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MEASUREMENTS O F DRAG COEFFICIENTS

FOR FALLING AND RISING SPHERES IN F R E E MOTION

T h e s i s b y

A . W e r n e r P r e u k s c h a t

In P a r t i a l Fulfillment of the R e q u i r e m e n t s F o r the D e g r e e of

Aeronautical Engineer

California Institute of Technology P a s a d e n a , California

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ACKNOWLEDGEMENTS

The author wishes to e x p r e s s h i s deep gratitude to P r o f e s s o r H. W. Liepmann under whose e x p e r t guidance the p r e s e n t work was p e r f o r m e d and f o r his advice and the m a n y helpful suggestions, and to P r o f e s s o r s Millikan and Zukoski, m e m b e r s of h i s supervising committee.

The author would a l s o like to acknowledge his indebtedness to P r o f e s s o r Wu f o r placing t h e t e s t f a c i l i t i e s i n the Hydrodynamics L a b o r a t o r y a t the author's disposal.

Thanks a r e a l s o extended to M r s . Geraldine K r e n t l e r f o r typing the m a n u s c r i p t and to the staff of the Guggenheim Aeronautical L a b o r a t o r y and the Hydrodynamics L a b o r a t o r y f o r t h e i r kind help with the e x p e r i m ents

.

(3)

ABSTRACT

The purpose of this t h e s i s is to investigate the d r a g coefficient

of s p h e r e s i n f r e e motion, falling and r i s i n g , i n w a t e r .

Following the introduction the t e s t set-up, s p h e r e r e l e a s e m e c h a n i s m , s p h e r e s and timing device and r e c o r d e r a r e d e s c r i b e d .

Section 3 gives the r e s u l t s of the d r a g coefficient m e a s u r e m e n t s f o r f r e e falling s p h e r e s which show good a g r e e m e n t with the known m e a s u r e m e n t s , quoted, f o r instance, i n the Handbuch d e r E x p e r i - m entalphysik.

Section M e a l s with the m e a s u r e m e n t of d r a g coefficients of f r e e - r i s i n g s p h e r e s . It was found t h a t the f r e e l y r i s i n g s p h e r e s move

i n a n o s c i l l a t o r y path of which wavelength and amplitude depend on the r a t i o of s p h e r e density and w a t e r density.

T h e l o c a l d r a g coefficient of the s p h e r e s was m e a s u r e d to be the s a m e a s f o r falling s p h e r e s . It was found to b e independent of the s p h e r e motion.

No c r i t i c a l Reynolds n u m b e r was found f o r the onset of the o s c i l l a t o r y motion of the s p h e r e . The oscilPatory motion a p p e a r e d to be independent of initial d i s t u r b a n c e s of the s p h e r e motion.

F r o m photographs of the s p h e r e paths Strouhal n u m b e r s w e r e f o r m e d which a r e about one t h i r d the value given f o r c i r c u l a r c y l i n d e r s i n the s a m e Reynolds number range.

(4)

oscillatory f o r c e coefficient on a c i r c u l a r cylinder.

(5)

TABLE O F CONTENTS

Acknowledgements Abstract

Table of Contents L i s t of Symbols 1. Introduction 2 . Equipment

2. 1. T e s t Set-up

2. 2. Sphere Release Mec&ai?ism 2 . 3 . Spheres

2 . 4 . Timing Device and Recorder 3 . Drag Measurements of F r e e -Falling Spheres

3 . l . General Remarks 3 . 2 . Results

3 . 3 . Discussion of Results 3 . 4 . Conclusion

4. Drag Measurements of Free-Rising Spheres 4.1. General Remarks

4. 2. Spheres Rising i n 6-inch Diameter Tube 4. 2. 1. Results

4. 2. 2 , Discussion of Results 4. 2. 3 . Conclusion

4 . 3 . Spheres Rising i n L a r g e Water Tank

(6)

References Appendix A Appendix B 'Tables F i g u r e s

v

TABLE

OF

CONTENTS (Contd. )

PAGE

4 . 3 . Spheres Rising in Large Water Tank (contd. )

