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Melnikov, D. (2004). Development of numerical code for the study of marangoni convection (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences appliquées – Chimie, Bruxelles.

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DEVELOPMENT OF NUMERICAL CODE FOR THE STUDY OF MARANGONI CONVECTION

CO-PROMOTEUR: DR. V. M. SHEVTSOVA

THèSE PRÉSENTÉ!

PAR DENIS MELNIKOV POUF

OBTENIR LE GRADE DE DOCTEUP

EN SCIENCES APPLIQUÉES

CODE FOR THE STUDY OF MARANGONI CONVECTION

A Thesis

Presented to The Faculty Of Applied Sciences

by

Denis Melnikov

In Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in Applied Sciences

Université Libre de Bruxelles

May 2004

To my parents

ment of Physical Chemistry of Pree University of Brussels. Headed by Professor Jean-Claude Legros for already more than twenty years, the MRC gave me an excellent opportunity to ac- complish my PhD degree work and présent the thesis. Thanks to the friendly atmosphère, irreplaceable support from every staff member and facilities that I hâve got the work was very fruitful and exciting, and my stay enjoyable. The expérience I hâve got for the years spent at the MRC is really huge and valuable.

I would like to personally thank Jean-Claude Legros, who through almost six years was the promoter of my thesis, providing an extraordinary assistance during ail the time, funding the Project, providing the opportunity for foreign travels for participating at conférences, and helping me in realizing my research potential. Without Jean-Claude this project would not hâve been possible.

I am further grateful for the extraordinary support and input from my supervisor Valonfei»

Shevtsova, for her continuai generosity and tolérance in allowing me the time I needed to com­

plété this Project. Without the supervision, management, help and ideas I hâve got from Valentina the work could not be as productive and interesting. I could not even wish a better supervisor and without her I could hardly manage to write and hâve published as many papers.

Also, I would like to express my appréciation for the computers System administrator Patrick Queeckers who always helped me in resolving the problems coming from computers, software and supplying me with assistance and helpful advises. Despite his permanent occupation with computer stufî and giving assistance to ail the experimental works carrying out in the laboratory, he always found out some time for helping me.

My work expérience could not hâve been as pleasurable without friendship of my fellow re- searchers, Carlo Saverio lorio, Natacha Callens, Andrei Vedernikov. Thank you ail for enjoyable lunch times and coflFee breaks on afternoons. Spécial thanks go for Carlo for making me a com­

pany for midnight coffee breaks at the Brussels airport. Also, I would like to convey my sincerest gratitude to Frank Dubois, Pierre Colinet, Stefan Van Vaerenbergh, Christophe Minetti, Mar­

cel Hennenberg, Philippe De Gieter, Mohamed Mojahed, Benoit Scheid, Jean-Charles Dupin, Rachid Naji for your help and friendly atmosphère you hâve created for me.

The emotional support of my family, mother (Tamara Melnikova) and father (Euguenii

Melnikov), is known without saying. Being far away from me, you are always deeply concerned

about your son. I love you and I will always do, and this work is dedicated for you.

0.1 ACKNOWLEDGMENTS... i

0.2 LIST OF SYMBOLS... xiii

0.3 ABSTRACT... xv

0.4 PREFACE... xvii

1 SUMMARY 1 2 INTRODUCTION 3 2.1 CRYSTAL GROWTH ... 3

2.2 LIQUID BRIDGE (LB) ... 6

2.3 MATHEMATICAL MODELS ... 13

2.4 SCALING ANALYSIS ... 16

3 NUMERICAL APPROACH 20 3.1 PREVIOUS WORKS ON THE LIQUID BRIDGE PROBLEM . 20 3.2 DISCRETIZATION OF PARTIAL DIFFERENTIAL EQUA­ TIONS (PDE’s). GENERAL REMARKS... 21

3.3 MODEL PROBLEM... 21

3.3.1 Finite Différences... 21

3.3.2 Methods of solving linear Systems... 24

3.3.3 Finite volume method... 25

3.3.4 Unsteady PDE’s... 26

3.3.5 Stability of itérative methods... 27

3.4 NUMERICAL TECHNIQUE OF SOLVING THE SYSTEM OF NAVIER-STOKES EQUATIONS IN CYLINDRICAL GEOME- TRY... 30

3.4.1 The grid System... 33

3.4.2 Discretization and algorithm used for the présent study... 33

3.5 THE DISCRETE FOURIER TRANSFORM OF REAL FONC­ TIONS ... 35

3.6 THE ALGORITHM OF DISCRETE FAST FOURIER TRANS­ FORM OF REAL PERIODIC FONCTION ... 37

3.7 APPLICATION OF THE DISCRETE FOURIER TRANSFORM FOR SOLVING THE LAPLACE EQUATION... 39

3.8 THE ALTERNATING DIRECTION IMPLICIT METHOD (ADI) 40 3.9 THE THOMAS ALGORITHM... 41 4 INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON

ONSET OF INSTABILITY IN LIQUID BRIDGE 43

4.4 CODE VALIDATION ... 46

4.4.1 Medium Prandtl number liquids with constant viscosity . ... 46

4.4.2 High Prandtl number liquids with constant viscosity... 48

4.5 CONDITIONS OF GENERATING THE STANDING AND TRAVELING WAVES IN THE LIQUID BRIDGE ... 50

4.6 TEMPERATURE DEPENDENT VISCOSITY... 52

4.6.1 On the influence of the choice of the reference température... 52

4.6.2 Parametric study of the onset of instability for liquids with temperature-dependent viscosity... 57

4.6.3 The influence of viscosity on the wave properties of the flow... 60

4.6.4 High Prandtl number liquids with temperature-dependent viscosity . 65 4.6.5 Hydrothermal waves in liquid bridge with a buoyancy force, Pr = 35,Gr^0... 65

4.6.6 Hydrothermal waves in liquid bridge without buoyancy force, Pr = 35, Gr = 0... 70

4.7 CONCLUSIONS TO PART 4... 72

5 MULTISTABILITY OF OSCILLATORY THERMOCAPILLARY CON­ VECTION IN LIQUID BRIDGE 75 5.1 INTRODUCTION TO PART 5 75

5.2 CODE VALIDATION IN CASE OF Pr = 4, c/ - 0 76

5.3 BACKGROUNDS... 77

5.4 RESULTS ... 79

5.4.1 Multistability of the oscillatory flow ... 79

5.4.2 Spatiotemporal properties of TW m = 2 82

5.4.3 Spatiotemporal properties of TW m = 3 ... 87

5.4.4 Comparison of the solutions with different wave numbers... 95

5.4.5 Route to aperiodic oscillatory state... 98

5.5 CONCLUSIONS TO PART 5...100

6 ONSET OF TEMPORAL APERIODICITY IN HIGH PRANDTL NUMBER LIQUID BRIDGE UNDER TERRESTRIAL CONDITIONS 102 6.1 INTRODUCTION TO PART 6... 102

