Thermo-Fluid-Dynamic Modeling and

Simulations of the Drawing Process of

Photonic Crystal Fibers

Der Technischen Fakultät der Universität

Erlangen-Nürnberg

zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von

Giovanni Luzi

Erlangen, 2014

Als Dissertation genehmigt von der Technischen

Fakultät der Universität Erlangen-Nürnberg

Tag der Einreichung: 21.01.2013

Tag der Promotion: 21.02.2013

Dekanin: Prof. Dr.-Ing. Marion Merklein

Berichterstatter: Prof. Dr.-Ing. Antonio Delgado

Prof. Dr.-Ing. Philipp Epple

Prof. Dr. Michael Dreyer

i

AKNOWLEDGEMENTS

I would like to express my deepest appreciation to all the people who have supported and assisted me

during these four and half years of research at LSTM. First of all, I would like to especially thank my

professor and head of the institute Prof. Dr.-Ing. Antonio Delgado for bringing me this research

project. I would like to thank him for the continuous support, his highly valuable advices and his

constant encouragement during the whole period of my research. I would like also to express my

gratitude to my former supervisor Prof. Dr.-Ing. Philipp Epple for numerous technical discussions and

his guidance through this project. His enthusiasm and energy has helped me to towards the propel

completion of this work, from the beginning to the end. I cannot forget to thank also Mr. Ken

Fujimoto who intensively worked with me during this research period. Without his support, many of

the present results would probably not have been achieved. Furthermore, I would like to thank my

former and present colleagues who helped me along the way with discussions and contributions. They

are in no particular order: Jose Rodriguez Agudo, Alexandre Wolf, Vojislav Jovicic, Malik Al-

Amayreh, Henrik Smith, Matthias Semel, Oliver Litfin, Bettina Willinger (my present group leader)

and Alessandro Cubeddu, and PD Dr. Jovan Jovanovic, a highly calibre expert in fluid mechanics.

In order to carry out most of the numerical simulations, the powerful high-performance cluster (HPC)

of the university Erlangen-Nürnberg has been extensively used. Therefore my gratitude goes also to

Dr. Thomas Zeiser and Dr. George Hager for they support and their useful contributions. I want to

express my sincere gratitude to the support team of ANSYS Germany in Darmstadt, especially to Dr.

Mourad Lotfey and Oliver Zühlke for their determining contributions regarding the numerical

computations. Without their help, ten percent of the numerical results, presented in this thesis, would

probably have been achieved.

I gratefully acknowledge also the funding of the German Research Council (DFG), for supporting this

research project within the framework Cluster of Excellence “Engineering of Advanced Materials” at

the university Erlangen-Nürnberg. I thank especially Dr. Michael Scharrer for inviting me to see

experiments of fiber drawing performed at the Max-Planck Institute of Erlangen. Without his highly

valuable contributions and involving technical discussions, the quality of the paper published would

have been certainly much lower, and many of the results, presented in this thesis would not have been

obtained.

I am deeply grateful to my family, my mother Eleonora, my brothers Riccardo and Francesco, their

life companion, respectively Alessandra and Eugenia, and my three beautiful nieces Ana Sofia, Maria

Victoria and Martina, for supporting and encouraging me during the difficult periods of my research.

ii

Abstract

Microstructured optical fibers (so called “photonic crystal” or “holey” fibers) nowadays arouse great

interest, due to the many optical effects they permit to achieve. The hole-pattern allows the light

guidance within a solid or hollow core. These novel types of fibers may represent the future in

telecommunications, optics, gas-laser devices and precision sensing applications. Such fibers consist

of air holes arranged around a solid or hollow core. They are manufactured by heating and drawing

down an initial preform in several steps. During drawing, the size of the external and the internal hole-

diameters is greatly reduced. Several parameters have great influence on the fabrication process like,

for instance, the feed and the draw speed, the internal hole-pressurization and the surface tension.

Therefore, a complex overlapping of thermo-fluid-dynamical, constitutive and capillary effects occurs.

In this study, mathematical analysis and numerical simulations of drawing processes have been

performed in order to understand the thermo-fluid-dynamic properties, occurring during the drawing

of heated preforms. First of all, single capillaries have been considered, since the drawing process can

be treated in a completely analytical manner. An existing simplified asymptotic model presented in the

literature has been revisited and improved, by forcing the numerical solution to obey to the mass

conservation equation. This simplified model takes into account the effects of internal pressure and

surface tension, which are extremely important to determine the final fiber cross-section. Surface

tension tends to close the inner hole, while internal pressure acts in the opposite way to keep it opened.

The viscosity dependence of glass upon the temperature has been chosen in literature. The numerical

results of the simplified equation of motion have been compared with experiments, obtaining a good

agreement for the final fiber diameters both without internal-hole pressurization and when internal

pressure is applied. Second, the final diameters have been compared with those computed by solving

the full 3D set of N.-St. equations. The geometry has been created and meshed with the aid of the

software ANSYS Gambit. The corresponding N.-St. equations have been solved with the finite

elements method with the aid of the software ANSYS Polyflow. The shape of the final fiber computed

with both the simplified model and the full 3D N.-St. equations has been compared. A careful

comparison of the radial and the axial stress, the radial and the axial velocities, explains where and

why the two models match or not along the drawing direction. This comparison has been done for the

case of constant viscosity and it elucidates the interplay between surface tension and internal pressure.

Furthermore, it can be regarded as a basis to model more complex geometries by a commercial tool.

Finally, a 3D six-hole geometry has been also modelled numerically. As before, the commercial

software ANSYS Gambit has been employed to create the geometry and the mesh, while the

commercial software ANSYS Polyflow has been used to solve the discretized N.-St. equations. The

numerical results are found to be in good agreement with experiments even in case of high values of

internal pressure, where the final cross-section result highly deformed. In this study, thermal

equilibrium among the furnace gas, the fiber glass and the pressure gas has been assumed by guessing

iii

temperature profiles, which are found to be suitable to represent the temperature distribution inside the

furnace. The agreement between experiments and numerical simulations is good in a large variety of

cases; only under extreme drawing conditions, that is, high values of inner pressure and high peak

temperatures deviations are noticeable. In these cases, the temperature profile of the fiber has to be

correctly computed by solving the temperature equation, in order to calculate precisely the fluid flow

stresses.

iv

Table of Contents

ACKNOWLEDGEMENTS I

ABSTRACT II

LIST OF FIGURES AND TABLES VI

NOMENCLATURE IX

Chapter 1: Introduction 1

1.1 Introductory remarks on Photonic Crystal Fibers 1

1.2 State of the art and open questions in literature 2

1.3 The subject and the structure of the thesis 5

1.4 The aims and the achievements of the thesis 7

References 8

Chapter 2: Basics of Conventional and Photonic Crystal Fibers 10

2.1 General description 10

2.2 Some basics of optical fibers 11

2.3 Single and multimode fibers 12

2.4 Basics of PCFs 13

2.5 Optical properties 15

References 16

Chapter 3: Basic Equations and Main Techniques 17

3.1 Basic equations 17

3.2 The perturbation theory 18

3.3 Main techniques 18

3.4 Main procedures 19

References 20

Chapter 4: Results 21

4.1 Problem formulation 21

4.2 Problem solution 22

4.3 Asymptotic solutions and their implementation into a Matlab program 24

4.3.1 The simple case 27

4.3.2 Inertia and gravity effects 28

4.3.3 Surface tension effects 30

4.3.4 Limit of small surface tension 35

4.3.5 Internal pressure effects 35

4.3.6 The interplay between internal pressure and surface tension 36

4.3.7 Closure of a small hole 40

4.3.8 The general case 41

4.4 Experimental results 43

4.5 Numerical models 50

4.6 Comparison among solutions of the set of asymptotic equations with the full N.-St.

equation ones: the ideal case with constant viscosity 56

4.6.1 Comparison between the asymptotic equations and the N.-St. ones: the simple case 56

4.6.2 The case with surface tension effects 57

4.6.3 Internal pressure effects: p0=120 [Pa] and p0=135 [Pa] 65

4.6.4 The case with internal pressure and surface tension 74

4.7 Numerical solution and experimental validation of the drawing process of six-holes

optical fibers 82

References 87

Chapter 5: Conclusions 88

5.1 Conclusions and future works 88

v

References 90 Appendix A 91

References 93

Appendix B 94

References 96

Appendix C 97

References 98

Appendix D 99

References 103

Appendix E 104

References 108

Appendix F 109

References 111

Titel der Dissertation 112

Zusammenfassung 113

vi

List of Figures and Tables

Fig 1.1: Experimental setup: (1) Pulling device, (2) Preform, (3) Top opening, (4) Electrical oven,

(5) Pipe which provides the protective gas (Ar), (6) Glass Fibre. Image courtesy of the Max Planck

Institute for the science of the light (MPL) Erlangen. 2

Fig.2.1: Bundle of optical fibers, Mellish [4] 10 Fig. 2.2: The sketch of the structure of a SMF: the core, the cladding, the buffer and the jacket with

their respective dimensions [4] 12

Fig. 2.3: The structure of a conventional optical fiber: single-mode fiber (left), and multi-mode fiber

(right), Paschotta, Encyclopedia of Laser Physics and Technology [9] 13

Fig.2.4: Sketch of a solid-core PCF with a triangular lattice of air-holes a). Microscope picture of a

real solid-core triangular PCF b) Poli et al. [13]. 14

Fig.2.5: Sketch of the cross-section of a photonic band-gap, hollow-core fiber with a triangular

lattice of air-holes a). Microscope picture of a real hollow-core PCF b) [13]. 15

Fig. 4.1: Schematic diagram of drawing processes, Fitt et al. [1] 21 Fig. 4.2: Head of the program Fiberspinning.m, in which it is shown how to choose among different

cases, in order to select the physical effect(s) one wants to analyse 26

Fig. 4.3: Evolution of the internal and external radii of a preform, in case that only the effects of a

variable viscosity are considered. The viscosity varies along the axial position. Matlab image. 27

Fig. 4.4: Evolution of the internal and external radius of a preform, when only the effects of surface

tension are taken into account and the viscosity is assumed constant with the temperature 32

Fig. 4.5: Evolution of the internal and external radius of a preform. Comparison between the simple

case and the one in which the surface tension effects are included 32

Fig. 4.6: Influence of different values of surface tension on the internal diameter for different ar : a)

0.2ar , b) 0.3ar , c) 0.4ar , d) 0.6ar 33

Fig. 4.7: Evolution of the internal and the external radius of a preform for the case taking into

account only the effects of surface tension. In this case the viscosity varies with the temperature,

1950peakT [°C] 34

Fig. 4.8: Evolution of the initial preform as internal pressure and surface tension effects are

considered. The viscosity is here considered constant with the temperature: a) 0 120p [Pa], b)

0 140p [Pa], c) 0 160p [Pa], d)

0 175p [Pa]

38

Fig. 4.9: Evolution of the initial preform as internal pressure and surface tension effects are

considered. The viscosity is here considered non constant with the temperature, which reaches its

peak ( 1950peakT [°C]) at the center of the furnace: a) 0 0p [mbar], b) 0 5p [mbar], c) 0 20p

[mbar], d) 0 25p [mbar]

40

Table 1: Graphical visualization between the equations and the solutions of [1] and those obtained

by the author of this thesis for different cases 42

Fig. 4.10: Glass preform being introduced into the furnace 43 Fig. 4.11: Fiber drawing process. Left: Fiber being drawn out of the bottom of the furnace. Right:

Shape of the glass preform after drawing 43

Fig. 4.12: Possible temperature profiles along the x axis of the furnace. The peak temperature is in

this case 2050peakT [°C] 44

Fig. 4.13: Comparison between theoretical computations and experimental results for three different

“drawing speeds”, that is, 3.6 1f dU U , 5.4 1.5f dU U and 7.2 2f dU U : external diameter

a1), b1) and c1), and the ratio internal/external diameter a2), b2), and c2) , when no pressure is

applied.

46

Fig. 4.14: Comparison between experimental results and theoretical computations for the “drawing

speed” 5.4 1.5f dU U when pressure is applied. First, the temperature varies while the pressure is

kept constant at 0 9p [mbar]: external diameter a1), and ratio internal/external diameter a2).

Second, the temperature peak is kept constant at 1950peakT [°C] and the pressure varies from

0 0p [mbar] to 0 25p [mbar]:b1) external diameter, and b2) ratio internal/external diameter

47

Fig. 4.15: Comparison between theoretical computations and experimental results for two different

“drawing speeds” a) 1 10.2f dU U and b) 3 30.6f dU U , when no internal pressure is applied.

External diameters a1) and b1), ratio internal/external diameters a2) and b2). The final diameters

obtained are of the order of few hundreds of micro-meters

48

vii

Fig. 4.16: Comparison between experiments and theory for the “drawing speed” 2 20.4f dU U .

The pressure is varied between0 5p [mbar] and

0 40p [mbar]. External diameters a1) and ratio

internal/external diameters a2).

49

Fig. 4.17: a) The global view of the meshed geometry; b) front view 50 Fig. 4.18: (a) Optical microscope image of the micro-structured silica preform used in the fiber

drawing experiments. (b) Meshed geometry, frontal view. 51

Fig. 4.19: Screenshot of the pre-processor window of Polydata, Fujimoto [13] 52 Fig. 4.20: Schematic diagram of the procedure of the evolution parameter “ s ” [13] 54 Fig. 4.21: Grid deformation at different steps ( s ), [13] 55 Fig. 4.22: Comparison of the shape of the radii between the asymptotic model and the N.-St. one for

the simple case, when internal hole pressure, surface tension, inertial and gravity effects are

neglected and the viscosity is assumed constant with the temperature 57

Fig. 4.23: Comparison of the shape of the radii between the asymptotic model and the N.-St. one,

when only the effects of surface tension are taken into account 57

Fig. 4.24: rr at three different position along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension

effects are considered.

60

Fig. 4.25: xx at three different position along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension

effects are considered.

62

Fig. 4.26: u at three different position along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension

effects are considered. 63

Fig. 4.27: w at three different position along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension

effects are considered. 64

Fig. 4.28: Comparison of the shape of the radii between the asymptotic model and the N.-St. one,

when only the effects of internal pressure are taken into account and the viscosity is assumed

constant with the temperature: a)0 120p [Pa], b)

0 135p [Pa] 66

Fig. 4.29: rr at three different positions along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure. 67

Fig. 4.30: xx at three different positions along the drawing direction in the furnace: a)

0.02x [m], b1) and b2) 0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure. 71

Fig. 4.31: u at three different positions along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure. 72

Fig. 4.32: w at three different positions along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure. 73

Fig. 4.33: Comparison of the shape of the radii between the asymptotic model and the N.-St. one,

when both the effects of internal pressure and surface tension are taken into account and the

viscosity is assumed constant with the temperature: a), 0 150p [Pa], b),

0 215p [Pa] 74

Fig. 4.34: rr at three different positions along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. Case with both effects of surface tension and

internal pressure.

76

Fig. 4.35: xx at three different positions along the drawing direction in the furnace: a)

0.02x [m], b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. Case with both effects of surface

tension and internal pressure. 78

Fig. 4.36: u at three different positions along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. Case with both effects of surface tension and

internal pressure 79

Fig. 4.37: w at three different positions along the drawing direction in the furnace: a) 0.02x [m],

b1) and b2) 0.11x [m], c1) and c2) 0.28x [m]. Case with both effects of surface tension and

internal pressure. 80

Fig. 4.38: Temperature profile along the x axis of the furnace used in the simulations. The peak

temperature is in this case 1890 [°C] 83

viii

Fig. 4.39: Comparison of the SEM images of the final fiber cross-sections (grey-scale background)

with numerical simulations (blue overlaid contours) for applied internal pressure of a) 0 0p

[mbar], b) 0 150p [mbar], c)

0 250p [mbar], and d) 0 300p [mbar].

85

Fig. 4.40: Comparison of the final external diameter between experiments and simulations. The inner

pressure varies between 00 300p [mbar] 86

Fig. 4.41: Comparison of the final maximum sizes of the internal holes measured in the radial

direction between experiments and simulations for the big and the small hole in the upper half part of

the fiber. The inner pressure varies between 00 300p [mbar]

87

Fig. A.1: Evolution of the internal and external radius of a preform for the simplest case, when

internal hole pressure, surface tension, inertial and gravity effects are neglected and the viscosity is

assumed constant with the temperature 92

Fig. A.2: Evolution of the internal and external radii of a preform for the simplest case, when the

viscosity dependence upon the temperature (and therefore on the position) is considered. 93

Fig. B.1: Evolution of the internal and external radius of a preform. Comparison between the

simplest case and the case in which the inertia effects are taken into account. 95

Fig. D.1: Influence of different values of surface tension on the preform geometry: a) 0.3 [N/m],

b) 0.1 [N/m], c) 0.05 [N/m], d) 0.001 [N/m] 102

Fig. E.1: Evolution of the initial preform as different values of internal pressure are applied. Here

only internal pressure effects are considered: a) 0 60p [Pa], b)

0 80p [Pa], c) 0 100p [Pa] and

d) 0 137p [Pa]

106

Fig. E.2: Coordinate values at which the explosion occurs for different values of internal pressure:

a) 0 150p [Pa], b)

0 158p [Pa], c) 0 164p [Pa] and d)

0 190p [Pa] 107

Fig. F.1: Evolution of the initial preform in case of different values of surface tension: a)

0.3 [N/m], b) 0.1 [N/m], c) 0.01 [N/m], d) 0.001 [N/m] 110

Fig. F.2: Influence of internal pressure on the initial preform with 0.001 [N/m]: a)0 10p [Pa],

0 25p [Pa], 0 75p [Pa],

0 100p [Pa]. 111

ix

NOMENCLATURE

Symbol Definition

t time

x distance along capillary axis

r distance perpendicular to axis

0u leading order axial fluid velocity

u

axial component of the velocity vector

w

radial component of the velocity vector

Hp hole pressure

g acceleration due to gravity

ig vector of gravitational acceleration

if body force per unit volume

density

dynamic viscosity

0 reference dynamic viscosity

surface tension

L furnace length

1h inner diameter

2h outer diameter

ap ambient pressure

op hole overpressure

U typical axial draw speed

20h outer diameter preform

10h inner diameter preform

x

h radius

fU feed velocity

dU draw velocity

convergence parameter

log( d fU U )

Re Reynolds Number

slenderness parameter

it̂ tangent unit vector

stress tensor

in̂ normal unit vector

inm initial mass flow rate

outm final mass flow rate

. .N St Navier Stokes

..MA asymptotic method

0w leading order radial fluid velocity

ix coordinate vector

p thermodynamic pressure

ij extra stress tensor

1u first order axial component of the velocity vector

1w first order radial component of the velocity vector

P first order component of the pressure term in the asymptotic expansion

ij

Cauchy stress tensor

T furnace temperature

ij

Kronecher delta

K sum of the principal curvature of a free surface

xi

f function which describes a spatial position of a free surface

CFD Computational Fluid Dynamics

TIR Total Internal Reflection

PCFs Photonic Crystal Fibers

NA Numerical Aperture

V V number

LMA Large mode area

PBG Photonic Band Gap

Dt Substantial derivative

t Partial differentiation with respect to time

ix Partial differentiation with respect to space

3D Three dimensional

2D Two dimensional

MTIR Modified Total Internal Reflection

SMFs Single-Mode-Fibers

MMFs Multi-Mode-Fibers

summation symbol

o “small oh”

O “big oh”

infinite

f Function which depends on the slenderness parameter

Gauge function which depends on the slenderness parameter

lim limit

mc coefficient of the asymptotic expansion

iu velocity vector

0k free-space propagation constant

xii

a peak of the temperature profile

b position along the x coordinate where the peak is located

c parameter which determines the “shape” of the Gaussian distribution

RAM Random Access Memory

CPU Control Processing Unit

GB Gigabyte

GHz Gigahertz

s evolution parameter

finals final value of the evolution parameter

inits initial value of the evolution parameter

maxs maximum difference in the value of s between two steps

mins minimum difference in the value of s between two steps

iR radius of curvature

AMF Algebraic Multi-Frontal

SEM Scanning Electron Microscope

1

Chapter 1:

Introduction

1.1 Introductory remarks on Photonic Crystal Fibers

A new class of optical fibres, called photonic crystal fibers (PCFs), have rose in the recent two

decades, as described in the publication of Russell [1]. These fibers are able to guide light mainly by

two physical mechanisms, that is, the modified total internal reflection (MTIR) and the photonic band-

gap (PBG) effects. The first physical principle leans on the index difference between the solid core

and the air-holes cladding, while the second one is due to the formation of a photonic band-gap in the

cladding, see for instance Cerqueira [2]. PCFs are usually made from pure silica, but polymers or

materials which contain impurities are also used. Outstanding properties of PCF stem from the fact

that the lattice pitch of air holes can be freely designed. For instance, PCFs can be manufactured to be

endlessly single mode at any wavelength [1]. Furthermore, the mode area size can be controlled by the

presence of air holes. If the mode area is made very small, nonlinear fibers are obtained. On the other

hand, large mode area fibers find applications in high power delivery, for an extensive treatment on

this matter see the book of Poli et al. [3].

To fabricate PCFs, one has first to create a preform which contains the structure of interest but on a

macroscopic scale. For silica-based PCFs, the preform is usually made by stacking capillaries and rods

by hand: a typical preform would be a meter long, 20 [mm] diameter, and contains dozens to hundreds

capillaries. This stack is successively drawn into a preform called “cane” in a fiber-drawing tower. In

Fig. 1.1, a preform is shown while being inserted into the oven, which heats it up

2

Fig 1.1: Experimental setup: (1) Pulling device, (2) Preform, (3) Top opening, (4) Electrical oven, (5) Pipe

which provides the protective gas (Ar), (6) Glass Fiber. Image courtesy of the Max Planck Institute for the

science of the light (MPL) Erlangen.

The tube is then pulled from below and it assumes an elongated form. During the drawing process this

stack extends its length while reducing its outer diameter to several millimeters, and fuses the

individual capillaries. Finally, the cane is then drawn into a fiber, normally using internal pressure to

control the hole-size in the fiber. A wide range of PCF structures has been fabricated, each with

different optical properties [1], [2]. On the other hand, numerous fiber designs have been suggested in

the literature based solely on their optical properties, and determining the feasibility from a fabrication

standpoint requires a combination of theoretical, numerical and experimental work.

The processes of drawing microfibers and micro-capillaries out of molten optical materials are at the

moment poorly understood. A complex overlapping of thermo-fluid-dynamical, constitutive and

capillary effects in connection with solidification occurs. Thus, systematic optimization and control of

fiber manufacturing is far from being achievable.

1.2 State of the art and open questions in literature

The main purpose of theoretical and numerical modeling is not only to try to recover the shape and the

dimension of the final fiber or of the final fiber cross-section, but it is also necessary to try to quantify

the effects of some parameter during different stages of the drawing process, in order to be able to

control the manufacturing process. Some of these parameters can be manually set, that is, the feed and

the drawing speeds, the peak temperature of the furnace and the internal over-pressure. Other

parameters, like, for instance, the viscosity, the density, or the surface tension are properties of the

material and they all vary with the temperature.

3

The spinning of molten thread-lines has been analyzed for textile industry. For instance, Matovich et

al. [4] examined the spinning process of a continuous filament. The authors based their analysis on the

assumption that the derivative of the radius with respect to the axial direction is much smaller than

unity. This allowed them to simplify the equation of motion and to obtain a solution, both for

Newtonian fluid, taking into account gravity, inertia, viscous and surface tension forces and for more

complex ones, that is, Coleman and Noll fluids. Successively, Burgman [5] and Manfre [6] applied the

same ideas to optical fiber drawing processes. A lot of fiber-drawing models employ asymptotic

analysis based on the small aspect ratio of capillaries, see for instance Dewynne et al. [7], who derived

a simple model for fiber tapering. Fitt et al. [8] proposed a general model for axis-symmetric annular

fiber, which is capable of including the effects of inertia, gravity, surface tension and internal pressure.

Some years later, Voyce et al. [9] developed a similar model, which is capable to take into account the

effects of the fiber rotation. Afterwards, Voyce et al. [10] completed their model by adding an

equation for the viscosity dependence upon the temperature. They also performed some measurements

of the temperature profile of the furnace, at least, in a restricted part of the furnace which they termed

“hot zone”. In this region, the temperature reaches so high values, and the viscosity of the glass is so

low, that the material itself can be treated as a fluid. Moreover, a comparison of the final fiber

geometry between the theoretical model and experiments is presented for different rotation rates [10].

Nevertheless, an exhaustive comparison between a theoretical model and experiments, for different

drawing regimes, different peak temperatures of the furnace and different values of the inner applied

pressure is not yet present in the literature.