4 . 3 . P. General Remarks 17

4 . 3 . 2. Results 18

4. 3. 3 . Discussion of Results 19

4 . 3 . 4 . Summary 2 3

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LIST

O F SYMBOLS

r a d i u s of s p h e r e

amplitude of s p h e r e path buoyant f o r c e

d r a g coefficient

d r a g coefficient, quoted by the Handbuch d e r Experimentalphysik ( 11)

l a t e r a l o s c i l l a t o r y f o r c e coefficient d i a m e t e r of s p h e r e

d i a m e t e r of t e s t tube function (Appendix IS) i n e r t i a f o r c e

gravitational constant ( c . g. s ) function (Appendix

B)

function (Appendix B)

p r e s s u r e f o r c e

wall c o r r e c t i o n f a c t o r total m a s s

frequency

f o r c e (Appendix B) Reynolds n u m b e r

l o c a l d i r e c t i o n of s p h e r e motion Strouhal n u m b e r

tirn e

(8)

S u b s c r i p t s m e a s MS o S U v w vii

LIST

O F

SYMBOLS

(Contd. )

velocity velocity

coordinate i n v-direction (Appendix

B)

weight

horizontal coordinate of s p h e r e motion v e r t i c a l coordinate of s p h e r e motion angle of s p h e r e path with v e r t i c a l z angle (Appendix B)

weight density, g r . wt/crn 3 wavelength of s p h e r e path kinematic viscosity

m a s s density, g r / c m 3

m e a s u r e d m e a n s q u a r e

i n v e r t i c a l ( z ) direction s p h e r e

(9)

In some experiments made in the Mercury Tank a t the Ghaggenheim Aeronautical Laboratory to dete rmine magne to-fluid dynamic effects on the drag on spheres, i t was noticed that the measured drag of f r e e l y rising spheres, with no magnetic field,

-

could be a s much a s twice the "acceptedw value, quoted, for instance, by the Handbuch d e r Experimentalphysik (Ref. 1). The Handbuch values

a r e based on experiments with falling or fixed spheres; and differences in drag for rising and falling spheres have been previously observed by Hes selberg and Birkeland (Ref. 2) and others.

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2. EQUIPMENT

2. l . T e s t Set-Up.

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2 . 2 . Sphere Release Mechanism.

The sphere r e l e a s e mechanism, shown in Figure 2, could be

(12)

2 . 3 . Spheres.

In o r d e r to vary Reynolds number and density ratios p / p

S W

different spheres were used.

Table 1 gives data on spheres with p s i P w g r e a t e r than one, Table 2 on those with p s

/

Pw l e s s than one.

The diameters of the spheres were measured using a m i c r o - m e t e r . A mean diameter was found by averaging the diameters of ten spheres. The calculation of the sphere volumes was based on this average diameter. The weight of the spheres was determined using a milligram balance.

As i t appeared to be very difficult to buy commercial spheres lighter than water i n different diameters, s e v e r a l hollow spheres

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2.4. Timing Device and Recorder.

Several preliminary velocity measurements were made using a s timing devices a stopwatch and a magnetic-type indicator. These, however, gave no satisfactory results s o that a third timing device had to be built.

Using the principle that a change in light intensity can be

detected by photocells, a phototube timing device was constructed. In this device the sphere c r o s s e s a light beam which changes the light intensity a t the phototube, thus giving r i s e to a change i n electrical potential a t the amplifier tube (Fig. 4). The e l e c t r i c circuit i s shown i n Figure 4a. In o r d e r to obtain a l a r g e deflection of the galvanometer

of the Visicorder, on which the two time signals were recorded, a Heiland M $0

-

l 20 galvanometer connected to a zero-balancing output c i r c u i t (also shown in Fig. 4a) was used.
(14)

3. DRAG MEASUREMENTS O F FREE-FALLING SPHERES

3 . l . G e n e r a l R e m a r k s .

T h e d r a g coefficient was computed by a s s u m i n g t h a t the s p h e r e w a s moving steadily a t its t e r m i n a l velocity when e n t e r i n g the timing distance

.

Then

Ud

F u r t h e r m o r e , the Reynolds n u m b e r is Re

=

-

,

w h e r e vw, v w

the kinematic v i s c o s i t y of w a t e r , is a function of t e m p e r a t u r e and w a s obtained f r o m r e f e r e n c e 3 . F i g u r e 5 gives a, a s a function of

w t e m p e r a t u r e .

