ORNL-TM-3528.txt

 

ORNL-TM-13528

Contract No. W-TL05-eng-26

Reactor Division

SOLUTION OF THE EQUATION DESCRIBING THE INTERFACE BETWEEN TWO

FLUIDS FOR THE VOLUME AND PRESSURE WITHIN ATTACHED,

SESSILE SHAPED, BUBBLES AND DROPS

J. W. Cocke

AUGUST 1971

 

 

This report was prepared as an accoun
t o

sgonsm_'ed by the United States Government %e‘;(})x‘::-
é e qutgd States nor the United States Atomi.c Energy
th{;l;rrlmlsslon, nor any of their employees, nor any of
makesco:tractors, subcontractors, or their employees,
mmax l-a y warranty, express or implied, or assumes any
eg iability or responsibility for the accuracy, com-
pleteness or usefulness of any information app':n‘atus
product of process disclosed, or represents ’that its use’
would not infringe ptivately owned rights,

 

OAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee

Operated by

UNION CARBIDE CORPORATION

for the

U. S. ATOMIC ENERGY COMMISSION
iii

CONTENTS
A.bs-tract . . - . * . a - - - . - L] - » - - . . *
Introduction « o« ¢« ¢ & v 4 4 ¢ o o o & o s s e .

Derivation of the Interfacial Equation . . .
Numerical Solution of the Interfacial Equation .
Computer Program . . « « « o & o o o &+ & &
Solution of the Differential Equation . .
Solution for h/a and V/a® versus B and ¢ . .
Solution for h/a and V/a® versus B and r/a .
Solution for h/a and V/a® versus r/a . . . .
Range, Running Time, and Accuracy . . .
Results « ¢ v & ¢ v v v v v v 6 v 0 o 4 o
Discussion of Results . « + 4 ¢ « ¢ & « o
Estimation of the Accuracy . . « . . .

Comparison with Previous Results . . . . . .

ConclusionsS v ¢ ¢ « o o e 2 o o o o o o o s o
References .« v v v ¢ 4 6 o ¢ o 2 s o o « o o
Nomenclature . o o o & o & o o s o s o o o o o 4

AppendiCesS o o ¢« ¢« 4 4 4 s e e s s s s 4 e e e s
A, Parametric Crossplots . . . « . « « . .
B. Tabulated Results « ¢« « ¢ ¢ « o o « « &

C. Computer Program « « « + o « o« o o « « &

10

10

LT
17
21
21
22
25
27

43
SOLUTION OF THE EQUATION DESCRIBING THE INTERFACE BETWEEN TWO
FLUIDS FOR THE VOLUME AND PRESSURE WITHIN ATTACHED,
SESSILE SHAPED, BUBBLES AND DROPS

J. W. Cooke

ABSTRACT

A numerical computer program was written to solve the equa-
tion describing the interface between two immiscible fluids to
obtain the shape, size, volume, and pressure of attached bubbles
and droplets. These relationships are important to the study
of three-phase heat transfer, superheat, critical constants,
and interfacial energies. Previous solutions have been obtained
with limited accuracy for a restricted number and range of vari-
ables. The present results are given in both graphical and
tabular form for a wide range and number of parameters, and the
computer program is included so that an even broader range and
number of variables, as well as specific values, can be obtained
as required. In particular, an expression for the maximum-
bubble-pressure was derived which is considerably more accurate
over a wider range than previous expressions.

Keywords. Surface tension, interfacial tension, bubbles,
drops, contact angle, maximum-bubble-pressure, filuid-vapor in-
terface.

INTRODUCTION

A knowledge of relationships among volume, pressure, shape, and size
of attached bubbles and droplets are important in the study of boiling
and condensing heat transfer, superheat and critical point phenomena,
mass transfer studies, and in the measurement of contact angles and sur-

% These relationships can be obtained from the solution

face tension.'-
of the second-order, nonlinear differential equation describing the inter-
face between two fluids. This equation has been solved by perturbation
analysis for very small bubble sizes by Rayleigh® and Schrddinger,® by
numerical hand calculations for a wide range of discrete drop shapes by
Bashforth and Adams,7 and by numerical computer calculations for droplet
volumes and heights by Baumeister and Hamill.® However, the Rayleigh-
Schrodinger solution is no longer used extensively because of concern

regarding its accuracy; and the Bashforth and Adams (as well as the

1
Baumeister and Hamill) solutions were obtained for a limited number and

range of variables.

In order to check the accuracy and to extend the usefulness of the
above solutions, a numerical computer program was written to solve the
interfacial equation. The subject program solves the interfacial equation
for any value of the dimensionless shape parameter, B, for positive values
of the term 87 (sessile drops or above-attached bufibles)* and extends the
previous sclutions for ¢ > 180 degrees. The solution not only provides
the dimensionless X, 7, ¢ coordinates describing the profiles of the inter-
face, but also gives the bubble volume within and the pressure difference
across the interface as a function of profile shape and, in addition, the
maximim pressure difference across the interface for a given radius of
attachment (which is required for the calculation of the surface tension

by the maximum-bubble-pressure technique).

This report descrites the derivation of the interfacial equation,
its numerical solution, and the computer program employed. The re-
sults and an estimation of their accuracy are presented and compared

with the previous solutions.

DERIVATION OF THE INTERFACIAL EQUATION

At the interface separating two immiscible fluids (usually a liquid
and its vapor), a force imbalance exists which results in an interfacial
tension. The tension is a force per unit length pulling uniformly in all
directions tangent to the interface., Neglecting gravitational forces, a
force balance on an infinitesimal segment of a free surface is shown in
Fig. l1-a. The surface is bowed by a uniform differential pressure, P,
where r; and r, are the principal radii of curvature. The force balance
would thus be:

AZ X

P Az =2y | X — + Az —— , (1)
21‘1 21'2

 

*
The case for negative values of the term BZ (pendant drops or
below-attached bubbles) will be presented in a later report.
ORNL-DWG 7{- 7609

yAX

PAZAX

 

 

LIQUID

 

 

BUBBLE (8)

Fig. 1. Schematic View of a Sessile-Type Drop and Bubble
and the Force Balance on an Infinitesimal Surface Element.
or

P = ‘Y(L + }"—) . (2)

If gravitational forces are present, the influence of the dif-
ferential fluid density, p, must be considered. For an interface assumed
To be symmetrical about an axis of revolution, the force balance equation

for the bubble shown in Fig. 1-b will be

1 1 Pg
P_y(——+—-—)——(A—Z)=O s (3)

ry Ta g

c
where p = Pp, — Py At the orgin, ry = rz3 = b and thus
2y g
P=—+A— . (4)
b =

Furthermore, with r; = x/sin ¢, Eq. (3) becomes

2y  pgz sin ¢ 1
——+-———y< +-—-—)=o ! (5)
b g X To

 

c
A similar derivation for the drop shown in Fig. 1-b would also result
in Eq. (5). This equation can now be made dimensionless with respect to

©, and after rearranging:

 

 

1 sin ¢
- + =2+ Bz , (6)
R X
where .
gpb
B = (7)
Y&,
and
To X z
R =— " X ==, and Z2 = — .
b b b

Equation (6) can be transposed into Cartesian coordinates by

substituting
 

 

1 d*7/dx®
R [1+ (az/ax)*]1¥ =
and
. dz/ax
sin = -
[1 + (az/ax)® ¥~
to obtain
a2z, Az az azel¥ e
— + l+(—-—- — = (2 + 82) 1+<—-—-) , (8)
axe dx Xdx ax

which is a second-order nonlinear differential equation, with boundary

conditions:
X =0 3 2z =0
X =0 ; 4dz/XaX =1

Although Eq. (8) cannot be solved analytically in terms of ordinary

functions, its numerical solution is described in the next section.