Several authors included the energy equation in their asymptotic model, see for instance Meyers [11].

He coupled a simplified one-dimensional model for extensional flow with approximate expressions for

radiative heat exchange. Gupta et al. [12] presented a more sophisticated model by performing a

rigorous asymptotic analysis of the energy equation. Huang et al. [13] also considered in their model

the difference in the dopant concentration between the core and the cladding glass, due to diffusion

and convective transport. Howell et al. [14] analyzed the stretching of axis-symmetric heated threads,

examining the breakage as well. Wylie et al. [15] examined the flows of threads, considering the

effects of viscous heating. A comprehensive asymptotic analysis of the energy equation which takes

into account the effects of conductive, convective and radiative heat exchange has not been done yet.

The first contributions to stability analysis come back to Pearson et al. [16], who examined the

simplified equation provided by Matovich et al. [4], just for the case of a Newtonian fluid, noting a

resonant effect. Yarin et al. [17] studied the drawing process of a glass fiber in the unstable range. The

resonant instabilities give rise to oscillations, which are responsible of a periodic variation of the

cross-sectional size of the fiber. Stability analysis of more complex models, which includes the energy

equation, has been performed by Shah et al. [18]. They first considered a case of a non-isothermal

problem in which only the viscous forces are considered, and then a more general case in which also

the effects of inertia, gravity and surface tension forces are taken into account [19]. Shah et al. [19]

concluded that inertia forces are the most important under the point of view of the stability.

4

Gospodinov et al. [20] compared the draw resonance phenomenon for both isothermal and non-

isothermal conditions. They show that draw instability remains a quasi-periodic phenomenon. Gupta et

al. [21] assert that the cooling process strongly stabilizes the drawing process. Moreover, they

extended their analysis to viscoelastic flows. A fiber drawing process is characterized by the

competition among surface tension, internal pressure and viscous forces. A stability analysis, which

takes into account all these effect is at present absent in the literature.

There are a lot of mathematical models that make use of asymptotic analysis to describe extensional

flows in non-axis-symmetric geometries. A systematic derivation of the governing leading order

equations for elongating flows in arbitrary cross-sectional geometries is presented in the manuscript of

Dewynne et al. [22]. In a subsequent manuscript, Dewynne et al. [23] refined their model by adding

the effects of inertia and gravity. Following the ideas of [22], Cummings et al. [24] employed a

conformal map to describe the shape of the cross-section, including the effects of surface tension,

inertia and gravity. Ribe [25] examined the problem of bending and stretching of viscous sheets,

whose thickness is much smaller than the length.

Griffiths et al. [26] proposed an asymptotic model for thin annular viscous non-axis-symmetric

geometries by taking into account the effects of inner pressure and surface tension, in order to model

the Vello process. In the Vello method, molten glass is supplied to a die of required shape, that is, not

necessarily cylindrical, and then it is pulled down from below [26]. In a subsequent publication,

Griffiths et al. [27] focused their analysis on some particular shapes, that is, the square and the

rectangular ones. Panda et al. [28] presented a quasi-one-dimensional model for the motion of curved

viscous fibers. This type of fibers may be produced in a typical rotational spinning process for the

production of glass wool. Marheineke et al. [29] improved the model of [28], by adding the effects of

surface tension. Nevertheless, a detailed asymptotic analysis for a cross-section containing an off-axis

or more than one hole has not been proposed yet.

In literature, there is a large amount of examples of numerical simulations of two-dimensional axis-

symmetric geometries. Many of them present a detailed analysis of the heat transfer between the

furnace and the glass, see for instance Lee et al. [30]. They solved the full problem in terms of the

stream function, in order to evaluate the possibility to introduce simplifying expressions. Their

simulations revealed the presence of velocity and temperature gradients, and a crucial role of heat

transfer coefficients and furnace temperature profiles. Choudhury et al. [31] included in their model

the presence of a peripheral inert gas flow, and that of the top and the bottom iris. The final fiber

geometry resulted from a force balance, comprising viscous, gravitational, inertia and shear force.

Surface and draw tension have also been included in the model. Yin et al. [32] discussed the influence

of the purge gas region, which serves to maintain the environment inert on the whole drawing process.

They concluded that the drawing speed and the furnace temperature have the major impact during the

drawing, while the purge gas effects can be neglected. In a two series contribution, Xue et al. [33] first

compared the transient draw process between a non-isothermal, quasi-one-dimensional model with the

full three dimensional Navier-Stokes equations (3D N.-St.), for the case of an axis-symmetric fiber.

5

They also model the isothermal drawing of a five-hole structure. Second, they [34] treated the

isothermal fiber drawing of different materials, examining different cross-sectional shapes. They

performed a force balance on a small volume of the fiber, showing which forces are responsible for the

hole-shape changes. Experimental results agree qualitatively well with the numerical prediction of the

hole shape variations. Xue et al. [35] examined the transient heating process of a preform, including

radiative heat transfer across the internal holes. A detailed comparison between a numerical model for

both axis and non-axis-symmetric geometries and experiments, performed for different drawing

regimes, different values of internal pressure and peak temperature, is not yet available in the

literature.

1.3 The subject and the structure of the thesis

The investigations presented in this thesis represent a result of a closed interdisciplinary cooperation.

Experimental, mathematical analytical and numerical works have been performed in close cooperation

between the Max Planck Institute (in German Max Planck Erlangen Gesellschaft (MPEG)) and the

Institute of Fluid Mechanics (in German Leherstuhl für Strömungsmechanik (LSTM)). The activities

of MPEG have their focus on the built-up and improvement of an experimental set up for fabrication

of single micro-fibers, micro-capillaries as well as fiber bundles. The fabrication process of the fiber

has been observed with the aid of a CCD-camera. The experiments concern particularly photonic

materials, heating temperature of the furnace, feeding and drawing velocities of the preform.

This thesis is dedicated to understand the thermo-fluid-dynamic processes during drawing of a heated

preform. Concretely, preform modeling, mathematical analytical analysis and numerical simulations

of the fluid-dynamic and capillary effects, which occur during the process of fiber drawing, have been

performed. For the sake of simplicity, the connection between thermodynamics and fluid mechanics is

in general modelled by assuming known temperature distributions, e.g., that of the furnace.

The remainder of this thesis is divided into four chapters.

In Chapter 2, a general description of optical fibres is given, focussing on their main application. The

waveguide principle is briefly explained. Based on the number of supported modes, optical fibres can

be divided in two types: single mode fibres, that is, fibres that support one confined transverse mode

by which light can propagate along the fibre, and multimode fibres, that is, fibres that support multiple

transverse guided modes for a given optical frequency and polarization, as described in the book of

Paschotta [36]. After that, PCFs are introduced. They offer high design flexibility: it is possible to

obtain fibres with different optical properties by simply changing the geometry of the fibre cross-

section. Furthermore, one can successfully design fibres with the desired guiding, dispersion, and

nonlinear properties. The two light-guiding mechanisms are explained, i.e., the modified total internal

reflection (MTIR) and the photonic band gap effect (PBG), see for instance Poli et al. [3].

6

In chapter 3, the basic equations of Fluid Dynamics are introduced in their Euler formulation. In

particular, the mass conservation and the momentum equation are presented, both in Cartesian and in

cylindrical coordinates. By introducing the Stokes hypothesis, i.e. making the assumption that the

stresses in the fluid are proportional to the gradients of the velocity, the six components of the stress

tensor can be written in term of velocity gradients times a viscous term, as deeply described in the

book of Schlichting et al. [37]. Taking into account the incompressibility of the flow, a closed system

of four equations in four unknowns is therefore obtained. Nevertheless, the basic equations governing

the fluid motion are essentially nonlinear. Therefore it is complicated to find exacts solution almost in

any branch of fluid mechanics. Sometimes, the order of the partial differential equations can be

reduced, if particular transformations are applied. Often, the resulting (partial) differential equations

are easier to integrate numerically. Alternatively, approximate solutions to a particular problem can be

found with the aid of perturbation methods. Perturbation theory consists of finding solutions in terms

of power series in some “small” parameter that quantify the deviation from the exactly solvable

problem. The perturbation theory is widely described in the famous book of Van Dyke [38]. In order

to find a solution of either a single equation or a system of equations, there are generally two

systematic procedures, i.e., the substitution of an assumed series and the iteration upon a basic

approximate solution. Each of the two methods has advantages and drawbacks, which are shortly

described [38]. Generally, one looks for an approximation for either small or large values of the

coordinates. In the first case one speaks of direct coordinate expansion, while in the second case one

speaks of inverse coordinate expansion [38].

In chapter 4, the most important results obtained in this research work are reported. The drawing

process of a single capillary can be described analytically by simplified equations of the fluid flow,

found by Fitt et al. [8], focussing on the physical insight these simplified equation of motion provide,

as described in the manuscript of Luzi et al.1 [39]. In particular, the effects of internal pressure and

surface tension are deeply investigated, since these two parameters have the major influence on the

final fibre cross-section. In fact, the optical properties of a micro-structured fibre depend upon the size

and the location of the holes in the cladding. Then, this simplified model has been compared with

experimental trials, finding a good agreement between the theoretical results and the experiments, for

the final size of the inner and the outer radius, see Luzi et al. [40]. Then, the asymptotic model has

been compared with the full 3D set of N.-St. equations, for the case of an axis-symmetric capillary, for

the ideal case of constant viscosity, see Luzi et al. [41]. The full 3D model allows for modelling of

more complex geometries, which is the case of photonic crystal fibres. Once a good agreement

between the two models has been achieved, a 3D six-hole geometry has been constructed, and

simulated numerically. The numerical results have been found in good agreement with experiments,

even in case of high values of internal pressure, see Luzi et al. [42].

1 Contributions of the present author have been cited in cursive.

7

Finally, in chapter 5 some conclusions are elaborated, giving an outlook for future investigations. It is

worth to emphasize even in this introductory chapter that the real temperature distribution inside the

furnace is still unknown, and it constitutes the major source of discrepancy between experiments and

simulations. In fact, under extreme conditions of high peak temperatures and internal pressure, the

fluid flow stresses, which determine the hole-shape and position in the cross-section, assume very high

values. Since they depend on the viscosity, and therefore on the temperature, an incorrect temperature

profile prescribed along the drawing direction may lead to a not precise computation of the stresses

themselves [42].

1.4 The aims and the achievements of the thesis

First, the model proposed by Fitt et al. [8] has been re-examined, centering on the physical

understanding of the simplified equation of motion [39]. Furthermore, the solution of the asymptotic

equations takes the mass conservation equation into account. The interplay between internal applied

pressure and surface tension effects needs particular attention, either when they both are considered in

the equations or when they separately appear in them, providing an analytical description of the fluid

flow during the drawing process of single hollow capillaries [39]. Since the optical properties of a

micro-structured fiber critically depend upon the size and the location of the holes in the cladding, it is

important to predict how the fabrication parameters influence the final fiber cross-section. In the

model considered here, mostly the long thin geometry of the draw region is profited.

The viscosity of high-purity glass has been selected from the data published by Urbain et al. [43]. This

information has been used to predict the results of experimental trials, demonstrating a good

agreement between the theoretical and the experimental data (the deviation range between 1% and

5%), not only when no internal pressure is applied, but also when the internal hole is pressurized. The

theoretical model deviates up to 20% in values from the experiments under severe conditions, that is,

high values of the peak temperature and internal applied pressure [40].

In a second step, the final diameters obtained by solving the full 3D set of N.-St. equations have been

compared to those obtained by solving the asymptotic ones and to those measured experimentally for

different peak temperatures. Moreover, the shape of the inner and the outer radius, the radial and the

axial stress, the radial and the axial velocities have been computed with both the N.-St. equations and

the asymptotic ones at three sections along the drawing. Thus, the two sets of solutions can be

compared for different cases; it is possible to show in detail where both solutions are in good

agreement, and where and why the results differ. This has been done considering a constant value for

the viscosity [41].

This comparison not only shed more light on the interplay between surface tension and internal

pressure, having the possibility to look inside the fiber by post-processing the results, but also serves

as a basis to model more complex geometries by a commercial tool. Asymptotic solutions have the

8

advantage that they can be obtained very fast, whereas the full N.-St. solution is very time consuming

and expensive. The shape of the inner and the outer radius is seen to deviate between the two models

when the asymptotic stresses differ significantly from the N.-St. ones. Indeed, the shape of the fiber

radii, computed with the two models, matches when the asymptotic stresses are very close to the N.-St.

ones, and the deviation is very small [41].

Once a good agreement among the final diameters, obtained by solving the two sets of equations, has

been obtained, a 3D six-hole geometry has been modeled numerically. Then, the numerical results

have been compared with real, experimentally drawn fiber structures. First of all, the final diameters

obtained by the numerical simulations have been compared to those measured experimentally in

absence of internal pressure. Then, internal pressure has been applied and the gradual hole-

deformation with increasing pressure have been observed and compared between the numerical

solutions and the experiments. Again, a good agreement both for the size of the final cross-section and

the shape of the holes has been attained, which results particularly deformed in case of high values of

internal pressure. The results obtained numerically starts to deviate from the experimental ones under

severe conditions, that is, high values of applied inner pressure [42].

In the present work, thermal equilibrium among the furnace gas, the fiber glass and pressure gas (in

case of inner applied pressure) has been assumed by guessing a temperature profile which has been

found to be suitable to represent the temperature distribution inside the furnace, providing good

agreement between experiments and simulations in most of the cases. The assumption of thermal

equilibrium seems to work well, unless under severe conditions, say, high values of peak temperatures

together with inner applied pressure are reached. For high values of internal pressure, when the

difference among experiments, numerical computations and asymptotic equations starts to be visible,

heat exchange among the furnace gas, the fiber glass and the nitrogen gas should be taken into

account. The real temperature profile is needed to exactly compute the fluid flow stresses which act on

each cross-section along the drawing direction.

REFERENCES

[1] Philip St.J. Russell, “Photonic-crystal fibers”, Journal of Lightwave Technology, Vol. 24, No. 12, pp. 4729-4749, 2006 [2] Arismar Cerqueira S. Jr, “Recent progress and novel applications of photonic crystal fibers”, Rep. Prog.Phys 73 (2010) 024401

doi:10.1088/0034-4885/73/2/024401

[3] Poli F., Cucinotta A., Selleri S., “Photonic Crystal Fibers: Properties and Applications”, Springer Series in Material Science, Vol. 102, 2007 (236 pp.).

[4] M.R. Matovich and J.R.A. Pearson, “Spinning a molten threadline-Steady-state isothermal viscous flows”, Ind. Eng. Chem. Fund,

8(3), (1969), pp. 512-520. [5] J.A. Burgman, “Liquid glass jets in the forming of continuous fibers”, Glass Technol. 11 (1970), pp. 110-116

[6] G. Manfre, “Forces acting in the continuous drawing of glass fibres”, Glass Technol. 10 (1969), pp. 99-106

[7] J.N. Dewynne, J.R. Ockendon and P. Wilmott, “On a mathematical model for fibre tapering”, SIAM J. Appl. Math. 49 (1989), pp. 983-990

[8] A.D.Fitt, K.Furusawa, T.M. Monro, C.P. Please and D.J. Richardson, “The mathematical modelling of capillary drawing for

holey fibre manufacture”, Journal of Engineering Mathematics, Vol. 43, Issue 7, pp. 201-227, 2002. [9] C.J. Voyce, A.D. Fitt, T.M. Monro, “The mathematical modelling of rotating capillary tubes for holey-fibre manufacture”, J. Eng.

Math., 2008, pp. 69-87, DOI 10.1007/s10665-006-9133-3

[10] Christopher. J. Voyce, Alistair.D. Fitt, and Tanya.M. Monro, “Mathematical Modeling as an Accurate Predicting Tool in Capillary and Microstructured Fiber Manufacture: The Effects of Preform Rotation”, Journal of Lightwave Technology, Vol. 26, No.

7, April 1, 2008, pp. 791-798

9

[11] Matthew, R. Myers, “A Model for Unsteady Analysis of Preform Drawing”, AIChE Journal, April 1989, Vo. 35, No. 4, pp. 592-602

[12] G. Gupta, W. W, Schultz, “Non-Isothermal Flows of Newtonian Slender Glass Fibers”, Int. J. Non-Linear Mechanics, Vol. 33, No

1, pp. 151-163, 1998

[13] H. Huang, R.M. Miura, and J.J. Wylie, “Optical Fiber Drawing and Dopant Transport”, SIAM J. Appl. Math, 69 (2), pp. 330-347, 2008

[14] P.D. Howell, J.J. Wylie, H.Huang, and R.M. Miura, “Stretching of Heated Threads with Temperature-Dependent Viscosity:

Asymptotic Analysis”, Discrete and continuous dynamical systems. Series B., Volume 7, Issue 3, pp. 553-572, 2007 [15] Jonathan J. Wylie, Huaxiong Huang, and Robert M. Miura, “Thermal Instabilities in drawing viscous threads”, J. Flui Mech.,

(2007), vol. 570, pp. 1-16, doi: 10.1017/S0022112006002709

[16] J.R.A. Pearson and M.A. Matovich, “Spinning a molten threadline-Stability”, Ind. Eng. Chem. Fundam. 8(4), (1969), pp. 605-609 [17] A.L.Yarin, P. Gospodinov, O. Gottlieb, M.D. Graham, “Newtonian glass fiber drawing: Chaotic variations of the cross-sectional

radius”, Physics of Fluids, Volume 11, No. 11, pp. 3201-3207, November 1999.

[18] Y.T. Shah and J.R.A. Pearson, “On the stability of nonisothermal fiber spinning”, Ind. Eng. Chem. Fundam. 11(2), (1972), pp. 145-149.

[19] Y.T. Shah and J.R.A. Pearson, “On the stability of nonisothermal fiber spinning-general case”, Ind. Eng. Chem. Fundam. 11 (2),

(1972), pp. 150-153. [20] P.Gospodinov, A.L.Yarin, “Draw resonance of optical microcapillaries in non-isothermal drawing”, International Journal of

Multifase Flow, Volume 25, Issue 5, September 1997, pp. 967-976

[21] Dr. Gyanesh K. Gupta, Dr. William W. Schultz, Dr. Ellen M. Arruda, Xiaoyong Lu, “Nonisothermal model of glass fiber

drawing stability”, Rheologica Acta, November/December 1996, Volume 35, Issue 6, pp. 584-596

[22] J.N.Dewynne, J.R.Ockendon &P.Wilmott, “A systematic derivation of the leading-order equations for extensional flows in slender

geometries”, J.Fluid Mech,244, pp. 323-338, 1992 [23] J.N.Dewynne, P.D.Howell &P.Wilmott, “Slender viscous fibres with inertia and gravity”, Quart. J. Mech. Appl. Math.,47, pp. 541-

555, 1994

[24] L.J.Cummings and P.D. Howell, “On the evolution of non-axis-symmetric viscous fibres with surface tension, inertia and gravity”, J. Fluid Mech.,(1999), vol 389, pp.361-389

[25] N.M.Ribe, “Bending and stretching of viscous sheets”, J. Fluid Mech., (2001), vol. 433, pp.135-160

[26] I.M.Griffiths and P.D.Howell, “The surface-tension-driven evolution of a two dimensional annular viscous tube”, J. Fuid Mech, (2007), vol. 593, pp. 181-208, doi: 10.1017/S0022112007008683

[27] I.M.Griffiths and P.D.Howell, “Mathematical modelling of non-axis-symmetric capillary tube drawing”, J. Fuid Mech, (2008), vol.

605, pp. 181-206, doi: 10.1017/S002211200800147X [28] Satyananda Panda, Nicole Marheineke and Raimund Wegener, “Systematic derivation of an asymptotic model for dynamics of

curved viscous fibers”, Math, Meth, Appl. Sci., Vol. 31, Issue 10, pp. 1153-1173, 10 July 2008

[29] Nicole Marheineke and Raimund Wegener, “Asymptotic model for the dynamic of curved viscous fibres with surface tension”, J. Fuid Mech, (2009), vol. 622, pp. 345-369, doi: 10.1017/S0022112008005259

[30] S. H.-K. Lee and Y. Jaluria, “Simulation of the transport processes in the neck-down region of a furnace drawn optical fiber”, Int. J.

Heat Mass Transfer, Vol. 40, No 4, pp. 843-856, 1997 [31] S. Roy Choudhury and Y. Jaluria, “Practical Aspects in the drawing of an optical fiber”, J. Mater. Res. Vol 13, No 2, pp. 483-493,

Feb 1998

[32] Zhilong Yin and Y. Jaluria, “Thermal Transport and Flow in High-Speed Optical Fiber Drawing”, J. Heat Transfer 120 (4), pp. 916-930, (Nov 01, 1998), (15 pages), doi:10.1115/1.2825911

[33] S. C. Xue, R.I. Tanner, G.W. Barton, R. Lwin, M.C.J. Large and L. Poladian, “Fabrication of Microstructured Optical Fibres-

Part I: Problem Formulation and Numerical Modelling of Transient Draw Process”, Journal of Lightwave Technology, Vol. 23, Issue 7, pp. 2245-2254, 2005, doi: 10.1109/JLT.2005.850055

[34] S. C. Xue, R.I. Tanner, G.W. Barton, R. Lwin, M.C.J. Large and L. Poladian, “Fabrication of Microstructured Optical Fibres

Part II: Numerical Modeling and Steady-State Draw Process”, Journal of Lightwave Technology, Vol. 23, Issue 7, pp. 2255-2266, 2005, doi: 10.1109/JLT.2005.850058

[35] S.-C. Xue, L. Poladian, G.W. Barton, M.C.J. Large, “Radiative heat transfer in preforms for microstructured optical fibres”,

International Journal of Heat and Mass Transfer, Vol. 50, Issues 7-8, April 2007, pp. 1569-1576 [36] Rüdiger Paschotta, “Field Guide to Optical Fiber Technology”, SPIE Press Book, January 8, 2010, (128 pp.).

[37] H. Schlichting, K. Gersten, “Boundary Layer Theory”. Springer; Auflage; 8th ed. 2000 Corr. 2nd printing 2003 (Dezember 2003)

ISBN-10: 3540662797 (802 pp.). [38] M. Van Dyke, “Perturbation Methods in Fluid Mechanics”, Applied Mathematics and Mechanics Series, Vol. 8, Academic Press,

New York, London, 1964 (230 pp.). [39] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Asymptotic analysis of

flow processes at drawing of single optical microfibres”, International Journal of Chemical Reactor Engineering, Vol 9,Issue 1, No

A65, 2011, pp. 1-26 . [40] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Influence of surface

tension and inner pressure on the process of fiber drawing”, Journal of Lightwave Technology, Vol 28, No 13, pp. 1882-1888, July 1,

2010 [41] Giovanni Luzi, Philipp Epple, Cornelia Rauh, Antonio Delgado, “Study of the effects of inner pressure and surface tension on the

fibre drawing process with the aid of an analytical asymptotic fibre drawing model and the numerical solution of the full Navier-

Stokes equations”, Archive of Applied Mechanics, Vol. 83, Issue 11, pp. 1607-1636, 2013

[42] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Numerical solution and

experimental validation of the drawing process of six-hole optical fibers including the effects of inner pressure and surface tension”,

Journal of Lightwave Technology, Vol 30, No 9, May 1, 2012, pp. 1306-1311 [43] G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates”, Geochimica Cosmochimica Acta,

vol. 46, no. 6, pp. 1061-1072, 1982

10

Chapter 2:

Basics of Conventional and Photonic Crystal Fibers

2.1 General description

In this chapter, a general description of optical fibers (conventional and photonic crystal ones), the

principles upon which they work and their main applications is given, since this thesis aims at

understanding the thermo-fluid-dynamic properties occurring in fiber drawing processes. In literature,

the properties, design and application of optical fibers are treated extensively. Furthermore, excellent

overviews are provided by different authors, see for instance Tyagarajan et al. [1],Ajoy et al. [2] or

Bagad [3]. Therefore, a complete treatment of these contributions is neither possible here nor it is

aimed to.

Optical fibers are waveguides which allow the transmission of light in form of electromagnetic waves

[3]. They typically consist of a core surrounded by a cladding, whose refractive index is lower than

that of the core. The light is guided within the core of higher refractive index by a physical mechanism

called total internal reflection (TIR), see for instance [1], [2] or [3]. Optical fibers find applications in

various fields: for instance, they are widely used in communications, since light can be transmitted

over distances of hundredths kilometers. If fibers are made of silica glass, a long transmission is

accomplished without any additional amplification.