The w a t e r t e m p e r a t u r e was m e a s u r e d using a - P O to + I 0 8 d e g r e e C e l s i u s t h e r m o m e t e r . F r a c t i o n s of a d e g r e e could b e

0

e s t i m a t e d to

+

-

0. 1

C.

The density of the w a t e r , p w

,

h a s been taken 3

constant a t 1 g r / c m

.

The influence of the tube walls on the s p h e r e velocity was taken into account by multiplying the m e a s u r e d velocity by a wall c o r r e c t i o n f a c t o r Kw given by Lunnon (Ref. 4). F i g u r e 6 gives

K

a s a function of the r a t i o of s p h e r e d i a m e t e r to tube

w d i a m e t e r .

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3 . 2. Results.

Using the described photoelectric timing device the drag coefficients of several spheres of densities g r e a t e r than that of water have been determined. The results a r e given i n Table 3 and Figure 7.

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3 . 3 . Discussion of Results.

A,s can be seen from Figure '9 the drag coefficients f o r the different spheres scatter up to 18 per cent around their mean values, which agrees fairly well with the respective values of C

DH' This s c a t t e r has been observed by others, too, a s shown i n

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3.4. Conclusion.

P r o m the results of these measurements i t was concluded t h a t t h e timing device works satisfactorily. F u r t h e r m o r e i t showed that the drag coefficient for spheres with density ratios down to pslpw

=

l . 1 i s given by the value of

C

DH'

Therefore, i f there occur any changes i n sphere drag

(18)

18

4.

DRAG MEASUREMENTS

OF FREE-RISING

SPHERES

4 . 1 . General Remarks.

(19)

4. 2 . Spheres Rising in &>-inch Diameter Tube.

(20)

4. 2 . 1. Results.

(21)

4. 2 . 2 . Discussion of Results.

The oscillatory motion of the rising spheres s e e m s to be a typical feature of the sphere motion. The wavy motion becomes m o r e significant a s the density ratio p / p decreases. This wavy

S W

motion seems to be independent of even l a r g e l a t e r a l disturbances given initially to the sphere, a s i n the c a s e where the spheres r o s e freely through the plastic tube.

The displacement of the sphere out of i t s straight vertical path can be attributed to a low m a s s to force ratio, where the

force results from non symmetric vortex shedding from the sphere. F o r a falling sphere the m a s s to force ratio is g r e a t e r *than that for a rising sphere, s o one would expect l e s s significant o r no oscillatory motion of the heavy sphere. This was actually observed: the falling spheres moved in a straight path. F r o m the oscillatory sphere motion i t can be concluded that the point at which the force acts oscillates geometrically over the sphere. It was furthermore observed that the sphere traveled f i r s t along a two-dimensional, wavy path which l a t e r changed often into a m o r e o r l e s s helical path. No evident reason f o r this change of motion could be observed. Once i n a

helical motion, the sphere would never go back into the two-dimensional motion. It sometimes happened, however, that the change from two- dimensional to helical motion did not occur a t all, and the sphere traveled only along a sinusoidal path.

(22)

than CDH. The d r a g coefficient of the s p h e r e s with density r a t i o s

s lpw rr 8.93 approach

C

DH independently of Reynolds n u m b e r , with the simultaneous disappearance of the o s c i l l a t o r y motion.

The d r a g coefficients of the s p h e r e s with d i a m e t e r

d

=

0.725 inch and p s / p w

-

0.59 w e r e extensively examined. T h e s c a t t e r of C i s p a r t i c u l a r l y high f o r s

n

s 2 inch, but i s m u c h l e s s f o r s ;i24 inch. F u r t h e r m o r e , the lowest values of CD a p p r o a c h 'DH* T h i s suggests that the "local d r a g coefficientti of the s p h e r e might not b e significantly diffekemt f r o m the c o r r e s p o n d e n t value of

'DH which would have been m e a s u r e d f o r a falling s p h e r e with s a m e d i a m e t e r a t the s a m e Reynolds number.

Hence, i t was a s s u m e d that the "apparent d r a g coefficient" which was actually m e a s u r e d by our method, is the l o c a l d r a g

coefficient multiplied by a f a c t o r which depends m a i n l y on the location

of the wavy s p h e r e path with r e s p e c t to the timing distance ( F i g . 91, and t h a t the l o c a l d r a g coefficient would have been obtained with the s p h e r e velocity U

=

d s / d t .