NUMERICAL SOLUTICN OF THE INTERFACTAI, EQUATION

Equation (6) can be rearranged to read

1
R = (9)
2 + BZ — (sin ¢)/X

 

and by definition:

ag

— =1

dg

dX

— =Rcos ¢ =F (X, Z, @) , (10)
dg

dz

— B R Sin¢ = G (X, Z, ¢) - (ll)
d¢

As long as BZ > 0, R will always be finite,.
A numerical technique of fourth-order accuracy developed by Runge-
Kutta” was selected to solve the set of simultaneous Egs. (9), (10),
and (11); the iterative equations decribing this technique are:

1

X . =X +Z (k, + 2k + 2kp + ka) + O(08)° (12)

1

Zas =Bt 7 (mg + 2my + 2mp + my) + 0(ag)® (13)

o
I
e
+
8

“

n+l

where

O n gt
1 1 1
ky = 8 F(X, + =k, 2 +—m,, ¢ +—=09) ,
2 2 2
1 1 1
kg = M@ F(X, +—ky, 2 +—m, ¢ +—208) ,
2 2 2

m =A¢G(Xn+;—ko, zn+;mo, q)+-2-a¢) ,

1 1 1
mp = AP G(X, +—ky, Dyt —m, ¢ +—08)
2 2 D

mg = AP G(Xn + Ka, Zn +my, ¢+ OP)

and where the symbol 0(Ag)° represents a term which is small, of the
order (Ag)®, when Ap is small.

Equations (12) and (13) are of a form that can be readily trans-

formed into a computer program.
COMPUTER PROGRAM

The computer program for the solution of the interfacial equation
and for the calculation of the various output parameters is listed in
Appendix C. The program consisted of four parts which are discussed

below.

Solution of the Differential Equation. Two subroutines in both

 

single and double precision and consisting of four iteration loops were
written to solve Egs. (12) and (13). The subroutines RHOS and RHOD
calculate the values of R and supply the values of the functions F and
G to the subroutines RUNGKS and RUNGKD. These latter subroutines cal-
culate the values of the coefficients ki and m. and the new values of
X, Z, and ¢ for reintroduction in RHOS and RHOD to continue the
iterative procedure. The iteration procedure is initiated by equating

X, Z, and ¢ to zero and R to one.

Solution for h/a and V/a® versus B and ¢. The pressure and volume

 

within the attached bubble are calculated from X, Z, and ¢. The pressure

relation is given by Eg. (4) and can be simplified by using Eq. (7) and

the definition of the specific cohesion,

 

2vg
e = —= (14)
PE
to obtain
h/a = /2/B + Z.J/B/2 (15)
where
Z = Z(X = I‘) = A 3
T
and
h = ch/pg .

The volume relation can be obtained from the integration of

v
d(-—) =X az , (16)
b '

where X and dZ can be cobtained from Eq. (6). Integrating the resulting

eguation by parts gives the relationship:
- -

v TX° 2 sin ¢
- =—|2+pz-—] . (17)
b° B X

Upon substituting Eqs. (6), (14), and (15) into Eq. (17), the final

expression for the dimesionless volume 1s obtained:

Y T r h
~ == (—)(—) — sin ¢ | , (18)
a 8 a a

where
T
- =XV
a

is the dimensionless radius of attachment.

Solution for h/a and V/a® versus § and r/a. To obtain the pres-

 

gure and volume within attached bubbles for a given radius of attachment
from the numerical solutions X, Z, ¢, of the interfacial equation,
several conditional "IF" statements were required. The interrelation-
ship of h/a, 8, r/a, and ¢ are shown schematically in Fig. 2. The
iterative solution of the interfacial equation proceeds along constant

B lines for given steps of Ap. A conditional check is made to note the
intersection of the given r/a curve (denoted by L), and the values at
the point of crossing are determined by linear interpolation. ©Since

r/a is multi-valued, an additional check is necessary to note when the
increasing values of r/a start to decrease at ¢ = nn/2, n =1, 5 ...
(denoted by O), or when decreasing values start at ¢ = nn/2, n = 3,

7 .... In this manner, solutions for many values of r/a can be obtained

for a given value of 3.

Solutions for h/a and V/a® versus r/a. These solutions were some-

 

what more difficult to cbtain than those described above since it was
necessary to increment B as well as @¢. Both of the conditional checks
described above for crossing a given value of r/a and a change from in-
creasing to decreasing r/a values were required as well as a check for
the change from increasing to decreasing values of h/a (denoted by A in
Fig. 2).
ORNL-DWG 71- 7610

 

h/fa —»

 

B

 

 

 

 

Fig. 2.
Between h/a,

'rr/2

¢ —»

Schematic Representation of the Relationships
B, r/a, and 3.-
10

For & given value of 8, h/a is a multi-valued function so that the
choice of increasing or decreasing B to approach H/a must be carefully
considered. Furthermore, the direction of approach to E/a along the
given value of r/a must also be carefully considered. To insure the
most trouble-free solution over the entire h/a versus ¢ field, a
decremental approach from right to left (as shown by the arrows in

Fig. 2) was chosen.

To reduce the number of iterations required to locate h/a, an esti-
mate of the value of g is calculated from equations fitted to a few
preliminary results. The initial B value is then decremented along
first a coarse grid, and finally along an extra fine grid to obtain the

final solution using double precision.

Range, Running Time, and Accuracy

 

The ranges of the computer program as presently written are
0 < ¢ < 360° 0.1 €£r/a £2.0, and 0.02 £ B £ 150; however, these can
be easily extended at some sacrifice of either the running time or
accuracy. The average running time fof the program to obtain a value
of h/a for a given value of r/a is approximately 10 seconds on the
IBM 2360-91 Computer system. The average running time for the other

programs is considerably shorter per solution.

An estimate of the accuracy (to be given later) was obtained by
comparing the values of h/a as a function of ¢ for various values of B,

ANg, and for single and double precision.

The results of the computer solution are discussed in the next

section.

RESULTS
The results in both tabular and graphical form are presented in
this section and in the Appendix.

The profile of a bubble for B8 = 0.8 is shown in Fig. 3. The pro-
file is shown extended to ¢ = 360°, which would be possible if a suitable
11

> ORNL-DWG 74-7641
{.

ol LT TN /T IN

y ) N

\
\
)

/

 

 

 

0.9

0.8

 

0.6

0.4 N

0.2
0 ——*//

{0 08 06 04 02 0 02 04 06 08 10

x/b

 

z/b

\

-
/\/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3. Profile of a Bubble with B = 0.8 for 0° € ¢ < 360°
12

form of attachment were provided, and the center of bouyancy were to

remain on the z axis.,.

A tabuletion of B, x/b, z/b, h/u, and V/a® for various values of ¢
are presented in Tables B.l through B.6 (in Appendix B). Plots of h/a
and V/a® versus ¢ for various r/a are cshown in Figs. 4 and 5, respectively.
Figure 4 cleaurly shows the attaimment of a maximum value of h/a (h/a) for
a given radius of attachment. (This is the basis of the maximum-bubble-
pressure technique® for the measurement of surface tension.) In addition,

. .o . 3
there is a minimum value of h/a as well as a maximum value for V/a®.

The values of 1i/a and the corresponding values of V/a®, B, ¢, x/b,
E/b for various values of r/a are given in Table 1, and various cross-
plots of these variables are given in Figs. A.]l through A.4 (Appendix A).
These plots show that both Eya and E& approach asymptotic values of
Vfig and 180°,* respectively. Thus, the maximum pressure difference
(h/a) that a large tube can sustain will be very nearly independent of

tube diameter.

A least-squares, polynomial fit of the computer solutions for vari-

ous formulations of h/a and r/a were made. The forumlation that gave

the best fit was
a a a r
- - = T - - ) (19)
h r h a

which is of the same form as the perturbation solutions of Rayleigh-
Schrbdinger. For this reason, Eq. (19) was fitted to the data of Table 1
by the relationship:

£(y) =i + L1y + 1)

F(y') = ; = i3 + i4y + isyz + e » (20)
Y

 

where io’ i,, and i, are the coefficients of the Rayleigh-Schrddinger
solution and i, i,, 1g, «.. were determined by the least-sguares pro-

cedure and are listed in Table 2.

 

*
These values can be obtained by the solution of Eg. (3) with
1/ry =0 and 1/b = O,
13

ORNL-DWG 71-T76t2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6
r/a =0.2 3o
3 =
N
4 \
0.3 0.2
.fl—-<
33 0.4 \.\\\\
2 _ \Q\L‘_‘
. — . N\\, 10.0
° 0 40 80 120 160 200 240 280 320 360

¢,

Fig. 4. Variation of the Dimensionless Pressure Difference, h/a,
as a Function of @. for Various Values of 8 and the Dimensionless Radius
of Attachment, r/a.
1h

ORNL-DWG 74-7613

 

0 50 100 450 200 250 300 350
#:

Fig. 5. Variation of the Dimensionless Volume, V/aa, as a Function
of ¢, for Various Values of the Dimensionless Radius of Attachment, r/a.
Table 1.