Fig. 2.1: Bundle of optical fibers, Mellish [4]

Fibers are also widely used for illumination purposes; for instance, in medicine, an endoscope contains

many optical fibers, which are able to produce clear images of the insight of a human body. They are

also used to display road signs, which can be updated at any time, or they may be also used under

11

severe conditions, like, for instance, the illumination of ships, where the corrosive salt water makes

arduous the installation of electrical components, or the illumination of ambient in presence of

explosive gases [3].

2.2 Some basics of optical fibers

As already mentioned in the previous chapter, the light propagates as a wave along the optical fiber,

specifically by a principle called TIR. The light is usually launched into the core. If the light hits the

boundary between the core and the cladding with an angle which is greater than a “critical angle”, then

it bounces from the surface without propagating into the cladding, and it remains confined into the

core [3].

The light distribution inside the fiber can be computed by using the beam propagation method, as

described in the book of Paschotta [5]. If an initial electric field distribution is known at the inlet of an

optical fiber, then one can compute the field distribution in the whole fiber. In case of simple

geometries and small differences between the refractive index of the core and that of the cladding, this

can be computed analytically, while for complex geometries this has to be done numerically.

The field distributions whose shape does not vary in a cross-sectional plane perpendicular to the

propagation direction are termed modes [5].

Different types of modes exist, that is, guided modes, leaky modes and cladding modes. The firsts

ones are confined into the core with small losses, the second ones are also mostly confined into the

core but they show some losses into the cladding, while the third ones are confined in the cladding.

Two important parameters in the design of optical fibers are the numerical aperture NAand the V

number: the first one depends on the refractive indexes of the core and that of the cladding and it is

mainly related to the propagation losses, while the second one is associated with the number of guided

modes.

If the refractive index of the core and the one of the cladding are constant along the radial and the

propagation directions, one speaks of step-index fibers, while if the refractive index of the core

diminishes from the fiber axis toward core outer radius, one speaks of graded-index fibers.

To give an example, in case of step-index multimode fibers the NAreads

2 2

core claddingNA n n

(2.2.1)

while the V number is

2222claddingcore nnaaNAV

(2.2.2)

for more details, see [5]

12

2.3 Single and multimode fibers

Single-mode fibers (SMFs) convey only one single guided mode. A typical arrangement of a SMF is

sketched in Fig. 2.2

Fig. 2.2: The sketch of the structure of a SMF: the core, the cladding, the buffer and the jacket with their

respective dimensions [4]

The diameter of the core is very small and it varies between 8.3 [μm] and 10 [μm], see for instance the

website of the ARC Electronics [6]. Therefore, in order to obtain an efficient launch of the light inside

the fiber, strict conditions regarding the light source, the light direction and the beam profile and

alignment must be observed, as described in the website RP Photonics Encyclopedia of Paschotta [7].

A buffer layer is usually added to the cladding and everything is jacketed in order to protect the fiber

mechanically. SMFs are usually employed for transmissions over long distances, since the intermodal

dispersion is absent and the propagation losses are low.

In general, an optical fiber is multimode, since not only one but many modes can be supported, but it

becomes single-mode within a specific wavelength range.

In case of step-index fibers, the condition 2.405V ensures the single-mode propagation, see for

instance the website RP Photonics Consulting GmbH of Paschotta [8].

Multi-mode fibers (MMFs) convey several or many guided modes. Usually, MMFs have larger core

areas in comparison to SMFs, see Fig. 2.3, and higher numerical aperture, as explained in the website

RP Photonics Encyclopedia of Paschotta [8]. Due to that, the efficient launching conditions inside the

fiber are not restricted as in the case of SMFs, and it is just necessary that the light beam is directed

into the core with proper angle. Therefore, MMFs find application in short-distance transmissions,

where the light beam is generated by cheaper devices. The distance is constrained by the phenomenon

of the intermodal dispersion.

13

MMFs are denoted by the core and the outer diameter: for instance, a pair of values like 50/125

denotes the core (50 [μm]) and the cladding (125 [μm]) diameters, for more details see [8].

Fig. 2.3: The structure of a conventional optical fiber: single-mode fiber (left), and multi-mode fiber (right),

Paschotta, Encyclopedia of Laser Physics and Technology [9]

2.4 Basics of PCFs

PCFs are micro-structured fibers, possessing a two-dimensional (2D) photonic crystal structure at each

cross-section of the fiber axis, as described in the manuscript of Knight [10]. This structure is made of

air holes which extend throughout the whole length of the fiber.

In order to manufacture PCFs, usually a single material is used, like pure silica, but doped material is

also employed. These fibers provide light guidance in two ways: either by confining the light in a

central solid core, or by enclosing it in a central air-hole, see for instance Russell et al. [11]. In the first

case, the light travels along the fiber due to physical principle known as modified total internal

refraction (MTIR), which is quite similar to the guiding mechanism of standard optical fibers. In fact,

the average refractive index of the cladding is lower than that of the solid core. An example of a solid

core PCF is shown in Fig. 2.4: a sketch is presented in Fig. 2.4 a), while an example of a real solid-

core PCF is shown in Fig. 2.4 b). The small triangular lattice air holes have a diameter d of about 300

[nm] and a hole-to-hole spacing of 2.3 [μm]. Birks et al. [12] observed experimentally that only a

single mode can propagate though this fiber, even at very short wavelengths, in the form of a single

14

circular lobe. Higher order modes are not trapped by photonic crystal structure, and they leak out

though the gap of the glass.

Therefore, a solid-core PCF can be designed in such a way that only one or few modes are guided.

This can be accomplished by increasing the mode area of the fiber, that is the dimension (and the

number) of the air holes. Therefore a high power can be concentrated and transmitted in a small

Fig. 2.4: Sketch of a solid-core PCF with a triangular lattice of air-holes a). Microscope picture of a real solid-

core triangular PCF b) Poli et al. [13].

solid region. On the other hand, small mode area fibers present flat dispersion curves [13]. In the

second case, light is confined in a core with a lower refractive index than that of the cladding. The stop

bands, that is, the modes which cannot propagate through the cladding remain trapped inside the air-

core and propagate. Therefore the presence of a photonic band gap (PBG) is necessary for light

propagation [11]. The light has to be launched inside the fiber with a proper value of the propagation

constant , such that the transmission through the cladding region is forbidden. describes the

changes of the amplitude and phase of a wave in the direction of the propagation, see for example

Paschotta [14]. The condition0 1k , where

0k is the free-space propagation constant, ensures the

light propagation within the hollow core [13]. A sketch of the cross-section of a hollow core fiber is

presented in Fig. 2.5 a), while an example of a real hollow core fiber is given in Fig. 2.5 b). A large air

filling fraction, that is, the presence of big air holes is necessary for the hollow core guidance. If

properly designed, a hollow-core PCF exhibit a robust guidance of the propagating modes.

15

Fig. 2.5: Sketch of the cross-section of a photonic band-gap, hollow-core fiber with a triangular lattice of air-

holes a). Microscope picture of a real hollow-core PCF b) [13].

2.5 Optical properties

Solid-core PCFs have new optical properties related to the birefringence, dispersion, nonlinearities and

number of guided modes.

PCFs can be easily made birefringent, for instance, by introducing in the first ring of the structure two

diametrically opposite capillaries of different size. The level of birefringence achieves higher values

with respect to conventional fibers. Furthermore, birefringence is unaffected by the temperature unlike

in conventional birefringent fibers, which are made of different glasses with distinct thermal expansion

coefficients [13].

Dispersion can be tailored in several ways: for instance, the zero-dispersion wavelength moves to the

visible range if the central core area results very small [13], or ultra-flattened dispersion curves are

obtained with very small holes, see for instance the manuscript of Reeves et al .[15]. The dispersion

curves of two PCFs with the hole-diameters in the sub-micron range are reported, showing that they

remain confined near the zero-dispersion.

Nonlinear effects are obtained by concentrating the light in a very small core compared to the

dimension of the air-holes. The resulting field intensity is very high, giving rise to numerous nonlinear

effects which are responsible for the enlargement of the frequency spectrum of the light pulses [11].

This phenomenon is called “super-continuum generation”, and it is exploited, for instance, in

spectroscopy, interferometry and microscopy, due to the higher brightness achievable in comparison to

common supercontinuum sources. Furthermore, the problem of coupling the light source with an

optical fiber is automatically solved, see for instance the article of NKT Photonics [16].

16

Hollow-core PCFs are characterized by very low nonlinearities in comparison to solid-core silica

fibers, see for instance the manuscript of Ouzounov et al [17]. Furthermore, they are able to transmit a

higher optical power with respect to solid fibers. Therefore, they may find application both in

telecommunications and in high power delivery processes, as described in the paper of Humbert et al.

[18]. They compared the performances of a 7-unit-cell and a 19-unit cell hollow core PCFs. The

authors concluded that the choice of the right structure for laser beam delivery depends on the

wavelength of the pulses. Further applications are, for instance, the stimulated Raman scattering in

gases and the particle guidance within the hollow core [11].

REFERENCES

[1] K. Thyagarajan, Ajoy K. Ghatak, “Fiber Optic Essentials”, Wiley-Interscience, September 10, 2007, pp 242 [2] Ajoy K. Ghatak, K. Thyagarajan, “An Introduction to Fiber Optics”, Cambridge University Press, June 28, 1998 pp. 584

[3] V.S. Bagad, “Optical Fiber Communications”, Technical Publications, January 2008, pp. 246

[4] Bob Mellish, “Wikipedia - Optical Fibres” http://en.wikipedia.org/wiki/Fibre_optics

[5] Rüdiger Paschotta, “Field Guide to Optical Fiber Technology”, SPIE Press Book, January 8, 2010, (128 pp.).

[6] ARC Electronics, "Fiber Optic Cable Tutorial", (2007-10-01)

[7] Dr. Rüdiger Paschotta, “RP Photonics Encyclopedia”, http://www.rp-photonics.com/single_mode_fibers.html [8] Dr. Rüdiger Paschotta, “RP Photonics Encyclopedia”, http://www.rp-photonics.com/passive_fiber_optics3/html

[9] Dr. Rüdiger Paschotta, “RP Photonics Encyclopedia”, http://www.rp-photonics.com/multimode_fibers.html?s=ak

[10] J. C. Knight, “Photonic crystal fibres”, Nature, vol. 424, pp. 847–851, Aug. 2003 [11] P. St. J. Russell et al, “Photonic Crystal Fibers”, Science 299, 388 (2003), pp. 358-362 (Review article) DOI:

10.1126/science.1079280

[12] T.A. Birks, J.C. Knight and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber”, Optics letter, Vol. 22, Issue 13, pp. 961-963, (1997)

[13] Poli F., Cucinotta A., Selleri S., “Photonic Crystal Fibers: Properties and Applications”, Springer Series in Material Science, Vol.

102, 2007 (236 pp.). [14] Dr. Rüdiger Paschotta, “RP Photonics Encyclopedia”, http://www.rp-photonics.com/propagation_constant.html

[15] W. H. Reeves, J. C. Knight, P. St. J. Russel, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal

fibers”, Optics Express, Vol. 10, pp 609-613, July 2002. [16] NKT Photonics, “Supercontinuum Generation in Photonic Crystal Fibers”, V 2.0, July 2009

[17] D. G. Ouzounov, F.R. Ahmad, D. Müller, N. Venkatamaran, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch and A. L.

Gaeta, “Generation of megawatt optical solitons in hollow-core photonic bandgap fibers”, Science, Vol. 301, pp. 1702-1704, Sept. 2003

[18] G. Humbert, J. C. Knight, G. Bowmans, P. St. James Russell, D.P. Williams, P.J. Roberts and B.J. Mangan, “Hollow core

photonic crystal fibers for beam delivery”, Optics Express, Vol. 12, pp 1477-1484, Apr. 2004, http://www.opticsexpress.org/abstract.cfm?URI=oe-12-8-1477

17

Chapter 3:

Basic Equations and Main Techniques

3.1 Basic equations

The basic equations of fluid mechanics are the continuity, momentum and energy equation. The “Euler

form” of the continuity equation, for the general case of compressible flows, reads in Cartesian

coordinates

0

i

i

u

t x

(3.1.1)

See, for instance Schlichting et al. [1]. Herein, is the density, iu is the velocity vector, t is the time

and ix represent the spatial coordinate vector of the fluid flow.

In case of incompressible fluid flows, equation (3.1.1) simply reads

0i

i

u

x

(3.1.2)

The momentum equations of fluid mechanic express the Newton´s second law, which states that the

time variations of the momentum in the i direction is balanced by the sum of the external forces acting

in the same direction on the fluid element. The “Euler form” of the momentum equations reads

ij

i i j i

j i j

pu u u f

t x x x

(3.1.3)

Herein, i if g represents the body force per unit volume, in which ig is the vector of gravitational

acceleration, p is the scalar pressure field, and ij is the viscous stress tensor, for more details see [1].

If the fluid is assumed to be Newtonian, the unknown terms ofij are expressed in terms of the

derivative of the velocity field with respect to the space coordinates, and they are linearly proportional

to the dynamic viscosity , i.e.

2

3

ji kij ij

j i k

uu u

x x x

(3.1.4)

Under the assumption of incompressibility, the corresponding viscous stress tensor simplifies, that is

jiij

j i

uu

x x

(3.1.5)

In cylindrical coordinates, for axis-symmetric flows, (3.1.2) and (3.1.3) become [1]

18

1

0x ru rw

r

(3.1.6)

grwr

urur

pwuuuu xxxrrxrxt 1

)2(1

(3.1.7)

2

2t x r r x r r rr x xr

w uw ww p rw w u wr

(3.1.8)

in which equation (3.1.5) has been used. In equations (3.1.6), (3.1.7) and (3.1.8), u and w represent the

axial and the radial component of the velocity, respectively. The subscripts denote the derivatives,

precisely, t is the time, x is the axial coordinate and r is the radial one.

3.2 The perturbation theory

As it can be seen from the previous chapter, the basic equations of fluid mechanics, (3.1.1) and (3.1.3)

are non-linear, or precisely, quasi linear. Thus, it is difficult to find exact solutions of such equations.

One usually looks for self-similar solutions; in most of the cases, the partial differential equations

reduce to ordinary differential equations, which are easier to integrate numerically, see for instance

Hiemenz [2], for the case of plane stagnation-point flow or Gersten [3] and [4], for the case of suction

or blowing in a porous wall.

On the other hand, approximations are usually valid when one or more of the parameters or variables

of the problem are either large or small. The latter case represents a core subject of the thesis. In

literature, there are a large number of publications and books regarding this field, see for instance

Simmonds et al. [5], Jordan et al. [6], or Van Dyke [7]. Therefore perturbation theory is treated only to

the extent required in this thesis. If the perturbation quantity is a parameter, one speaks of a parameter

perturbation. Instead, if the perturbation quantity is one of the independent variable, one speaks of a

coordinate perturbation [7].

Usually, the perturbation quantity is denoted by ε. As ε approaches zero, the limit of the flow is the

basic solution or the “zeroth” approximation. The first term of the perturbation is called the first

approximation or the first order solution.

A perturbation quantity is never uniquely defined; for instance, the thickness parameter for a slender

body may be taken as it thickness ratio or its mean slope. Furthermore, it may be changed by a

constant multiplier [7].

3.2.1 Main Techniques

In perturbation theory, several techniques allow for finding approximate solutions of the equations

(3.1.1)-(3.1.3). Herein, only approximations which depend on a limit process are taken into account.

The result becomes exact when a perturbation quantity approaches zero or some other values.

19

Therefore, the behavior of the solution is considered by varying such perturbation quantity, leaving the

other coordinates and parameters fixed, that is, one sees how the function f behaves

as approaches zero. This is possible in many ways, and six of them are described in [7], with an

increasing order of refinement. In this thesis, the most precise one has been used, that is, the sixth one.

It consists of constructing an asymptotic expansion or expansion series to N terms of the unknown of

the equations (3.1.1)-(3.1.3). In general, a function f is expressed in terms of gauge

functions n of the form

01

21

asccccfN

n

nnNN (3.2.1)

defined by

01

asocf N

N

n

nn (3.2.2)

If the function f and the gauge functions n are known, the coefficient mc of the asymptotic

expansion can therefore be computed

m

M

n

nn

m

cf

c

1

1

0lim

(3.2.3)

for more details see [7]. If the number N of terms is infinite, then the asymptotic series is infinite. If

the gauge functions are all integral positive powers of , the series is called an asymptotic power

series. The simplest way to find approximate solutions of fluid mechanic problems by means of a

perturbation technique is to guess an asymptotic sequence nn o 1 consisting of integral powers,

say, n , bearing in mind that fractional powers or logarithms, may also occur. The asymptotic

sequence and the asymptotic expansion become unique once the perturbation quantity and the gauge

function are specified. In general, there are no rules which dictate how to choose the proper asymptotic

sequence; in general if the solution progresses properly, all terms match.

The resulting series may be either convergent or not, but this issue has not much practical value, since

one computes the first few terms of the series [7].

3.2.2 Main Procedures

In order to find an approximate solution by successive approximation of systems of ordinary or partial

differential equations, together with appropriate initial and boundary conditions, there exist two main

systematic procedures, that is

1. Substitution of an assumed series

2. Iteration upon a basic approximate solution

20

for more details see [7]. Usually, one often works with combinations of the two methods, as it has

been done in this thesis. If an initial approximate solution is known, then iterations can be started,

producing a group of terms in a single step. Therefore, there is no need to guess an asymptotic

sequence. On the other hand, an assumed series expansion is more automatic, and the unspecified

sequence may be determined by comparing terms of like powers of . Even if beyond the second term

it only generates significant results, it requires several steps in order to generate a group of terms [7].

REFERENCES

[1] H. Schlichting, K. Gersten, “Boundary Layer Theory”, Springer; Auflage; 8th ed. 2000 Corr. 2nd printing 2003 (Dezember 2003) ISBN-10: 3540662797 (802 pp.).

[2] Hiemenz K. (1911), “Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsström eingetauchten geraden Kreiszylinder”,

Dinglers Polytech Journals, Vol. 326, pp.321-410 [3] Gersten K., Gross, J.F. (1974a), “Flow and heat transfer along a plane wall with periodic suction”, Z. angew. Math. Phys. (ZAMP),

25.5.1974, Volume 25, Issue 3, pp. 399-408

[4] Gersten K., Gross, J.F. (1974b), “The flow over a porous body; a singular perturbation problem with two parameters”, L´Aerotecnica Missili e Spazio, Vol. 53, August 1974, pp. 238-250

[5] Simmonds J. G., Mann J. E. Jr. , “A first look at perturbation theory”, Dover Books on Physics, Dover publications; 2nd edition

(July 10, 1997), (160 pp.), ISBN-10: 0486675513 [6] Jordan D.W., Smith P, “Nonlinear Ordinary Differential Equations: an introduction for scientists and engineers”, Oxford Texts in

Applied and Engineering Mathematics, Oxford University Press; 4nd edition (October 11, 2007), (560 pp.), ISBN-10: 0199208255

[7] M. Van Dyke, “Perturbation Methods in Fluid Mechanics”, Applied Mathematics and Mechanics Series, Vol. 8, Academic Press, New York, London, 1964 (230 pp.).

21

Chapter 4:

Results

4.1 Problem formulation

The present thesis aims at studying the motion of fibers during the manufacturing process, starting

from the simplest case of axis-symmetric capillaries: Fig. 4.1 schematically represents the problem.

Fig. 4.1: Schematic diagram of drawing processes, Fitt et al. [1]

The drawing process leads to lengthening and, at the same time, to local contraction of the fiber. Thus,

the radius h at the local distance x of the fiber depends on the time t and on the drawing rate.

The material most commonly used is glass silica of various grades. Some common properties of the

glass can be found in [1]; the surface tension is of the order of 0.3 [N/m], the density is 2800

[kg/m3], and the dynamic viscosity of the molten glass is 37000 [Kg/(m*sec)].

The problem can be treated adequately by starting from the continuity and the momentum equations of

fluid mechanics, written in cylindrical coordinates, for incompressible and axis-symmetric flow fields,

that is, equations (3.1.6), (3.1.7) and (3.1.8). In addition, the molten glass can be treated as a

Newtonian fluid at low rates of deformations, as described in the book of Uhlmann et al. [2].

22

There is no doubt that the temperature has the most significant impact on the material parameters.

Nevertheless, temperature variations over the range of interest only affect significantly the viscosity,

which changes by several orders of magnitude. Other parameters do not vary critically with the

temperature within this range, for more details see the work of Lee et al. [3].

On the other hand, since the characteristic diffusive thermal time of the molten glass lies substantially

above than that of the convective momentum transport, it is a reasonable approximation to solve first

the momentum equations independently from the energy balance, and then correcting them afterwards.

Equations (3.1.6), (3.1.7) and (3.1.8) describe the behaviour of the fluid flow in the region comprised

within the inner and the outer region of the capillary, termed ),(1 txhr and ),(2 txhr , respectively,

see Fig. 4.1 above. In order to solve the problem in a closed form, boundary and initial conditions

must be formulated adequately. This is presented in the following chapter.

4.2 Problem solution

The solution of the problem consists of three basic steps. The equations of motion (3.1.6)-(3.1.8) are

first transformed into a set of non-dimensional equations by applying a suitable scaling, and then the

unknowns, i.e. velocities and pressure, are expanded in power series of the so called “slenderness

parameter” Lh / , that is,

2

0 1, , ,u u x t u x r t

2

0 1, , , ,w w x r t w x r t

2 , ,ap p P x r t

where the over-bars denote non-dimensional quantities [1]. Herein, h indicates the difference between

the outer and the inner radii of a typical drawn capillary, and L denotes the length of a typical “hot

zone”, as it will be specified later on. u and w are the axial and the radial component of the velocity

vector, respectively, and p is the pressure. 0u and 0w are the axial and the radial leading order

component of the velocity vector in the asymptotic expansion, while 1u and 1w are the axial and the

radial first order component of the velocity vector in the asymptotic expansion. ap represents the

ambient pressure and P is the first order term of the asymptotic expansion of the pressure variable.

t indicates the time, while x and r indicate the spatial and the radial position, respectively, for more

details see [1].

The second step makes use of the small value of in order to simplify the equations of motions

considering the kinematic and the dynamic boundary conditions on the free boundaries.

The kinematic boundary conditions states that, at the boundary, a surface is always composed by the

same fluid particles. For the two free surfaces, in cylindrical coordinates, they read

23

0 1 0 1 1t xw h u h at r h

(4.2.1)

0 2 0 2 2t xw h u h at r h

(4.2.2)

The dynamic boundary conditions constitutes a force balance across a surface, in the normal

0 01 1 12

1

T HU p Un n at r h

h L L

(4.2.3)

0 02 2 22

2

T aU p Un n at r h

h L L

(4.2.4)

and in the tangential direction,

1 1 10Tt n at r h

(4.2.5)

2 2 20Tt n at r h

(4.2.6)

respectively, for more details see [1]. In the equations above, in and it are the unit vectors normal and

tangential to the surface, and i assumes the values of 1 and 2. is the stress tensor, Hp is the hole

pressure, ap is the ambient pressure, is the surface tension, 0 andU are typical reference dynamic

viscosity and axial velocity, respectively. The subscripts denote derivatives.

For brevity, the first two steps that lead to the solution of the problem will not be discussed here, since

one can find them described in details in [1]. The simplified equations of motion, found by [1], in the

final dimensional form read

2 2 2 2

2 1 0 0 0 2 1 0 1 23t x xx

h h u u u g h h u h h

(4.2.7)

2 2

0 1 2 1 2 1 22 2

1 1 0 2 2

2 1t x

p h h h h h hh h u

h h

(4.2.8)

2 2

0 1 2 1 2 1 22 2

2 2 0 2 2

2 1t x

p h h h h h hh h u

h h

(4.2.9)

where 0 H ap p p . Those equations represent the starting point of the present work. Equation

(4.2.7) is a simplified momentum equation, while equations (4.2.8) and (4.2.9) represent evolution

equations for the radii 1h and 2h . Adequate initial and boundary conditions for the system of equations

(4.2.7)-(4.2.9) are

1 10, 2 20, 0 00, 0, 0, , , ,f dh x t h h x t h u x t U u x L t U

(4.2.10)

[1]. The geometry of the preform is known, i.e. the values of the inner and of the outer radii

at 0x [m], as well as the feeding and the drawing velocities, fU , and dU , respectively.

Equations (4.2.7), (4.2.8) and (4.2.9) offer the possibility to reduce the enormous mathematical and

numerical efforts often required for solving the general equation model (3.1.6)–(3.1.8). Additionally,

basic physical features of the flow become better accessible. In this context, studying asymptotic cases

offers unique possibilities for optimizing the control of the drawing process.