Considering a sinusoidal s p h e r e path, the r a t i o of / C is e q u a l t o 1 / c o s 3 a (Appendix A ) . The ' ~ , r n a x D, m i n m a x

calculation was b a s e d upon the a s s u m p t i o n of a steady s t a t e motion along the sinusoidal path. If we take CD, min a s the value of

C

DH and 'D, m a x a s the maximura.1 a p p a r e n t d r a g coefficient m e a s u r e d , we find the maximum angle between v e r t i c a l and s p h e r e path i s about 30'. T h i s is a value which s e e m e d not to b e too high c o m p a r e d with the o b s e r v e d motion. F u r t h e r m o r e , a calculated m e a n s q u a r e a v e r a g e

C D

b a s e d on this a showed good a g r e e m e n t with the a v e r a g e value

m a x

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4. 2 . 3 . Conclusion.

(24)

4 . 3 . Spheres Rising i n Large Water Tank.

(25)

4 . 3 . 1 . General Remarks.

The drag coefficients of the spheres with density ratios l e s s than one were evaluated b y measuring the local velocity of the spheres with the aid of a Voightlgnder Bessarnatic c a m e r a , of which the used shutter time was measured. The shutter time measurements were made by using the photoelectric timing device (Fig.

41,

with a light beam which was f i r s t sent through the camera. The m e a s u r e d

shutter time was 0.190 t

-

0.002 sec.

The spheres were photographed when passing a certain point in the water through a +ndow situated a t the side of the tank below the water surface, leaving thus a streak on the film corresponding to the traveled distance during the shutter opening. A m i r r o r was

0

placed in the water a t an angle of 45 with respect to the c a m e r a - sphere plane in o r d e r to obtain the three-dimensional path of the sphere.

(26)

4. 3 . 2. Results.

Figure 10 shows the reproduction of one of the pictures

(sphere No. 47) from which the local sphere velocities were determined which a r e given in Table 5. Figures 11 to 14 give repro&aLnctions of the photographs of the different sphere paths, Figures 15 and 16 reproductions of the l a t e r a l displacements of the spheres.

Table 5 gives also the drag coefficients obtained during these tests; these a r e plotted on Figure 8.

Table 6 gives the minimum, maximum and average wavelength of the motion of the different spheres; these a r e non dimensionalized with the sphere diameter and plotted in Figure 17 versus the density ratio. In Table 7 the maximum l a t e r a l amplitudes of the sphere path a r e given and plotted in Figure 18; they a r e determined either from the sphere path pictures of the Figures 11 to 14 o r from the Figures

(27)

4. 3 . 3 . Discussion of Results.

The experiments c a r r i e d out i n the water tank show that the local drag coefficients of the rising spheres, for the densities

considered, a r e indeed those given by the values of CDE, within the usual s c a t t e r observed for sphere drag coefficients. This i s t r u e i n d e p e d e n t l y of the oscillatory motion with which the sphere r i s e s .

A distinct wavelength of the sphere motion i s observed, which i s constant along the sphePe path within an average maximum s c a t t e r of 11.41 per cent of a l l spheres (Table 6 ) . The two-dimensionality of the photographs has h e r e no effect on the measurement.

The evaluation of the l a t e r a l amplitudes of the sphere motion has been c a r r i e d out, but i t could not be done very accurately because

of either the two-dimensionality of the Figures 11 to 14 or of the changing scale of the Figures 15 and 14; this may account f o r the

large s c a t t e r of data in Figure 18. The sphere sometimes experiences sudden displacements Prom i t s general axis of motion, for which no obvious reason could be observed.

It was again observed that the sphere traveled f i r s t i n a two-dimensional motion which went over l a t e r into a helical motion. This was especially well seen f o r sphere No. 39 (Fig. 16). It was however noticed that the sphere did not remain during a l l i t s r i s e

i n this two-dimensional motion, a s i t sometimes did i n the experiments i n the 6 -inch diameter tube.

(28)

independently of the timing distance. This was observed to be true independently of Reynolds number, too (Fig. 8 , spheres No. 31 and 38), which indicates that there i s no c r i t i c a l Reynolds number for the onset of significant oscillations of the rigid spheres.

