A numerical study of a drop on a vertical wall

A Numerical Study of a Drop on a Vertical Wall 1 F. M I L I N A Z Z O * AND M A R V I N S H I N B R O T t *Royal Roads Military College, FMO Victoria, Victoria, British Columbia, Canada; and tDepartment of Mathematics, University of Victoria, Victoria, British Columbia, Canada V8W 2 Y2

Received October 10, 1986;accepted January 21, 1987 The shape of a drop on a vertical plane is studied numerically, under the hypothesisthat the resistance forces are sufficient to keep the wetted area unchanged as Bond number increases from zero. The aim is to see, first, if this hypothesisis consistent and, second, whether bifurcation can occur at certain Bond numbers. It is shownthat there is no bifurcation, that increasingBond number sufficientlyalwaysproduces a change in wetted area, but that it is not inconsistent to take the wetted area as fixed for small enough Bond number.

© 1988 Academic Press, Inc.

I. INTRODUCTION The problem of the shape of a raindrop on a windowpane is intriguing, and surprisingly difficult. First of all, it may be altogether impossible for a drop to stick to a clean, smooth, vertical plane. (See (3, 4), for a discussion.) It seems to us that although such a possibility must be considered, the view that it describes reality is academic, for drops do stick to ordinarily clean, ordinarily smooth panes of glass, and the problem of the shape of such drops deserves explication. But even given that a drop does stick to a vertical plane, it is still not clear how the problem is to be stated in mathematical form, for it is not certain what the appropriate boundary conditions should be. The first possibility would seem to be to treat the question as a free boundary problem, with the wetted area adjusting itself in such a way that the contact angle remains constant. This, however, is impossible, as was proved in (7, 15): the contact angle of a drop on a plane can only be constant if the plane is horizontal. Given then that the contact angle cannot remain constant, but that the drop does someResearch supported by the Natural Science and Engineering Research Council of Canada under Grant A8560. 0021-9797/88 $3.00 Copyright © 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

how stick to the surface, the simplest remaining hypothesis is that the angle somehow adjusts itself to keep the wetted area fixed. This is the tack taken, more or less explicitly, in a number of recent publications (6, 7, 11, 12) (see also (2)). In this paper, we make a systematic numerical study of the problem of a drop on a vertical plane, partly to address the question of whether the wetted area can remain fixed, and partly because of the intrinsic interest of the problem. Consider a drop on a vertical plane II, wetting a circular area of radius a. Define Bond number by the formula pga 2 B-

,

[1.1]

O"

where p is the density of the fluid in the drop, g is the acceleration due to gravity, and tr is the surface tension of the gas/drop interface. When B = 0, the surface of the drop is necessarily a spherical cap; accordingly, when B = 0, the contact angle is a constant, say q'0, a number that is determined by the given volume of the drop. Our analysis proceeds by systematically varying "Yobetween 0 and 7r and, for each value of 3'0, allowing B to increase as far as it will go. There are again several apparent possibilities as to what can happen when B is large.

254 Journal of Colloid and Interface Science, Vol. 121, No. 1, January 1988

DROP

The first and obvious one is that either the receding contact angle 3"R (at the top of the drop) goes to zero, or the advancing angle 3'A (at the bottom of the drop) goes to 7r. In either case, increasing the Bond number beyond this point causes the wetted area to change and the original hypothesis to be violated. Another possibility might be that some sort of instability occurs, possibly a bifurcation, as conjectured in (7). (See also (11, 12).) We show here that in no case does this happen, that in all cases solutions can be found for all values of B up to the point where either 3'R = 0 or 3"a = 7r. Moreover, we show that if the calculation is continued beyond this point, the surface necessarily penetrates the plane H, indicating that at this point the hypothesis that the wetted area remains unchanged is necessarily violated. A corollary is that the hypothesis of constant wetted area is consistent as long as B is less than one of the limiting values and the force holding the drop on the plane is sufficient (cf. (7)). We present also a graph of maximum Bond number vs 3'o and point out that this graph is rather close to the graph obtained by making an O(B 2) calculation and using this result to calculate the Bond number at which either 3"R = 0 or 3"A = 7r. That these results are so close supports the principal hypothesis of Finn and Shinbrot (6, 7) that the resistance force supporting a drop on a plane has the form F = -XT~p,