15

Radius of Attachment (r/a)

Maximum Values of the Pressure (h/a) and Corresponding
Values of the Size, Shape, and Volume of Bubbles as a
Function of the

 

 

r/a ? B x/b 2/b h/a v/a®

0.12 90 .646 0.0290810 0.995156 0.999705 8.413519 0.003651
0.1k 90.795 0.0397216 0.993412 0.998120 7.236L65 0.005806
0.16 91.4ho7 0.0521086 0.991243 1.00397 6.357323 0.008787
0.18 91.906 0.0662774 0.988791 1.00649 5.676507 0.012624
0.20 92. 411 0.082286 0.986012 1.00865 5.134652 0.017439
0.20 92.906 0.100192 0.982929 1.00988 4.693892 0.023404
0.24 93.420 0.120066 0.979526 1.0106k4 L. 328985 0.030656
0.26 93.945 0.141985 0.975814 1.01077 h.opolls 0.0392273
0.28 94,488 0.166037 0.971786 1.01039 3.761790 0.049570
0.30 95 .47k 0.192565 0.966824 1.01574 3.537926 0.062105
0.225 96,472 0.229022 0.960L16 1.01752 3.299450 0.0802973
0.350  97.476 0.269500 0.953463 1.01785 3.097815 0.101911
0.375  98.987 0.315035 0.944859 1.02327 2.,9257k4L 0.128876
0.400 100.225 0.365084 0.936222 1.02319 2.T7TTL3 0.158475
0.4b25 101.985 0.421688 0.925568 1.02746 2.6Lg595 0.197373
0.450 103.483 0.483756 0.914990 1.02640 2.538105 0.239818
0.475 104.821 0.551689 0.904402 1.02173 2. khobes 0.286173
0.50 107.164 0.631617 0.889730 1.02464 2.35527% 0.346254
0.55 111.986 0.822982 0.857398 1.02159 2.214232 0.501889
0.60 117.488 1.072133 0.819487 1.00938 2.104842 0.708170
0.65  123.990 1.407182 0.774913 0.986809 2.01991k 0.987762
0.70  130.993 1.853431 0.727151 0.950582 1.953875 1.347727
0.75 137.489 2.41%075 0.68265L4 0.9002832 1.90210k 1.768822
0.80 1Lko.66k4 2.977900 0.655617 0.853059 1.860k45  2.136163
0.90 152.491 L,841565 0.5784L48 0.742939 1.7986L9 3.270626
1.00 158.hko2 7.100181 0.530738 0.648972 1.753511 L. 356567
1.10 162.989  10.070060 0.490221 0.567348 1.718720 5.521721
1.20 165.994 13.866110 0.L455742 0.4g7922 1.690847 6.726L07
1.30 167.994  18.715220 0.42kg72 0.438376 1.667904 8.005399
1.0 169.991  25.16100 0.394711 0.385323 1.6486L40 9.386485
1.50 171.489  33.48783 0.366575 0.339160 1.632205  10.839000
1.60  172.4904  Lh,16k29 0.340486 0.299033 1.618007 12.356187
1.70  173.488 s58.14618 0.315285 0.263383 1.605608  13.971941
1.80  173.998  75.94175 0.292110 0.232455 1.564682  15.640580
1.90  174.987  99.7358 0.268978 0.204340 1.584k975  17.453834
2.00 175.489 130.1581 0.247919 0.180031 1.576295  19,314144

 
16

Table 2. Coefficients for the Polynomial Equations
Fitted to the Computer Solution

 

 

Coefficients Eq. (19) Eg. (21)
ig 1.00000 -0.00090
iq -0.66667 1.04439
ig -0.66667 -0.47175
ia 0.03230 1.43283
ig -5.52833 -4.59801
ig 61.1913k 5.38228
ig -351.38141 -2.,73720
1 1099.76625 0.51837
ig -1930.93994
ig 1913.36384
110 -1003.22519

111 216.93848

 
17

The main disadvantage of Eq. (19) is that an iterative procedure is
necessary to calculate a/h. A simpler formulation, but less accurate,

was also fitted to the computer results:

a r
- f(-—) . (21)
h a

The coefficients for the polynomial fit of Egq. (21) are also listed in

Table 2.

The accuracy of these two polynomial fits is discussed in the next

section.

DISCUSSION OF RESULTS
An estimate of the accuracy of the computer results and compari-
sons with previous results are presented in this section.

Estimation of the Accuracy. Results were obtained for h/a versus ¢

for B = 0.1 and 100.0 with three values of Agp = 1°, 1/2°, and 1/4° using

 

both single and double precision. There was no change in h/a (to the
seventh decimal place) as /Ay was decreased from 1° to 1/4° when double
precision was used for B = 0.1 and only 0.00008% change for B = 100.0.
The single precision results are plotted in Fig. 6, where the percent
difference is with reference to the double precision, A¢ = 1/4°, values.
As can be seen, an interval less than Ap = 1° decreased the accuracy of

the results because of rounding errors when single precision was used.

Except for the maximum pressure results listed in Table 1, all the
tabulated results were computed using single precision and interval
NP = 1°. The values listed in Table 1 were calculated using double pre-
cision and are good to at least the sevnth decimal place. The other

tabulated results are good to at least the fifth decimal place.

Comparison with Previous Results. As anticipated, the careful,

 

tedious hand calculations of Bashforth and Adams (which required a number
of years to complete) were in good agreement (to the fifth decimal place)
with the present computer sclution. Trouble, however, develops when the

“Bashforth and Adams tables are singly and doubly interpolated to apply
18

ORNL-DWG 74— 7644

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.005
A B
O |r——— _ .00 04
o 0.25 0.
£~
< \\\\\\, ::\\\\\k
= -0.005 \
2 \ \ 100 100
w
a
W —0.010 N N\
| '8
& \
> 0.50 100
-0.015
< \\
Ll
a
-0.020
\ 0.25 100
-0.025
0 50 100 450 200 250 300 350
¢

Fig. 6. Comparison of the Single Precision Results for h/a with the
Double Precision, Agp = 1/4, Values as a Function of ¢, Ap, and 8.
19

their results to practical calculations. Sudgen,lo "oy careful inter-
polation" of Bashforth's tables, constructed a table for calculating
surface tension by the maximum-bubble-pressure method. (This table is
the one most often referred to in current literature on surface tension.)
The percent difference between our solution and Sudgen's as a function of

r/a is shown in Fig. 7. The maximum difference is -0.1%.

Although an error of 0.1% can be neglected for some studies at
elevated temperatures (where other errors are more significant), this
magnitude of error can be significant for many measurements made at room
temperature, where theoretical studies of small changes in the molecular
structure of the interface are being conducted. Futhermore, this error
can be magnified by as much as 20 times when the two tube, differential

technique is used to measure surface tension.

To be particularly noted in Fig. 7 is that the Rayleigh-Schrddinger
solution is in better agreement with the computer solution than Sudgen's
results all the way from O € r/a < 0.45. This range of r/a covers a
large portion of the surface tension studies that have been conducted in
the past. In fact, the Rayleigh-Schrodinger equation is in error by less
than 1.05% all the way to r/a = 1.0. Thus, this much simpler analytic
solution can be used in many cases where precise surface tension values

are not needed.

Also shown in Fig. 7 are the deviations of Egs. (19) and (21) from
the computer solutions. Equation (19) agrees to within x0.05% all the
way to r/a = 1.5. The simpler Eq. (21) agrees to within +0.07% from
0.2 < r/a < 1.5; but Schrddinger's equation is recommended for r/a < 0.2.
The main disadvantages of Egs. (19) and (21) is that double precision

should be used in their solution, especially at the larger values of r/a.