6.2 CODE VALIDATION IN CASE OF Pr = 18, g = 103 6.3 ANALYSIS OF DATA...104

6.4 RESULTS AND DISCUSSION... 104

6.4.1 Possible flow régimes...104

6.4.2 Bifurcations of the time dépendent periodic flow... 105

6.4.3 Non-linear properties of the flow... 109

6.4.4 Onset of aperiodic oscillatory state... 116

6.5 CONCLUSIONS TO PART 6... 121

7 EFFECT OF AMBIENT CONDITIONS NEAR THE INTERFACE ON

FLOW INSTABILITY 123

7.1 INTRODUCTION TO PART 7... 123

7.4 MODELLING OF HEAT EXCHANGE ON THE FREE SUR­

FACE ...126

7.5 RESULTS... 129

7.5.1 The Rôle of the Biot Number...129

7.5.2 The Rôle of the Température Distribution in the Ambient Gas . . . 130

7.5.3 Modeling of the Shielding...131

7.6 RESULTS OF NUMERICAL SIMULATIONS... 133

7.7 CONCLUSIONS TO PART 7...135

8 ONSET OF INSTABILITY IN LOW PRANDTL NUMBER LIQUID BRIDGE WITH DEFORMABLE FREE SURFACE 136 8.1 INTRODUCTION TO PART 8...136

8.2 PROBLEM DESCRIPTION...137

8.3 BASIC ASSOMPTIONS ...139

8.4 SOLUTION METHOD ...141

8.5 NUMERICAL ASPECTS...142

9 FUTURE OUTLOOK 147

10 DELETERIOUS CONVECTIVE FLOW ARISING IN MICROGRAV-

ITY EXPERIMENTS 149

11 REFERENCES 165

2.1 Floating zone. On the left: a sketch of the method; on the right: a photo of the real technological process of a silicone crystal production by Topsil. The bar is melted by the ring heater and pulled down for the melted silicone to re-crystallize. 5

2.2 Liquid bridge (half zone) model... 7

2.3 Transition Phenomena of Thermocapillary Flow in Liquid Bridge (after D. Schwabe et al.). Mac nieans critical Marangoni number... 12

2.4 Schematic illustration of flow in liquid bridge... 14

2.5 Boundary layers in liquid bridge... 17

3.1 Non-uniform ID mesh... 22

3.2 Geometry of the System. Liquid bridge with cylindrical free surface... 32

3.3 Numerical algorithm of solving governing équations... 35

4.1 Pr = 4, Re = 1300, F = 1, r/ = const. Température distribution (a) and disturbance flow (b) at the cross section z=0.5... 47

4.2 Pr = 4, Re = 1300, F = 1, v = const. Température distribution on the free surface... 48

4.3 Pr = 4, Re = 1300, F = 1, n = const. Phase shift between température and velocity for the traveling wave m = 2... 49

4.4 Mixed mode, Pr — 30, Re = 1000, P = 1. (a) The température signais from two thermocouples at different azimuthal positions (ip = 0 and p> =

) and (b) power spectrum for the beginning of the process {t < 4.0) conflrming existence of the mixed mode... 50

4.5 Pr = 30, Re = 1000, P = 1, n = const. Température distribution on the free surface... 51

4.6 Pr — 30, Re — 1000, P — 2, n — const. Température distribution (a) and disturbance flow (b) at the cross section z=1.0... 52

4.7 Pr = 30, Re — 1000, F = 2, n = const. Température distribution on the free surface... 53

4.8 Pr = 30, Re = 1000, F = 2, n = const. Data of four thermocouples... 54

4.9 Schematic phase plane. Stable state is traveling wave (TW)... 55

4.10 Transition from standing to traveling wave. Time profiles of température in two different azimuthal positions. Pr — 4, Re = 1300, Gr = 0, F = 1... 56

4.11 Steady-state distribution of température and isolines of stream functions when

the reference température is (a) the température of the cold disk T

= Tcoid,

(b) the mean température in the System T

= (L^ot + Tcoid)/‘^- BuA a-nd dashed

Unes correspond to constant and variable viscosity respectively. Pr = 4, Re =

1000,F = 1... 57

4.13 Surface température distribution for different Prandtl numbers near onset of in- stability (a) Pr = 4, m = 2 and (b) Pr = 35, m — 1... 60 4.14 Température disturbance field in a z = 0.5 horizontal cross section for Pr = 4,

Re = 3000, P = 1. (a)constant viscosity, iîj, = 0 and (b) temperature-dependent viscosity, Ri, = —0.5. The axisymmetric part is subtracted from the total tem­

pérature distribution... 61 4.15 Dependence of (a) température amplitude upon y/e and (b) net azimuthal flow

upon e, Pr=4... 62 4.16 Isolines of mean azimuthal velocity, V^^meam for (a) Pr = A, Re = 1030, Gr =

0, iîj/ = 0 (lines) and Ri, = —0.9 (shadows), and (b) Pr = 35,iîe = 370, Gr = 491,iî^ = 0... 64 4.17 The time-periodic température profiles and their power spectra for Pr = 35, Bo =

1.221.{a)Re = 345, {h)Re = 400, {c)Re = 488 ... 67 4.18 Obliquity of TW on the free surface. The axial phase différence cf) at Re = 400

and 488 calculated by using maxima of oscillatory component of the température on the free surface... 68 4.19 Pr = 35, Re = 345, .6o = 1.227, R^, = —0.083. Standing wave, m=l. Tem­

pérature distributions in horizontal cross-section z=0.5 and the surfaces of equal température... 69 4.20 The température signais from two thermocouples at different azimuthal positions

(p = 0,ip = TT and (b) power spectrum when Pr — 35, Re — 620, R^ = —0.21, Bo = 0. 71 4.21 Pr = 35, Re — 620, Bo — 0.0, = —0.21. Traveling wave, m=2. Température

distributions in horizontal cross-section z=0.5 and the surfaces of equal température. 73 5.1 Phase plane far beyond the oscillatory bifurcation; Pr = 4, m — 2, Re =

4800. Closed, thin but deformed trajectory indicates presence of one fundamental frequency and harmonies in the spectrum... 78 5.2 Ascertainment of stable oscillatory solution with wave number m = 2. Initial guess

is a flow field with a symmetry m = 3 which is unstable for this set of parameters and finally decays. The température profiles correspond to Re = 700, Pr = 4, T = 1. 80 5.3 Température disturbance fields in a z = 0.5 horizontal cross section(upper part)

and on the free surface (lower part) for Pr = 4, Re = 1500, T = 1, R,y = —0.5.