24

4.3 Asymptotic solutions and their implementation into a Matlab

program

In the forthcoming sections, several asymptotic limits of the system of equations (4.2.7)-(4.2.9) are

presented. They provide meaningful information about the physical effects which influence the fiber

drawing process. Even if they are already discussed in [1], they are briefly re-proposed here, in order

to introduce the reader to the topic and clarify the treatise of the subject. Precisely, the results

presented in the sections (4.3.1), (4.3.2), (4.3.4), (4.3.5) and (4.3.6) can be found in [1]. Some

derivations of the final results, which are not given in [1], are not straightforward or they are even

formidable and laborious. For completeness, they are available in the corresponding appendices, along

with a discussion about the domain of applicability and graphical visualizations of the results. The

results of the sections (4.3.3), (4.3.7) and (4.3.8) are exposed in peer reviewed publications by the

author of this thesis, and they constitute a salient part of the achievement of this thesis, as it will be

clarified later on. Therefore, they are described here in details. Moreover, a table, called Table I,

containing the equations of the sections (4.3.1)-(4.3.8) and their corresponding solution, is presented at

the end of this section, in order to have a graphical distinction between the results obtained by [1] and

those obtained by the author.

Common values of feeding velocities are 0.0001fU [m/sec], while those of drawing velocities are

0.04fU [m/sec]. In the present thesis, the length of the furnace 0.3L [m], that is, the total length

of the furnace, has been used for ideal cases with constant viscosity, while the length 0.1L [m],

centred in the middle of the furnace, has been chosen for real cases of variable viscosity, as described

in details in paragraph 4.4.

In order to systematically examine the asymptotic limits of interest in the system of equations (4.2.7)-

(4.2.9), a computer program, named “Fiberspinning.m”, has been written in Matlab 7.8. Matlab 7.8 is

a high-level programming language mainly intended for numerical computations. It is possible to

deeply investigate the physical effects which affect a fiber drawing process by retaining the effects of

inertia, gravity, surface tension, internal pressure and viscosity in the equations, first separately and

then together. “Fiberspinning.m” not only reproduces the plot of the results already derived by [1], but

it is able to simulate real axis-symmetric fiber drawing cases, thus allowing the comparison with

experiments. In Fig. 4.2 it is possible to see the head of the program “Fiberspinning.m”. By typing the

name “Fiberspinning(n)”, where n is an integer between 1 and 10, the program starts to run. Each case

that can be selected is fully investigated in the next subsections, specifically:

Fiberspinning(1): Hollow fiber: the effects of surface tension, inertial and gravitational forces

and internal hole pressure are neglected. Both cases with constant and variable viscosity are

examined; see subsection 4.3.1.

25

Fiberspinning(2): Only inertial forces are considered. The viscosity is constant; see subsection

4.3.2.

Fiberspinning(3): Only gravitational forces are considered. The viscosity is constant; see

subsection 4.3.2.

Fiberspinning(4): Effects of surface tension. The viscosity is constant; see subsection 4.3.3.

Fiberspinning(5): Only effects of small viscosities are considered, where <<1. The viscosity

is constant; see subsection 4.3.4.

Fiberspinning(6): Only internal pressure is considered. The viscosity is constant; see

subsection 4.3.5.

Fiberspinning(7): Interplay between internal pressure and surface tension. The inertial and the

gravitational forces are neglected, and the viscosity is assumed constant; see subsection 4.3.6.

Fiberspinning(8): Interplay between internal pressure and surface tension. The inertial and the

gravitational forces are neglected, but the viscosity is variable; see subsection 4.3.6.

Fiberspinning(9): Interplay between internal pressure and surface tension, when a very small

hole is considered. The viscosity is constant; see subsection 4.3.7.

Fiberspinning(10): Interplay among internal pressure, surface tension, inertial and

gravitational effects with variable viscosity. The general case is considered; see subsection

4.3.8.

26

Fig. 4.2: Head of the program Fiberspinning.m, in which it is shown how to choose among different cases, in order

to select the physical effect(s) one wants to analyse.

27

Considering, for instance, only the effects of a variable viscosity along the drawing direction, by

typing “Fiberspinning(1)” in the prompt of Matlab 7.8, the following result for the evolution of the

two radii is obtained, see Fig. 4.3.

Fig. 4.3: Evolution of the internal and external radii of a preform, in case that only the effects of a

variable viscosity are considered. The viscosity varies along the axial position. Matlab image.

Furthermore, “Fiberspinning(n)” displays in the prompt of Matlab 7.8, where requested, several

results. For instance, it prints the value of the radii1h and

2h , the one of the asymptotic velocity0u , the

one of the radial stress rr etc.

All the results presented in the next sections of this chapter, and many of them presented in the

following chapters (where appropriately specified) have been obtained by the program

Fiberspinning.m. For a better visualization, the program Origin 8.0 has been employed to reproduce

the graphs.

4.3.1 The simple case

The simplest possible situation is to consider the steady state case, neglecting the internal hole-

pressure, surface tension, inertial and gravitational effects, and assuming the viscosity constant with

the temperature. Therefore, one obtains from (4.2.7), (4.2.8) and (4.2.9), after introducing the initial

and the final conditions (4.2.10)

28

L

xUu f exp0

(4.3.1)

0

101u

Uhh

f

(4.3.2)

0

202u

Uhh

f

(4.3.3)

where )/log( fd UU , for more details see [1].

Taking into account the viscosity dependence on the temperature, one obtains a slightly different

equation for the axial velocity, i.e.

00

0

( ( ))exp

( ( ))

x

Lf

d

Tu U

d

T

(4.3.4)

while the evolution equations for the radii 1h and 2h are the same as (4.3.2) and (4.3.3), [1].

A consequence of the fact that no force acts in the plane of the fibre cross section is that the initial

ratio between the two radii, that is, 10 20h h is preserved. A complete derivation of the final equations

(4.3.1), (4.3.2), (4.3.3) and (4.2.4) is given in Appendix A, where it is also shown that the viscosity

term simplifies throughout the computations, for the case of constant viscosity.

4.3.2 Inertia and gravity effects

A slightly more complicated case is obtained by adding the effects of inertia to the previous case.

Therefore, one remains with

2 2 2 2

2 1 0 0 2 1 03x xx

h h u u h h u

(4.3.5)

2

1 0 0x

h u

(4.3.6)

2

2 0 0x

h u

(4.3.7)

By using the conditions (4.2.10), one obtains

29

0

3

3 1

Cx

f

Cx

f

U Ceu

C U e

(4.3.8)

1 10

3 1

3

Cx

f

Cx

k U eh h

ke

(4.3.9)

2 20

3 1

3

Cx

f

Cx

k U eh h

ke

(4.3.10)

where the constantC satisfies the transcendental equation [1].

1 13

CL

d

f CLf

U e

UUe

C

(4.3.11)

Equations (4.3.8)-(4.3.10) are fully derived in Appendix B. From equations (4.3.9) and (4.3.10), it can

be noted that the ratio 1 2h h remains constant along the drawing direction. Furthermore, due to the low

velocities involved in the fiber drawing problem, it is clear that inertia effects are negligible compared

to others, like for example those of surface tension.

The effects of gravity in the drawing process can be investigated by including them in the simple case.

Therefore, the system of governing equations reads

2 2 2 2

2 1 2 1 03 xx

h h g h h u

(4.3.12)

2

1 0 0x

h u

(4.3.13)

2

2 0 0x

h u

(4.3.14)

With the help of equations (4.2.10), one gets

2 2

0

2

2

B x C B x C

B x C

Be a a Beu

B Be

(4.3.15)

[1], where the constants B andC satisfy simultaneously the following transcendental equations

2 22 2C B C B C B

fB Be U Be a a Be

(4.3.16)

2 22 2B L C B L C B L C

dB Be U Be a a Be

(4.3.17)

The final equations for the two radii 1h and 2h are the same as (4.3.2) and (4.3.3) but now equation

(4.3.15) for the asymptotic velocity has to be used, for more details see [1]. Equations (4.3.15)-

(4.3.17) are derived in Appendix C.

30

4.3.3 Surface tension effects

One can easily notice that in the asymptotic limit considered in the previous sections the initial shape

of the radius is preserved. In contrast to that, it will be shown here how the effects of surface tension

can cause dramatic changes in the shape of the radius, when fibres are drawn. Keeping the effects of

surface tension in the equations (4.2.7)-(4.2.9), one has

2 2

2 1 0 1 23 0xx

h h u h h

(4.3.18)

1 2 1 22

1 0 2 2

2 1x

h h h hh u

h h

(4.3.19)

1 2 1 22

2 0 2 2

2 1x

h h h hh u

h h

(4.3.20)

Integrating the first equation and rearranging, it gives

1 2

0 2 2

2 13x

C h hu

h h

(4.3.21)

where C is an arbitrary constant which arises from the integration, [1]. The quantity 2 2

2 1 0h h u ,

which can be regarded as the mass flux of the draw, can be obtained by subtracting (4.3.20) from

(4.3.19). When (4.2.10) apply, one gets

2 2

20 102 2

2 1

0

fh h Uh h

u

(4.3.22)

[1]. Inserting it in the expression for the derivative of the velocity (4.3.21), one has

1 2

0 0 02 2 2 2

20 10 20 103 3x

f f

h h Cu u u

h h U h h U

(4.3.23)

Expanding the derivatives in the equation for the radii (4.3.19) and (4.3.20) one obtains

2 1 2 01

1 2 2020 10

22

xx

f

h h h uhh

uh h U

(4.3.24)

1 1 2 02

2 2 2020 10

22

xx

f

h h h uhh

uh h U

(4.3.25)

where an expression for 0

0

xuu

can be obtained from (4.3.23).

The evolution equations for the internal and the external radii therefore read

2 1 2 1 21

1 2 2 2 2 2 2

20 10 20 10 20 1022 3 3

x

f f f

h h h h hh Ch

h h U h h U h h U

(4.3.26)

1 1 2 1 22

2 2 2 2 2 2 2

20 10 20 10 20 1022 3 3

x

f f f

h h h h hh Ch

h h U h h U h h U

(4.3.27)

31

Thus, a system of three equations in three unknown can be solved numerically, but first the value of

the constant C arising in the equations has to be fixed. The solution strategy used in order to solve the

problem is to guess a velocity profile in the following form

0 expfu U xL

(4.3.28)

where is the ratio log d fU U and is a convergence parameter. The derivative of (4.3.28) reads

0 expx fu U xL L

(4.3.29)

Now, an expression for the constant C can be obtained when equation (4.3.23) is evaluated at 0x ,

that is,

2 2

20 10 10 203 fC h h U h hL

(4.3.30)

Inserting now (4.3.30) in (4.3.23) a complete system of equations for the asymptotic velocity and the

radii is derived,

0

1020

02

10

2

20

210

33u

UhhLu

Uhh

hhu

ff

x

(4.3.31)

fff

xUhhLUhh

hhh

Uhh

hhhh

1020

2

10

2

20

211

2

10

2

20

2121

3322

(4.3.32)

fff

xUhhLUhh

hhh

Uhh

hhhh

1020

2

10

2

20

212

2

10

2

20

2112

3322

(4.3.33)

for more details, see Luzi et al. [4]. In order to solve numerically the system of equations (4.3.31)-

(4.3.33) a fourth order Runge-Kutta Merson method has been used. Choosing a good initial guess for

the parameter and a sufficient fine grid size, the numerical integration is performed and the mass

conservation equation in outm m is checked; if the latter is not satisfied, then is increased by a

certain small amount and the integration is repeated. The process repeats until the condition

1110in outm m is satisfied. A results for such a case, only when surface tension is considered, is

presented below

32

Fig. 4.4: Evolution of the internal and external radius of a preform, when only the effects of

surface tension are taken into account and the viscosity is assumed constant with the temperature

[4]. In Fig. 4.4, 10 0.004h [m], 20 0.01h [m] and 0.25 [N/m]. As expected, the radii undergo a

contraction, since the effects of surface tension tend to “close” the internal hole. Thus, surface tension

effects clearly affect the fiber geometry ratio, see, for instance, Fig. 4.5 below

Fig. 4.5: Evolution of the internal and external radius of a preform. Comparison between the

simple case and the one in which the surface tension effects are included

[4]. In some cases, the surface tension can lead to hole-collapse and this must be prevented. In order to

provide a better insight into the surface tension effects, some examples regarding the evolution of the

internal radius of preforms are presented, when different ratios 10 20/ar h h are tested. 20 0.02h [m]

will be kept constant and 10h will be varied.

33

In Fig. 4.6, the value of surface tension is varied between 0.01 [N/m] and 0.3 [N/m]. The first

of the two values has practically no influence on the inner radius, while the situation changes

dramatically if the latter one is used. The value of 0.3 [N/m] almost produces the collapses of the

structure, in particular for very small values of ar , i.e. 0.2ar , see Fig. 4.6a).

Fig. 4.6: Influence of different values of surface tension on the internal diameter for different

ar : a) 0.2ar , b)

0.3ar , c) 0.4ar , d) 0.6ar

for more details, see [4]. Herein, the values of 1 20h h have been plotted instead of those of 1h . The

case with non-constant viscosity is also considered. Following the same strategy, one begins to assume

an expression for the velocity as in (4.3.28), now taking into account that the viscosity is not constant

anymore. Performing the necessary derivations and substituting it into (4.3.23) one gets a new

expression for the constant C , that is

2010

2

10

2

2003 hhL

UhhTC f

(4.3.34)

The equation for the velocity now becomes

34

1 2

0 0 02 220 1020 10

3 ( )3x

ff

h hu u D u

L T x h h Uh h U

(4.3.35)

where D is

xT

TD

0

(4.3.36)

The evolution equations for the two radii are

2 1 2 01

1 2 2020 10

22

xx

f

h h h uhh

uT x h h U

(4.3.37)

1 1 2 02

2 2 2020 10

22 ( )

xx

f

h h h uhh

uT x h h U

(4.3.38)

where the expression for 00 /uu x has to be calculated from (4.3.35).

In Fig. 4.7 the effects of variable viscosity are shown when a preform is drawn. The temperature

dependent viscosity profile used here is given in the paper of Voyce et al. [5], and the temperature

profile is the one given in Fig. 4.12, with 0.05c , as it will be clarified later on.

Fig. 4.7: Evolution of the internal and the external radius of a preform for the case taking into

account only the effects of

surface tension. In this case the viscosity varies with the temperature, 1950peakT [°C]

[4]. Herein, 20 0.01h [m], 10 0.00365h [m], 0.00009fU [m/sec] and 0.025dU [m/sec]. The

initial values of the two radii are those of a real preform, whose dimensions have been measured by a

scanning electronic microscope (SEM).

35

4.3.4 Limit of small surface tension

In this section, the surface tension effects are held in the equations (4.2.7)-(4.2.9), but their effect is

considered as “small”. This means that the non-dimensional ratio is much less than unity, or

that the effects of the surface tension are much smaller compared to those of the viscosity. The

governing equations are still (4.3.18), (4.3.19) and (4.3.20), but now the solution is sought by

expanding the unknown of those equations in power series of .

By employing (4.2.10), after lengthy computations, one finally arrive at

22

0

20 10

21 1

3

xxL x

L Lf

Le xu U e e e

h h L

(4.3.39)

2102 2 2

1 10 20 10

20 10

3 1 13

xx L x xL L L

f

h xLeh h e h h e e e

U h h L

(4.3.40)

22 2 2

2 20 10 20 20

20 10

e 1 e 3 e 13

x x x xL L L L

f

L xh h e h h h e

U h h L

(4.3.41)

for additional details see [1]. Hole-collapse can be investigated by setting 1 0h in (4.3.40), and it

arises at x L when the following condition is satisfied

20

10 20 10

logf d f

LhU U U

h h h

(4.3.42)

[1]. A detailed derivation of the equations (4.3.39)-(4.3.41) is presented in Appendix D.

4.3.5 Internal pressure effects

During the process of capillary drawing, the closure of a capillary can often be avoided by means of

internal pressurization. Retaining the effects of internal pressure in the equations (4.2.7)-(4.2.9), one

has

2 2

2 1 03 0xx

h h u

(4.3.43)

2 22 0 1 2

1 0 2 2

2 1x

p h hh u

h h

(4.3.44)

2 22 0 1 22 0 2 2

2 1x

p h hh u

h h

(4.3.45)

With the aid of (4.2.10), after cumbersome computations, one finally obtain

36

0

x

Lfu U e

(4.3.46)

12

2 2 2

10 20 10

1

1

2 2

20 10

x

o L

f

x

L

p Le

U

h h h eh

h e h

(4.3.47)

12

2 2 2

20 20 10

2

1

2 2

10 20

x

o L

f

x

L

p Le

U

h h h eh

h e h

(4.3.48)

[1]. The system of equations (4.3.46)-(4.3.48) is fully derived in Appendix E.

4.3.6 The interplay between internal pressure and surface tension

In real cases, during the drawing, one may have to take into account the combined effects of internal

pressure and surface tension. The internal pressure is sometimes necessary in order to prevent hole-

collapse. Again, starting from equations (4.2.7), (4.2.8) and (4.2.9) and considering both effects of

internal pressure and surface tension, one deals with the following system of equations

2 2

2 1 0 2 13 0xx

h h u h h

(4.3.49)

2 2

1 2 1 2 1 22

1 0 2 2

2 1

o

x

p h h h h h hh u

h h

(4.3.50)

2 2

1 2 1 2 1 22

2 0 2 2

2 1

o

x

p h h h h h hh u

h h

(4.3.51)

It can now be manipulated as (4.3.18), (4.3.19) and (4.3.20), achieving the following final system of

equations for the steady isothermal case

37

1 2

0 0 0 02 220 1020 10

33x

ff

h hu u u u

L h h Uh h U

(4.3.52)

2

0 1 2 2 1 2 1 211 2 2 2 2

20 1020 10 20 102 32 3

x

ff f

p h h h h h h hhh

L h h Uh h U h h U

(4.3.53)

2

0 1 2 1 1 2 1 222 2 2 2 2

20 1020 10 20 102 32 3

x

ff f

p h h h h h h hhh

L h h Uh h U h h U

(4.3.54)

for more details see [4]. If the surface tension coefficient is set equal to zero in (4.3.52), (4.3.53) and

(4.3.54), one immediately obtains (4.3.43)-(4.3.45), with the help of (4.3.22) and that of the boundary

conditions (4.2.10). Furthermore, one finds 1 . If the pressure term is set equal to zero in (4.3.53)-

(4.3.54) one readily obtains (4.3.31)-(4.3.33). This means that all the formulas derived are consistent

to each other. Another interesting point is that if the derivatives of the diametersxh and that of

the velocityxu tend to zero, which means that the radii

1h and2h , and the velocity

0u become

constants.

One immediately notes the effects of the interplay between internal pressure and surface tension,

which leads to a higher value of the internal applied pressure which can be tolerated by the fiber.

While internal pressure, roughly speaking, tends to open the hole, the surface tension tries to close it,

see as instance Fig. 4.8.

Here, 20 0.01h [m], 10 0.00365h [m], 0.0001fU [m/sec] and 0.04dU [m/sec],

38

Fig. 4.8: Evolution of the initial preform as internal pressure and surface tension effects are considered. The

viscosity is here considered constant with the temperature: a) 0 120p [Pa], b)

0 140p [Pa], c) 0 160p [Pa],

d) 0 175p [Pa]

[4]. The non-isothermal case, in which the viscosity is let to vary with the temperature, has also been

analyzed and it represents a more appropriate model of fiber drawing. The system of the governing

equations of the process reads

39

1 2

0 0 0 02 220 1020 10

33x

ff

h hu u D u u

L T x h h UT x h h U

(4.3.55)

1 2

2 2220 100 1 2 2 1 2 1

1 2 2

20 10

0

20 10

3

22

3

f

x

f

f

h hD

LT x h h Up h h h h h hh

T x h h Uu

T x h h U

(4.3.56)

1 2

2 2220 100 1 2 1 1 2 2

2 2 2

20 10

0

20 10

3

22

3

f

x

f

f

h hD

LT x h h Up h h h h h hh

T x h h Uu

T x h h U

(4.3.57)

where the constant D is given by (4.3.36), [4]. The shape of the two radii is thus modified by the

viscosity profile, in accordance with [5], see Fig. 4.9. The temperature profile is given in Fig. 4.12,

with 0.05c .

Here, 20 0.01h [m], 10 0.00365h [m], 0.00009fU [m/sec] and 0.025dU [m/sec],

for additional details see [4].

40

Fig. 4.9: Evolution of the initial preform as internal pressure and surface tension effects are considered. The

viscosity is here considered non constant with the temperature, which reaches its peak ( 1950peakT [°C]) at the

center of the furnace: a) 0 0p [mbar], b)

0 5p [mbar], c) 0 20p [mbar], d)

0 25p [mbar]

4.3.7 Closure of a small hole

In this section both effects of internal pressure and surface tension are considered in the asymptotic

equations (4.2.7), (4.2.8) and (4.2.9), but the internal radius 1h is considered “small”. By introducing an

appropriate scaling in the equations (4.3.49), (4.3.50) and (4.3.51), as well as the initial and boundary

conditions (4.2.10), the following analytical solution can be derived

41

0

x

L

fu U e

(4.3.58)

1 10

0

exp exp2 2

xx

PL Lx

h Pe h e G Pe duL L

(4.3.59)

2 20

x

Lh h e

(4.3.60)

Herein, 0 2 fP p L U and 2 fG U , for additional details see [1]. A criterion for the

collapse of the fibre reads

10

0

exp2

L

u L PuG Pe du h e

L

(4.3.61)

and similarly, one for the expansion of the inner radius, i.e.

0

10

fUp

h L

(4.3.62)

[1]. The system (4.3.58)-(4.3.60) and equations (4.3.61) and (4.3.642) are derived in details in [1]. In

Appendix F, a discussion about the applicability of these equations along with practical examples is

given.

4.3.8 The general case

Now, all the effects of inertia, gravity, surface tension, internal pressure and those of a temperature

dependent viscosity are analysed together. Starting from the system of equations (4.2.7)-(4.2.9), and

following the same strategy utilized to obtains the systems (4.3.31)-(4.3.33), one obtains the following

system of equations

0 expx fy u U xL L

(4.3.63)

21 2 0 0

0 02 20 20 10

33

x x xx x

f

h h u uy Ty u u g

u T xT x T xT x h h U

(4.3.64)

2

0 1 2 2 1 2 011 2 2

020 1022 ( )

xx

f

p h h h h h uhh

uT x h h U

(4.3.65)

2

0 1 2 1 1 2 022 2 2

020 1022 ( )

xx

f

p h h h h h uhh

uT x h h U

(4.3.66)

where the steady state case has been considered. In (4.3.65) and (4.3.66), the ratio 00 /uu x has to be

computed from (4.3.63) and (4.3.28), for more details see the manuscript of Luzi et al. [6].

42

4.3

.8

4.3

.7

4.3

.6

4.3

.5

4.3

.4

4.3

.3

4.3

.2

4.3

.1

Sectio

n

Gen

eral case (4.2

.7), (4

.2.8

), (4.2

.9)

Clo

sure o

f a small h

ole (4

.3.4

9),

(4.3

.50), (4

.3.5

1)

Intern

al pressu

re and

surface ten

sion

effects (4.3

.49), (4

.3.5

0), (4

.3.5

1)

Intern

al pressu

re effects (4.3

.43

),

(4.3

.44), (4

.3.4

5)

Sm

all surface ten

sion

limit (4

.3.1

8),

(4.3

.19), (4

.3.2

0)

Su

rface tensio

n effects (4

.3.1

8),

(4.3

.19), (4

.3.2

0)

Inertia effects (4

.3.5

), (4.3

.6), (4

.3.7

)

Grav

ity effects (4

.3.1

2), (4

.3.1

3),

(4.3

.14)

Sim

plificatio

n fro

m th

e gen

eral case

(4.2

.7), (4

.2.8

), (4.2

.9)

Eq

uatio

ns

Fitt &

al. [1]

Fitt &

al. [1]

Fitt &

al. [1]

Fitt &

al. [1]

Fitt &

al. [1]

Fitt &

al. [1]

Fitt &

al. [1]

Fitt &

al. [1]

Auth

or

Equatio

ns (4

.3.6

3), (4

.3.6

4),

(4.3

.65), (4

.3.6

6)

Equatio

ns (4

.3.5

8), (4

.3.5

9),

(4.3

.60) an

d (4

.3.6

1), (4

.3.6

2)

Constan

t visco

sity (4

.3.5

2), (4

.3.5

3),

(4.3

.54)

Variab

le visco

sity (4

.3.5

5), (4

.3.5

6),

(4.3

.57)

Equatio

ns (4

.3.4

6), (4

.3.4

7),

(4.3

.48)

Equatio

ns (4

.3.3

9), (4

.3.4

0),

(4.3

.41) an

d (4

.3.4

2)

Constan

t visco

sity (4

.3.3

1), (4

.3.3

2),

(4.3

.33)

Variab

le Visco

sity (4

.3.3

5), (4

.3.3

7)

(4.3

.38)

Inertia E

ffects

(4.3

.8), (4

.3.9

), (4.3

.10), (4

.3.1

1)

Grav

ity E

ffects

(4.3

.15), (4

.3.1

6), (4

.3.1

7)

Equatio

ns (4

.3.1

), (4.3

.2), (4

.3.3

) an

d (4

.3.4

)

Solu

tions

Luzi &

al. [6

]

Fitt &

al. [1]

Luzi &

al. [4

]

Fitt &

al. [1]

Fitt &

al. [1]

Luzi &

al. [4

]

Fitt &

al. [1]

Fitt &

al. [1]

Au

tho

r

Table 1: Graphical visualization between the equations and the solutions of [1] and those obtained by the author

of this thesis for different cases

43

4.4 Experimental results

As described in the previous section, during experiments a glass preform is introduced into a hot

furnace, which heats it up, see Fig 4.10. The tube is then pulled from below and it assumes an

elongated form, see Fig. 4.11.