255

ON A VERTICAL WALL

the drop to penetrate the plane II and so introduces numerical artifacts into the results. Offsetting the origin out of II, as suggested by Lawal and Brown (12), overcomes the problem, at least partially. As for arc length continuation, this is the use of the idea introduced by Keller (10) and is particularly useful for the discovery of bifurcations. If there is a bifurcation from the branch of solutions obtained by continuing on Bond number, the use of arc length continuation will display it and, since we observe no such thing, we conclude that there is none. Our method seems to apply quite well also to drops on a plane that is not vertical. Work on this problem is proceeding, and will be reported elsewhere. 2. T H E P R O B L E M

We consider a coordinate system with the xy-plane parallel to II and the y-axis vertical. (See Fig. 1.) The distance between II and the plane z = 0 is called the offset. Suppose, as indicated in the Introduction, that the drop wets a circle of radius a in II. We begin by replacing x, y, and z by ax, ay, and az to obtain dimensionless coordinates (x, y, z), and then introduce polar coordinates (r, 0, ~p) with 0 the longitude and ~ the latitude, according to the formulas

[1.2]

a hypothesis made in (6, 7) on the dual grounds that [1.2] is consistent with both the classical Young-Laplace diagram and the principle of virtual work. Our analysis is related to that of Milinazzo (14), with two changes: the use of arc length continuation and of what we call an offset. We use polar coordinates (0, tp), with 0 the polar and ~p the azimuthal angle, and the drop is described by a function R(O, ~p) giving the distance of the surface of the drop from the origin of coordinates. In (14), the origin is taken at the center of the wetted circle. However, this choice of origin and coordinates does not allow

/

/

FIG, 1. T h e c o o r d i n a t e system. Journal of Colloid and Interface Science, Vol. 121, No. 1, January 1988

256

M!LINAZZO

AND

SHINBROT

x = r s i n 0 s i n ~p,

y = r cos 0 sin ~p,

[2.1]

/

Z = r c o s ~o.

The problem is to find a function r = R(O, ~o)describing the surface of the drop. Define Bond n u m b e r B by [1.1 ]. When B -- 0, the drop surface is a sector of a sphere, as indicated in Fig. 2, and the contact angle is constant, say, 3"0. We always choose the offset so

that the origin is at the center of the sphere formed by the surface of the drop when B - O. This choice limits the contact angle (when B # 0) to the interval (3'0 7r/2, 3"0 + 7r/2), since there necessarily is a coordinate singularity at the ends of this interval (see Fig. 3). However, this is not a restriction in our calculations, since either the receding contact angle 3"R goes to zero or the advancing angle 3"A goes to 7r before the limit 3"0 - r / 2 or 3,0 + rr/2 is encountered. Moreover, allowing the Bond n u m b e r to increase beyond the critical value where the drop penetrates the plane shows that our numerical scheme resolves the solution very well even near the values 3"0 - 7r/2 and 3"0 + ~-/2 where the coordinate singularities occur. (See the discussion below on Fig. 37.) When B - 0, the surface of the drop is given by the formula r = csc 3"0, 0 ~< 0 ~< 2~-, 0 ~< ~p

3

/

ro

FIG. 3. R e s t r i c t i o n o n c o n t a c t angles.

~< 3'0, in the coordinate system just defined. Let D denote the wetted area when B = 0: D = {(0,~o): 0 < 0 ~ 27r, 0 < ~o~<~,o}. Since, by assumption, the wetted area remains fixed when B changes, the function R(O, ~o) is always defined on D. O u r problem can be stated as finding a function R making stationary the functional

I= f f {R[(R2 + R2)sin2~o+ R2]l/2 D

+ B R 4 c o s 0 sinZ~o)dOd~p. [2.2] Z

II

The first term in [2.2] represents the surface energy, the second, the gravitational energy of the drop. On the boundary of D, we require (Fig. 2)

R(O, 3"0) = csc 3"0.

[2.3]

We also assume the volume of the drop to be given, so that R satisfies

f f R3sin ~dO&o= 3 V +~r cot 3"o,

[2.4]

D

FIG, 2. B o n d n u m b e r zero. Journal of Colloid and Interface Science, Vol. 121, No. 1, January 1988

where V is the preassigned dimensionless volume. The second term on the right of [2.4] is

DROP

the volume of the cone subtended at the origin by the domain D in the offset plane. The Euler-Lagrange equation associated with the problem ofextremizing [2.2] with the side condition [2.4] becomes, after a little simplification, ~ {[(R2 + R~'sin2~ -1 R2)sin2~ + R2],/2j

3

Ro

2R sin2~o [(R2 + R2)sin2~ + R 2] 1/2 + BR2cos 0 sin2~o+ #R sin ~.

