blob
-,{+ o_> rt / 2
/
Edge spread
¥+(x_<_/2
Figure 4. Fluid ¢:onfigurations. Ut_p('r two rows: nonw(,tting liqui,l,_: h_w_r tw(, rows: wetting liquids.
654


Equilibrium Fluid Interface Behavior Under Low- and Zero-Gravity Conditions

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Equilibrium Fluid Interface Behavior Under Low- and Zero-Gravity Conditions

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