Annular jets

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tq fs lv3

ANNULAR JE ^S

'

_Michael

Stewart

_Borgas

B.Sc.

(Hons.), University of

^Adelaide

Thesis submitted

for the

^Degree

of

Master

of

Science

in

^the

University of

Adelaide

Departnent

of Applied

Mathematics.

by

September ^1982.

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TABLE OF CONTENTS

STJMMARY

SIGNED STATEI"ÍENT ACKNOWLEDGEMENTS

INTRODUCTION

CFTAPTER

1.

ANNULAR ^NOZZLES

2

3 4

fntroduction

Equations

for Initial

^Slope

Separation

of Variables

Equations Numerical Procedure

Page

1 11 111

4.

6.

8.

10.

L4.

15.

17.

18.

19.

1

I

5.

Computing ^Kappa

6.

Numerical Results

7. Large ß Limit 8. Offset

Pipes

9.

^Conclusion

CHAPTER

2.

^ANNULARJETS IVITH SURFACE TENSION

2I.

1. Introduction

2L.

2.

Equations

for Thick,

Slender Annular Jets

3. Solution of the

Boundary Value Problem

4.

^Computed

Results

and Discussion

5.

Thin-Slender Annular Jets

6. Solutions for

Thin-Slender Annular Jets

7. Thin

Annular Sheets

of

Water

8.

^Equations

for

a

Thin

Jet

9.

The Dynamic Equations

10.

Conclusions

23.

25.

28.

37.

39.

4r.

43.

49.

53.

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CHAPTER

3.

STABILITY OF ANNULAR COLUMNS OF WATER

1. Introduction

2.

^Temporal

fnstability of

an Annular Colunn

3.

^Equations

of

Motion

4. Kinenatic Boundary Conditions

5.

Dynaníc Boundary Conditions

6.

The

Dispersion

Relations

7.

Results

8. Small ß Limit 9. Thin Jet Línits 10. fnterrnediate Modes

BIBLIOGRAPFTY

55.

57.

s7.

58.

60.

62.

69.

7L.

74.

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STJMIvIARY

In this thesis annular jets, falling vertically

^(when

gravity is included), are considered.

Thus

in

any

horizontal

plane

the jet lies between two concentric circles. The three

nain jet pararneters examined are surface tension, jet thickness and a pressure difference across the annulus. Various types of

dynamic behaviour

are also

considered,

including formation of

jets

from noz,zles and

stability of jets.

Techniques

are

developed where

the

behaviour

of

such

jets

may be described

mathematically,

and

solutions for

a wide spectrum

of jet

paraneters presented.

1

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SIGNED ^STATEMENT

I

^hereby

declare that this thesis contains

no

material

which has been accepted

for the

award

of

any other degree

or diplona in

any

University

and,

to the best of

^my

knowledge,

it contains

no

material previously

^published^by

any

other person,

except where due

reference is

made

in

the

text of the thesis.

M.

S.

^BORGAS.

11

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ACKNOWLEDGEMENTS

I

^would

like to

^thank

ny supervisor,

Professor

E.0.

Tuck,

for his

advice and guidance

in

completing

this thesis.

I

would

also like to thank Ms. E. Henderson for her

excellent

t1rying.

The work

for thís thesis was carried out fron January 1982

to

^September^1982,

during

which

ti¡ne I was financed by an

Adelaide

University

Research Grant.

111