Baumeister and Hamill presented their results as plots of droplet
volumes and heights as functions of droplet radii and contact angles.
In this form, their results could not be conveniently compared with the
present results. In addition, their results were given to only the

third significant figure.
20

ORNL-DWG 74-7645

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 .4
// \ 410
0.4 / ‘\ 0.9
/ ~ = RAYLEIGH-SCHRODINGER
. | / \ o SUGDEN - o8
= 03 | \ EQUATION (19) 07
z [ 1] { T ——— EQUATION (21) 1°
" \ \ 4 06
9 [ 4 |
Z 02 ‘ 0.5
i [}
o / \ ~ 0.4
z \ / AT~ ‘ 'y / /
L \ ® \ C / \
E 0 v ) ‘5/f\ \‘\"7{, .d\ * A \\ b7
o ° / ® \ / ' \ /'7, * \ /
P \ 7 \ *L
° L -~
-0.1 %o
-0.2
0 0.2 0.4 0.6 0.8 1.0 .2 1.4 1.6 1.8

r/a

Fig. 7. Deviations of Previous Sclutions and Present Polynomial
Equations from the Present Computer Solution of the Maximum-Bubble
Pressure.
21

CONCLUSIONS

A computer program was written to solve the second-order, nonlinear,
differential equation describing the axially symmetric interface between
two immiscible fluids. The present solution is limited to sessile-shaped
drops and bubbles; the solution for pendant-shaped drops and bubbles
requires a different approach and will be given in a later report. All
of the present results are accurate to at least the fifth decimal place

with some results accurate to the seventh decimal place.

The results are presented in a form that is useful in the analysis
of boiling and condensing heat transfer, superheat, critical constants,
and in the measurements of contact angles and surface tension. In addi-
tion, these results may be of use in the design of equi-stressed shells
for containment vessels (i.e., above-ground water tanks, under-water
storage of liquids and gases, lighter-than-air ballons, and submerged
marine laboratories). The computer program is listed so that a wider

range and number of variables can be obtained as desired.

REFERENCES

1. F. Kreith, Principles of Heat Transfer, pp. 308-437, International
Textbook Company, Scranton, Pennsylvania, 1lst ed., 1959.

2. J. A. Edvards and H., W. Hoffman, Superheat Correlation for Boiling
Alkali Metals, Proceedings of the Fourth International Heat Transfer
Conference, Versailles, September 1970 (under the book title '"Heat
Transfer 1970"), Elsevier Publishing Ccmpany, Amsterdam, Netherlands,

1970,

 

 

3. A. V. Grosse, The Relationship Between the Surface Tension and FEnergies
of Liquid Metals and Their Critical Temperatures, J. Inorg. Nucl. Chem.,

Ph:1l7-156 (1962).

 

4. D. W. G. White, Theory and Experiment in Methods for the Precision
Measurement of Surface Tension, Trans. ASME, 55: 757 (1962).

5. Lord Rayleigh, On the Theory of the Capillary Tube, Proc. Roy. Soc.,
92 (Series A): 184-195 (1915).

 

6. FErwin Schrddinger, Notizilber den Kapillardruck in Gasblasen, Ann.
Physik., 46: 413-418 (1915) T
T

9.

10.

0
@]

|

22

F. Bashforth and J. C. Adams, An Attempt to Test the Theories of
Capillary Action by Comparing the Theoretical and Measured Forms of
Drops of Fluids, University Press, Cambridge, Massachusetts, 1833.

 

 

K. J. Baumeister and T. D. Hamill, Liquid Drops: Numerical and
Asymptotic Solutions of their Shapes, NASA-TN-D-4779, National
Advisory Committee for Aeronautics, September 1968.

F. B. Hildebrand, Introduction to Numerical Analysis, pp. 236-239,
McGraw-Hill, New York, 1956.

S. Sugden, The Determination of Surface Tension from Maximum Pressure
in Bubbles, J. Chem. Soc., 1: 858-866 (1922).

NOMENCLATURE

distance from the origin to the plane of attachment, cm

specific cohesion 2ygc/pg, enf

radius of curvature at the vertex of the bubble, cm
local acceleration due to gravity, cm/sec”
dimensional constant, dyne.sec®/g.cm

pressure differential across interface, cm of fluid with density p

meximum value of h for given value of r/a, cm of fluid with
density p

coefficients of polynomial equations
coefficients of numerical eguations
coefficients of numerical equations
pressure differential across interface, dyne/cm?
radius of circle of attachment, cm
dimensionless radius of attachment
r; Pprincipal radii of curvature of the interface, cm
dimensionless value of ry, rg/b

volume enclosed by the interface, em®
<

23

volume enclosed by the interface for a given value of r/a at
h/a = h/a, cm®

horizontal coordinate, cm

value of x for a given value of r/a at h/a = h/a, cm
dimensionless x, x/b

value of X at x =r

independent variable in polynomial equation

vertical coordinate, cm

value of z for a given value of r/a at h/a = h/a, cm
dimensionless z, z/b

value of Zat x =1

Greek Letters

al &6 BB <

dimensionless parameter = gpbs/gcy = 2(b/a)?

value of 8 for a given r/a at h/a = h/a

interfacial tension, dyne/cm

angle between axis of revolution and the normal to the surface
value of ¢ at the radius of attachment

value of ¢ for a given r/a at h/a = h/a

positive density difference between the two fluids, g/cm3
APPENDICES
A. PARAMETRIC CROSSPLOTS

Various crossplots of the parameters shown in Figs. 4 and 5 are
given in Figs. A.l, A.2, and A.3. The maximum bubble radius as a
function of the angle qbr for various radii of attachment is shown
plotted in Fig. A.4. TFor 3., < 0%, the maximum bubble radius would
be the radius of attachment. Also shown in Fig. A.4 is the locil of
the maximum bubble radius where the maximum bubble pressure 1is

reached.

27
28

ORNL-DWG 7{— 7646

 

 

 

 

 

 

7

 

 

N

2 —~—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O 02 04 06 08 10 12 44 16 8 2.0
f/a
Fig. A.l. Dimensionless Maximum-Pressure-Difference Across the

Interface Separating Two Fluids as a Function of the Dimensionless Radius
of Attachment.
29

ORNL-DWG 74-T7617

 

80

 

70

 

60

 

50

 

 

 

30 /
20

m —
T

90 {00 #10 120 130_ 440 1150 160 4170 480
$r

 

 

 

 

 

 

 

 

 

 

 

 

4__——/

 

 

Fig. A.2. The Value of the Parameter B at h/a = h/a as a Function of ¢ .
r
30

ORNL—DWG T7i-T7648

20 r
1.5 ' ///

o 1.0 //

 

 

 

 

 

 

 

05 /

/

90 100 HO 120

 

 

 

 

 

 

 

 

 

 

 

 

{70 180

0o
130 140 {50 160

ér

Fig. A.3. The Dimensionless Radius of Attachment as a Function of
¢. at h/a = h/a.
31

ORNL-DWG 71-7619

2.2

2.0 Ofi:::f \\\\

\LOCI OF (%/0)yax

 

 

 

{.8

SN

 

7 1.0
4 / N\

/ | ‘\\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x 1.2
s /—\
= 0.6
3 ‘///,f" “\\\\\
1.0
N TN
0.8 ’ l e \
/ /03 N \
’—\
0.4 |= // //
" /
0.2 Hm=]
0
90 120 150 180 210 240 270 300 330 360

¢

Fig. A.4. Maximum Radius of a Bubble (x/a at ¢ = 90°) as a Function
of ¢_ for Various Radii of Attachment (r/a). TFor ¢_ < 90°, (x/a) = r/a.
T T ? max
B. TABULATED RESULTS

The computer results for the size, shape, pressure, and volume
of attached bubbles and drops for given radii of attachment are listed

in Table B.l through B.6. The Results are arranged in ascending values

of ¢r.