Fig 4.10: Glass preform being introduced into the furnace

Fig 4.11: Fiber drawing process. Left: Fiber being drawn out of the bottom of the furnace. Right: Shape of the

glass preform after drawing

It is known that the temperature in the furnace is not constant, but it varies strongly along the x

coordinate. From the experiments carried out by [5], it is known that the temperature profile first

increases, then it reaches a maximum, and finally it decreases further on. A Gaussian distribution is

found to be suitable to describe the temperature profile inside the furnace, see for example [5] and

44

[6], at least in the important part, the so called “hot zone”. In the present study the function

2

22

( ) exp

x b

c

T x a

has been used, where a , b and c are parameters which assume certain values.

More precisely, a indicates the peak of the profile,b the position along the x coordinate at which the

peak is located (in our case 2b L ), and c the “shape” of the distribution itself, i.e. the flatness or the

steepness of the curve. In Fig. 4.12, the temperature profiles which have been used are shown, see also

[6].

Fig. 4.12: Possible temperature profiles along the x axis of the furnace. The peak temperature is in this case

2050peakT [°C]

The real furnace temperature profile for high temperatures, i.e. 1800T [°C], is not known, since it

is very difficult to measure experimentally and it critically depends on the gas flow conditions inside

the furnace (for example the opening of the top and bottom iris), as discussed in Chodhury et al. [7],

and in Yin et al. [8]. Therefore a temperature (and the resulting viscosity) profile of the glass in

thermal equilibrium with the furnace is here simply assumed. The shapes shown in Fig. 4.12 are

adequate to represent the temperature profile in the region between 0.1x [m] and 0.2x [m]. One

can regard this part as the “hot zone length”. Outside this zone, the temperature decreases

continuously and one can assume that the viscosity is high enough so that the glass does not deform.

Once accurate data become available for the furnace temperature profile (which is currently under

investigation both numerically and experimentally) the effects of heat transfer to the preform can be

incorporated into the model in a straightforward fashion.

The concentration of water and other impurities can dramatically affect the viscosity curve of the silica

glasses and great care must therefore be taken when choosing an appropriate viscosity curve.

Furthermore, the standard methods used to measure viscosity experimentally have several degrees of

accuracy, depending on the temperature range of interest. The functional form of the viscosity as a

function of the temperature used in the present numerical simulations is given in [5]

45

7 5154000.1 5.8 10 exp s

8.3145 2271.10567T Pa

T

(4.4.1)

based on the work of Urbain et al. [9]. Herein, the temperatureT is expressed in Celsius degree [°C].

The numerical solutions of the system of equations (4.3.63)-(4.3.66), obtained by the program

“Fiberspinning.m”, have been compared directly with experimental results. During the experiments

three glass preforms have been used. The exact value of external and internal diameter has been

measured accurately for each preform by a SEM. First of all, three different values of “drawing

speeds”f dU U with no applied pressure have been tested, 3.6 1f dU U , 5.4 1.5f dU U and

7.2 2f dU U respectively, and the temperature varied gradually from 2050peakT [°C] to

1850peakT [°C] in steps of 25peakT [°C]. Here, fU is given in [mm/min] while dU is given in

[m/min], so the “drawing speeds” are not strictly non dimensional, but their units is [mm/m]. Graphs

showing the comparisons between theoretical and experimental values of the final external diameter

are presented below in Fig. 4.13,

46

Fig. 4.13: Comparison between theoretical computations and experimental results for three different “drawing

speeds”, that is, 3.6 1f dU U , 5.4 1.5f dU U and 7.2 2f dU U : external diameter a1), b1) and c1), and

the ratio internal/external diameter a2), b2), and c2) , when no pressure is applied.

The graphs show that there is a good agreement between theory and experimental studies, even in case

that the temperature reaches very high values (e.g. 2050peakT [°C]), for more details see [6]. In a

further step, inner pressure has been applied to preforms with the drawing ratio of 5.4 1.5f dU U ;

47

first the pressure has been kept constant and the temperature varied from 2050peakT [°C] to

1850peakT [°C] in steps of 25peakT [°C], and

Fig. 4.14: Comparison between experimental results and theoretical computations for the “drawing

speed” 5.4 1.5f dU U when pressure is applied. First, the temperature varies while the pressure is kept constant

at 0 9p [mbar]: external diameter a1), and ratio internal/external diameter a2). Second, the temperature peak is

kept constant at 1950peakT [°C] and the pressure varies from 0 0p [mbar] to 0 25p [mbar]:b1) external

diameter, and b2) ratio internal/external diameter

later on the temperature was kept constant at 1950peakT [°C] and the pressure has been varied

between 0 5p [mbar] and 0 25p [mbar], in steps of 0 2.5p [mbar], see Fig. 4.14. Experiments

and theory seem to match better when the value of 0.0952c instead of 0.0752c in Fig. 4.14 a1)

and a2), while the opposite occurs in Fig. 4.14 b1) and b2), [6]. This is probably due to the fact that in

the present analysis the heat exchange is not taken into account, and when the temperatures reach high

values (i.e. 2000peakT [°C]) the temperature profile with the value of 0.0952c better describes

the fibre surface temperature. In case of lower temperatures (i.e. 1950peakT [°C]) the temperature

48

profile with the value of 0.0752c gives accurate results also in case of applied internal pressure,

which indicates that it is probably the one closer to the real fibre temperature profile.

It is worth to point out that at lower temperatures and at lower values of the pressure the experimental

results practically coincide with theoretical calculations and the functional form of the viscosity as a

function of the temperature is well known, while it is not so for high temperatures (e.g.

2000T [°C]).

Until now, final diameters which range from 22 1h [mm] to

22 2h [mm] have been obtained with

the above “drawing speeds”. Increasing significantly the “drawing speed” from the previous used to

1 10.2f dU U , 2 20.4f dU U and 3 30.6f dU U , one obtains diameters of a few hundreds

of micro-meters. First, no pressure is applied and the temperature is varied as before from

2050peakT [°C] to 1850peakT [°C], in steps of 25peakT [°C], as it is shown in Fig. 4.15 below

Fig. 4.15: Comparison between theoretical computations and experimental results for two different “drawing

speeds” a) 1 10.2f dU U and b) 3 30.6f dU U , when no internal pressure is applied. External diameters

a1) and b1), ratio internal/external diameters a2) and b2). The final diameters obtained are of the order of few

hundreds of micro-meters

49

Also here the agreement between theory and experiments is good, even for such small fibers, [6]. As

before, a better agreement is obtained when the value of 0.0952c instead of 0.0752c , when the

temperatures peaks fall in the range between 2000peakT [°C] and 2050peakT [°C], see Fig. 4.15

a1) and a2), b1) and b2), while for lower temperature (i.e. 2000peakT [°C]) the difference between the

temperature profiles with the two values of c is not noticeable. Now, pressure is applied to preforms

with 2 20.4f dU U , starting from 0 0p [mbar] up to 0 25p [mbar], in steps of

0 2.5p

[mbar], and the temperature is maintained fixed at 1950peakT [°C], see Fig. 4.16.

A different scenario occurs when the temperature is fixed (i.e. 1950peakT [°C]) and internal pressure

is applied. In this situation the temperature profile with the value of 0.0752c gives good accuracy

when theoretical computations and experiments are compared, [6]. When the two viscosity profiles

(i.e. the viscosity profiles related to the two temperature profiles with the two values of c are compared

for the peak temperature 1950peakT and 2000peakT [°C], there is a difference of one order of

magnitude in the values. If the value of c is lower, the distribution of temperature and the viscosity is

narrower near the peak, so higher pressure can be achieved.

Fig. 4.16: Comparison between experiments and theory for the “drawing speed” 2 20.4f dU U . The

pressure is varied between 0 5p [mbar] and 0 40p [mbar]. External diameters a1) and ratio

internal/external diameters a2).

50

4.5 Numerical models

In order to solve numerically the full 3D set of the governing equations in Cartesian coordinates, that

is, (3.1.2), (3.1.3) and (3.1.5), the commercial packages Ansys Gambit and Ansys Polyflow have been

used.

The geometry and the mesh have been created by the first of the two softwares, that is, Ansys Gambit.

In case of an axis-symmetric capillary, an orthogonal curvilinear structured 3D mesh has been found

suitable to solve the problem. In ideal cases with constant viscosity, the following mesh has been used,

see Fig. 4.17, where only a quarter of the initial circular cylinder is considered

Fig. 4.17: a) The global view of the meshed geometry; b) front view

51

Fig. 4.18: (a) Optical microscope image of the micro-structured silica preform used in the fiber drawing

experiments. (b) Meshed geometry, frontal view.

52

In case of a 3D six-hole geometry, whose image of the cane used in the experiments is given in Fig.

4.18a) with the relative dimensions, an unstructured tetrahedral grid (1200x70) has been chosen. The

grid has been refined in the vicinity of the holes, so that the shape deformation can be accurately

captured, see Fig. 4.18b). In the axial direction, the grid is equally spaced.

Similar structures with three to six holes have applications as e.g. low-loss hole-assisted fibers, see for

instance Nakajima et al. [10], or suspended-core fibers for chemical sensing, see, for instance, the

paper of Euser et al. [11].

Afterwards, the meshed geometries have been exported with the necessary extension .msh, and they

have been imported in Ansys Polyflow, which is a finite-element CFD software. The latter is

composed of a pre-processor, Polydata, see Fig. 4.19, and a solver. It has been chosen because it is

particularly suitable for extrusion, thermoforming, blow molding, glass forming and fiber drawing

processes, as described in the Polyflow user´s manual [12]. Furthermore, it is possible to define in

details the properties of the material.

Fig. 4.19: Screenshot of the pre-processor window of Polydata, Fujimoto [13]

In Polydata, it is possible to impose appropriate initial and boundary conditions, to set the adequate

remeshing technique and to choose the most suitable numerical method. The initial conditions are set

as follow: at the beginning of the draw zone, i.e. at z=0, the axial velocity is constant and equal to the

feed speed fU . At the end of the draw zone, that is at z=L, a constant draw speed dU is imposed. At

the interfaces the kinematic boundary conditions are

0

i

ix

fv

dt

df

Dt

Df

(4.5.1)

and the dynamic boundary conditions are, in the normal direction,

53

Hijij pKnn

(4.5.2)

aijij pKnn

(4.5.3)

And in the tangential directions

0ijij tn

(4.5.4)

where K is the sum of the principal curvature of the free surface andij is the stress tensor, for more

details see the manuscript of Xue et al. [14]. Equation (4.5.2) applies for an inner surface, while

equation (4.5.3) applies for an outer surface. Equations (4.5.1), (4.5.2), (4.5.3) and (4.5.4) are written

in Cartesian coordinates, since Polyflow works with this coordinate system.

The boundary conditions are set by specifying the forces acting on a free surface, in the normal and in

the tangential direction. Along the normal direction, the value of the overpressure 0p (force per unit

area) and that of the surface tension (force per unit length) have been set. Along the tangential

direction, a zero force has been specified [13].

During the drawing, large deformations of the flow along one direction occur. At each position along

the drawing direction, the grid is computed, and the nodes are allocated in order to satisfy the physical

problem. Thus, the old grid is substituted by a new one, in which the position of the nodes is re-

computed [13]. The remeshing technique “Optimesh” has been selected for the computation of new

grids. The whole grid is cut into planes, perpendicular to the position along the drawing direction, and

the position of the node is calculated by minimizing the energy of deformation of the mesh [12]. The

beginning of the draw zone has been specified as “Inlet”, while the end of the draw zone has been

specified as “Outlet” in the option menu of “Optimesh”. The kinematic boundary condition termed

“line kinematic condition” has been chosen. It allows the displacements of the nodes in the plane to

which they pertain [12].

The computation of the new position of the nodes is computationally expensive, and sometimes is not

possible to simulate processes, in which the fluid flow result highly deformed.

An “evolution-parameter” s allows the final grid deformation to be achieved after several steps [12].

The following parameters inits , finals , maxs and mins can be set. The first one indicates how big is the

deformation in the first step, finals , fixes the upper limit for the deformation, maxs and mins limit the

amount of maximum and minimum admissible deformation between two steps, respectively [12]. In

Fig. 4.20, a schematic diagram shows how a simulation is computed, if s is specified.

54

Fig. 4.20: Schematic diagram of the procedure of the evolution parameter “ s ” [13]

For the case of axis-symmetric capillaries, inits =0, finals =1, maxs =0.1 and mins =0.001, while for the

case of the six hole geometry inits =0, finals =1, maxs =0.001 and mins =0.0001. As already explained,

the final deformation of the geometry and that of the grid is attained through several steps, see Fig.

4.21 below

55

Fig. 4.21: Grid deformation at different steps ( s ), [13]

In order to compute all the simulations, the Algebraic Multi-Frontal (AMF) direct solver, which is

based on the Gauss´s elimination method [12], has been selected.

All the simulations of the axis-symmetric capillaries have been run in a 8 Intel Xeon E5345 cores

machine, which has 2.33 Gigahertz (GHz) Control Processing Unit (CPU), and 16 Gigabyte (GB)

Random Access Memory (RAM). Even if the total amount of elements is not high, the amount of

RAM requested to perform the simulations is high (around 14 GB).

All the simulations of the six-hole geometry have been run in a 16 AMD Opteron 6134 cores machine,

which has 128 GB RAM, since a high amount of RAM is requested.

56

4.6 Comparison among solutions of the set of asymptotic equations with the full N.-St. equation ones: the ideal case with constant viscosity

The present paragraph focusses on an exhaustive comparison between the numerical solution of the

full 3D set of N.-St. equations, i.e. equations (3.1.2), (3.1.3) and (3.1.5), and the solution of the

asymptotic equations (4.2.7)-(4.2.9), as described in the manuscript of Luzi et al. [15]. First of all, the

shapes of the internal and external radii, obtained by solving the full 3D set of the N.-St. equations, are

compared to those obtained by solving the asymptotic ones. Afterwards, the radial and the axial stress

and the radial and the axial velocities are computed with both the N.-St. equations and the asymptotic

ones at three sections along the drawing direction. In this way, it is possible to localize the positions

where both solutions are in good agreement and those in which the results diverge.

Therefore, more light can be shed on the interplay between surface tension and internal pressure,

having the possibility to look inside the fiber by post-processing the results.

The main advantage of the asymptotic equation is that they can be solved very quickly, while the full

N.-St. solution results very time consuming and expensive. On the other hand, solving the full 3D set

of N.-St. equation allows for modelling of real complex geometries, which are important in the

fabrication of PCFs.

In this paragraph an example of a common practical case in which the preform has an outer radius

20 0.01h [m] and an inner radius 10 0.004h [m] is considered. Typical values of the velocities have

been considered; 0.0001fU [m/sec], 0.04dU [m/sec]. The length of the furnace has been set to

0.3L [m], since ideal cases with constant viscosity are now investigated.

4.6.1 Comparison between the asymptotic equations and the N.-St. ones: the simple

case

In this section, equations (4.3.1), (4.3.2) and (4.3.3) are directly compared with the results obtained by

solving the N.-St. ones, in case that surface tension, internal pressure, inertia and gravity effects are

not taken into account.

In Fig. 4.22, the comparison of the shape of the internal and the external radii between the asymptotic

model and the N.-St. one is presented

57

Fig. 4.22: Comparison of the shape of the radii between the asymptotic model and the N.-St. one for the simple

case, when internal hole pressure, surface tension, inertial and gravity effects are neglected and the viscosity is

assumed constant with the temperature

The asymptotic solution and the N.-St. ones practically coincide at each point along the drawing

direction, and the discrepancies between the two models reach the maximum value of 3%.

4.6.2 The case with surface tension effects

A comparison between the shapes of the radii, obtained by solving the system of equations (4.3.31)-

(4.3.33) and the N.-St ones is presented below in Fig. 4.23

Fig. 4.23: Comparison of the shape of the radii between the asymptotic model and the N.-St. one, when

only the effects of surface tension are taken into account

58

The solutions of the asymptotic equations match with those obtained by solving the full set of N.-St.

ones, and the contraction, to which undergo the diameters due to the surface tension effects, is

correctly predicted.

The percentage difference between the two models can be computed for each of the two radii. In

particular, three sections along the drawing direction are chosen for the analysis, that is, 02.0x [m],

11.0x [m] and 28.0x [m]. By analyzing the normal stress variation, as well as the velocity field

distribution, both in the radial and in the axial direction, it is possible to explain where and why the

two models match or not.

The normal stress in the radial direction caused by the viscous action, say, rr is defined by

2rr

w

r

(4.6.1)

The derivative of the radial velocity with respect to the radius in the asymptotic model reads

2

00

2 r

Au

r

w x

(4.6.2)

where A is

2 2

0 1 2 1 2 1 2

2 2

2 12

p h h h h h hA

h h

(4.6.3)

in dimensional form whose units are [m2/sec], for more details see [1]. In Fig. 4.24, rr is plotted along

the radial direction, when computed by both of the two methods.

rr varies by several orders of magnitude along the drawing direction, as it is shown in Fig. 4.24a),

4.24b1) and 4.24b2), 4.24c1) and 4.24c2). It is also possible to notice the good agreement between

the rr computed by the asymptotic method A.M, and the full set of N.-St. equations, for the three

sections along the drawing direction. The constant trend along the radial direction, for the simple case,

is correctly reproduced by the A.M., since A results zero in (4.6.2) and therefore the derivative of the

radial velocity with respect to the radius is only function of the axial velocity, 0u , which depends only

on x and t .

In case that the surface tension effects are taken into account, the radial stress reduces from the outer

radius toward the inner one, and this can also be explained analytically. As already discussed, rr is

defined in (4.6.1), while the derivative of the asymptotic radial velocity with respect to the radius is

defined in (4.6.2). For the case in which only the effects of surface tension are considered, it reads

B

A

ff

hh

hhhh

r

uUhhL

uUhh

hh

r

w

2

1

2

2

2121

2

0

1020

02

10

2

20

210

2

1

626

(4.6.4)

59

in which (4.3.31)-(4.3.33) have been used. The two terms A and B in eq. (4.6.4) are constant along the

radial direction, therefore their difference is higher if 2r is greater and vice-versa.

The radial stress caused by the viscous action results bigger along the drawing direction in case that

the surface tension effects come into play, see Fig. 4.24b2). Therefore, the inner radius reduces more

with respect to the simple case in which the surface tension is not taken into account.

At the beginning of the drawing, the radial stress computed for the simple case assumes slightly higher

values than that computed for the case with surface tension. In the latter case, the trend of the radial

stress computed by the A.M. can be explained by analyzing the order of magnitude of the terms in

(4.6.2). At the position 02.0x [m], the first term in (4.6.2) is within the order of magnitude of

)10( 3O , while A is within )10( 8O . On the other hand, r results within )10( 3O , and the second term

in (4.6.2) is also within )10( 3O , giving to rr the nonlinear trend along the radial direction.

At the center of the furnace, the rr calculated for the case which includes the surface tension effects is

almost three times bigger than the one calculated for the simple case. Hence, the inner surface reduces

more in size, due to the surface tension effects which act in order to close the inner hole. At this

position of the drawing, both terms of (4.6.2) are within )10( 3O , but the first one is about 5-6 times

bigger than the second one, thus explaining the almost constant trend of rr along the radial direction,

see Fig. 4.24b1) and Fig. 4.24b2).

At the end of the drawing, the radial stress gets closer between the two cases, see Fig. 4.24c1) and Fig.

4.24c2). The nearly constant trend of rr along the radial direction is due to the fact that the first term

of (4.6.2) is within 110O , while the second one is within 410O

. Near the outer surface, the

radial stress assumes higher values, when computed with the N.-St. equations, since a lot of points are

concentrated there and the distance between two of them becomes very small.

In the same way, it is possible to consider the normal component of the stress along the x-direction,

that is xx , which is defined by

2xx

up

x

(4.6.5)

The value of the scalar pressure p is readily obtained by post-processing the results in case of the N.-

St. equations, while the asymptotic scalar (relative) pressure P reads

2

1 2 0 1 002 2

20 10 f

h h p h uP u

xh h U

(4.6.6)

in dimensional coordinates, for more details see [1].

P depends only on x and t , thus it is constant along the radial coordinate.

60

Simple case, A.M. h2=8.19 [mm], h1=3.28 [mm] N.-St. h2=8.35 [mm] h1=3.43 [mm].

=0.25 [N/m], A.M. h2=8.092 [mm], h1=2.55 [mm] N.-St. h2=8.28 [mm] h1=2.75 [mm].

b1) x=110 [mm] A.M. h2=3.33 [mm], h1=1.33 [mm] N.-St. h2=3.38 [mm] h1=1.40 [mm].

b2) x=110 [mm] A.M. h2=3.25 [mm], h1=0.338 [mm] N.-St. h2=3.28 [mm] h1=0.358 [mm].

c1) A.M. h2=0.611 [mm], h1=0.244 [mm] N.-St. h2=0.614 [mm] h1=0.254 [mm].

c2) A.M. h2=0.565 [mm], h1=5.45 [μm] N.-St. h2=0.554 [mm] h1=5.39 [μm].

Fig. 4.24: rr at three different position along the drawing direction in the furnace: a) 0.02x [m], b1) and b2)

0.11x [m], c1) and c2) 0.28x [m].

The simple case and the one with surface tension effects are considered.

61

The axial stress is almost constant along the radial direction, varying from [mbar] to [mbar], see Fig.

4.25a), Fig. 4.25b1) and Fig. 4.25b2) and Fig. 4.25c1) and Fig. 4.25c2), as the cross-section becomes

smaller and smaller along the drawing direction. The agreement between the A.M. and the N.-St.

equations is very good, since the discrepancies between the two methods range between 3% and 8%,

for both the two cases. At the end of the drawing, the values of the axial stress become very close

between the two models, see Fig. 4.25c1) and Fig. 4.25c2).

At each stage of the drawing, the values of the axial component of the velocity match very well, when

computed by both of the two methods, with a maximum discrepancy of 5%-6%, for both of the two

cases, see Fig. 4.26a), Fig. 4.26b1) and Fig. 4.26b2), Fig. 4.26c1) and Fig. 4.26c2). The trend of

u along the drawing direction is very similar to that of xx . For instance, at the end of the drawing,

where the values of the axial stress assume similar values, also the values of u become closer, see Fig.

4.26c1) and Fig. 4.26c2). The constant trend of the axial velocity along the radial direction is correctly

predicted by the A.M., therefore the assumption of a leading order axial velocity, which is radial

independent, is correct.

The leading order, asymptotic, radial component of the velocity w reads

r

Ar

uw x

2

00

(4.6.7)

where A has been already defined in (4.6.3). The radial velocity decreases along the radial direction,

see Fig. 4.27a), Fig 4.27b1) and Fig. 4.27b2), Fig. 4.27c1) and Fig. 4.27c2), where the minus sign

indicates that the velocity vector is pointing inward. w decreases from the outer surface 2h toward the

inner surface 1h , therefore the outer surface contracts much more and much faster than the inner one.

At the beginning of the drawing, that is, at 02.0x [m], w decreases linearly along the radial

direction for the simple case, see Fig 4.27a). The linear trend of the radial velocity is also obtained by

solving the asymptotic equations, since A is zero in (4.6.7). In case that the surface tension effects are

present, the radial velocity decreases in a nonlinear fashion, see Fig. 4.27a). The asymptotic velocity

also presents this trend, since in the vicinity of the inner radius both terms of (4.6.7) are

within )10( 6O , but the second is bigger than the first one, thus it dominates. On the contrary, in the

vicinity of the outer radius, the first term is within )10( 5O , while the second one is within )10( 6O ,

therefore the trend of the curve becomes linear.