A s i t has been observed by Winny (Ref. 5) there exists a typical vortex shedding from a sphere in the Reynolds number range considered. He observed furthermore that the average vortex frequency on the

surface of the sphere i s not the same a s in the wake but i s about four times higher. Due to this vortex motion, superimposed on the straight fluid motion, the fluid will e x e r t additional fluctuating p r e s s u r e s n the sphere which will influence the sphere path if the inertia of the sphere i s not too high.

Therefore the motion of the sphere will be determined primarily by the sphere m a s s and the sphere will respond significantly only to the lowest frequencies with the highest energy. A s Fung (Ref. 6 ) has shown the maximum of the power spectrum occurs a t a StrouhaE number

S = : - which i s quite low and almost constant a t S ii 0 . 0 5 for Reynolds

uo

4

6

numbers from 0 . 3 3 x 10 to 1 . 2 7 x 10

.

Assuming this to be true for lower Reynolds numbers, too, this would explain the low Strouhal numbers obtained from the photographs, given in Figure 19, in comparison with those observed for cylinders in the s a m e Reynolds number range, cf. A . Roshko (Ref. 71, who gives S

=

0 . 2 1 .
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varying p r e s s u r e force (i. e . , the p r e s s u r e integral over the sphere) by considering the force balance on the sphere under some simplifying assumptions

.

In Appendix B the necessary p r e s s u r e integral which m u s t balance the buoyant force and the inertia force has been calculated under the following a s sumptions:

1 . The drag of the sphere i s equal to the buoyant force times the cosine of the sphere path angle with the vertical,

2. the sphere path i s sinusoidal and two-dimensional, 3. the velocity i s constant i n vertical direction.

The calculations give a time dependent l a t e r a l force

K

which i s given in Figure 20 a s a function of a or time. F o r sphere No. 47, for which an example has been calculated through, the maximum force i s %ax

=

1.44 g r , wt. This corresponds to a maximum force coefficient

'max

=

a.

16,

which corresponds to a force coefficient for a c i r c u l a r cylinder of

C

max, cyl.

=

0. 786. This i s of the same o r d e r of magnitude a s H. Drescher (Ref. 8) has actually m e a s u r e d for

5

c i r c u l a r cylinders in water a t Re =: l . 1 3 x 10

,

a s he obtains oscillating force coefficients between 0 . 6 and 1.3.

Hence, although the approwimation is quite crude, i t gives reasonable results for the maximum force coefficient of the sphere in connection with the observed data f o r the sphere path.

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created i n which mainly the frequencies 2f and 4f appear, where f is the frequency of the sphere motion. A s the amplitudes of the 48- force a r e much higher than any other, the theoretical force coefficient would therefore give a four times higher Strouhal number, which

would correspond to the observations Winny (Ref. 5) has made. Hence, based on the calculations of the necessary f o r c e coefficient, a theoretical Strouhal number i s found with a value between 0.1496 and 0.354

depending on the density ratio p / p

.

S W

The Pact that the Strouhal number seems to be a function of

P,/P, i s due to the method of determining the Strouhal number, i. e.

,

by evaluating S from the response of the sphere to a certain oscillatory force. As the ratio decreases the sphere will respond m o r e to higher frequencies and therefore the Strauhal number, formed with the observed data will. increase. F o r fixed bodies, on which the

(31)

' 4 . 4 Summary.

The drag coefficients of rigid, freely rising spheres i n water 3

with densities i n the range 0.526 < p s < 0 . 9 3 g r / c m were found to be the s a m e a s those for rigid, free-falling spheres of densities l a r g e r than one g r / c m 3 with the same diameter, a t the s a m e Reynolds number.

The drag coefficients were found to be independent of the oscillatory sphere motion which dies out when the density ratio

s / p w approaches one. This oscillatory motion was practically non-existent f o r

=

0 . 9 3 independently of the Reynolds number, so that a c r i t i c a l Reynolds number for the oscillations of the sphere path could not be found.

"Apparent drag coefficients" were obtained when the drag coefficient of the sphere was tried to be determined by a distance

-

per time measurement of the sphere velocity. This apparent drag

coefficient was found to be a function only of the geometry of the sphere path along which the sphere traveled.