(a) m = 2 and (b) m = 3 solutions. The axisymmetric part is subtracted from the total température distribution... 81 5.4 Surfaces of constant température disturbance fields for different symmetry pat­

terns in three-dimensional représentation for Pr — 4, Re — 1500, T = 1, Ri, —

—0.5. (a) m = 2 and (b) m = 3 solutions... 82 5.5 Evolution of temporal power spectrum with increase of the Reynolds number for

the m = 2 solution. Square root of amplitude is shown. The spectra always hâve one fundamental frequency and harmonies. No broadband noise is generated. . . 83 5.6 Ratios of the amplitudes of harmonies to the fundamental frequency in the appro-

priate powers (a) and global entropy (b), Eq.(8.6), as functions of the Reynolds

number for m = 2 traveling wave. Ai,i = 1,2 etc., means the amplitude of the

i-th harmonie, A

is the fundamental frequency. The rhombs correspond to the

calculated points, and the solid line is the resuit of spline interpolation 84

5.8 The dependence of the logarithms of the first harmonie amplitude A\ on the amplitude of main frequency A

when the Reynolds number increases from Re = 700 up to Re = 6000. Despite the strictly time-periodic oscillations of © and V the three different régimes are seen for the m = 2 solution... 86 5.9 Evolution of spectra of température oscillations with the increase of Re (values

of Re are shown in upper right corners) for the m = 3 solution... 88 5.10 Evolution of temporal power spectrum with increase of the Reynolds number for

the m = 3 solution. Square root of amplitude is drawn. The second incommen- surate frequency exists for 3300 < Re < 5300. The broadband noise is developed at 4200 < Re < 5000 causing the aperiodic oscillations... 89 5.11 Main frequencies vs. the Reynolds number for the m = 3 solution. The frequency

u>i exists only in quasi-periodic and aperiodic phases and slightly beyond the onset of the second periodic dynamics (3300 < Re < 5300)... 90 5.12 Amplitudes of the main frequencies (fundamental and the subfrequency) in spec­

trum for the m — 3 solution as a fonction of the Reynolds number... 90 5.13 Return maps of axial velocity for different Reynolds numbers for m = 3 solution.

Re — 3000 - periodic one frequency oscillations, Re = 3500 - two incommensurate frequencies quasi-periodic oscillations, Re = 3950 - period doubling, Re = 4500 - aperiodic oscillations... 91 5.14 Phase planes of axial velocity for different Reynolds numbers, m = 3 solution.

Re = 3000 - periodic one frequency oscillations, Re = 3500 - two incommensurate frequencies quasi-periodic oscillations, Re — 3950 - period doubling, Re — 4500 - aperiodic oscillations... 92 5.15 Température record and its power spectrum for m = 3 solution in the quasi-

periodic régime, Re = 4000.

— 65.19, u\ = 32.21... 93 5.16 Température temporal power spectrum for m = 3 solution in the aperiodic régime,

Re = 4500. Two characteristic frequencies

= 70.56 and u\ — 32.36... 93 5.17 Ratios of the amplitudes of harmonies to the fundamental frequency in the appro-

priate powers (a) and global entropy (b), Eq.(8.6), as fonctions of the Reynolds number for m = 3 traveling wave. Ai,i — 1,2 etc., means the amplitude of the i-th harmonie, A

is the fundamental frequency. The rhombs correspond to the calculated points, and the solid line is the resuit of spline interpolation... 94 5.18 The évolution of the maxima of axial velocity signais as the Reynolds number in­

creases for m = 3 oscillatory solution. One-maximum oscillations undergo multi- maxima ones at Re « 3300 and then hâve only one maxima after Re « 5500. . . 95 5.19 Amplitudes of température oscillations vs. the Reynolds number, A

=

0.5(©max — ©mm)- Solid line and astéries represent m = 2, while the dashed line with rhombs dénotés m = 3 mode... 96 5.20 Dependence of the fundamental frequency upon the Reynolds number. Solid line

and astéries represent m = 2, while the dashed line with rhombs dénoté m — 3 mode... 97 5.21 Net azimuthal flow, defined by Eq.(8.7), vs. the Reynolds number. Solid line

and astéries represent m = 2 solution, while rhombs dénoté m = 3 solution and

dotted line corresponds to the regular branch along which the m = 3 solution is

periodic... 98

always periodic. The solution described by the traveling wave with m = 3 wave- number undergoes aperiodic bifurcation preceded by the quasi-periodic dynamics.

The letters inside the bars dénoté; S - stationary, P - periodic, QP

~ two frequencies quasi-periodic and N P - non-periodic...100 6.1 Two different types of symmetry of the solutions. Snapshots of température field

disturbances in z — 0.5 transversal section (upper drawing) and on the free surface (lower drawing). (a) Traveling wave at AT = 18.0RT {Re = 1904); m = 1 mode;

(b) mixed standing wave at AT = 18.5RT {Re = 1957); m = 1 + 2 mode... 106 6.2 Evolution of spatial power spectrum with the increase of AT. Spectra of the

solutions are represented during one oscillatory period. Results show presence of the mixed mode when parameter values are; (a) - AT = 18.7K {Re = 1978), (b) - AT = 33K {Re = 3492), (c) - AT = 38K {Re = 4021)... 108 6.3 Net azimuthal flow, defined by eq. 6.1, vs. the température différence. In the

insertion the région of the standing wave near the onset of instability is shown. . 109 6.4 Fundamental frequency of température oscillations. Two frequency skips occur

with the increasing AT. The dimensional frequency in Hz can be calculated as / = uol2-nTch... 110 6.5 Evolution of température oscillations with the increase AT; (a) AT Ri

8.03R: {Re « 850), (b) AT Ri 18.30R {Re = 1936) and AT Ri 20.00RT {Re = 2116).112 6.6 Evolution of spectra of température oscillations with the increase of AT (values

of AT are shown in upper right corners)...113 6.7 Splitting of the maxima of the température oscillations. Following rhombs one

may identify the values of maxima and their amount. Insertion shows the région where only even or odd harmonies die... 114 6.8 Scaled ratios of the amplitudes of the first three harmonies to the fundamental

frequency in appropriate powers vs. the température différence. PW - periodic window... 115 6.9 Global entropy S, eq. 5, calculated for the température time sériés... 115 6.10 Phase portrait showing the transition from periodic to chaotic response via qua-

siperiodic and periodic sequence; (a) thin closed orbit at AT = 30.0; (b) a quasiperiodic orbit, AT = 35.0; (c) back to periodic orbit, AT = 36.5; (d) funnel-shaped chaotic orbit at AT == 37.0...117 6.11 Evolution of temporal power spectra made for the température time sériés with

température différence, (a) - plotted for the whole range of AT € [6,40], (b) - shown for AT G [35,40]...119 6.12 Schematic bar-graph represents transitions of the liquid bridge System on the

way to chaos under the normal gravity conditions. S - 2D stationary régime, P - periodic, QP - quasi-periodic and NP - non-periodic, SW and TW mean standing and traveling waves. The mode m — 2 is dominant while both, m = 1 -|- 2, are présent... 121 7.1 Experimental set-up used for study of thermocapillary convection in liquid bridge

by Shevtsova and Mojahed. It corresponds to the sériés of experiments in shielded

liquid bridge... 124

3-D calculations for Pr — 108, T = 1.2...129 7.4 Disturbance surface température distributions for different profiles in the ambient

gas Tamb = Tiin. 3D calculations for Pr — 108, T = 1.2. Steady solution is subtracted... 130 7.5 The dependence of température upon time. 3-D calculations when Pr=108, Re =