33
34

 

 

Table B.1l. ©Size, Shape, Pressure, and Volume of Bubbles and
Drops with a Given Radius of Attachment
r/a = 0.2

3

?.. x/b z/b h/a V/a
30.00 2.99 0.05168 0.00135 0.26341 0.00030
18.00 3.86 0.06667 0.00225 0.34009 0.00042
8.00 5.80 0.10000 0.00505 0.51009 0.00063
5.50 7.00 0.12061 0.00734 0.61519 0.00077
k.00 8.21 0.14142 0.01013 0.72143 0.00090
3.50 8.7 0.15119 0.01158 0.77125 0.00096
2.90 9.66 0.16609 0.01400 0.84731 0.00106
2.30 10.86 0.18650 0.01765 0.95144 0.00119
2.00 11.66 0.20000 0.02317 1.02032 0.00128
1.80 12.30 0.21082 0.02262 1.07556 0.00135
1.50 13.49 0.23094 0.02717 1.17823 0.00149
1.10 15.81 0.26968 0.03727 1.37604 0.00175
0.95 17.05 0.29019 0.04327 1.48078 0.00189
¢.80 18.63 0.31623 0.05161 1.61378 0.00207
0.66 20.59 0.34816 0.06295 1.77694 0.00229
0.58 22.0k 0.37139 0.07194 1.89569 0.00247
0.50 23.84 0.40000 0.08399 2.04200 0.00268
0.46 2L .92 0.41703 0.09166 2.12910 0.00281
0.Lh2 26.17 0.4364Y 0.10089 2.22841 0.00296
0.38 27.62 0.45883 0.11220 2.34306 0.003173
0.34 29.35 0.48507 0.12636 247745 0.00334
0.30 31.46 0.51640 0.14467 2.63801 0.00361
0.25 34.87 0.56569 0.17662 2.89087 0.00h05
0.23 36.59 0.58977 0.19386 2.01458 0.00k27
0.21 38.60 0.61721 0.21486 3.15569 0.00455
0.19 40.99 0.64889 0.24102 3.31872 0.00489
0.17 L3.01 0.68599 0.274 7k 3.51007 0.00531
0.15 47.60 0.73030 0.31993 3.73910 0.00585
0.14 49.86 0.75593 0.34889 3.87195 0.00622
0.13 52.51 0.78446 0.38410 L.02025 0.00664
0.12 55.69 0.81650 0.42796 4.,18731 0.00719
0.11 59.65 0.85280 0.48479 L.37771 0.00792
0.10 6L.87 0.89443 0.56306 4,5980L 0.00898
0.09 72.70 0.94281 0.6861L 4.85960 0.01079
0.0828 84.85 0.98295 0.88466 5.09473 0.01h4k
35

 

 

Table B.1 (Continued)

8 ¢ x/b z/b h/a v/a®
0.0828 95.17 0.98295 1.05331 5.12905 0.01878
0.09 107.58 0.94281 1.24471 4,97809 0.02660
0.10 115.80 0.89443 1.35781 L.77575 0.03445
0.11 121.43 0.85280 1.42604 L .59845 0.04173
0.12 125.82 0.81650 1.47280 b.hhy30kh 0.04890
0.13 129.46 0. 78446 1.50654 4.30642 0.05606
0.14 132.59 0.75593 1.53162 4.18487 0.06328
0.15 135434 0.73030 1.55053 L.07611 0.07058
0.17 140.07 0.68599 1.57565 3.88935 0.08542
0.19 144,07 0.64889 1.58951 3.73435 0.,10060
0.21 147.60 0.61721 1.59609 3.60326 0.11610
0.23 150.78 0.58977 1.59777 3.49067 0.13189
0.25 153.70 0.56569 1.59596 3.39269 0.14794
0.34 164 .94 0.48507 1.56461 3.07046 0.22259
0.38 169. 3k 0.45883 1.54451 2.96739 0.25668
0.h2 173.53 0.43644 1.52274 2.87998 0.29109
0.46 177.57 0.41703 1.50000 2.80452 0.3257h
0.58 189.15 0.37139 1.42956 2.62679 0.43005
0.66 196.74 0.34816 1.38256 2.53500 0.49949
0.70 200.55 0.33806 1.35938 2.40453 0.53405
0.80 210.35 0.31623 1.30143 2.404h23 0.61955
0.90 £220.90 0.2981h 1.24333 2.32476 0.70350
1.00 233.12 0.28284 1.18282 2.25059 0.78543
1.10 250.16 0.26968 1.11197 2.17306 0.86409
1.10 289.74 0.26068 1.01358 2.10009 0.85531
1.00 306.55 0.28284 1.00345 2.12376 O.77164
0.90 318.54 0.29814 1.01038 2.16850 0.68854
0.80 328.88 0.31623 1.02769 2.23111 0.60513

 
36

 

 

Table B.2. Size, Shape, Pressure, and Volume of Bubbles and
Drops with Given Radius of Attachment
r/a = 0.3
B ® x /b 2 /b h/a v/a®
30.0 4.55 0.07746 0.00307 0.27011 0.00165
18.0 5.87 0.10000 0.00509 0.34859 0.00215
12.0 7.20 0.12247 0.00764 0.42700 0.00264
8.0 8.83 0.15000 0.01145 0.52201 0.00325
5.5 10.66 0.18091 0.01672 0.63076 0.00393
4.0 12.53 0.21213 0.02303 0.73967 0.0046k4
3.5 13.41 0.22678 0.02640 0.79085 0.00496
2.9 14,77 0.2491k 0.03194 0.86891 0.00547
2.3 16.63 0.27975 0.04045 0.97588 0.00618
2.0 17.87 0. 30000 0.04663 1.04663 0.00667
1.8 18.88 0.31623 0.05197 1.10340 0.00745
1.7 19.45 0.32540 0.05512 1.13547 0.00727
1.5 20.76 0.34641 0.06271 1.20901 0.00779
1.25 22.85 0.37947 0.07579 1.32483 0.00861
1.2 234+35 0.38730 0.07908 1.35225 0.00881
1.1 24 46 0.40k452 0.08662 1.41264 0.00925
1.0 25 .7k 0.42426 0.0957h4 1.48191 0.00977
0.95 26.46 0.43529 0.10110 1.52063 0.01005
0.9 27.24 0.44721 0.1070k 1.56252 0.01038
0.8 29.05 0.47434 0.12135 1.65789 0.01113
0.7 31.28 0.50709 0.14017 1.77323 0.01206
0.66 32.32 0.52223 C.14943 1.82662 0.01250
0.58 3L.79 0.55709 0.17222 1.94970 0.01358
0.5 37.92 0.60000 0.20334 2.10167 0.01499
0.46 39.86 0.62554 0.22365 2.19240 0.01588
0.42 42,13 0.65465 0.24857 2.29609 0.01695
0.38 h4.87 0.68825 0.27994 2.41618 0.01829
0.34 48,25 0.72761 0.32082 2,55764 0.02000
0.3 52.62 0.77460 0.37669 2.72788 0.02236
0.25 60.61 0.84853 0.48659 3.00046 0.02715
0.23 65.36 0.88465 0455577 3.13731 0.03036
0.21 72 .20 0.92582 0.65884 3.29956 0.03559
0.195 81.53 0.96077 0.80327 3.45338 0.04423
37

 

 

Table B.2 (Continued)

B ¢ x/o z /b h/a v/a®
0.1925 95.40 0.96694 1.01465 3.53793 0.06203
0.21 108.40 0.92582 1.19288 3.47260 0.08758
0.23 115.90 0.88465 1.27945 3.38272 0.1086k%
0.25 121 .34 0.84853 1.33222 3.2994L 0.12796
0.3 131.17 0.77460 1.4ho212 3.12503 0.1741k4
0.34 137.09 0.7276 1.42704 3.0137h 0.21045
0.38 142.08 0.68825 1.43800 2.92095 0.24666
0.42 146,46 0.65465 1.44035 2.84223 0.28285
0.46 150.41 0.62554 1.43727 2. 77443 0.31905
0.50 154,05 0.60000 1.43052 2.71526 0.35524
0.58 160.65 0.55709 1.41014 2.61634 0.42743
0.66 166.63 0.52223 1.38465 2.53620 0.49919
0.7 169.45 0.50709 1.37088 2.50133 0.53471
0.8 176.18 0. 47434 1.33487 2.42538 0.62303
0.9 182.56 0.44721 1.29805 2.36147 0.7098%
1.0 188,74 0.42426 1.26124 2.30604 0.79517
1.1 194.79 0.4ok52 1.22489 2.25680 0.87863
1.2 200.82 0.38730 1.18910 2.21206 0.96038
1.5 219.70 0.34641 1.08403 2.09350 1.19399
1.7 234.58 0.32540 1.01218 2.01784 1.33856
1.7 304.88 0.32540 0.83893 1.85810 1.29849
1.2 337.48 0.38730 0.87818 1.97123 0.91837
1.1 343,30 0.40452 0.89730 2.01386 0.84032
1.0 349,15 0.h2k26 0.92136 2.06571 0.761k2
0.9 355.15 0.4k721 0.,95155 2.12903 0.68168

 
Table B.3.