[2.5]

Here, ~ is the Lagrange multiplier associated with the volume constraint [2.4]. We want to solve the partial differential equation [2.5] along with the boundary condition [2.3]. 3. S O L U T I O N

The computation starts from Bond number zero, where the solution is known, and continues to positive Bond numbers using the pseudo-arc length method introduced by Keller (10). The computations were done with from 15 to 21 modes in the direction of 0 and from 71 to 99 grid points in ~p. The results using these different grids are indistinguishable. 4. R E S U L T S A N D C O N C L U S I O N S

] - O0 - - {[(Rc2 q- R2)s-~n2qoq- R2] 1/2}

-

257

ON A VERTICAL WALL

OF THE PROBLEM

An approximation to the solution of the problem [2.5], [2.3], [2.4] is obtained by making the equations discrete using a combination of finite difference and pseudo-spectral estimates on a uniform grid. Centered finite differences are used to approximate derivatives with respect to the azimuthal angle ¢. The solution R is of course an even function of 0. This fact is used to expand R in a cosine series in 0. Derivatives with respect to 0 are approximated using a pseudo-spectral method. The coordinate singularity at ¢ = 0 is handled by taking the value of R at ¢ = 0 to be the average of R over the first row of grid points adjacent to the singularity. The volume constraint [2.4] is approximated by using the trapezoidal rule in both 0 and ¢. For a more detailed description of the algorithm, see (14). The resulting system of nonlinear algebraic equations is solved iteratively using Newton's method, with the Jacobian of the differenced equations calculated exactly. When the sum of the absolute values of the correction and the right-hand sides is less than 10 -7, the iteration is terminated.

When B = 0, the drop surface is a sector of a sphere. Accordingly, the contact angle, which is the angle between the drop surface and the plane 1I, is a constant, which we call 3,o. The relation between the given volume V and % is easily computed to be V= 7r(2 + cos 3,0)(1 - cos 3,0)2 3 sin370

[3.1]

It is more convenient to treat 3'0 as given, instead of V. To aid the reader in conversion to V, we append Table I, which shows V as a function of 3,0. We remind the reader that V is dimensionless. The actual volume of the drop is obtained by multiplying the value in the table by a 3. When B > 0, the contact angle, which we denote by % is no longer constant, but it is a function of 0, the angle around the boundary of the circle D. The most useful outputs of the TABLE I 3'0 vs V(3"0) Vo (°)

v

0 15 30 45 60 75 90 105 120 135 150 165 180

0 0.2080 0.4310 0.6879 1.008 1.442 2.094 3.206 5.441 11.16 33.08 241.4

JournalofColloidandInterfaceScience,Vol. 121,No. 1, January 1988

258

MILINAZZO AND SHINBROT

c o m p u t a t i o n described in Section 3 are the values o f the receding angle '¥R and the advancing angle 3'A as a function o f B o n d n u m ber. These results are presented in Figs. 4 - 1 4 for values o f 3'0 b e t w e e n 15 and 165 ° . T h e

m a i n conclusion to be drawn from the figures is that essentially in all cases either 3'g is zero or q~h is 7r w h e n B is large enough. The only exceptions to this rule are the curves associated with 3'0 o f 75 and 90 ° where,

180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150-

150-

120-

120 'to = 1 5 °

90-

90

60-

60

30-

30

o.'25

o.5o

o~5

1.()0

0.25 0.50 0.75 1.001.251.501.75 2 . 0 0 2 . 2 5 2.50 2.75

BOND N U M B E R (B) BOND NUMBER (B)

"7 v s . B

"7 v s . B

FIG. 4. 3"vs B (3"0= 15°)7. 3' v s B (3"0 =

FIG.

180 ...................................................................

60°)-

180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150150120-

'70 =

30 °

90-

o

120-

"70

90-

60-

J

30-

60_ ~

-

~

30-

~

0.;,5 o.&o o.V5 i.'00 1.'25 1.~o 1.75 2.bo

0.25 0 . ; 0

BOND NUMBER (B)

0.-15 1.C)0 1.'~5 1 . ; 0

I.~'5 2.£)0 2.75

BOND NUMBER (B)

q vs.B

"7 vs, B

FIe. 5.3" vs B (3"0= 30°).

FIG. 8. 3" VS B

(3'5 = 75°).

180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

..................................................... 150-

150

120120

"7o =

45

o 9

0

-

~

'7o

~

90°

90 7

60-

~

.