120, Ru = -0.38148, Bi = 0.5 (a) Tamb = T

^ (b) Tamb = Tcoid...131 7.6 Sketch of the gas circulation in the case of shielding. Two convective vortexes

appear in gas phase... 132 7.7 Resuit of the experiment in non-shielded liquid bridge, température vs. time;

r = 1.2, V = 0.9...133 7.8 The dependence of the critical Reynolds number on the Prandtl number, V =

0.9, r = 1.2. The crosses correspond to the experimental points and solid line draws the linear interpolation of them; the stars and the dashed line correspond to the numerical results for Bi = 0.48; the circle indicates the numerical resuit for Bi = 5.0... 134 8.1 Experimental dependence of the critical température différence ATcr upon liquid

bridge volume, obtained by Mojahed and Shevtsova for P = 1.2, Pr — 105, Bodyn = 2.3. The branches hâve different azimuthal wave numbers, m = 1 and m = 2... 137 8.2 Deformed liquid bridge... 138 8.3 Dependence of (a) relative volume upon the contact angle near the hot disk and

of (b) the pressure jump upon the relative volume...142 8.4 Température field disturbances in mid-cross section and on the free surface for

Pr = 0.017, r = 1, Re = 3500, Ro = 0, = 60°... 144 8.5 Température time-series for Pr = 0.017, P = 1, Re — 3500, Bo = 0, ah — 60°. . 145 8.6 Distributions of Vr along the free surface for different contact angles. Pr —

0.017, P = 1, Re = 2500, Bo = 0...145 8.7 Distributions of Vz along the free surface for different contact angles. Pr =

0.017, P = 1, Re = 2500, Bo = 0... 146 10.1 Geometry of the problem... 150 10.2 Velocity field (a) and température isolines (b) in horizontal XY-cross section for

pure buoyant fiow when Rox = 32.97, Roy = 86.6, Roz — 0, (AT/Ax = 3 K/rnm).154 10.3 Velocity field (a) and isolines of the déviation of the température from linear

profile (b) in XZ-cross section for pure buoyant fiow when Ra^ = 32.97, Roy = 86.6, Rüz = 0. The particular cross section is shown in the small cell... 155 10.4 Velocity field (a) and température isotherms (b) in YZ-cross section for pure

buoyant fiow when Rox — 32.97, Roy = 86.6, Roz — 0. The particular cross section is shown in the small cell...156 10.5 Température profiles along the numerical thermocouples, curve 1 corresponds to

T(x, y=0.5, z=0.3) and curve 2 corresponds to T(x, y=0.5, z=0.7); (a) Pure

buoyant fiow. (b) Mixed buoyant and Marangoni convection... 157

10.6 Distribution of the Marangoni number along the bubble. Large bubble case. . . . 159

liquid-gas interface where the bubble is situated... 160 10.8 Combined thermocapillary and buoyant convection. Small bubble case, 5.5mm,

Ma = 74 000. Température fields in mid-crossection Z = 25mm and on the liquid-gas interface where the bubble is situated... 161 10.9 Combined Thermocapillary and Buoyant Convection. Small bubble case, 5.5mm.

Température fields in mid-crossection Z — 25mm and on the liquid-gas interface where the bubble is situated...161 lO.lOCombined thermocapillary and buoyant convection. Small bubble case, 5.5mm,

Ma — 32 300. Température fields in mid-crossection Z = 25mm and on the

liquid-gas interface where the bubble is situated... 162

3.1 Finite-difference schemes Ref. [30]... 31

4.1 Results of the tests on the convergence on grid for Pr = 1, F = 1... 46

4.2 Comparison with linear stability analysis for Pr = 1,3 and 4, F = 1... 47

4.3 Influence of R^, on Rccr and on the critical frequency for medium Prandtl number liquids... 58

4.4 . The rate of decreasing of Recr with Prandtl number due to variable viscosity. . 58

4.5 Properties of tetradecamethylhexasiloxane... 65

4.6 . Comparison of simulations with experimental observations, (f.f. means funda- mental frequency)... 66

4.7 . Critical Reynolds number, when Pr = 35, Gr ^ 0... 70

4.8 . Critical Reynolds number, when Pr = 35, Gr = 0... 70

4.9 . Comparison of critical Reynolds and wave numbers... 72

5.1 Effect of the grid resolution on the parameters of supercritical flow, Re ^ SRccr- 77 5.2 Study of the final flow symmetry on different grids, Re ^ lORef... 80

5.3 Spatial disorganization of the flow 5$, eq.( 8.8), as a function of the Re, 3300 < Re < 5000, m = 3... 99

6.1 Effect of the grid resolution on the parameters of supercritical flow, Re = 3.06Pccr, while AT is increased... 103

6.2 Effect of the grid resolution on the parameters of supercritical flow, Re = b-SRe^r, while the System is cooled... 103

6.3 The change of flow régimes with the increase of AT... 120

7.1 Physical properties of the silicone oil 10 cSt... 125

8.1 Critical Reynolds number for unit aspect ratio when Pr = 0.01... 143

8.2 Critical Reynolds number for the aspect ratio F = 1.2 when Pr = 0.01...143

8.3 Critical Reynolds number for deformed liquid bridge and low Prantdl numbers. . 143

10.1 Physical properties of a mixture of Ethylene-glycol and water...152

10.2 The values of [V'maxl Rf = 0.71, Fi = 1, Fy = 1... 153

10.3 Velocity and the température on the free surface ai x — y = 0.5, z = 1. Results of 3D calculation: Pr = 1, Fx = l,Fy = 1... 153