38

Size, Shape, Pressure, and Volume of Bubbles and
Drops with Given Radius of Attachment

 

 

r/a = O.h
¢ x/b z/b h/a v/a®
30.0 6.17 0.10328 0.00548 0.27941 0.00533
18.0 7.98 0.13333 0.00912 0.36068 0.00692
12.0 9.79 0.16330 0.01372 0.44187 0.00850
8.0 12.02 0.20000 0.02063 0.54126 0.01047
5.5 14.54 0.24121 0.03019 0.65309 0.01270
4.0 17.13 0.28284 0.04173 0.76612 0.01504
3.5 18.35 0.30237 0.04787 0.81926 0.01613
2.9 20 .2k 0.33218 0.05808 0.90040 0.01786
2.3 22.86 0.37300 0.07385 1.01170 0.02030
2.0 2L ,63 0.40000 0.08546 1.08546 0.02196
1.8 26.06 0.42164 0.09548 1.14468 0.02332
1.5 28.78 0.46188 0.11589 1.25507 0.02594
1.25 21.84 0.50596 0.14108 1.3764kL 0.02897
1.1 3k.23 0.53936 0.16228 1.46875 0.03139
1.0 36.17 0.56569 0.18038 1.54176 0.03341
0.8 41.32 0.62246 0.23257 1.72823 0.03897
0.66 W6.69 0.69631 0.29255 1.90883 0.04512
0.58 50.97 0.7h278 0.34404 2.04222 0.05041
0.5 56.89 0.80000 0.41997 2.20998 0.05828
0.46 60.93 0.83L406 0. 47434 2.31263 0.06L4173
0.42 66.32 0.87287 0.54540 2.43395 0.07262
0.38 74.69 0.91766 0.66956 2.58601 0.08785
0.36 97.25 0.94281 0.98565 2.77520 0.14837
0.38 105.99 0.91766 1.09093 2.76968 0.18416
0.k2 115.44 0.87287 1.18463 2.72505 0.23502
0.46 121.96 0.83406 1.23412 2.67701 0.27950
0.5 127.16 0.80000 1.26371 2.63185 0.32143
0.58 135.45 0.74278 1.12924 2.55291 0.40169
0.66 142.15 0.69631 1.29953 2.48730 0.47925
0.7 145.12 0.67612 1.29854 2.45854 0.51723
0.8 151.82 0.63246 1.2880L4 2.39577 0.61073
0.9 157.75 0.59628 1.27074 2.34315 0.70194
1.0 163.17 0.56569 1.24977 2.2979k4 0.79124
1.1 168.21 0.53936 1.22692 2.25831 0.87839
39

 

 

Table B.3 (Continued)
¢ x/b z /b h/a V/a®
1.2 172.98 0.51640 1.20312 2.22293 0.96372
1.2 175.29 0.50596 1.19105 2.20652 1.00587
1.5 186.20 0.46188 1.13075 2.13396 1.208L46
1.7 194 .51 0.43386 1.08367 2.08375 1.36230
.0 206,81 0.40000 1.01557 2.01557 1.57998
.5 229,01 0.35777 0.90542 1.90671 1.90695
2.5 310.00 0.35777 0.714k42 1.69317 1.81377
2.0 331.06 0.40000  0.7340L 1.73k401 1.47971
1.7 32,67 0.43386 0.76316 1.78825 1.27326
1.5 250.54 0.46188 0.79225 1.84081 1.13175

 
Table B.k.

Lo

Size, Shape, Pressure, and Volume of Bubbles and
Drops with Given Radius of Attachment

 

 

r/a = 0.6
B 3. x/b z /b h/a. V/a®
60.0 6.98 0.11094 0.00647 0.21801 0.01748
30.0 9.75 0.15492 0.01266 0.3072k 0.02813
18.0 12,64 0.20000 0.02120 0.39692 0.03656
12.0 15.54 0.24495 0.03198 0.48658 0.04515
8.0 19.17 0. 30000 0.04838 0.59676 0.05604
5.5 23.35 0.36181 0.07133 0.72131 0.06879
4.0 27.72 0.42426 0.09970 0.84811 0.08255
3.5 29.83 0.45356 0.11501 0.90807 0.08935
2.9 33.16 0.49827 0.14101 1.00025 0.10029
2.3 37.95 0.55950 0.18242 1.12813 0.11668
2.0 41.31 0.60000 0.21408 1.21407 0.12865
1.8 L4 .15 0.63246 0.24231 1.28397 0.13910
1.7 45.82 0.65079 0.25954 1.32394 0.1454,
1.5 49.88 0.69282 0.30298 1.41709 0.16135
1.2 57.15 0.75895 0.38608 1.57014 0.19222
1.2 59.11 0.7T460 0.40935 1.60808 0.20112
1.1 63.92 0.80904 0.46757 1.69516 0.22420
1.0 70.86 0.84853 0.55331 1.80546 0.26115
1.0 110.89 0.84853 0.97076 2.10065 0.61473
1.1 119.62 0.80904 1.01944 2,104k 0.7h1L7
1.2 126.22 0.77460 1.04278 2.,09873 0.85283
1.2 129.07 0.75895 1.04934 2.09449 0.90534
1.5 1L0.75 0.69282 1.05581 2.06906 1.14754
1.7 148.28 0.65079 1.04488 2.04798 1.32506
1.8 151.67 0.63246 1.03679 2.37679 1.40998
2.0 157.90 0.60000 1.01784 2.01784 1.57280
2.3 166.26 0.55950 0.98579 1.98964 1.80255
3.0 182.08 0.48990 0.90845 1.92912 2.28289
3.5 193.87 0.45356  0.8557h 1.88796 2,58703
.0 204,25 0.42426 0.80597 1.84692 2.86294
6.0 258.64 0.34641 0.60326 1.62222 3.68271
6.0 281.23 0.34641 0.55639 1.54104 3.59171
4,0 332.13 0.42426 0.55032 1.48537 2.56096
2.0 251.28 0.48990 0.59863 1.54966 2.03835

 
b1

Table B.5. Size, Shape, Pressure, and Volume of Bubbles and
Drops with Given Radius of Attachment

 

 

r/a = 1.0
8 3. x/b z /b h/a v/a®
60.0 13.46 0.18257 0.01914 0.28739 0.17182
30.0 19.254 0.25820 0.03877 0.40835 0.24755
18.0 25,2k 0.33333 0.06585 0.53089 0.32807
16.0 26.91 0.35355 0.0745k4 0.56L438 0.35097
12,0 31.60 0.40825 0.10137 0.65655 0.41635
8.0 40.18 0.50000 0.15888 0.81776 0.54200
6.5 45,96 0.55470 0.20262 0.91998 0.63182
5.5 51.73 0.60302 0.24929 1.01642 0.72663
4.5 61.02 0.66667 0.32813 1.15886 0.89237
) 69.09 0.70711 0.39734 1.26903 1.05200
3.7 77.81 0.73522  0.46919 1.37338 1.24390
3.7 103.15 0.73522 0.63176 1.59450 1.95007
b,o 113.93 0.70711 0.67052 1.65536 2.32888
4.5 125.33 0.66667 0.69002 1.T70L70 2.78302
5.5 140.90 0.60302 0.68546 1.73973 3.48414
6.5 152.52 0.55470 0.66411 1.75195 L.oskhe
8.0 166.51 0.50000 0,62550 1.75100 L, 76824
10.0 181.82 0. 44721 0.57493 1.73279 5,54361
16.0 220 .40 0.35355 0. 44901 1.62354 7.13667
20.0 253,01 0.31623 0.36935 1.48421 7.66733
20.0 286.54 0.31623 0.32147 1.33281 7.19870
16.0 316.08 0.35355 0.31169 1.23514 6.05930
10.0 348,47 0.kh721 0.35006 1.22997 b o208

 
42

Teble B.6. Size, Shape, Pressure, and Volume of Bubbles and
Drops with Given Radius of Attachment