62

Simple case, A.M. h2=8.19 [mm], h1=3.28 [mm] N.-St. h2=8.35 [mm] h1=3.43 [mm].

=0.25 [N/m], A.M. h2=8.092 [mm], h1=2.55 [mm] N.-St. h2=8.28 [mm] h1=2.75 [mm].

b1) x=110 [mm] A.M. h2=3.33 [mm], h1=1.33 [mm] N.-St. h2=3.38 [mm] h1=1.40 [mm].

b2) x=110 [mm] A.M. h2=3.25 [mm], h1=0.338 [mm] N.-St. h2=3.28 [mm] h1=0.358 [mm].

c1) A.M. h2=0.611 [mm], h1=0.244 [mm] N.-St. h2=0.614 [mm] h1=0.254 [mm].

c2) A.M. h2=0.565 [mm], h1=5.45 [μm] N.-St. h2=0.554 [mm] h1=5.39 [μm].

Fig. 4.25: xx at three different position along the drawing direction in the furnace: a) 0.02x [m], b1) and b2)

0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension effects are considered.

63

Simple case, A.M. h2=8.19 [mm], h1=3.28 [mm], N.-St. h2=8.35 [mm], h1=3.43 [mm].

=0.25 [N/m], A.M. h2=8.092 [mm], h1=2.55 [mm], N.-St. h2=8.28 [mm], h1=2.75 [mm].

b1) x=110 [mm] A.M. h2=3.33 [mm], h1=1.33 [mm] N.-St. h2=3.38 [mm] h1=1.40 [mm].

b2) x=110 [mm] A.M. h2=3.25 [mm], h1=0.338 [mm] N.-St. h2=3.28 [mm] h1=0.358 [mm].

c1) A.M. h2=0.611 [mm], h1=0.244 [mm] N.-St. h2=0.614 [mm] h1=0.254 [mm].

c2) A.M. h2=0.565 [mm], h1=5.45 [μm] N.-St. h2=0.554 [mm] h1=5.39 [μm].

Fig. 4.26: u at three different position along the drawing direction in the furnace: a) 0.02x [m], b1) and b2)

0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension effects are considered.

64

Simple case, A.M. h2=8.19 [mm], h1=3.28 [mm] N.-St. h2=8.35 [mm] h1=3.43 [mm].

=0.25 [N/m], A.M. h2=8.092 [mm], h1=2.55 [mm] N.-St. h2=8.28 [mm] h1=2.75 [mm].

b1) x=110 [mm] A.M. h2=3.33 [mm], h1=1.33 [mm] N.-St. h2=3.38 [mm] h1=1.40 [mm].

b2) x=110 [mm] A.M. h2=3.25 [mm], h1=0.338 [mm] N.-St. h2=3.28 [mm] h1=0.358 [mm].

c1) A.M. h2=0.611 [mm], h1=0.244 [mm] N.-St. h2=0.614 [mm] h1=0.254 [mm].

c2) A.M. h2=0.565 [mm], h1=5.45 [μm] N.-St. h2=0.554 [mm] h1=5.39 [μm].

Fig. 4.27: w at three different position along the drawing direction in the furnace: a) 0.02x [m], b1) and b2)

0.11x [m], c1) and c2) 0.28x [m]. The simple case and the one with surface tension effects are considered.

65

At the center of the drawing, i.e., at 11.0x [m], w assumes a linear profile for both cases, which is

correctly predicted by the asymptotic equations, see Fig. 4.27b1) and Fig. 4.27b2). As far as the simple

case concerns, A is zero in (4.6.7), thus the asymptotic radial velocity is linear along the radius, see

Fig 4.27b1). In case that the surface tension effects are considered, the linear term in (4.6.7) is slightly

higher than the nonlinear one near the inner surface, while in the vicinity of the outer surface, the first

term in (4.6.7) is much bigger than the second one, therefore the velocity profile becomes strongly

linear, see Fig. 4.27b2). At the position 28.0x [m], the trend of the radial velocity is completely

linear in both cases, see Fig. 4.27c1) and Fig. 4.27c2). This trend is also correctly computed by the

A.M., since the first term of (4.6.7) is always bigger than the second one. In particular, in the vicinity

of h2, the first term is found to be within )10( 4O while the second one is found to be within )10( 8O .

Instead, in the vicinity of h1, both of terms are within )10( 6O , but the first one is greater than the

second one, [15].

4.6.3 Internal pressure effects: p0=120 [Pa] and p0=135 [Pa]

In this section, only the effects of internal pressure are considered. The draw is considered steady and

isothermal, and the effects of inertia, gravity and surface tension are neglected. In Fig. 4.28, the

comparison of the shape of the internal and the external radius between the asymptotic model,

equations (4.3.46), (4.3.47), (4.3.48), and the N.-St one is presented for two values of the internal

pressure, i.e. p0=120 [Pa] and p0=135 [Pa].

66

Fig. 4.28: Comparison of the shape of the radii between the asymptotic model and the N.-St. one, when only the

effects of internal pressure are taken into account and the viscosity is assumed constant with the temperature:

a) 0 120p [Pa], b) 0 135p [Pa]

67

0p =120 [Pa], x=0.02 [m], A.M. h2=8.80 [mm], h1=4.60 [mm], N.-St. h2=8.89 [mm] h1=4.63 [mm].

0p =135 [Pa], x=0.02 [m], A.M. h2=8.92 [mm], h1=4.82 [mm], N.-St. h2=8.98 [mm] h1=4.81 [mm].

b1) x=0.11 [m] A.M. h2=5.38 [mm], h1=4.43 [mm], N.-St. h2=5.44 [mm] h1=4.49 [mm].

b2) x=0.11 [m] A.M. h2=7.04 [mm], h1=6.34 [mm] N.-St. h2=6.83 [mm] h1=6.10 [mm].

c1) x=0.21 [m] A.M. h2=2.47 [mm], h1=2.20 [mm], N.-St. h2=2.50 [mm] h1=2.23 [mm].

c2) x=0.21 [m] A.M. h2=6.23 [mm], h1=6.13 [mm], N.-St. h2=4.97 [mm] h1=4.84 [mm].

Fig. 4.29: rr at three different positions along the drawing direction in the furnace: a) x=0.02 [m], b1) and b2)

x=0.11 [m], c1) and c2) x=0.21 [m]. Case with only internal pressure.

68

It can be seen that in case of moderate values of inner pressure, the asymptotic solutions agree well

with the full 3D N.-St. ones, see Fig. 4.28a), while in extreme cases, when the fluid flow is largely

deformed in the radial direction, they clearly diverge, see Fig. 4.28b).

The key to understand where and why the two models match or not relies again on the comparison of

the normal stresses and the velocity field.

The values of rr vary by several orders of magnitude along the drawing direction, see for instance

Fig. 4.29a), Fig. 4.29b1) and Fig. 4.29b2), Fig. 4.29c1) and Fig. 4.29c2). The higher the inner applied

pressure, the bigger the radial stress caused by the viscous action results along the drawing direction.

The radial stress caused by the viscous action naturally increases along the drawing direction, since the

molten glass reduces its size, see Fig. 4.29a), Fig. 4.29b1) and Fig. 4.29b2), Fig. 4.29c1) and Fig.

4.29c2). Now, the radial stress increases from the outer toward the inner surface, since the overpressure

is applied directly inside the fiber. Therefore, the inner surface expands more than the outer one. This

fact finds also an analytic explanation: the radial stress caused by the viscous action is calculated in

(4.6.1), and the derivative of the leading order radial velocity can be calculated from (4.6.2). In case

that only internal pressure is applied, it reads

B

eU

Lp

xL

B

eU

Lp

xL

xL

A

xL

f

heh

ehhh

heh

ehhh

hh

ep

re

LU

r

w

L

x

f

L

x

f

2

20

1

2

10

2

10

2

20

2

20

2

10

1

2

20

2

10

2

20

2

10

2

10

2

20

0

2

0

0

0

1

2

(4.6.8)

where (4.3.46), (4.3.47) and (4.3.48) have been used. The two terms A and B in eq. (4.6.8) are

constant along the radial direction, therefore their sum is higher if 2r is smaller and vice-versa. Thus,

rr decreases from the inner toward the outer surface.

At the beginning of the drawing, the radial stress assumes similar values for both of the two cases

1200 p [Pa] and 1350 p [Pa], see Fig. 4.29a). In the latter, the values are slightly higher than in

the former. The rr computed by the A.M. agrees well with those obtained by solving the full set of N.-

St. equations for both of the two values of internal pressure. The slightly non-linear behaviour of rr ,

correctly computed by the A.M., is explained by inspecting eq. (4.6.2): for the case 0 135p [Pa],

both the first and the second terms are within )10( 3O , but in the vicinity of the inner surface the

second term is greater than the first one. Instead, for the case 0 120p [Pa], in the vicinity of the outer

69

surface the first term of (4.6.2) is found to be within )10( 3O , while the second one is found to be

within )10( 4O , thus the trend of the asymptotic stress becomes linear.

In the center of the furnace, the rr calculated for the case 1350 p [Pa] are higher than those

calculated for the case 1200 p [Pa]. A higher value of the inner pressure induces higher radial

stresses, since the cross–section area is lower, see Fig. 4.29b1) and Fig. 4.29b2). The fiber radii assume

also higher values, being the fiber blown up.

The radial stress computed by the A.M. agrees well with those computed by the N.-St. equations, even

if, at this position along the drawing, the fiber shape starts to diverge, see Fig. 4.29b1) and Fig. 4.29b2).

The radial stress computed by solving both the N.-St. and the asymptotic equations presents a small

nonlinear variation with the radius, since both terms of (4.6.2) are within )10( 3O , for both cases.

Finally, at the end of the drawing, rr results much higher for the case with higher internal pressure,

being the cross-sectional area almost completely shrank. The radial stress agrees remarkably well

between the two methods, see Fig. 4.29c1), for the case 1200 p [Pa], where the fiber radii also

match, see Fig. 4.28a). For the case 1350 p [Pa], the fiber shape does not match between the two

methods, and the radii computed by the A.M. result higher than those computed by the N.-St.

equations, see Fig. 4.28b). The almost linear trend of rr along the radial direction is due to the fact that

the first term of (4.6.2) is within 210O , while the second one is within )10( 3O , for the

case 1200 p [Pa]. For the case 1350 p [Pa], the second term of (4.6.2) results within )10( 2O , but

smaller than the first one.

A careful analysis of the normal component of the stress along the x direction merits also

consideration: xx is defined in (4.6.5), the scalar pressure p is carried out from the post-processing

tool for the N.-St. equations, and the asymptotic scalar pressure P has already been defined in (4.6.6).

The axial stress shows an almost constant trend along the radial direction, at each of the three position

of the drawing, see Fig. 4.30a), Fig. 4.30b1) and Fig. 4.30b2) and Fig. 4.30c1) and Fig. 4.30c2). Its

values vary from mbar to bar, as the cross section reduces its size.

The agreement between the A.M. and the N.-St. equations is very good at the all three stage of the

drawing. In fact, the discrepancies range from 0.6% to 6% for both of the two cases, even if the shape

of the fiber does not match between the two methods.

Now, the velocity field needs to be analyzed, that is, the axial component u of the velocity and the

radial one w . It is possible to notice that, at each stage of the drawing, the values of the axial

component of the velocity agree well between the two methods, for the two cases of the inner

pressure, see Fig. 4.31a), Fig. 4.31b1) and Fig. 4.31b2), Fig. 4.31c1) and Fig. 4.31c2). The constant

trend of u along the radial direction, shown by the N.-St. equations, is correctly reproduced by the

A.M. As far as the radial velocity is concerned, one sees that it decreases from the outer

70

surface 2h toward the inner surface 1h . Also in this case, 2h contracts much more and much faster

than 1h , at least at the beginning of the drawing. Toward the end of the furnace, w slowly decreases

from 2h to 1h , since the two surfaces get close to each other and the outer radius does not contract

anymore.

At the beginning of the drawing, the values of w match between the two methods, for both cases of

internal pressure. In fact, if w is computed by the A.M., its slightly nonlinear trend along the radial

direction finds an explanation in (4.6.7). In correspondence of the inner surface, the first term of

(4.6.7) is within )10( 6O , while the second one is within )10( 5O , determining the slightly nonlinear

trend of the velocity along the radial direction. Instead, toward the outer surface, the first term of

(4.6.7) is within )10( 5O , while the second one is within )10( 6O , thus the linear part has the major

influence. This happens for both cases of internal overpressure, see Fig. 4.32a). At the center of the

drawing, the values of w match between the two methods only for the case 1200 p [Pa], while there

is a big discrepancy, for the case 1350 p [Pa], see Fig. 4.32b1) and Fig. 4.32b2). The linear trend

along the radial direction, shown by the N.-St. equations is reproduced by the A.M. since the first term

of (4.6.) is much bigger than the second one. Nevertheless, both terms are within )10( 5O .

Analogously, at the end of the drawing, there is a good matching between the values of w calculated

with the two methods only for the case 1200 p [Pa]. For the case 1350 p [Pa], the discrepancies

overcame the 50% in value, see Fig. 4.32c1) and Fig. 4.32c2). The radial velocity shows a linear trend,

both when computed by solving the N.-St. equations and when computed by the A.M. For the

case 0 120p [Pa], the second term of (4.6.7) is within 510O , and the first one is

within )10( 4O and thus it dominates. For the case 0 120p [Pa], both terms of (4.6.7) are

within )10( 4O , but the first one is bigger, and therefore it dominates [15].

71

0p =120 [Pa], x=0.02 [m], A.M. h2=8.80 [mm], h1=4.60 [mm], N.-St. h2=8.89 [mm] h1=4.63 [mm].

0p =135 [Pa], x=0.02 [m], A.M. h2=8.92 [mm], h1=4.82 [mm], N.-St. h2=8.98 [mm] h1=4.81 [mm].

b1) x=0.11 [m] A.M. h2=5.38 [mm], h1=4.43 [mm], N.-St. h2=5.44 [mm] h1=4.49 [mm].

b2) x=0.11 [m] A.M. h2=7.04 [mm], h1=6.34 [mm] N.-St. h2=6.83 [mm] h1=6.10 [mm].

c1) x=0.21 [m] A.M. h2=2.47 [mm], h1=2.20 [mm], N.-St. h2=2.50 [mm] h1=2.23 [mm].

c2) x=0.21 [m] A.M. h2=6.23 [mm], h1=6.13 [mm], N.-St. h2=4.97 [mm] h1=4.84 [mm].

Fig. 4.30: xx at three different positions along the drawing direction in the furnace: a) 0.02x [m], b1) and

b2) 0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure.

72

0p =120 [Pa], x=0.02 [m], A.M. h2=8.80 [mm], h1=4.60 [mm], N.-St. h2=8.89 [mm] h1=4.63 [mm].

0p =135 [Pa], x=0.02 [m], A.M. h2=8.92 [mm], h1=4.82 [mm], N.-St. h2=8.98 [mm] h1=4.81 [mm].

b1) x=0.11 [m] A.M. h2=5.38 [mm], h1=4.43 [mm], N.-St. h2=5.44 [mm] h1=4.49 [mm].

b2) x=0.11 [m] A.M. h2=7.04 [mm], h1=6.34 [mm] N.-St. h2=6.83 [mm] h1=6.10 [mm].

c1) x=0.21 [m] A.M. h2=2.47 [mm], h1=2.20 [mm], N.-St. h2=2.50 [mm] h1=2.23 [mm].

c2) x=0.21 [m] A.M. h2=6.23 [mm], h1=6.13 [mm], N.-St. h2=4.97 [mm] h1=4.84 [mm].

Fig. 4.31: u at three different positions along the drawing direction in the furnace: a) 0.02x [m], b1) and b2)

0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure.

73

0p =120 [Pa], x=0.02 [m], A.M. h2=8.80 [mm], h1=4.60 [mm], N.-St. h2=8.89 [mm] h1=4.63 [mm].

0p =135 [Pa], x=0.02 [m], A.M. h2=8.92 [mm], h1=4.82 [mm], N.-St. h2=8.98 [mm] h1=4.81 [mm].

b1) x=0.11 [m] A.M. h2=5.38 [mm], h1=4.43 [mm], N.-St. h2=5.44 [mm] h1=4.49 [mm].

b2) x=0.11 [m] A.M. h2=7.04 [mm], h1=6.34 [mm] N.-St. h2=6.83 [mm] h1=6.10 [mm].

c1) x=0.21 [m] A.M. h2=2.47 [mm], h1=2.20 [mm], N.-St. h2=2.50 [mm] h1=2.23 [mm].

c2a) x=0.21 [m], A.M. h2=7.04 [mm], h1=6.34 [mm]. c2b) x=0.21 [m], N.-St. h2=6.83 [mm], h1=6.10 [mm].

Fig. 4.32: w at three different positions along the drawing direction in the furnace: a) 0.02x [m], b1) and b2)

0.11x [m], c1) and c2) 0.21x [m]. Case with only internal pressure.

74

4.6.4 The case with internal pressure and surface tension

The comparison between the shapes of the fibre is shown in Fig. 4.33 below, for two values of internal

pressure, that is, 1500 p [Pa] and 2150 p [Pa]. The system of equations (4.3.52), (4.3.53) and

(4.3.54) correctly computes the shape of the fiber also in case of high values of internal pressure, when

the fluid flow is widely deformed in the radial direction.

Fig. 4.33: Comparison of the shape of the radii between the asymptotic model and the N.-St. one, when both the

effects of internal pressure and surface tension are taken into account and the viscosity is assumed constant with

the temperature: a) 0 150p [Pa], b)

0 215p [Pa]

Again, an analysis of the stresses generated inside the fluid flow exhibit where and why the shape of

the two radii matches. Once more, starting with the radial stress caused by the viscous action, i.e. rr ,

one may observe that it increases its values from some mbar to bar along the drawing direction see

Fig. 4.34a), Fig. 4.34b1) and Fig. 4.34b2), Fig. 4.34c1) and Fig. 4.34c2).

75

At the beginning of the drawing, rr is to some extent linear along the radial direction for the case

1500 p [Pa], both when computed by the N.-St. equations and by the A.M, see Fig. 4.34a). By

inspecting eq. (4.6.2), one finds out that the first term is within )10( 3O , while the second one is

within )10( 4O , therefore it dominates. The radial stress increases from the inner toward the outer

surface, indicating that the surface tension effects prevail against the internal pressure ones, see Fig.

4.34a).

Also for the case 2150 p [Pa], rr agrees well between the two models. Now, the first and the

second term of (4.6.2) are within )10( 3O , inducing a strong nonlinear behaviour on the stress along

the radial direction. The radial stress now decreases from 1h toward 2h , suggesting that the inner

pressure effects become more relevant compared to the surface tension ones, see Fig. 4.34a).

In order to explain this effect, one should again examine the derivative of the asymptotic radial

velocity with respect to the radial coordinate, which now reads

B

A

ff

hh

hhphhhh

r

uUhhL

uUhh

hh

r

w

2

1

2

2

2

2

2

102121

2

0

1020

02

10

2

20

210

2

1

626

(4.6.8)

Again, the two terms A and B in eq. (4.6.8) are constant along the radial direction, but now the term B

contains the closing effects of the surface tension and the opening ones of the internal pressure.

Therefore, the resulting algebraic sum of these two effects determines the sign (positive or negative) of

the term B, and thus the trend of rr along the radial direction.

At the center of the drawing also, rr shows a non-constant trend along the radial direction, for both

values of the inner pressure. The agreement between the two models is good for the two values of

inner pressure, see Fig. 4.34b1) and Fig. 4.34b2). The nonlinear trend of rr along the radial direction is

due to the second term of (4.6.2), being of the same order of magnitude of the first one, that is,

within )10( 3O .

The radial stress now diminishes from the outer toward the inner surface, denoting that the surface

tension effects are stronger than the inner pressure ones

76

0p =150 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.58 [mm] h1=3.76 [mm], N.-St. h2=8.71 [mm] h1=3.87 [mm].

0p =215 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.91 [mm] h1=4.60 [mm], N.-St. h2=8.99 [mm] h1=4.62 [mm].

b1) x=0.11 [m], A.M. h2=3.68 [mm] h1=1.59 [mm], N.-St. h2=3.73 [mm] h1=1.65 [mm].

b2) x=0.11 [m], A.M. h2=5.95 [mm] h1=4.96 [mm], N.-St. h2=6.09 [mm] h1=5.11 [mm].

c1) x=0.28 [m], A.M. h2=0.600 [mm] h1=0.197 [mm], N.-St. h2=0.590 [mm], h1=0.195 [mm].

c2) x=0.28 [m], A.M. h2=1.17 [mm] h1=1.02 [mm], N.-St. h2=1.196 [mm], h1=1.06 [mm].

Fig. 4.34: rr at three different positions along the drawing direction in the furnace: a) 0.02x [m], b1) and

b2) 0.11x [m], c1) and c2) 0.28x [m]. Case with both effects of surface tension and internal pressure.

77

Similar considerations hold also at the end of the drawing. In fact, rr does not vary linearly along the

radial direction, increasing from the inner toward the outer surface.

It is much higher than that at the two other stages along the drawing, and it also approaches similar

values for the two different cases of the inner overpressure, see Fig. 4.34c1) and Fig. 4.34c2). The

slightly non-linear trend of the asymptotic stress is again explained by looking at the order of

magnitude of the two terms in (4.6.2).

For the two cases of internal pressure, the constant part is within )10( 1O , while the nonlinear one is

within )10( 2O , thus being the latter almost comparable with the former.

An analysis of the normal component of the stress along the x direction is also interesting. As

previously discussed, xx is defined in (4.6.5), the asymptotic scalar pressure P is defined in (4.6.6),

and the scalar pressure p is computed by the commercial software.

78

0p =150 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.58 [mm], h1=3.76 [mm], N.-St. h2=8.71 [mm] h1=3.87 [mm].

0p =215 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.91 [mm], h1=4.60 [mm], N.-St. h2=8.99 [mm] h1=4.62 [mm].

b1) x=0.11 [m] A.M. h2=3.68 [mm], h1=1.59 [mm], N.-St. h2=3.73 [mm] h1=1.65 [mm].

b2) x=0.11 [m] A.M. h2=5.95 [mm], h1=4.96 [mm] N.-St. h2=6.09 [mm] h1=5.11 [mm].

c1) x=0.28 [m] A.M. h2=0.600 [mm], h1=0.197 [mm], N.-St. h2=0.590 [mm], h1=0.195 [mm].

c2) x=0.28 [m] A.M. h2=1.17 [mm], h1=1.02 [mm], N.-St. h2=1.196 [mm], h1=1.06 [mm].

Fig. 4.35: xx at three different positions along the drawing direction in the furnace: a) 0.02x [m], b1) and

b2) 0.11x [m], c1) and c2) 0.28x [m]. Case with both effects of surface tension and internal pressure

79

0p =150 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.58 [mm], h1=3.76 [mm], N.-St. h2=8.71 [mm] h1=3.87 [mm].

0p =215 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.91 [mm], h1=4.60 [mm], N.-St. h2=8.99 [mm] h1=4.62 [mm].

b1) x=0.11 [m] A.M. h2=3.68 [mm], h1=1.59 [mm], N.-St. h2=3.73 [mm] h1=1.65 [mm].

b2) x=0.11 [m] A.M. h2=5.95 [mm], h1=4.96 [mm] N.-St. h2=6.09 [mm] h1=5.11 [mm].

c1) x=0.28 [m] A.M. h2=0.600 [mm], h1=0.197 [mm], N.-St. h2=0.590 [mm], h1=0.195 [mm].

c2) x=0.28 [m] A.M. h2=1.17 [mm], h1=1.02 [mm], N.-St. h2=1.196 [mm], h1=1.06 [mm].

Fig. 4.36: u at three different positions along the drawing direction in the furnace: a) x=0.02 [m], b1) and b2)

x=0.11 [m], c1) and c2) x=0.28 [m]. Case with both effects of surface tension and internal pressure

80

0p =150 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.58 [mm], h1=3.76 [mm], N.-St. h2=8.71 [mm] h1=3.87 [mm].