10.4 Numerical results for pure buoyancy induced flow... 155

10.5 Numerical results for buoyancy induced velocities in future space experiments. lg,| = 10-^go,AT = 60K,... 157

10.6 Numerical results for combined Rayleigh and Marangoni convection. Pr =

20, Fx = 2.5, Fy = 1, V = 0.0468 [V] mm/s...162

A,B A^

Bi Bo d g G Gr k m Ma N

P,

P

■Pc/l

Pr

Q

r Po Rv Ra Re S SW t

^ch T T To

'Rcold Thot

TW P V*

Vch

Vr V,

amplitudes of disturbances

amplitudes of harmonies in spectrum Biot number

Bond number bridge height

unit vector in axial direction accélération due to gravity physical domain

Grashof number thermal diffusivity azimuthal wave number Marangoni number

outward normal unit to the boundary of physical domain pressure

pressure scale Prandtl number

heat flux on free surface radial coordinate

radius of endwalls

relative variation of viscosity Rayleigh number

Reynolds number entropy of signal standing wave state time variable time scale température

period of oscillations reference température température of cold endwall température of hot endwall traveling wave state

velocity vector

” provisional” velocity fleld velocity scale

radial component of velocity

axial component of velocity

azimuthal component of velocity

axial coordinate

7 r Ar At Ax Az Aif AT e A ly u' l/j' ÜJ P PO â a o-Q (7X '^th 0 00 T

0 cr ch i j k left mean r right 7

n

coefficient of thermal expansion boundary of physical domain aspect ratio

mesh step in r-direction time step

spatial mesh step mesh step in z-direction mesh step in (/^-direction

température différence between rods distance from critical point

thermal conductivity

dimensionless kinematic viscosity kinematic viscosity

coefficient in the linearized, kinematic-viscosity équation of state frequency of oscillations

fluid density

fluid density at the reference température stress tensor

surface tension

surface tension at the reference température T

coefficient in the linearized, surface-tension équation of state time of thermal equilibrium achieving

déviation from linear dimensionless température profile dimensionless température with respect to the cold endwall azimuthal coordinate

net azimuthal flow stream function

Subscripts

estimated at the reference fluid température critical value

characteristic value

current number of mesh point with respect to r current number of mesh point with respect to (p current number of mesh point with respect to z wave spreading in counter clockwise direction averaged in azimuthal direction

différentiation with respect to r wave spreading in clockwise direction value on the boundary of physical domain

Superscripts

current time

0.3 ABSTRACT

A numerical code for solving the time-dependent incompressible three-dimensional Navier- Stokes équations with finite volumes on overlapping staggered grids in cylindrical and rectangular geometry is developed. In the code, written in FORTRAN, the momentum équation for the velocity is solved by projection method and Poisson équation for the pressure is solved by ADI implicit method in two directions combined with discrète fast Fourier transform in the third direction. A spécial technique for overcoming the singularity on the cylinder’s axis is developed.

This code, taking into account dependence upon température of the viscosity, density and surface tension of the liquid, is used to study the fluid motion in a cylinder with free cylindrical surface (under normal and zero-gravity conditions); and in a rectangular closed cell with a source of thermocapillary convection (bubble inside attached to one of the cell’s faces). They are significant problems in crystal growth and in general experiments in fluid dynamics respectively.

Nevertheless, the main study is dedicated to the liquid bridge problem.

In Part 4, the development of thermocapillary convection inside a cylindrical liquid bridge is investigated by using a direct numerical simulation of the 3D, time-dependent problem for a wide range of Prandtl numbers, Pr = 1 — 35. For Pr > 0.08 (e.g. silicone oils), above the critical value of température différence between the supporting disks, two counter propagating hydrothermal waves bifurcate from the 2D steady State. The existence of standing and traveling waves is discussed. The dependence of viscosity upon température is taken into account. The critical Reynolds number and critical frequency at which the System undergoes a transition from a 2D steady State to a 3D oscillatory flow decreases if the viscosity diminishes with température.

The stability boundary is determined for Pr — 3 — 5 with a viscosity contrast {r'max/’^min) up to a factor 10. Near the threshold of instability the flow organization is similar for the constant and variable viscosity cases despite the large différence in critical Reynolds numbers. The influence of variable viscosity on the flow pattern is increased when going into the supercritical région. The study of spatial-temporal behavior of oscillatory convection for the high Prandtl number, Pr = 35, demonstrates a good agreement with previously published experimental results. For this high Prandtl number liquids instability begins as a standing wave with an azimuthal wa,venumber m = 1 which then switches to an oblique traveling wave si 4 — 5% above the onset of instability.

To see the dynamics of the System in range of high values of the Reynolds number and trace the scénario of onset of chaos, two problems were considered: Pr = A,0 — g and Pr — 18.8,1 — g cases.

In Part 5, for a medium Prandtl number liquid bridge, Pr = 4, with unit aspect ratio under

zéro gravity conditions, a parametric investigation of the onset of chaos was numerically carried

out. Spatiotemporal patterns of thermocapillary flow were successively studied beginning from

the onset of instability up to the appearance of the non-periodic flow and further on. Two-

dimensional steady flow becomes oscillatory with azimuthal wave number m — 2 as a resuit of

Hopf bifurcation at Ref — 630. A second independent solution with wave number m = 3 was

found to appear at Re^ ^ 810. Two branches of three-dimensional periodic orbits, traveling

waves with m — 2 and m — 3, coexist for Re > Re^. Additional stable branches do not

connect them. The different flow organizations reveal different behaviors in the supercritical

area. The m — 2 traveling wave (TW) always remains periodic, but the mode m — 3 starts

exhibiting chaotic features at Re

4200. The onset of temporal non-periodicity was shown to be

associated with development of broadband noise in spectra and preceded by a quasi-periodicity.

The flow stabilizes back to periodic with single frequency when Re exceeds a value Re « 5100.

The window of periodicity exists up to at least Re = 6000, the largest investigated value of the Reynolds number.

In Part 6, following closely to the geometry and conditions of the laboratory experiments, a silicone oil IcSt, Pr = 18.8, is chosen as the test liquid. The simulations are done at normal gravity conditions and unit aspect ratio. Dépendance of viscosity of the fluid upon température allows us to be doser to the real phenomenon. Both spatial and temporal changes occurring in the System are analyzed. The results are compared to experimental data. The following sequence of well-defined dynamic régimes was detected when température différence between the supporting disks is increasing: steady, periodic, quasiperiodic, periodic and chaotic. The observed succession of bifurcations on the way to the aperiodicity (chaos) is absolutely similar to the one coming from experiments. Except these dynamic bifurcations the System exhibits numerous transitions in spatial organization of the flow, which is difflcult to identify in the experiments unambiguously. Two-dimensional steady-state flow undergoes standing wave (SW) with azimuthal wavenumber m = 1 as a resuit of supercritical Hopf bifurcation. Moving above the critical point the following succession of the flow States has been numerically found; SW(m

= 1) —> TW (m = 1) ^ SW(m = 1 + 2) —+ TW(m = 1 + 2). TW to be read as traveling wave.