The oscillatory motion of the sphere was f i r s t a two-dimensional motion, then changed sometimes into a helical one, for which no

reason could be observed. .

F o r both the two-dimensional and the helical motion a distinct wavelength could be found which remains constant along the sphere path within a certain amount of s c a t t e r . The amplitudes of the sphere

motion a r e given only approximately.

(32)

24

I . have been found which depend on the sphere densities and a r e about

three times s m a l l e r than those obtained for oscillating flow over c i r c u l a r cylinders.

Under certain simplifying assumptions the n e c e s s a r y

p r e s s u r e force for a force balance on the sphere has been calculated. This gives a maximum force coefficient of the same o r d e r of magnitude a s observed for fluctuating force coefficients on c i r c u l a r cylinders.

The theoretical p r e s s u r e force fluctuates with a main

(33)

R e f e r e n c e s

1. Handbuch d e r Experimentalphysik, Band 4, 2. T e i l (1932). 2. Th. Hes s e l b e r g und B.

J.

Birkeland: Steiggeschwindigkeit

d e r Pilotballone, B e i t r . z. P h y s . d. f r e i e n Atrnosph., 4, 196 (1912).

3 . 0 , W. Eshbach: Handbook of Engineering Fundamentals

,

Vo1. 1 (1936).

4. R. G. Lunnon: F l u i d R e s i s t a n c e to Moving S p h e r e s , P s o e . Roy. Soc. London ( A ) 118, 680 (1928).

5. M. F. Winny: Vortex System Behind a S p h e r e Moving Through Viscous Fluid, A . R . C . R e p o r t s and Memoranda No. 1531,

T,

3305 ( 1 9 3 2 ) .

6.

Y

.

C

.

F u g : F l u e tuating Lift and D r a g Acting on a Cylinder i n a Flow a t S u p e r c r i t i c a l Reynolds Numbers. IAS-Paper No. 60-6 (1960).

7. A . Roshko: On the Development of Turbulent Wakes f r o m V o r t e x S t r e e t s , T h e s i s , California Institute of Technology, (1952).

(34)

APPENDIX

A

CAL

CUL

A

T/QM

G!!

SPHERE DRAG COEFFICIENT.

Assum

p

f

ions

:

1.

DRAG

of

SPHERE

k/he/e

(35)
(36)

EXAMUL

E

Sphere

N o 4 6

4-058

..
(37)

APPENDIX B

CAL C

UL

A T/OM O f

m/?CES

On/ SPHERE .

A s s u m p t i o n s :

.

Drag

of sphere

D

=

a,

coS

3 d's w h e r e 130 =

9 %

; / T O ( / -

-

)

YN

2 r z

2,

sphere

path

=

A

a s

-

/I

3 .

Jh

verflca/ c / / h c t i + ~ 2 =

uvt

C/,

=

consf-.

From . :

(38)

A .

Force B c / / ~ r , c e U - cv'/'rechbr;,:

D r a j

+

Jnerh'aforce =

A*

G U S

oc

+ Pressure f o r c e )

U

(39)

5.

Force

B a / ~ n c e in

v-

d r e c h b n :

Jnerhuforce

+

B,sinot

=

pressure orce

f

i v

From

t h e

f ~ i ~ u r e bc/ow M

+he

force

P

iu'h~ch

eyerfs

t h e

sphere O n +he

flu/$

from g e o m e f r / c a / c o n s ~ ' d e r -

a

hbns

.

P

rnu$/st

b e ba/anced

by

6-

pressure force

K

whrch e v e r t s +he f/u,'o/ on

f h e

S p h e r e .
(40)
(41)
(42)

E X A M P L E .

-

Sphere

N c .

47

we

O&+Q;H +hen :

(13

has been evo/uoied

g m i ~ ~ / / ~ )

;

K =

K

,i

given ,i,figure

20

(43)
(44)
(45)

T A B L E 2

Sphere No. d biT

PS"W

K w

CD

u2

in gr

-

-

i n /sec 2 2

uo.

24

153.11

72.05 mold

71.60

''

73.80

"

51.00 "

71.02 ar

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TABLE O F  CONTENTS  Acknowledgements  Abstract  Table  of  Contents  L i s t  of  Symbols  1
TABLE  OF  CONTENTS  (Contd.  )

References

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