0p =215 [Pa], =0.3 [N/m], x=0.02 [m], A.M. h2=8.91 [mm], h1=4.60 [mm], N.-St. h2=8.99 [mm] h1=4.62 [mm].

b1) x=0.11 [m] A.M. h2=3.68 [mm], h1=1.59 [mm], N.-St. h2=3.73 [mm] h1=1.65 [mm].

b2) x=0.11 [m] A.M. h2=5.95 [mm], h1=4.96 [mm] N.-St. h2=6.09 [mm] h1=5.11 [mm].

c1) x=0.28 [m] A.M. h2=0.600 [mm], h1=0.197 [mm], N.-St. h2=0.590 [mm], h1=0.195 [mm].

c2) x=0.28 [m] A.M. h2=1.17 [mm], h1=1.02 [mm], N.-St. h2=1.196 [mm], h1=1.06 [mm].

Fig. 4.37: w at three different positions along the drawing direction in the furnace: a) x=0.02 [m], b1) and b2)

x=0.11 [m], c1) and c2) x=0.28 [m]. Case with both effects of surface tension and internal pressure.

81

The axial stress shows a constant trend along the radial direction, at each of the three positions of the

drawing, and it is again correctly computed by the asymptotic equations, see Fig. 4.35a), Fig. 4.35b1)

and Fig. 4.35b2), Fig. 4.35c1) and Fig. 4.35c2).

Furthermore, the agreement between the A.M. and the N.-St. equations is very good at the three stages

of the drawing. In fact, the discrepancies range from 0.6%, see Fig. 4.35a), to 4%, see Fig. 4.35c1) and

Fig. 4.35c2), for both of the two cases of the inner pressure.

Moreover, the values of the axial component of the velocity agree well between the two methods, for

the two cases of the inner pressure, see Fig. 4.36a), Fig. 4.36b1) and Fig. 4.36b2), and Fig. 4.36c1) and

Fig. 4.36c2).

u is constant along the radial direction, and therefore the assumption of a radial independent leading

order axial velocity is correct.

The discrepancies between the two methods are only between 1% and 5%, see Fig. 4.36c1) and Fig.

4.36c2).

Analyzing the radial velocity field, one notices the good matching between the two methods, for all

the three stages of the drawing, see Fig. 4.37a), Fig. 4.37b1) and 4.37b2), Fig. 4.37c1) and Fig. 4.37c2).

w presents a linear trend along the radial direction, decreasing from 2h toward 1h .

At the beginning of the drawing, the linear trend of w is correctly reproduced by the A.M. In fact, the

first term of (4.6.7) is within5(10 )O

near the outer surface, while it is within6(10 )O

near the inner

one. Nevertheless it is always bigger than the second one, which is within6(10 )O

, and it prevails.

Furthermore, the radial velocity takes on similar values for the two different cases of internal pressure,

see Fig. 4.37a). At the center of the drawing, the linear trend of w is recovered by the A.M. For the

case 1500 p [Pa], the first term of (4.6.7) results within )10( 5O , while the second one results

within )10( 6O . For the case 2150 p [Pa] instead, both terms of (4.6.7) are within )10( 5O , but the

first results much bigger than the second one.

Additionally, the velocity profile at this stage of the drawing is steeper than that at the previous

location, revealing that the two radii shrink faster than before.

At the end of the drawing, the values of the radial velocity are again correctly predicted by the A.M.,

see Fig. 4.37c1) and 4.37c2), with a maximum discrepancy of 7%, see Fig. 4.37c2). For the case

1500 p [Pa], the first term of (4.6.7) is within )10( 4O close to the outer surface and

within )10( 5O close to the inner surface, while the second one is always within )10( 6O . Instead, for

the case 2150 p [Pa], the first term of (4.6.7) is always between )10( 4O and the second one is

always between )10( 5O . Since the linear part of (4.6.7) dominates over the non-linear one, the trend

of the radial velocity results linear along the radial direction.

82

4.7 Numerical solution and experimental validation of the drawing process of six-hole optical fibres

In this section, the fiber drawing of a 3D six-hole geometry is numerically modeled and the results

have been compared with real, experimentally drawn fiber structures. First the final diameters

obtained by the numerical simulations are compared to those measured experimentally in absence of

internal pressure. Secondly, internal pressure has been applied and the gradual hole-deformation with

increasing pressure have been observed and compared between the numerical solution and

experiments. In this way, it is possible to show in detail in which cases both experiments and

simulation are in good agreement, i.e. when the inner pressure is not applied or it does not reach high

values, and where and why the results differ.

During the experiments, a constant “drawing speed” was used, with a feed rate into the furnace

0.0003fU [m/sec] and a fiber draw speed 0.3117dU [m/sec].

The glass is drawn in a standard graphite resistance furnace under protective Ar atmosphere at the

temperature of 1890 [°C]. The length of the “hot zone”, where the temperature reaches so high values

that the fiber glass deforms, is again estimated to be approximately 10 [cm] for practical purposes and

it is set in the center of the furnace, for more details see [4]-[6].

The high temperatures and the influence of gas flow conditions inside the furnace render the exact

temperature profile particularly difficult to measure. Furthermore, the inner pressure is supplied by

clean dry nitrogen gas, in order to avoid moisture as it enters the glass, which is not pre-heated

separately. Due to the high reduction of the fiber in size along the drawing direction, the pressure gas

strongly accelerates since the mass flow rate remains constant in a steady state process. Therefore, a

further convective heat exchange between the nitrogen gas and the glass may have to be taken into

account. Nevertheless, the numerical results presented in this section do not take any such thermal

modeling into account.

Numerical simulations have been run with several shapes of temperature profiles. It has been found

that better agreement between the numerical model and the experimental results is achieved when T

decreases sharply in the second half of the hot zone (which also agrees with experimental temperature

measurements at the top and bottom iris of the furnace). Two Gaussian functions are therefore

considered, one for the first half of the hot zone, i.e. x b and another one for the second half of the

hot zone, that is x b . More precisely, if x b , then 0.046c , if x b then 0.028c , see Fig.

4.38

83

Fig. 4.38: Temperature profile along the x axis of the furnace used in the simulations. The peak temperature is

in this case 1890 [°C]

The functional form of the viscosity as a function of the temperature used in the present numerical

simulations is again given in (4.4.1).

In this section a direct comparison between numerical simulations and experiments is presented. The

geometry and the mesh are shown in Fig. 4.18b).

During the experiments one glass preform has been used. The exact values of the external diameter

and the internal ones have been accurately measured by a SEM, see Fig. 4.18a). The internal pressure

applied during drawing is varied from 0 0p [mbar] to 0 300p [mbar], in steps of 0 50p [mbar].

Representative graphs showing the comparison between the numerical simulations and experiments

are presented below.

Starting with the case when no internal pressure is applied, it is possible to notice that a very good

agreement has been obtained between simulations and experiments. The final diameters obtained by

the SEM images and the numerical calculations almost coincide. The shape of the holes in the final

cross-section is circular and no deformation occurs, see Fig. 4.39a).

In order to see some hole-deformation, internal pressure should be provided into the holes. A very

good agreement between experiments and simulations is again obtained when the value of the inner

pressure is risen up to 0p =150 [mbar]. It can be seen that not only the final dimension of the holes is

increased but also that some hole-deformation became visible, see Fig. 4.39b). This is due to the

interaction among the pressure difference across a surface, the surface tension effects in each of the six

holes and the fluid flow stresses which act on each cross section along the drawing direction. This

balance is strongly influenced by the vicinity and the position of the holes themselves, which act in

such a way that the holes assume an elongated form. For deformed (i.e. non-circular holes), the

maximum hole size was measured in the radial direction.

The balance between the pressure difference across a surface and the surface tension force can be

84

described by the Young-Laplace equation, that is

iRp

(4.7.1)

where p is the pressure difference across a surface of a hole, and iR is the radius of curvature of an

inner surface. Not only the size of the inner holes is increased, but also the external diameter of the

fiber becomes larger, see Fig. 4.39b). In order to see a more pronounced hole-deformation, the value

of the internal applied pressure should be increased. For instance, by applying an internal pressure

of0 250p [mbar], the holes become enlarged and elongated toward the center of the fiber. The size

of the internal holes increases considerably and the interaction between two adjacent holes of different

size is noticeable, as it is shown in Fig. 3.39c). This effect will dramatically affect the final shape of

the capillary as the internal pressure is increased.

In particular, the surface of the small holes is differently distorted if the adjacent hole is a small one or

a big one. In the former case the surface results rounded, which means that almost no hole-to-hole

interaction occurs between two different small holes. In the latter case, the surface of the small holes

results flattened, due to the high stresses between two close capillaries of considerably different size.

A good agreement between experiments and simulations is again obtained, even if the exact contour of

the simulated inner surfaces slightly differs from the experimental ones. The size of the surface of the

inner holes computed numerically result bigger than the one obtained experimentally after the

drawing.

85

Fig. 4.39: Comparison of the SEM images of the final fiber cross-sections (grey-scale background) with

numerical simulations (blue overlaid contours) for applied internal pressure of a) 0 0p [mbar], b)

0 150p

[mbar], c) 0 250p [mbar], and d)

0 300p [mbar].

Indeed, the shape and the size of the external fiber diameter are correctly predicted by the numerical

simulations.

The situation is even more dramatic in case of 0 300p [mbar]: here the surface of a small hole is

also a little bit flattened on the side close to another small hole, since the pressure is so high that some

interaction between two small adjacent holes begins to be visible. The small holes have to enlarge

backwards where the stresses are not so high, in order to satisfy the mass conservation equations.

Again, a good agreement between experiments and numerical simulation has been obtained, even if

the exact shape of the inner holes does not match, see Fig. 4.39d).

The final external diameters computed numerically have been compared to those measured

experimentally for different values of internal pressure. A very good agreement between experiments

and simulations has been obtained, even in case of very high pressures. The discrepancies between

experiments and simulations vary from 1.5% up to 5%, see Fig. 4.40.

The final maximum sizes of the internal holes in the radial direction measured experimentally can be

also directly compared with those obtained numerically for different values of internal pressure. It can

86

Fig. 4.40: Comparison of the final external diameter between experiments and simulations. The inner pressure

varies between 00 300p [mbar]

be noticed that a good agreement between experiments and simulations has been obtained also for

values of internal pressure up to 0 300p [mbar]. The percentage difference of the values measured

experimentally, to those obtained numerically varies from 0.06% to 14 %, see Fig. 4.41.

High values of the inner pressure lead to strong deformations of the glass and larger deviation between

the simulated results and the experimental ones, since small differences between the real and the

assumed temperature profile become important when strong deformation of the fused glass occur.

Indeed, the fluid flow stresses which act on each cross section along the drawing direction become

higher when high values of inner pressure are applied, determining the shape of the inner holes. Thus,

since they depend on the viscosity, and therefore on the temperature, the knowledge of the real

temperature profile becomes essential in order to accurately reproduce the exact shape of the inner

holes, as described in the manuscript of Luzi et al. [16].

87

Fig. 4.41: Comparison of the final maximum sizes of the internal holes measured in the radial direction between

experiments and simulations for the big and the small hole in the upper half part of the fiber. The inner pressure

varies between 00 300p [mbar]

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, Issue 7, pp. 201-227, 2002

[2] Uhlmann, D.R. and Kreidl, N. J. “Glass Science and Technology”, Academic Press, New York, 1984

[3] S.H-K. Lee and Y. Jaluria, “Simulation of the Transport Processes in the neck-down region of a furnace drawn optical fiber”, Int. Journal of Heat and Mass Transfer, Vol. 40, 1997, pp. 843-856

[4] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “ Asymptotic Analysis of

Flow Processes at Drawing of Single Optical Microfibres”, International Journal of Chemical reactor Engineering, Vol. 9, Issue 1, Article A65, 201, pp. 1-26

[5] Christopher J. Voyce, Alistair D. Fitt, and Tanya M. Monro, “Mathematical Modeling as an Accurate Predictive Tool in

Capillary and Microstructured Fibre Manufacture: The Effects of Preform Rotation”, Journal of Lightwave Technology, Vol. 26, No 7 ,April 1,2008, pp. 791-798

[6] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Influence of surface

tension and inner pressure on the process of fibre drawing”, Journal of Lightwave Technology, Vol 28, No 13, July 1, 2010, pp. 1882-1888

[7] S. Roy Chodhury and Y. Jaluria, “Practical Aspects in the Drawing of an Optical Fiber”, J. Mat. Res., Vol. 13, pp. 494-503

[8] Zhilong Yin, Y. Jaluria, “Thermal Transport and Flow in High-Speed Optical Fiber Drawing”, Transaction of the ASME Heat

Transfer Division, Vol. 120, pp. 916-930, November 1998

[9] G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and alumino-silicates”, Geochimica Cosmochimica Acta, vol. 46, no. 6, pp. 1061-1072, 1982

[10] Kazuhide Nakajima, Kazuo Hogari, Member, IEEE, Jian Zhou, Katsusuke Tajima, and Izumi Sankawa, “Hole-Assisted Fiber

Design for Small Bending and Splice Losses”, IEEE Photonics Technology Letters, Vol. 3, No 12, pp. 1737-1739, December 2003 [11] T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer et al., “Quantitative Broadband Chemical Sensing in Air-

Suspended Solid-Core fibers”, Journal of Applied Physics 103 (2008), pp. 103-108

[12] N.N., ANSYS POLYFLOW 12.1 User´s Guide, Ansys Inc. 2009, (pp. 859) [13] K.Fujimoto, “Numerische Modellierung von Optischen Kristallfasern”, Studienarbeit, 2011, (pp. 74).

[14] S.C.Xue, R.I. Tanner, G.W. Barton, R. Lwin, M.C. J. Large and L. Poladin, “Fabrication of Microstructured Optical Fibers-Part

I: Problem Formulation and Numerical Modelling of Transient Draw Process”, Journal of Lightwave Technology, Vol. 23,Issue 7, pp 2245-2254, 2005

[15] Giovanni Luzi, Philipp Epple, Cornelia Rauh, Antonio Delgado, “Study of the effects of inner pressure and surface tension on the

fibre drawing process with the aid of an analytical asymptotic fibre drawing model and the numerical solution of the full Navier-Stokes equations”, Archive of Applied Mechanics, Volume 83, Issue 11, 2013, pp. 1607-1636

[16] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Numerical solution and

experimental validation of the drawing process of six-hole optical fibers including the effects of inner pressure and surface tension ”, Journal of Lightwave Technology, Vol 30, No 09, May 1, 2012, pp. 1306-1311

88

Chapter 5:

Conclusions

5.1 Conclusions and future works

The present work aims at optimizing and controlling drawing processes, first on the basis of a suitable

mathematical model, which provides a better access to the basic feature of the flow process and

greatly reduces the simulation time, and then with the aid of numerical simulations.

In the beginning, the mathematical model proposed by Fitt et al. [1], which couples the effects of

surface tension and internal pressure in the process of fibre drawing, has been analysed and revisited.

For simple cases the solution can be treated completely on an analytical manner, and it provides basic

insights into the velocity profile and the fibre shape. Successively, the effects of internal-hole

pressurization and surface tension have been included in the model, first separately and then together.

Even for these more complex situations the simplified equations are solved with less numerical cost.

The results obtained elucidate how the surface tension acts in such a way that the radii undergo a

contraction and in some cases, when the internal radius of a preform is very small, it may lead to the

complete collapse of the structure. On the other hand, when internal pressure is applied, an

enlargement of the internal hole occurs. If the applied pressure exceeds certain values, it may lead to

the explosion of the structure. Finally, if both effects are considered, it can be seen how internal

pressure acts as a stabilizing mechanism, maintaining holes opened and vice-versa how surface tension

increases the value of the explosion pressure, see for instance the manuscripts of [1] and the one of

Luzi et al. [2].

Second, this simplified model has been compared with experiments. The viscosity of high-purity silica

glass has been carefully chosen from data published by Urbain et al. [3]. This information has been

used to predict the results of experiments trials, demonstrating a good agreement between the two sets

of data, not only when no internal pressure is applied, but also when the internal hole is pressurized.

The theoretical model proposed here is seen to deviate from the experiments under sever conditions,

i.e. high values of temperature and internal pressure applied, see Luzi et al. [4]. In this analysis

thermal equilibrium is assumed by guessing two temperature profiles which represent the temperature

distribution inside the furnace, and one of them seems to work well even at high temperatures (e.g.

T>2000 [°C]), where the lack of knowledge about the temperature data is high.

Third, the simplified mathematical model has been compared with the full 3D set of the N.-St.

equations, in some asymptotic limits of interest. The viscosity is assumed constant at this stage of the

analysis.

89

For the simple case and the one in which the surface tension effects are taken into account, the shape

of the internal and the external radii match very well between the two methods. The stresses and the

velocity field are also correctly computed by the asymptotic method, and the agreement with the N.-St.

equations is very good at the three stages of the drawing.

In case that only the effects of internal-hole pressurization are considered, the asymptotic model

provides good results for moderate values of internal pressure. If the internal pressure is too high, the

radial velocity, and consequently, the component of the stress in the radial direction fail to be correctly

computed by the A.M. The radial velocity and the radial stress computed by the A.M. result higher

than those obtained by solving the full set of N.-St. equations. Therefore, the radii obtained by solving

the asymptotic equations assume higher values than those computed by the N.-St. ones. Nevertheless,

the stress and the velocity in the axial direction agree well between the two methods.

Afterwards, both effects of surface tension and internal-hole pressurization have been considered

together. The asymptotic model agrees very well with the N.-St. one, and the fiber shape is correctly

reproduced, even in case of high values of internal pressure. As far as the stress and the velocity field

concerns, the agreement between the A.M. and the N.-St. is very good, see Luzi et al. [5].

Finally a six holes structured fibre has been simulated numerically. The numerical simulations have

been compared directly with the experiments, obtaining a good agreement not only in absence of

internal pressure, but also when the internal holes are pressurized. A good agreement has been

obtained both for the size of the final cross-section and the shape of the holes, which results

particularly deformed in case of high values of internal pressure. The results obtained numerically start

to deviate from the experimental ones under sever conditions, that is high values of applied inner

pressure. In the present analysis throughout the thesis, thermal equilibrium among the furnace gas, the

fibre glass and the pressure gas has been assumed by guessing a temperature profile which is found

suitable to represent the temperature distribution inside the furnace. The assumption of thermal

equilibrium seems to work well even up to high values of the pressure, i.e. 0p =300 [mbar], and the

agreement between numerical simulations and experiments is good, as shown in the paper of Luzi et

al. [6]. For high values of the internal pressure, when the differences between experiments and

numerical computations start to be visible, heat exchange among the furnace gas, the fiber glass and

the nitrogen gas should be taken into account. More accurate numerical simulations and a more

detailed model, which take heat exchange into account, will be presented in future works.

Nevertheless, these results demonstrate that an analytical asymptotic model based on the small aspect

ratio of capillaries and numerical simulations can be a powerful and accurate predictive tool for the

drawing of micro-structured optical fibres, and they will play an increasingly important role in the

development of new photonic crystal fibres.

90

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for

Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, Issue 7, pp. 201-227, 2002 [2] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “ Asymptotic Analysis of

Flow Processes at Drawing of Single Optical Microfibres”, International Journal of Chemical reactor Engineering, Vol. 9, Issue 1,

Article A65, 201, pp. 1-26 [3] G. Urbain, Y. Bottinga, and P. Richet, “Viscosity of liquid silica, silicates and allumino-silicates”, Geochimica Cosmochimica

Acta, Vol. 46, no. 6, pp. 1061-1072, 1982

[4] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Influence of surface tension and inner pressure on the process of fibre drawing”, Journal of Lightwave Technology, Vol 28, No 13, pp. 1882-1888, July 1,

2010

[5] Giovanni Luzi, Philipp Epple, Cornelia Rauh, Antonio Delgado, “Study of the effects of inner pressure and surface tension on the fibre drawing process with the aid of an analytical asymptotic fibre drawing model and the numerical solution of the full Navier-

Stokes equations”, Archive of Applied Mechanics, Vol. 83, Issue 11, pp. 1607-1636, 2013

[6] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “Numerical solution and experimental validation of the drawing process of six-hole optical fibers including the effects of inner pressure and surface tension ”,

Journal of Lightwave Technology, Vol 30, No 09, May 1, 2012, pp. 1306-1311

91

APPENDIX A

Beginning from equations (4.2.7), (4.2.8) and (4.2.9) and neglecting the effects of internal pressure,

surface tension, inertia and gravity and considering the steady state case, one obtains

2 2

2 1 03 0xx

h h u

(A.1)

2

1 0 0x

h u

(A.2)

2

2 0 0x

h u

(A.3)

for more details, see the manuscript of Fitt et al. [1].

Equation (A.1) can be directly integrated, and by using (4.3.22), one obtains

2 2 020 10

0

3 xf

uh h U C

u

(A.4)

Herein, the viscosity has been considered constant, andC is a constant which arises from integration.

By performing another integration, one obtains

2 220 103

0

f

Cx

h h Uu De

(A.5)

where D is another integration constant. Employing equation (4.2.10) at 0x , one readily gets

fU D

(A.6)

whileC can be found by employing the condition (4.2.10) at x L , that is,

2 2

20 103ln

f d

f

h h U UC

L U

(A.7)

Inserting (A.7) into (A.5), one obtains the final form for the equation of the velocity, i.e.,

0

xL

fu U e

(A.8)

where )/log( fd UU , for more details see [1]. It is possible to notice, that the viscosity

term cancels out during the integration. The final equations for the internal and the external radii can

be obtained by integrating (A.2) and (A.3), using (4.2.10) at 0x and (A.8). Therefore, one gets

21 10

xLh h e

(A.9)

22 20

xLh h e

(A.10)

[1]. The evolution of the two radii during the drawing process is shown in Fig. A.1 below. Here,

20 0.01h [m], 10 0.00365h [m], 0.0001fU [m/sec] and 0.04dU [m/sec].

92

Fig. A.1: Evolution of the internal and external radius of a preform for the simplest case, when

internal hole pressure, surface tension, inertial and gravity effects are neglected and the viscosity

is assumed constant with the temperature

for more details, see the paper of Luzi et al. [2].

In case that the viscosity dependence upon the temperature has to be taken into account, one can still

achieve an exact solution of the equation (A.1). By performing derivations similar to those shown

above to obtain (A.8), using (4.3.22) and (4.2.10), one finally obtain

0 0

0

x Ld d

T T

fu U e

(A.11)

for the velocity, while for the two radii

1 10

0

fUh h

u

(A.12)

2 20

0

fUh h

u

(A.13)

where (A.11) has to be used in (A.12) and (A.13), [1].

The evolution of an initial preform during the drawing process, for the case of a variable viscosity, is

shown below. The viscosity is related to the temperature via equation (4.4.1). The temperature profile

is given by a Gaussian distribution, see Fig. A.2. Herein, 1950a [°C] and 0.0752c .

93

Fig. A.2: Evolution of the internal and external radii of a preform for the simplest case, when the

viscosity dependence upon the temperature (and therefore on the position) is considered.

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for

Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43,Issue 7, pp. 201-227, 2002

[2] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “ Asymptotic Analysis of

Flow Processes at Drawing of Single Optical Microfibres”, International Journal of Chemical reactor Engineering, Vol. 9, Issue

1, Article A65, 2011, pp. 1-26

94

APPENDIX B

Starting from equations (4.3.6) and (4.3.7), and applying the boundary conditions (4.2.10), one obtains

for the inner and the outer radii

2 2

1 0 10 fh u h U

(B.1)

2 2

2 0 20 fh u h U

(B.2)

Now, inserting (B.1) and (B.2) into (4.3.5), one gets for the velocity

00

03

xx

x

uu

u

where the viscosity term is here considered constant. The above equation can be integrated to give

2

0 0 0 03

xu u Cu

(B.3)

whereC is an integration constant. Now, introducing the transformation

'

0

3 xwu

w

, and its first

derivative

2'' '

0 2

3 xx x

x

w w wu

w

in (B.3), the previous equation reduces to

''

'

xx

x

wC

w

(B.4)

Equation (B.4) can be integrated two times, and when returning to the original variable, one obtains

20

2

3 Cx

Cx

CC eu

C e CA

(B.5)

where 2C and A are two constants which arise from the integration. Now, by means of the boundary

condition (4.2.10), 0 0 fu x U , one finds A

2 23

f

C CA

U C

(B.6)

Substituting back into (B.5), one obtains for the velocity

0

3

3 1

Cx

f

Cx

f

U Ceu

C U e

(B.7)

Inserting now (B.7) into (B.1) and (B.2), the final equations for the two radii assume the following

form

95

1 10

3 1

3

Cx

f

Cx

C U eh h

Ce

(B.8)

2 20

3 1

3

Cx

f

Cx

C U eh h

Ce

(B.9)

as shown in Fitt & al [1]. Here the constant is termed C . This constant can be evaluated by solving

the following transcendental equation

1 13

CL

d

f CLf

U e

UUe

C

(B.10)

obtained by dividing evaluating equation (B.7) at 0x , over equation (B.7) at x L . From (4.2.10),

one has 0 du x L U , [1].