Transition to chaos occurs while the flow pattern represents a traveling wave with a mixed mode m = 1 + 2. The particular attention is paid to the examination of spécial properties of the flow:

entropy, net azimuthal flow, frequency skips, splitting of maxima and related phenomena.

In Part 7, for a liquid bridge fllled by 10 cSt silicone oil {Pr « 108), the appearance and the development of thermoconvective oscillatory flows were investigated for different ambient conditions around the free surface. The calculations were made under terrestrial conditions both with or without shielding of a liquid bridge. It allowed to croate different température distributions in the ambient gas near the interface. The different heat transfer possibilities at the free surface are analyzed. For the both cases the results are in an excellent agreement for the détermination of the critical température différence.

In Part 8, transition from two-dimensional thermoconvective steady flow to a 3D flow is con- sidered for a low Prandtl number fluids {Pr ^ 10“^) in a liquid bridge with a non-cylindrical free surface. For Pr < 0.08 (e.g. liquid metals), in supercritical région of parameters 3D but non-oscillatory convective flow is observed. The computer program developed for this simula­

tion transforms the original non-rectangular physical domain into a rectangular computational domain.

In Part 10, study of how presence of a bubble in experimental rectangular cell influences

the convective flow when carrying out microgravity experiments. As a model, a real experiment

called TRAMP is numerically simulated. The obtained results were very different from what

was expected. First, because of residual gravity taking place on board any spacecraft; second,

due to presence of a bubble having appeared on the experimental cell’s wall. Real data obtained

from experimental observations were taken for the calculations.

0.4 PREFACE

The présent thesis contains the results of study by means of direct numerical simulations of instabilities taking place in thermocapillary convection in liquid bridge. A Summary, giving a brief introduction to the thermocapillary convection, is followed by an introduction to the main subject of the work and an introduction to numerical technique of solving partial difîerential équations. The next and the main part of the thesis is composed of the following papers: already published in abridged and slightly modified versions in order to put them together; and the ones that are not published yet or the work on which is still in progress.

PART 4. THREE-DIMENSIONAL SIMULATIONS OF HYDROTHERMAL INSTABIL- ITY IN LIQUID BRIDGES. INFLUENCE OF TEMPERATURE DEPENDENT VISCOSITY.

V.M. Shevtsova, D.E. Melnikov and J.C.Legros, Physics of Fluids, Volume 13, Number 10, pp. 2851-2865 (2001).

PECULIARITIES OF THREE-DIMENSIONAL FLOW IN LIQUID BRIDGES AT HIGH PRANDTL NUMBERS. V.M. Shevtsova, D.E. Melnikov and J.C.Legros, Computational Fluid Dynamics Journal, Vol.9, No.l, pp.653-662 (2001).

INFLUENCE OF VARIABLE VISCOSITY ON CONVECTIVE FLOW IN LIQUID BRIDGES. 3D SIMULATIONS OF GROUND BASED EXPERIMENTS. V.M. Shevtsova, D.E.

Melnikov and J.C.Legros, Proceedings of the Ist Intl. Symp. on Microgravity Research and Ap­

plications in Physical Science and Biotechnology, Sorrento, Italy, Sept., pp. 101-109 (2000).

PART 5. MULTI STABILITY OF THE OSCILLATORY THERMOCAPILLARY CON­

VECTION IN LIQUID BRIDGE. V.M.Shevtsova, D.E.Melnikov and J.C.Legros, PHYSICAL REVIEW E 68, 066311, 13 pages (2003).

CHAPTER 6. ONSET OF TEMPORAL APERIODICITY IN HIGH PRANDTL NUMBER LIQUID BRIDGE UNDER TERRESTRIAL CONDITIONS. D.E.Melnikov, V.M.Shevtsova and J.C.Legros, 12 pages, Will appear in Physics of Fluids, Volume 16 (2004).

PART 7. THE CHOICE OF THE CRITICAL MODE OF HYDROTHERMAL INSTA- BILITY BY LIQUID BRIDGES. V.M.Shevtsova, M.Mojahed, D.E.Melnikov and J.C.Legros, Lecture Notes in Physics, Springer, pp. 240-262 (2003).

EFFECT OF AMBIENT CONDITIONS NEAR THE INTERFACE ON FLOW INSTABIL- ITY. V.M.Shevtsova, D.E.Melnikov and J.C.Legros, Adv. in Space Res., Vol 32/2 pp 155-161 (2003).

PART 8. ONSET OF INSTABILITY IN LOW PRANDTL NUMBER LIQUID BRIDGE WITH DEFORMABLE FREE SURFACE. D.E.Melnikov, V.M.Shevtsova and J.C.Legros, The work is in progress.

PART 10. THERMAL CONVECTION IN RECTANGULAR CAVITY. 3-D SIMULA­

TIONS OF TRAMP EXPERIMENT. V.M. Shevtsova, D.E. Melnikov and J.C.Legros, Post- Flight DATA Analysis on FOTON-12, ESA, Noordwijk, European Space Agency Contrac- tual Report 42 pages, March 2000.

THE STUDY OF STATIONARY AND OSCILLATORY WEAK FLOWS IN SPACE EX-

PERIMENTS. V.M. Shevtsova, D.E. Melnikov and J.C.Legros, Microgravity Science and

Technology, Volume XIV/2 (2004).

SUMMARY

Almost 150 years ago Weber [153], Thomson [145] and Marangoni [71] investigated the effect of spreading of a liquid with a lower surface tension over another liquid with a higher surface tension. So-called ”wine tears”, a phenomenon when in a wine glass, due to convection caused by variations of surface tension resulted from the évaporation of alcohol, the wine climbs up along the glass as a thin film, reaches a certain height and accumulâtes. Gravity causes the wine to flow down in the shape of ” tears” [145]. Surface tension - an important localized property - is represented as the magnitude of the force per unit length normal to a eut in the interface, or the free energy per unit area. It arises on the interface between two immiscible fiuids or at a fluid-gas interface and can be attributed to unbalanced molecular attraction which tends to pull molécules into the interior of a liquid phase, and hence to minimize the surface area [35].

This results in a higher potential energy for the molécules at the interface. Consequently the interface tends to get its area reduced in order to minimize the potential energy, as if it were in a State of tension like a stretched membrane [129].

The surface tension is a fonction of many properties of liquid, e.g. température and con­

centration. Any variations in température or composition resuit in local variations of surface tension along a liquid free surface and thus induce convective motion. This effect is called Marangoni convection. Thermodynamically the spreading of a liquid means that the interface will tend to assume a state of lower surface energy and does so by the spreading of areas of lower interfacial tension. Présent thesis considers only temperature-induced convection, often called thermocapillary convection. In a terrestrial environment, Marangoni convection is usually overshadowed by buoyancy-driven flow. In the reduced gravity environment, however, buoyancy is greatly reduced and Marangoni convection could become very important. There are cases when the thermocapillary fiows ruined experiments in fluid dynamics, e.g. due to presence of a bubble inside experimental cell [121], see detailed investigation of this problem in Appendix 10.