60

30-

~

30 i i ~ , , , . . . . 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

~/ v s . B BOND NUMBER (B)

FIG. 6. 3' vs B (3'0 = 45°). Journal of Colloid and Interface Science, Vol. 121, No. 1, January 1988

o.~5 o.~o o.~5

1.oo

BOND NUMBER (B) "-/ vs. B

FIG. 9.3" vs B (3"0= 90°).

1~5

1Ao

DROP ON A VERTICAL WALL 180

...............................................

180

7--

150

150-

120

120"

~o2~

~/

60-

90" •Vo

60"

"~"7 o =

259

=

150 o

105 ° 30"

30-

0.605 o.61o 0.0%

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 BOND NUMBER (B)

0.620

0.0'25

BOND NUMBER (B)

~/ v s . B

~/ v s . B

FIG. 10. 3' vs B (3"0= 105°) -

FIG. 13.3' vs B (3"0= 150°).

180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150-

112015080[ ~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120 90 t

90 60

"70

=

165 o

60 "7 °

=

120 o

80

30

0.65 0.~0 0.1'5 o.~o 0.~5 o,&o 0.~5 0.40 0.4t5

0.0602 0.0'004 o.obo8 0.0'008 0.0;10 BOND NUMBER (S)

BOND NUMBER (B)

7 vs.B

q, v s . B

FIG. 14. 3' vs B (3'0 = 165°).

FIG. 11. 3, vs B (~'o = 1 2 0 % ~8o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3150

_x. 3

120

+ : 74 =O ° ~4

290

80

~'o

~

135

o

: *&=180 o

x

= o.

X

3 ow

x

30-

x

0.;2

0.64

O.(}6

0.68

O.10

O. 12

O.114

BOND NUMBER (B) "7 vs. B

x

15

30

45

60 75 "7 o

90 1 0 5 1 2 0 1 3 5 1 5 0 1 6 5 1 8 0

(degrees)

FIG. 12. 3, VS B ('to = 1 3 5 %

FIG. 15. Maximum bond number.

however, 'X/Rand 3'A c o m e to w i t h i n 5 o o f the limiting values. T h e difficulty here is the proxi m i t y o f the critical values of'YRand 3'A to the coordinate singularity that occurs at 3'0 7r/2. This is discussed further below.

A n important d a t u m is the value o f the B o n d n u m b e r at w h i c h the c o m p u t a t i o n ends. This can be read off o f Figs. 4 - 1 4 and is plotted in Fig. 15. For the smaller values o f 3'0, the receding angle goes to zero. For the larger ValJournal of Colloid and Interface Science,

V o l . 1 2 1 , N o . 1, J a n u a r y

1988

260

MILINAZZO AND SHINBROT 1.0

1.0-

B=O. B=.612 B= 1.30 B= 1.88

0.5~ 7o

=

0.5-

~ ~ . . ~

30 °

COS~ 3'0

9'0

6~0

0

120

150

180

-0.5-

COS"/ 30

B=0.

90

60

120~180

B=.734

-0.5 cos

7

vs.

0

cOS "y

-1.0

vs.

-1.£

FIG. 16. COS3` VS O(3`0 = 30°) •

FIG. 19. COS3` vs 0 (3`0 = 75°) •

1.0

0.5

0.5

,,

cos

cos 30

90

60

O

120

I

1 5 0 ~ 1 8 0- " B=2.45

30

60

0

90

90 o~ -0.5

o61 cos

q

vs,

, 150

,e-B= O. 180

~ - ' ~

B =.515

~120

7°

. o8o

O -1.0

-1.£

~B=1.46

FIG. 20. cos 3' vs 8 (3'0 = 90°).

FIG. 17. COS3` VS ~ (3`0 = 45°) 1.0" 1.0 ¸

0.5 ~

~

0,5"

B= O. B = .765 cOS

c08~

i

30

,

i

60

0o

1 o\X1 o

-1

7o =

"~ 30

oB-1,7

o

13=0. ~~ '

0.5" -0.5 ¸

~

cos

q

vs.

~

B = .241

B= 2.30 B= 2.60

-1.0-

105°

8

cos

-1.0

q

vs.