 

 

rfa = 1.5

B 2, x /b z/b h/a v/a®
100.0 21.00 0.21213 0.03058 0.35762 0.83884
80.0 23.69 0.23717 0.03859 0.40220 0.94928
60.0 27471 0.27386 0.05210 0.46793 1.11632
40.0 34.97 0.33541 0.08054 0.58381 1.42593
30.0 41.80 0.38730 0.11119 0.68884 1.72815
2L.0 48.69 0.43301 0.14478 0.79022 2.04596
20.0 56.13 0. 47434 0.18271 0.894k02 2.40o6k49
18.0 61.81 0.50000 0.21185 0.96887 2.69503
16.0 70.85 0.53033 0.25677 1.07982 3.18127
18.0 124.73 0.50000 0439672 1.5234%9 6.89629
20.0 133.98 0.4743k 0.3944k 1.56354 7.6608L
24,0 148.02 0.43301 0.38033 1.60617 8.85724
20.0 163.84 0.38730 0.35409 1.62959 10.20717
50.0 200.61 0.30000 0.2791k4 1.59568 12.93770
6L .0 222,48 0.26517 0.23868 1.52697 13.97587
80.0 255.04 0.23717 0.19292 1.37824 14 .29483
80.0 284 .50 0.23717 0.16827 1.22238 13.20281
6k.0 313.1k 0.26517 0.16132 1.08935 11.13856
50.0 330.62 0.30000 0.16778 1.03892 9.65580
32.0 354,65 0.37500 0.19654 1.03617 7.76366

 
C. COMPUTER PROGRAM

The Fortran listing of the computer program for solving the inter-
facial equation and computing the various parameters presented in this
report is given in Table C.1. The ranges of ¢, B, and r/a can be
easlly extended at some sacrifice of either the running time or the
accuracy. Other parameters, such as maximum drop height and radius as
a function of drop volume and contact angle, can be obtained either by

modifying one of the MAIN subroutines or by adding another routine.

43
Ly

Table C.1., Computer Program

 

O

OO

SOLUTION DF INTERFACIAL EQ. FOR THE PRESS. AND VOL.
WITHIN ATTACHED BUBBLE. CASE 1-ATTACHED ABOVE (POSITIVE BETA)

IMPLICIT REAL*B (D) 4REAL*4(A-Cy E-H,40-2)
DIMENSIONA{(4) sB{4)3Cl4)sD(4)4E(8)R{4),G1(10}),4,P(4)
DIMENSIONDA(4) 4 DB(4)4DC(4)4DZ(4)4DE(8)4DP(4),DR(4)
9 FORMAT (I1,12)
11 READ 9 ,41,4N3
IF (I-1)401,510,501

SOLUTION AT MAX ,PRESS FOR GIVEN VALUES OF R/A(=G)

300 FORMAT(F15.7)
10 CONTINUE

DO 201 N&=1,N3
READ300,+6

ESTIMATE OF INITIAL BETA
IF(GeGE«0.099 .AND.G.LT.0.4)G0T01001
IF(GeGE«0+39.AND.G«LT.0.8)G0TO1002
IF(GCeGEW0e79 e ANDG oL T«3.01)G0T0O1003

1001 BETA=411175-,92434%G+3.92521%G%% 2

GOTO301

1002 BETA=2,87267-12.78143%GC+16.40021%G**x2,
GOTO301

1003 BETA=16 43044349 443188%G+40454628 %G 32,
GOTO301

C COURSE BETA GRID

301 BETA=BETA+O041%BETA
N20=0
P(2)=0
58 BETA=BETA-.O01=*BETA
CALL MAINS(A4BsCsZ 9EsH sJ4sP +BETA,G4R)
IF(P{1)-P(21))59,459,60
60 P(2)=P (1)
N20=N20+1
GOT0O58
53 TF(N20-1)301,301,57
57 PRINT 50,+BETAsH
PRINT 100
PRINT 2004,E(8)4E(6)4E(TIWR{1)4P(1),V
BETA=BETA+.,02*BETA
P(2)=0

C FINE BETA GRID

65 BETA=BETA-.O001*BETA
CALL MAINS(AyBsCyZsEsHyJyP+BETAGHR)
IF(P(1)-P (21161461462

62 P{2)=P (1)
GOTO65
C

OO0

OO,

45
Teble C.1 (Continued)

61 PRINT 504+BETA,H
PRINT 100
PRINT 2004E(8)yE(6)sE(T)sR{1)4P (1)4V
BETA=BETA+.002%BETA
bP(2)=0
EXTRA FINE BETA GRID WITH DOUBLE PRECI SION
DBETA=BETA
DG =G
66 DBETA=DBETA-.0001*DBETA
CALL MAIND(DA+DB+DC+DZ +DEsDH,J,DP,DBETA,DG,DR)
TF{DP (1) -DP (2)1634634,64
64 DP{2)=DP (1)
GOTO66
63 DV=3414159265%DG #*{DP (1) =*DG =DSIN(DE(5)}))
DE(4)=DE(1)%57 ,29577957
DE(8)=DE(5)=57 .29577957
PRINT 50,DBETA,DH
50 FORMAT(IHZ2 46HPETA =43E15,7 46X y3HH =,E15,7///)
100 FORMAT(1HO 95X 9 3HPHI #8X ¢ 3HX /By 13X 3 3HZ /By 13X, 3HR/ Ay 13X, 3HH/ A,
113X 44HV/AZ)
200 FORMAT(1IH ,0P1lF943,1P5E16 .7}
PRINT100
PRINTZ200,DE(8B),DE(6)}¢DE(7)+sDGC,DP (1) ,DV
201 CONTINUE
GOTO 11

SOLUTION FOR GIVEN VALUES CF BETA AND R/A(=G1)

400 FORMAT(B8F10.7)
401 CONTINUE

DO 202 N4=1,N3

READ 4004BETA,(G1(1),1=1,8)

C,’—\LL ”AINIS(A,B,C 72 7E 7H 9J ’P ,BETA,G].,R )
202 COCNTIMUE

GOTO 11

SOLUTION FOR GIVEN VALUES OF BETA AT INTERVALS (N1) OF PHI
UP TO N2

501 CONTINUE

DO 203 N4=1,4N3

READ 5004+BETAyNLI,N2
500 FORMAT(F1074214)

CALL MATINZ2S(A4BosCyZ yEsH 9 J 4P +sBETASZG4R yN1,N2)
203 CONTINUE

GOTO 11

END
L6

Table C.1 (Continued)

 

O D

O~

150

SUBROUTINE MAINS{A4ByCosZ yEgH 43 J9eP ¢BETA,G4R)

IMPLICIT REAL*8 (D) 4REAL*4(A-Cy E=Hy0~7)
DIMENSTONA(4)yB(4)yC{4) 2 {4)yE(8) 4P (4)4R(4)
H=0.1745329E-01

E(6)=0.

J=3

E{(1)=0,.

E(2)=0.

E(32)=0,

Nl=1

R(2)=0.

DO150M=14360

CALL RUNGKS(AyRsCosZ yEyH s JyBETA4G,P)
R{1)=E(2):SORT(BETA/2.,0)
GOTO(445)4N1

IFIGR(1))543,3

TF{(R{1)-R{2))8B,48,10

IF(GR(1))146,7

N1=2

R{2)=R (1)

E(5)=E(1)

E{6)=FE(2)

E(T)=E(3)

CONTINUE

F={G-R({(2)})Y/(R(1)-R(2))
E(B)=F==(E(1)-E(5))+E(5)

(6 )=F3({Z{2)~FE(6))+E(6)

LT ) =Fa(E(23)~E(7)Y)+E(T7)
P(1)=SCRTI(?/EETAY+E(T7)*=SQRT(BETA/2.)
GOTOS

P(1)=1.0

P{(2)1=2.0

R ETUR N

END

SUBROUTIME MAIND(DA,DByDCyDZ 4DEsDH +JsDPy DBETA, DG4 DR)

IMPLICIT REAL*B (D) REAL*4(A~-CyE-H,0-2)
DIMENSTONDA(4) ,DB(4),DC(4),DZ (4),DE(B)4sDP(4)43DR(4)
DH=0.2726646D=02

DE(6)=0.