The ratio between the two factors, the so-called dynamic Bond number, defines whether gravity or thermocapillary forces are dominant.

In fiuid convection with a free surface it is sometimes the case that surface tension efîects become important even under normal gravity environment. For instance in such configurations as crystal growth by Bridgeman or fioating-zone method the gravitational forces are negligible [17, 140]; Or, as in Bénard’s original experiments, the layer of fluid is very thin and the convection is surface-tension dominated [7, 88]. The criterion for the dominant thermocapillary convection over buoyancy is ratio between thermocapillary forces to buoyant forces written as:

Gr Do

where Bo = Gr/Re = pg(5H‘^/

is Bond number, définitions of the Grashof Gr and thermo- capillary Reynolds Re numbers and of the other ternis are given below. Kamotani et al. [46]

measured critical température différence using containers of different sizes filled by Pr = 27 liquid. They found an upper limit under which the flow is thermocapillary dominated, it is Bo < 0.24. Using liquids with different properties, the Bond number should be recalculated.

It has been recognized that thermocapillary Marangoni effects may be an important factor influencing convection instability in thin fiuid layers [134, 26, 95, 109, 128]. Surface tension effects in heated fiuid layers are becoming increasingly important in current research on micro- hydrodynamics where fiuid layers are thin and surface tension effects are significant, e.g. they are of industrial interest in applications to microelectronic fabrication (such as dewetting). Fur- ther, these Marangoni flows arise in many diverse application areas such as coating, electronic cooling, welding processes, wafer drying, biofluidic chip fabrication and medicine delivery. This is a reason why much attention has been given in recent years to Marangoni flows.

Three different approaches can be found in the literature for a theoretical description of the influence of the Marangoni convection on mass transfer [39]. A detailed review about experi­

mental studies concerning interfacial convections is given in [89, 105]. Here they are.

Analytical and numerical studies are based on linear or nonlinear differential équations which are solved for simplified boundary conditions. These théories predict the stability behavior of liquid-liquid and gas-liquid Systems. The best known model of this kind has been proposed by Sternling and Scriven in [136] for quiescent and infinitely extended phases with a plane interface.

Mass transfer coefficients are calculated by semi-empirical corrélations. Generally, these models modify the film theory of Lewis and Whitman in [67] or the surface renewal model of Higbie in [41]. A limited number of parameters is used in these models [96, 112].

Empirical corrélations describe the relationship between mass transfer coefficient and inter­

facial convection. These équations can only be applied to the corresponding test apparatus, Systems and conditions studied.

Concerning the crystal growth from melt, much work has been done in the past, both exper- imentally and theoretically. Most of the experimental investigations were carried out in normal gravity but some results from microgravity experiments are now available. High Prandtl num­

ber (Pr) fluids hâve been used in most experiments. Although Pr of crystal melts is generally

smaller than unity, much less experimental information is available for low Pr fluids due to

some experimental difficulties. The transport phenomena in the floating-zone melt hâve been

simulated by many investigators in the so-called liquid bridge configuration, in which a liquid

column is suspended vertically between two differentially heated rods.

INTRODUCTION

2.1 CRYSTAL GROWTH

The industrial crystal growth of a range of electronic materials is a core technology of the electronic industry. For example, growth from the melt of bulk single crystals of semiconductors provides the wafers for integrated circuits; oxide crystals are used for laser host lattices. The most utilized crystal growth techniques from the melt are:

1. The directional solidification (Bridgeman technique);

2. The crystal pulling technique named after Czochralski;

3. The zone melting method, the floating zone method is one of a variants of its vertical configuration.

Each method has its advantages and disadvantages. Before going to the main subject of the research, let us to briefly introduce and compare to each other the Czochralski and the floating zone technologies.

Refining of silicone by the Czochralski technology is the most common method of pro- ducing monocrystalline silicone. Large diameter monocrystalline silicone can be grown with the Czochralski technique. As in the case of floating zone technology growth of Czochralski monocrystalline silicone starts from a small diameter monocrystalline seed crystal. Czochralski silicone is pulled from a quartz crucible in which the silicone is kept in the molten phase by heating the rotating crucible either by induction or résistive heating. Ségrégation of impurities between the liquid phase and the solid phase is as good as is the case for fioat zone silicone, but care must be taken during Czochralski growth in order not to contaminate the molten silicone from the surrounding atmosphère. Because of the open furnace principle it is difficult to grow Czochralski silicone with resistivities above 1.000 Ohm — cm. Czochralski silicone is mainly grown with resistivities below 100 Ohm — cm which is sufficient for the making of most ICs, including low- to medium power ICs.

In comparison with fioat zone silicone, Czochralski silicone can be grown to very low resistiv­

ities in the order of 1 — 2 mOhm — cm. This is important for low and medium power applications, where the Czochralski silicone wafer serves as a carrier substrate for the actual components built on épitaxial material deposited on top of the Czochralski silicone wafer. When growing to these very low resistivities it becomes important to control the oxygen content in the silicone. Big efforts are being invested into oxygen control issues. Also, at these low resistivities issues resis- tivity control becomes important. The Czochralski technique is the one that gives the largest, roundest rods, and that is important. The larger the substrate the more chips can be made in a single step. The industry is tooled up to use certain standard diameters. The most com­

mon standard now is 200mm, but the state-of-the-art is 300mm. The Czochralski technique

accounts for approximately 90% of the world market of silicone crystals, mainly used for memory chips and other integrated circuits, the remaining 10% are float zone crystals used for power applications and sensors.

The floating zone method, one of the most fascinating purification techniques, is of technical importance for the growth of silicone. The main technological advantage of the method is containerless which aids to prevent contamination. Zone refining of crystals in order to increase their purity has been known for more than 50 years (was first patented in 1952 by Theuerer and used for the first time by SIEMENS AG). The purification method takes advantage of the concentration change by ségrégation for most impurities during the liquid - solid transition.

The floating zone is suspended between the melting and the freezing interfaces. The floating zone technology has been perfected for the growth of silicone monocrystals. Being freely floating with no contact to crucibles and other possible contaminant sources make it possible to grow monocrystalline silicone with the highest purity which is important for a number of electronic and optoelectronic applications. Floating zone monocrystalline growth starts from a high purity, small diameter seed crystal. The seed crystal is prepared in the right crystalline direction in order to grow pure silicone with no crystalline defects.