0

~'~---~

~~.B=.825

B= .454 B=.626 t3= .754 B=.874

FIG. 18. COS3` vS O(3`0 = 60°) •

FIG. 21. cos 3` vs 0 (3`o = 105°) .

u e s o f 3"0 the a d v a n c i n g angle goes to 7r. A n interesting fact is that t h e r e c e d i n g angle goes to zero a n d t h e a d v a n c i n g angle to 7r s i m u l t a n e o u s l y w h e n 3"0 = 7 5 ° , a p p r o x i m a t e l y . T h e largest v a l u e o f B o n d n u m b e r is a b o u t 2 . 6 5 a n d occurs at a v a l u e o f 3"0 s o m e w h a t less t h a n 60 ° . T h e v a l u e s of-), for other v a l u e s o f 0 are also o f interest a n d are p l o t t e d in Figs. 1 6 - 2 4 for v a l u e s o f 3"0 f r o m 30 to 150 °. ( T h e curves as-

sociated w i t h 3'0 o f 15 a n d 165 ° are o m i t t e d since the v a r i a t i o n w i t h 0 is so slight in these cases.) T h e shape o f the drop can also be calculated. T h e i n t e r s e c t i o n o f the d r o p surface w i t h the p l a n e o f s y m m e t r y (x = 0) is s h o w n in Figs. 2 5 - 3 5 . T o see w h a t h a p p e n s w h e n B o n d n u m b e r is a l l o w e d to e x t e n d b e y o n d the critical o n e , w e a d d Figs. 36 a n d 37. Figure 36 is typical o f w h a t h a p p e n s for the s m a l l e r v a l u e s

Journal of Colloid and Interface Science,

Vol.121,No.1,January1988

lO

DROP ON A VERTICAL WALL

0.5

'70 =

261

I

120 o

~B=O. ~B=.499 ~ B

oo,~ ~ \ \ s ' 0

0 Go

1~o

: 1,05

~o

-0.5

=. •997

FIG. 22. cos 3"vs 0 (~o = 120% q'o =

1.0-

o

FIG. 25. Drop cross section (3"0= 15°).

0.5-

•7o : COS"/

15

3'0

o'o

~o

~o

135 o

~&o

~&o

-0•5-

-1.0"

FIG. 23. COS3"vs 0 (3"0= 135°). 1.0-

: 1.30

R=1.88 0.5=

15Q o

9b O 1~o

15o

~o cos

"7

3'0

e'o

1~o

-0.5B=O. B-.007tl ~ B : . O 162 B =.0245

-1.(

"70 :

30 o

FIG. 24. cos 3' vs 0 (3'0 = 150°).

FIG. 26. Drop cross section (3"o= 30°).

o f 3'0: increasing B b e y o n d the critical value forces 3"R to turn negative, so the drop penetrates the plane. Similarly, Fig. 37 shows what happens in a typical situation for the larger values o f 3"o. The m o s t important conclusion to be drawn from these figures is that in all cases for Bond numbers larger than the critical ones, the drop penetrates the plane.

Figure 37 illustrates another point o f interest. The straight line portion o f the graph, between the points marked A and B in the figure, is due to the coordinate singularity. The straight line occurs exactly at 3' = 3'o + 7r/2, which is where the singularity occurs, It can be seen that the drop is resolved very well right up to the singularity. Journal of Colloid and Interface Science, Vol. 121, No. 1, January 1988

~

~

B= .932

t ~-~-B= ~

B

O.= . 5 1 5

1.81

=

2.45

--B=1.40

.......................~

........................

"7o

=

-~='-~-L----

45 o ~/o

=

90°

-B=2,09

FIG. 30. D r o p cross section ('to = 90°).

FIG. 27. D r o p cross section (3'o = 45°).

L4,

B=0.

B= S-.765

•3, °

__/B=.241

=

60

1

°

q~o

=

105

i

FIG. 31. D r o p cross section (3`0 = 105°) •

FIG. 28. D r o p cross section (3'0 = 60°).

B 2-,34 ~ B

.................... ~

B-O. B-O. = .0997

= 1,99

13 = . 4 1 3

~ - ~ : o ~ ....

i

r

"7o

=

I i i

75 o

=

120

o

FIG. 32. D r o p cross section (3,0 = 120°).

FIG. 29. D r o p cross section (3,0 = 75°). 262 Journal of Colloid and Interface Science, Vol. l 21, No. l, January 1988

")'o

i i

~ ,

~

=

'B=/3294 ~

~

B

~ , ~ B ~

B= .612

-.0711

/

=.107

B

/

=.138

......i!

~B= 1.30 B= 1.88

..........................

E= 2,58 B= 3.14

"fo =

135

= 3.97

o i

FIG. 36. The shifted drop (small 3'o).