J=3

DE(1)=0.

DE(2)=0.

DE(3)=0.

N1=1
b7

Table C.1 (Continued)

 

OO,

DO150M=1,720

CALL RUNGKD(DAWDByDCyDZ4DEyDHy JyDBEETALZ DG4 DP)
DR{1I=DE(2)*DSART(DBETA/2.0)

GOTO(445)4N1

IF(DG-DR({1}))5,150,150

TF(DG=DR (1)) 146,7

DR{Z)=DR (1)

DE(5)=DE(1)

DE(6)Y=DEL(2)

DELT7)Y=DE(3)

N1=2

CONTINUE

DF=(DG-DR (2))/(DR(1)-DR (2))
DE(5)=DF*{DE{1)-DE(5))+DE(5)
DE(6)Y=DF*={DE(2)-DE(6))+DE(6)
DE(7)=DF*{DE(3)-DE(7))+DE(T7)
DP(1)=DSQRT(2./DBETA)+DE(T)==DSQRTI(DBETA/2.)
RETURN

END

SUBROUTINE MAINIS{A,BsCyZ yE4H J4PBETA,G1,R)

IMPLICIT REAL=8 (D) REAL*4{A-CyE-H,0-7Z)
DIMENSIONA(G) ,B(4),C(4),Z(4),E(8)4P(4)4R(4),G1(10)
H=(6726646E-02

E{&6)=0.

OO0 OO0
. 4 9 e

n o #n

Zmmm>Pc

=
b
—

I=1

DO 150 M=1,720

CALL RUNGKS(AsByCsZ+EsHyJyBETALZG1,4P)
R{1)=E{z2)*SQRT(BETA/2.0)
GOTO(445422)4M1
ITF(GI(IY-R{1))T7 46,3
IF(RT1)-R{2))10,10,414C
I=I-1

M2=-1

hl=2
IF(GIIII-RI(1)}20 46,7
TF(RI1}Y-R(21)140421,21
I=1+1
L3

Table C.1 (Continued)

 

Oy Oy O

O

272 N2=1
N1=3
IF(GI(I)-R (1117 464140
T F={(GI(I)-R(2))/(R{1)R(2))
E(L)=F={E(L)-E(5))}+E(5)
E(2)=F*x{E(2)-E(6))+E(6)
E(3)=F=(E(3)}-E(T7)})+E(T)
6 P{1)=SQRTI{2./BETA)+E(3)*SQRT(BETA/2.)
V=3,14159265%G 1(I )= (P (1}*G1{I)=SIN(E(1)))
El4a)= E(1)%57 429577957
50 FORMAT(1HZ y6HBETA =4yE15.7 46X,y 3HH =yE15.7///)
PRINT 504BETAH
PRINT 100
PRINT 200,E(4)yE(2),E(3)sG1(I)4P(1),V
200 FORMAT(1H 40P1F9.3,1P5E16,7)
100 FORMAT(1HO,5Xy3HPHI 38X ¢3HX /By 13X y3HZ /By 13X43HR/A,13X43HH/A,
113X 44HV/A3)
I=1+NZ2
IF(I-8}130,130,3
130 TF(I-11)941404140C
140 R{2)=R (1)
E(5)=E(1)
E(6)=E(2)
E(7)=E(3)
150 COMTINUE
S RETURN
END

SUBROUTINE MAIN2S(As4ByCoeZ sEgH 9 J9yP +BETA,G4RyN1,N2)

IMPLICIT REAL®B (D) yREAL*4{A-CyEH ,0-2)
DIMENSTIONA(L) 3B (4) 3C(4)9Z(4),E(8)4P(4) 4R (4)
101 FORMAT(1HQ 35X y3HPHI 48X 33HX /By 13X y3HZ /By 13Xy 3HR/ Ay 13X 3HH/ A,
112X 44HV/A3)
50 FORMAT(1IH]1 sSHBETA=9E15.7 96X 92HH=yEL5.7///)
200 FORMAT(1H s0P1F9.341P5E16.7)
H=0s1745229E-01
PRINT 504BETA,H
PRINT 101
E(6)=0.
J=3
N2=N2/N1
M3=,1746E-01/H
M1=N2%N1
E(1)=0.
E(2)=0.
E(3)=0.
DO 150 HM=14N2
OO O

OO0

100

150

20

50
30

60

49

Teble C.1 (Continued)

DO 100 N=1,N1

CALL RUNGKS(A4ByCoZsEsHyJyBETALG,P)
CONTINUE

G=E(2)*SQRT(BETA/2.,0)

E(6)=E(2)

E(7)=E(3)
P(1)=SQRT(2./BETAY+E(T)*SQRT(BETA/2.)
V=3e1415933G*(P(1)*G-SIN(E(1)))
E(4)=E(1)%57,29577957

PRINT 2004+E(4)4E(6)4E(T)4G4P(1),V
CONTINUE

RETURN

END

SUBROUTINERUNGKD(DA,DB,DC,DZ ,DE,DH,J,DBETA,DG,DP)

IMPLICIT REAL*B (D) sREAL*4 (A-CyE-H,0-Z)
DIMENSIONDA(4)4DB(4)9DC(4),DZ(4)4DE(8)4DP(4),DR(4)
DA(1)=DH*0.16666666667
DA(2)=DH*0433333333333

DA(3)=DA(2)

DA(4)=DA(]1)

DB(4)=0.

DB(1)=DH*>0e5

DB(2)=DB(1)

DB (3}=DH

DO2CI=1,J

DC(I)=0.

DZ(I1)=DE(I)

DO30K=144

CALL RHOD{(DA,DB,DC,DZ yDE,DH,J,DBETA,DG,DP)
DZ{li=1.

DO501 =1,J

DC(I }=DC(I)+DA(K)=*=DZ (I )
DZ(I)=DE(I)+DB(K)=*DZ (I}

CONTINUE

DC(1)=DH

DO60I =144

DE(I)=DE(I}+DCI(I)

RETURN

END

SUBRGUTINE RHOD (DA,DByDC+DZ4DE,DH,J+DBETA,DG,DP)

IMPLICIT REAL*8B(D)ysREAL*4 (A=CyE-H,0-Z)
DIMENSTONDA(4) 4DB(4)4DC(4)yDZ(4)4DE(8),DP(4),DR(4)
50

Table C.1 (Continued)

 

OO

OO OO

™

70

75
76

20

10

75
16

IF(DZ(1))70475470

DRH=140/{2.0+DBETA=DZ (3)~DSIN(DZ(1))/DZ(2))
GOTOT76

DRH=140

DZ(2)=DRH=*DCOS(DZ (1))

DZ (3)=DRH*DSIN(DZ (1))

RETURN

END

SUBROUTINERUNGKS(AsByCsZ yEsH 9y J+BETA,G,P)

IMPLICIT REAL®B (D) yREAL*4(A-Cy E-H,0-2)
DIMENSIONA(4) 9B(&) o Cl4) 9Z {4) 3 E(8)4P (4) 4R (4)
All)=H*0.16666666667

A(2)=H*0433333333333

A(3)=A(2)

Al4)=A(1)

B{4)=0.

B(1l)=H*0.5

B(2)=B(1)

B{3)=H

DOZ201=14J

C(I}Y=0.

Z({I)=E(I)

DO30K=1,+4

CALLRHOS (A 4B 4CeZ yE9H s J9BETALG,4P)

Z(1l)=1.
DO50I=1,4
CII)=CIlI }Y+A(K)=Z{
Z(I)=E(I)+B(K)=Z |
CONTINUE

Cl1)=H

DO60CI =1,4J
E(IY=E(I)+C(I)
RETURN

END

I)
1)

SUBROUTINERHOS({AyBaCyZ sEsH s J4BETA,G4P)

IMPLICIT REAL*8 (D} sREAL*4(A-CyE-H,0-2)
DIMENSIONA(4) 9yB{4)3Cl4) 32 (4),E{8) 4P (4) 4R (4)
IF(Z(1))70475,470
RH=1.0/(2.,0+BETAXZ(3)-SIN(Z (1)) /Z(2))
GATO76

RH=1.0
Z(2)=RH=*=COS{(Z (1)
Z(3)=RH=SIN(Z (1)
RETURN

END

)
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