From equations (B.8) and (B.9), it can be noted that the ratio1 2h h always remains constant during the

drawing process. Furthermore, due to the low velocities involved in the problem, it is clear that inertia

effects are negligible compared to others, like for example those of surface tension. A comparison

between the shapes of the two radii, obtained by solving equations (A.8)-(A.10) for the simple case,

and equations (B7)-(B9) for the case in which inertia effects are taken into account, is presented

below.

Fig. B.1: Evolution of the internal and external radius of a preform. Comparison between the

simplest case and the case in which the inertia effects are taken into account.

Herein, 0.0001fU [m/sec], 0.04dU [m/sec], 1 0.004h [m] and 2 0.01h [m]. The results

practically coincide and no difference is noticeable in the shape of the radii between the two cases.

96

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, Issue 7, pp. 201-227, 2002

97

APPENDIX C

Beginning from equations (4.3.13) and (4.3.14), when the boundary conditions (4.2.10) apply,

one obtains for the inner and the outer radii

2 2

1 0 10 fh u h U

(C.1)

2 2

2 0 20 fh u h U

(C.2)

Inserting (C.1) and (C.2) into (4.3.12), one obtains

2

0

0

0

03

x

xx

u gu

u

(C.3)

Equation (C.3) can be reduced of one order by introducing the transformation 2

'

0 0xw u u , that is,

0

'

0

2 2u

ww a

u

(C.4)

where3

ga

. After integrating (C.4) one gets

2

0 02w Bu au

(C.5)

where B is an integration constant. Coming back to the original variable

2

' 2

0 0 02xu Bu au

(C.6)

Equation (C.6) can be rearranged in the following form

0

2

0 02

dudx

Bu au

(C.7)

In order to evaluate the integral on the left-hand-side of (C.7), the following substitution, which is

suggested in the website Youmath [1], is prompt

2

0 0 02t Bu Bu au

(C.8)

So that 0u can be found by squaring both sides of (C.8), that is,

2

0

2

tu

a t B

(C.9)

Its first derivative reads

2

0 2

2

2

t B atdu

a t B

(C.10)

The integral now simplifies considerably

98

1 1dt x C

aBt

B

(C.11)

Integrating and returning back to the original variable, one obtains, after some rearrangements, the

final form of the equation for the asymptotic velocity,

2 2

0

2

2

B x C B x C

B x C

Be a a Beu

B Be

(C.12)

as it is shown in Fitt & al. [2]. Here1C B , and

2C C .

In order to find the two constants, equation (C.12) has to be evaluated at 0x , and at x L ,

obtaining two transcendental equations which has to be solved simultaneously

2 22 2C B C B C B

fB Be U Be a a Be

(C.13)

and

2 22 2B L C B L C B L C

dB Be U Be a a Be

(C.14)

The final equations for the two radii read from (C.1) and (C.2)

1 10

0

fUh h

u

(C.15)

2 20

0

fUh h

u

(C.16)

where0u is expressed by (C.12). Also in this case, the ratio 1 2h h remains constant at all stages of the

drawing process, for more details see [2].

REFERENCES

[1] http://www.youmath.it/lezioni/analisi-matematica/integrali/607-sostituzioni-di-eulero-per-il-calcolo-di-integrali.html

[2] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for

Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, Issue 7, pp. 201-227, 2002

99

APPENDIX D

The starting point is now represented by the equations (4.3.18), (4.3.19) and (4.3.20). Integrating

(4.3.18) and employing (4.3.22), one obtains

1 2

0 0 02 2 2 2

20 10 20 103 3x

f f

h h Cu u u

h h U h h U

(D.1)

whereC is an integration constant. The solution is found by expanding in power series of

L hU the unknowns, that is,

0 01 02u u u

(D.2)

1 11 12h h h

(D.3)

2 21 22h h h

(D.4)

is a non-dimensional ratio between the surface tension and the viscosity. Now, inserting (D.2), (D.3)

and (D.4) in (D.1), one obtains

11 12 21 22

01 02 01 022 2

20 10

01 022 2

20 10

3

3

x

f

f

hU L h h h hu u u u

h h U

Cu u

h h U

(D.5)

By considering like powers of , the following system of equations for the velocity, correct to O ,

can be constructed

0

01 012 2

20 10

11 21 1

02 01 022 2 2 2

20 10 20 10

3

3 3

x

f

x

f f

Cu u at

h h U

hU L h h Cu u u at

h h U h h U

(D.6)

(D.7)

The integration of (D.6) is straightforward, and when (4.2.10) apply the following expression for the

leading order asymptotic axial velocity is obtained

01 ex

Lfu U

(D.8)

so the constant 2 2

20 103 fC h h U L .

By employing the same procedure, that is, substituting (D.2), (D.3) and (D.4) in equations (4.3.19) and

(4.3.20), one obtains after some algebra

100

2 0

11 01

2 2

11 21 11 21 012 1

11 12 01 11 02 2 2

20 10

2 0

21 01

2 2

11 21 11 212 1

21 22 01 02 21 012 2

20 10

0

2

0

2

x

xf

x

xf

h u at

h h h h uhUh h u h u at

L h h U

h u at

h h h hhUh h u u h u at

L h h U

(D.9)

(D.10)

(D.11)

(D.12)

correct to O . The equations (D.9) and (D.11) at0 can be readily solved, giving

211 10

xLh h e

(D.13)

and

221 20

xLh h e

(D.14)

where (4.2.10) and (D.8) have been used. In order to solve the equations (D.10) and (D.12) at1 , one

needs to calculate02u from (D.7), that is

2

02 02

20 10

03

xL

x

hUeu u

L L h h

(D.15)

where (D.8), (D.13) and (D.14) have been used. Integrating the latter, one obtains

2

02

20 10

2

3

x xL L

Lu e De

h h

(D.16)

where D is a constant which arise from integration, and the expression Uh L has been

utilized. D has to satisfy simultaneously two boundary conditions

20 10

2

20 10

20

3

2

3

LD at x

h h

LD e at x L

h h

(D.17)

(D.18)

provided that 02( 0) 0u x and 02( ) 0u x L . Therefore, the constant D can be written as

2

20 10

21 1

3

L xD e

h h L

(D.19)

in order to satisfy (D.17) and (D.18). Finally, one obtains

22

02

20 10

21 1

3

x xL L

L xu e e e

h h L

(D.20)

after having substituted (D.19) in (D.16). Now, using (D.20) in (D.2), one attains the final form of the

equation for the velocity, that is,

101

22

0

20 10

2e 1 1

3

x x xL L L

f

L xu U e e e

h h L

(D.21)

as shown in the manuscript of Fitt & al. [1]. Using (D.13) and (D.14) in (D.10), one finds after some

simplifications

20

01

12 01 10 02

01 01 20 10

2

f

f f

x

Uh

U U uh u h u

u u h h

(D.22)

Utilising (D.8), integrating the latter and rearranging, one obtains

3

220 102

12 02

20 10

ee e

2 2

xLx x

L L

f f f

L h hEh u

U h h U U

(D.23)

Now, the integration constant E can be evaluated at 0x

20

20 10

2L hE

h h

(D.24)

where (D.20) has been used. Substituting now (D.24) in (D.23), with the aid of (D.20), one obtains

after some algebra

22 2

12 20 10 10

20 10

e 3 1 e e 13

x x xL L L

f

L xh h h h e

U h h L

(D.25)

Therefore, the final equation for the internal radius reads

22 2 2

1 10 20 10 10

20 10

e 3 1 e e 13

x x x xL L L L

f

L xh h e h h h e

U h h L

(D.26)

[1], where (D.25) and (D.13) have been used in (D.3). Similarly, inserting (D.13) and (D.14) in (D.12),

one obtains

10

01

21 01 20 02

01 01 20 10

2

f

f f

x

Uh

U U uh u h u

u u h h

(D.27)

Inserting (D.8) in (D.27) and performing the integration, one obtains

3

210 20 022

21

20 10

ee e

2 2

xLx x

L L

f f f

Lh h uZh

U h h U U

(D.28)

The constant Z is found by evaluating (D.28) at 0x , that is,

10

20 10

2 LhZ

h h

(D.29)

where (D.20) has been utilized. Finally, by substituting (D.29) in (D.28) one gets

22 2

21 10 20 20

20 10

e 1 e 3 e 13

x x xL L L

f

L xh h h h e

U h h L

(D.30)

Thus, as far as the external radius concerns, one finally obtains

22 2 2

2 20 10 20 20

20 10

e 1 e 3 e 13

x x x xL L L L

f

L xh h e h h h e

U h h L

(D.31)

102

[1], where (D.14) and (D.30) have been used in (D.4).

Now, it may be very useful to understand under which conditions the system of equations (D.21),

(D.26) and (D.31) can be used. They have been derived by expanding the unknowns in power series of

a small parameter , which is a non-dimensional ratio between surface tension and viscosity. In the

figures below, the value of surface tension is varied again between 0.01 [N/m] and 0.3 [N/m],

and the mass conservation equationin outm m is checked.

Furthermore, the following criterion for the hole-collapse is derived in [1]

20

10 20 10

logf d f

LhU U U

h h h

(D.32)

Equation (D.32) and mass conservation equation need to be discussed together.

Fig. D.1: Influence of different values of surface tension on the preform geometry: a) 0.3 [N/m], b)

0.1 [N/m], c) 0.05 [N/m], d) 0.001 [N/m]

In Fig. D.1 the following values 0.0001fU [m/sec], 0.04dU [m/sec], 1 0.004h [m]

and 2 0.01h [m] have been chosen. The last value, that is 0.001 [N/m], keeps the mass

conserved, and it has a minimum influence on the shape of the radii. The criteria (D.32) is also

satisfied, therefore no collapse occurs, see Fig. D.1d).

103

The second and the third values tested, which are 0.1 [N/m] and 0.05 [N/m], do not keep the

mass conserved. Even if equation (D.32) is satisfied and no collapse occurs, see Fig. D.1b) and Fig.

D.1c), the system of equations (D.21), (D.26) and (D.31) does not deliver correct physical results.

The situation changes dramatically if the value 0.3 [N/m] is used. The mass conservation equation

and the criterion (D.32) result violated. In this case, collapse occurs and the inner diameter assumes

also negative values.

Thus, the system of equations (D.21), (D.26) and (D.31) provides reliable results only in the last case,

see Fig. D.1d), where a very small value of the surface tension parameter keeps the mass conserved.

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for

Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, pp. 201-227, 2002

104

APPENDIX E

The case in which only internal pressure effects are taken into account is governed by the system of

equations (4.3.43), (4.3.44) and (4.3.45). Subtracting (4.3.45) from (4.3.44) and applying the boundary

condition (4.2.10), one obtains equation (4.3.22). The substitution of equation (4.3.22) into equation

(4.3.43) allows the latter to be integrated, that is,

0

x

Lfu U e

(E.1)

where equation (4.2.10) has been used. Now, using (E.1) in (4.3.44) and (4.3.45) and expanding the

derivatives, one gets

2 2 2

0 1 2 20 10 1

1 2 2

20 102

f

x

f

p h h h h U hLh

h h U

(E.2)

2 2 2

0 1 2 20 10 2

2 2 2

20 102

f

x

f

p h h h h U hLh

h h U

(E.3)

where equation (4.3.22) has been used. Dividing now (E.3) by (E.2), one has

2 2 2

0 1 2 20 10 22

2 2 21

0 1 2 20 10 1

f

f

p h h h h U hdh L

dh p h h h h U hL

(E.4)

Setting now 2 2

20 10 fA h h UL

, equation (E.4) can be reformulated as follows

2

2 12

21 1 2

o

o

h p h Adh

dh h p h A

(E.5)

So that the variables can be separated, say

2 2

2 2 1 1

2 1

o op h A dh p h A dh

h h

(E.6)

and the latter equation can be integrated

2 2

2 12 1ln ln

2 2o o

h hp A h p A h C

(E.7)

whereC is an integration constant. Rearranging the previous equation, one obtains

2 2

2 12

1

2ln

oph h C

h

h A

(E.8)

Substituting the expression for A in the latter equation, one gets

2

2 21 20 10

exp2

xL

o

f f

Lp eh LC

h U h h U

(E.9)

105

Applying the boundary condition (4.2.10) at 0x , the constantC can be evaluated as

2 2

20 10 10

20

ln exp2

f o

f

h h U h LpC

L h U

(E.10)

Substituting back, rearranging and squaring the terms, one obtains

122 2 202 1 2

10

xo L

f

Lpe

Uhh h e

h

(E.11)

Finally, subtracting from each side2

1h and employing again equations (4.3.22) and (E.1), the final

expression for the internal radius reads

12

2 2 2

10 20 10

1

1

2 2

20 10

x

o L

f

x

L

p Le

U

h h h eh

h e h

(E.12)

The final expression for the external radius may be obtained by re-writing (E.11) in the following way,

1 22 2102 12

20

xo L

f

Lpe

U he h h

h

(E.13)

Now, subtracting2

2h from both sides, using (4.3.22) and (E.1), one obtains

12

2 2 2

20 20 10

2

1

2 2

10 20

x

o L

f

x

L

p Le

U

h h h eh

h e h

(E.14)

for more details see the publication of Fitt & al. [1].

It is possible to realize how the shape of the two radii is modified, in case of increasing applied

pressure, see for instance Fig. E.1. Herein, 20 0.01h [m], 10 0.00365h [m], 0.0001fU [m/sec]

and 0.04dU [m/sec].

106

Fig. E.1: Evolution of the initial preform as different values of internal pressure are applied. Here only internal

pressure effects are considered: a) 0 60p [Pa], b)

0 80p [Pa], c) 0 100p [Pa] and d)

0 137p [Pa]

as exposed in the manuscript of Luzi et al. [2]. Dividing now 1h by

2h , one obtains

22

201

2 2

2 10

exp 1 x Lo

f

h p Lhe

h h U

(E.15)

for additional details see [1]. The higher the pressure, the more the ratio 1 2h h increases till certain

point at which the fiber may finally explode and the expression above became unbounded. To clarify

this point, consider the mass conservation of the fluid flow taken between two different points A and B

i.e.

2

2 2 2 2 2 12 1 2 1 2 2

2

1 BA A B B B

B

hh h h h h

h

(E.16)

One easily sees that if 1h approaches 2h the mass is not conserved anymore. By considering 1 2h h in

(E.15), an expression for which (E.12) and (E.14) become unbounded can be worked out

10

20

2log 1 log

f

o

U hLx

p L h

(E.17)

for more details see [1].

107

One may claim that at x = L the fibre structure has the weakest part and so it is there that the

explosion begins to occur. Setting x = L in (E.17) and solving for the pressure, one obtains

10 20

max

2 log

1

f

o

U h hp

L e

(E.18)

where max0p is the maximum pressure the fibre can tolerate [1].

If now the value of the internal pressure is increased untill the fiber explodes, one can see that the

coordinate value along the furnace, at which the explosion occurs, decreases as the pressure increases,

see Fig. E.2

Fig. E.2: Coordinate values at which the explosion occurs for different values of internal pressure: a)

0 150p [Pa], b) 0 158p [Pa], c) 0 164p [Pa] and d) 0 190p [Pa]

[2]. The value of the pressure necessary to decrease the x-coordinate at which the explosion occurs

does not increase linearly. This happens because the amount of material to blow up increases as the

“beginning of the hot zone” is approached, and the thickness of the fibre does not decrease linearly

along the furnace as long as the draw takes place.

108

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for

Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, Issue 7, pp. 201-227, 2002

[2] Giovanni Luzi, Philipp Epple, Michael Scharrer, Ken Fujimoto, Cornelia Rauh, Antonio Delgado, “ Asymptotic Analysis of

Flow Processes at Drawing of Single Optical Microfibres”, International Journal of Chemical reactor Engineering, Vol. 9,Issue 1,

Article A65, pp. 1-26, 2011

109

APPENDIX F

It may be useful to understand under which conditions the system of equations (4.3.58), (4.3.59),

(4.3.60), (4.3.61) and (4.3.62) can be used. It has been derived by scaling the unknown1h and the

surface tension with the parameter , regarding that a small values of surface tension may close a

small hole.

Therefore, four values of the surface tension parameter have been tested, that is, 0.3 [N/m],

0.1 [N/m], 0.01 [N/m] and 0.001 [N/m], see Fig. F.1, in order to see under which

conditions the system of equations (4.3.58)-(4.3.60) provides physical results. The first of the four

values leads to a situation in which the internal hole almost collapses, see Fig. F.1a), and the mass

conservation equation is clearly violated, since there is a discrepancy between inm and outm of 16%.

Therefore, the present and higher values of surface tension cannot be used with the present system of

equations.

The second value of the surface tension presents the same problem, since the discrepancy

between inm and outm is of 9.31%, see for instance Fig. F.1b).

This problem does not occur with lower values of surface tension, see for instance Fig. F.1c) and.

F.1d). In case that 0.01 [N/m], the difference between inm and outm attains 1.19%, while for the last

case 0.001 [N/m] the difference is 0.12%. Thus, the latter value of surface tension provides the

best physical results.

110

Fig. F.1: Evolution of the initial preform in case of different values of surface tension: a) 0.3 [N/m], b)

0.1 [N/m], c) 0.01 [N/m], d) 0.001 [N/m]

Now, the value of the surface tension 0.001 [N/m] is maintained fixed, while the value of the

internal pressure is varied. Precisely, four values of the internal pressure have been used, that is,

0 10p [Pa], 0 25p [Pa], 0 75p [Pa] and 0 100p [Pa]. The result is shown in Fig. F.2. For

the first of the four cases, i.e. F.2a), the right-hand-side of (4.3.62) is lower than its left-hand-side,

therefore hole-expansion does not occurs. In this case the mass conservation equation holds with a

percentage difference between inm and outm of 2.62%. Analyzing the second case, i.e. the one proposed

in Fig. F.2b), it also happens that the right-hand-side of (4.3.62) is greater than its left-hand-side, and

no hole-expansion is visible. In this case, the mass conservation equation does not holds and the

discrepancy between inm and outm attains 7.5%. The case shown in Fig. F.2c) is different, since now

equation (4.3.62) holds, and hole-expansion takes place. However, the mass is not conserved, since the

difference between inm and outm is of 33.13%. Even worse is the case shown in Fig. F.2d), where the

difference between inm and outm reaches the 54.11%. Equation (4.3.62) is satisfied, and hole-expansion

111

occurs. In all of the four cases considered now, the right-hand-side of (4.3.61) is always greater than

its left-hand-side, so hole-collapse never happens.

Herein, 20 0.01h [m],

10 0.004h [m], 0.0001fU [m/sec] and 0.04dU [m/sec],

Fig. F.2: Influence of internal pressure on the initial preform with 0.001 [N/m]: a)

0 10p [Pa], 0 25p [Pa],

0 75p [Pa], 0 100p [Pa].

REFERENCES

[1] A.D.Fitt, K.Furusawa, T.M. Monro, C.P.Please and D.J.Richardson, “The Mathematical Modeling of Capillary Drawing for

Holey Fibre Manufacture”, Journal of Engineering Mathematics, Vol.43, Issue 7, pp. 201-227, 2002

112

Thermofluiddynamische Modellierung

und Simulationen des

Faserziehprozesses von photonischen

Kristallfasern

Der Technischen Fakultät der Universität

Erlangen-Nürnberg

zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von

Giovanni Luzi

Erlangen, 2014

113

Zusammenfassung

Mikrostrukturierte optische Fasern (sogenannte "Photonische Kristalle" oder "Hohlfasern“) stoßen

gegenwärtig auf überaus großes Interesse, da sie erlauben eine Vielzahl von optischen Effekten zu

generieren. Das Lochmuster erlaubt die Lichtführung innerhalb des festen oder des hohlen Kerns.

Diese neuartige Faserart besitzt zukünftig ein hohes Potential in einem breiten Anwendungsspektrum:

Telekommunikation, Optik, Gaslasergeräte und Präzisionssensorik. Solche Fasern bestehen aus einem

massiven oder hohlen Kern, der von luftgefüllten Löchern umgeben wird. Photonische Kristalle

werden durch die Erwärmung bis in die Nähe des Schmelzpunktes und Dehnen der Faser (Ziehen) aus

einer anfänglichen Preform in mehreren Schritten hergestellt. Während des Ziehens, reduziert sich die

Größe der äußeren Abmessungen und der inneren Lochdurchmesser sehr stark. Zahlreiche Parameter

beeinflussen das Herstellungsverfahren wie zum Beispiel die Materialzufuhr- und

Ziehgeschwindigkeit, der interne Lochdruckausgleich und die Oberflächenspannung. Daher tritt eine

komplexe Überlagerung von thermofluiddynamischen, konstitutiven und kapillaren Effekten ein. In

der vorliegenden Dissertation werden Ergebnisse von mathematischen Analysen und numerischen

Simulationen des Ziehprozesses mit dem Ziel vorgestellt und diskutiert, die während des

Ziehprozesses auftretenden thermofluiddynamischen Effekte besser zu verstehen sowie zu

kontrollieren. Zunächst sind einzelne Hohlkapillaren betrachtet worden, da sich der Ziehprozess für

diesen Fall vollständig analytisch behandeln lässt. Ein in der Literatur für andere Anwendungen

angegebenes, stark vereinfachtes asymptotisches Modell wurde überarbeitet und durch die numerische

Lösung der Gleichung der Massenerhaltung stark verbessert. Das verbesserte Modell berücksichtigt

die Auswirkungen von Innendruck und Oberflächenspannung, die den endgültigen Querschnitt der

Faser weitestgehend bestimmen. Die Oberflächenspannung führt zu der Tendenz eines möglichst

kleinen Lochdurchmessers. Im Unterschied hierzu sorgt der Innendruck dafür, dass sich der

Durchmesser nur bis zum gewünschten Durchmesser verringert. Die in den Berechnungen verwendete

Temperaturabhängigkeit der Viskosität entstammt der Literatur. Der Vergleich der erhaltenen

numerischen Ergebnisse der vereinfachten Bewegungsgleichung mit experimentellen Ergebnissen

zeigt eine gute Übereinstimmung hinsichtlich des Endfaserdurchmessers ohne und mit

Druckbeaufschlagung. Des Weiteren wurden die endgültigen Durchmesser mit denen verglichen, die

sich aus der Lösung der vollen 3D Bewegungsgleichung ergeben. Letztere wurden mit Hilfe

verschiedener kommerzieller Codes gewonnen. Der Geometrieabbildung und Vernetzung diente die

Software ANSYS Gambit. Die Lösung der Bewegungsgleichungen geschah sodann mit einer Finite-

Elemente-Methode, die ANSYS Polyflow bereitstellt. Auch hier zeigt sich eine sehr gute

Übereinstimmung der vom vorgeschlagenen asymptotischen Modell und von den 3D Navier-Stokes-

Gleichungen vorausgesagten Ergebnisse für den Endfaserdurchmesser. Weitere Untersuchungen

betreffen die axiale Faserform. Anhand eines sorgfältigen Vergleichs der radialen und axialen

Belastung und Geschwindigkeiten wird erläutert, an welchen Stellen und warum die beiden Modelle

114

entlang der Ziehrichtung einander nicht vollständig entsprechen. Dieser Vergleich beinhaltet den Fall

konstanter Viskosität. Gerade dieser Vergleich illustriert das Zusammenspiel von

Oberflächenspannung und Innendruck und bildet somit eine tragfähige Basis für die Behandlung

komplexerer Geometrien mit Hilfe eines kommerziellen Codes. Schließlich wurde eine Sechsloch-3D-

Geometrie numerisch simuliert. Ähnlich wie zuvor, geschah dies mit der kommerziellen Software

ANSYS Gambit für Geometrieapproximation und Vernetzung sowie mit ANSYS Polyflow als Löser.

Die erhaltenen numerischen Ergebnisse weisen eine gute Übereinstimmung mit Experimenten auch

bei hohen Werten des Innendrucks auf, bei welchen starke Querschnittsdeformationen vorliegen. In

dieser Teiluntersuchung wurde vom thermodynamischen Gleichgewicht zwischen Gasofen, Glasfaser

und Gasdruck ausgegangen. Dabei mussten geeignete Ofentemperaturprofile mangels direkter

Messbarkeit geschätzt werden. Dennoch ergeben sich in zahlreichen Fällen nur geringe Abweichungen

zwischen den berechneten und den gemessenen Profilen. Nur bei sehr hohen Temperatur- und

Druckwerten treten merkliche Abweichungen auf. Um weitere Verbesserungen zu erzielen, müsste in

diesen Fällen das Temperaturprofil der Faser durch zusätzliche Lösung der thermischen

Energiegleichung korrekt berechnet werden.

115

116

117