FIG. 33. Drop cross section ('go = 135°) •

B=O, .B = .00711

"/o =

150 o

PIG. 34. Drop cross section (3'0 = 150°). FIG. 37. The shifted drop (large 3'0).

O. B= .000449 . ~ R=.000890

=E 2

z

m

E=

15 30 45 60 75 90 105120135 150165 180 "/o =

165 °

o [degrees)

F;IG. 38. M a x i m u m bond n u m b e r via second-order calculations.

FIG. 35. Drop cross section (3'o = 165°). 263

Journal of Colloid and Interface Science, Vol. 121, No. 1, January 1988

264

MILINAZZO AND SHINBROT

More i m p o r t a n t than this, however, is the m a i n conclusion that for B o n d n u m b e r s in excess o f the critical values, the drop necessarily moves, either the wetted area changing or the entire drop r u n n i n g off the plane. For the smaller values o f "to, the top o f the drop moves down. F o r the larger values o f "t0, the b o t t o m o f the drop in a sense rolls, and the drop elongates. 2 F r o m Figs. 25-35, we also infer that, at all values of"to, it is consistent to assume that the wetted area remains a disk for small B. These predictions, that the top o f the drop moves d o w n when "to is small, but that the wetted area remains constant (as long as the resistance force is sufficient to keep the drop on the plane) can be verified and are subject to experiment. We k n o w o f no experiments that have been performed on a vertical plane, although s o m e work has been d o n e with nonhorizontal planes (1, 5, 8, 9, 13, 16). Finally, we note Fig. 38, which superposes on the critical B o n d n u m b e r in Fig. 15, the critical B o n d n u m b e r calculated by m e a n s o f a nonsingular O(B 2) calculation (see (7)). It can be seen that the error is never m o r e than a b o u t 12% and is usually m u c h less. Figure 38 corrects a figure in (7) based on a calculation in (14), m a d e without an offset. The lack o f offset did not allow penetration o f the plane and p r o d u c e d a numerical artifact, resulting 2 It should be added that the drop shapes for those Bond numbers for which the drop penetrates the plane are not to be relied on a representing the shape of the drop, since the volume of the drop lying above the plane in these cases is no longer the given volume V.

JournalofColloidandInterfaceScience,Vol.121,No. 1,January1988

in rather larger critical B o n d n u m b e r s than those shown in Figs. 15 a n d 38. T h e close a p p r o x i m a t i o n o f the curves in Fig. 38 to the calculated critical Bond numbers reinforces the hypothesis [1.2] o n the resistance exerted on the drop by the supporting plane II. We refer the reader to (7) for details o f this. REFERENCES 1. Ablett, R., Philos. Mag. 46, 244 (1923). 2. Brown, R. A., Orr, F. M., Jr., and Scriven, L. E., J. Colloid Interface Sci. 73, 76 (1980). 3. Dussan, E. B., V, and Chow, R. T. P., J. FluidMech. 137, 1 (1983). 4. Dussan, E. B., V, J. Fluid Mech. 151, l (1985). 5. Elliott, G. E. P., and Riddiford, A. C., J. Colloid Interface Sci. 23, 389 (1967). 6. Finn, R., and Shinbrot, M., J. Math. Anal, Appl. 123, 1 (1987). 7. Finn, R., and Shinbrot, M., Math. Methods Appl. Sci., in press. 8. Johnson, R. E., Jr., Dettre, R. H., and Brandreth, D. A., J. Colloid Interface Sci. 62, 201 (1977). 9. Jones, W. R., Contact angle and surface tension measurement of a 5-ring polyphenyl ether, NASA TM86927, May 1985. 10. Keller, H. B., in "Recent Advances in Numerical Analysis" (C. de Boor and G. Golub, Eds.), pp. 73-94. Academic Press, New York, 1978. 11. Lawal, A., and Brown, R. A., J. Colloid Interface Sei. 89, 332 (1982). 12. Lawal, A., and Brown, R. A. J. Colloid Interface Sci. 89, 346 (1982). 13. MacDougall, G., and Ockrent, C., Proc. R. Soc. London Ser. A 180, 151 (1980). 14. Milinazzo, F., Math. Methods Appl. Sci., in press. [Appendix] 15. Shinbrot, M., J. Math. Methods Appl. Sci. 7, 383 (1985). 16. Yarnold, G. D., and Mason, B. J., Proc. Phys. Soc. London B 52, 125 (1949).