Floating zone technology for growing silicone monocrystals is by far the most pure method and results in silicone with unique properties as opposed to any other growth technology. Float zone silicone can be grown with resistivities exceeding 100.000 Ohm — cm because of the intrin- sic process purity. Czochralski silicone does not hâve this high level of purity. Floating zone technique has a superior performance with respect to crystal defects like vacancies or interstitial agglomérâtes. In addition, material grown by this method contains basically no oxygen (concen­

tration of oxygen is lower by three orders of magnitude in float zone silicone in comparison with Czochralski silicone where crucibles needed for suspending the liquid silicone). Perhaps the most important feature of the floating zone technology is the ability to exactly control the resistivity of the crystal. This is particularly important for applications using the bulk of the silicone wafers for manufacturing devices. There are two practical ways of obtaining the very good resistivity control. One is by doping the crystals when they are pulled by introducing controllable amounts of gaseous dopants into the growth chamber. Most common dopants are phosphorus and boron for n— and p—type, respectively. This technique is called in-situ doping or gas phase doping.

The other technique is by doping the crystal after it has been pulled. This technique is called ex-situ doping and it is done in neutron irradiating reactors. The starting material for ex-situ doping is high resistivity silicone that after being irradiated with a controllable dose of neutrons changes its resistivity by transforming silicone atoms to dopant atoms. Crystals produced by this method in a terrestrial environment are not large in size due to the weight of the melt which tends to destroy the liquid zone held by surface tension. Silicone crystals are at présent industrially grown at diameters up to 150?nm, weighing more than 35kg.

Briefly, the floating zone method could be described as a long semiconductor (e.g. silicone) or metallic (e.g. GaAs alloy) rod which is at first locally melted and then re-crystallized. The floating zone method uses a small heater to locally melt a small portion of the bar (Fig. 2.1).

The bar is slowly pulled through the heater and thus the melted zone will be re-solidified.

Since the ring heater créâtes température gradients along the free surface of the melt, con­

vection of different types occurs in the melt and it influences the quality of the crystal made

by this method [36]. Among the known crystal defects, which were grown by the industrially-

used methods, are voids, inclusions, distortion, strain, dislocations, inhomogeneities, striations

[45]. Among the mechanisms causing the imperfections there are gravity-dependent and those

that take place both on ground and under microgravity conditions. The gravity-dependent are

strain fields dislocations caused by weight of crystal; grain boundaries and voids being a resuit

Figure 2.1: Floating zone. On the left: a sketch of the method; on the right: a photo of the real technological process of a silicone crystal production by Topsil. The bar is melted by the ring heater and pulled down for the melted silicone to re-crystallize.

of constitutional supercooling; macroscopie inhomogeneities and striations (resuit of steady and unsteady buoyancy convection respectively); too small crystal pressure caused by hydrostatic pressure. Strain fields dislocations are also a resuit of spatial and temporal température varia­

tions being caused by thermocapillary convection only, as well as the striations which are created by both steady and unsteady Marangoni convection. The processes caused by the interface ki- netics (microscopie mechanism), independent of gravity, lead to forming micro- and macroscopie inhomogeneities in crystal.

Imperfections induced by gravity.

Because of the différence between the compositions of crystal and co-existing melt even in State of equilibrium, macroscopie inhomogeneities with respect to the distribution of the dopants occur in crystal as a resuit of ségrégation phenomenon. Strong convective mixing makes dopant’s concentration profile being strongly growing with solidifled fraction. Camel and Favier [9] hâve made a systematic analysis of the dependence of longitudinal ségrégation profiles in the presence of buoyancy. They found ranges of different ségrégation behavior in directional solidification, and presented the resuit in tenus of dimensionless convective beat transport Gr x Sc {Sc is the Schmidt number) versus dimensionless growth rate.

Microscopie inhomogeneity with respect to the distribution of dopants is very common in mixed crystals. It is gravity driven unsteady convection and as a conséquence unsteady beat transport that cause température fluctuations on the solid-liquid interface and créâtes striations via the growth rate dependence of the ségrégation.

These imperfections along with the other gravity-caused ones, such as dislocations, strain fields, voids and grain boundaries, can be eliminated by réduction of gravity.

Gravity—independent imperfections.

Later researches, [31, 149], hâve reported that buoyancy is not the only mechanism of the

oscillatory instability and thermocapillary convection also may be a reason of the oscillations.

Eyer et al. [31] grew a silicone crystal of 8mm. in diameter on a sounding rocket using the floating zone technique. Investigations of the crystal revealed striations, even though the process took place under microgravity. Even if the microgravity environment on board the rocket is always far from absolute 0 — g, buoyancy can not overshadow the thermocapillarity, that made thinking of an important rôle of Marangoni convection in crystal growth.

Strain fields, dislocations, voids and striations can also occur in absence of gravity. Each in- homogeneity has its own mechanisms but it is always the Marangoni convection that stays behind ail them. It does not matter if the thermocapillary convective flow is steady or time-dependent, it is always undesirable. For example, microscopie striations are caused by time-dependent Marangoni convection while steady convection causes the increase in macrosegregation.

The oscillation phenomenon in floating-zone melt was first reported in 1979 by Chun and Wuest [19] and by Schwabe and Scharmann [110]. Later, several démonstrations of oscillatory thermocapillary flow in model floating zones hâve been given. Schwabe et al. [94] showed that stationary convection exists in the form of an axisymmetric roll bounded by the free surface.

Having taken a zone of length less than 3.5 mm they observed a transition to the oscillatory régime in the form of azimuthally traveling wave fov Ma = Re x Pr > 7 x 10^. In [20], Chun made a study of coupling of the oscillatory régime to the rotation of the crystal rods, observing transition to turbulence.

The flnal quality of the crystal is degraded resulted from the oscillatory convective motion leading to fluctuations in both beat- and mass-transfers at the liquid-solid interface where the melt re-solidifies and thus causing dopant striations in the crystal. Aiming at eliminating these defects and improving the homogeneity, investigations of the convection in melting zone, its stability and dépendance upon différent factors started. Originally, it is natural convection that was thought to cause the oscillatory régime. That is why the microgravity environment was chosen for industrial production of crystals. Beginning with the early seventies, a lot of microgravity experiments were performed. Since Apollo mission up to year 1995 at least 77 experiments were successfully done.

While in floating zone method there are a free surface and a longitudinal température gra^

dient, one can not suppress completely the thermocapillary convection, but it is possible at least to somehow weaken the oscillatory régime. Most of the works aiming at suppressing the oscillations use methods of altering the steady state and thus decreasing the effective Marangoni number to attenuate the fluctuations. Among the methods the most popular are to put the floating-zone formed by electromagnetically active melt under magnetic field (Dold et al. [25], and Croll et al. [24]); to croate the counter flow of the ambient gas (Dressler and Sivakumaran

[29]); to impose vibrations of the end-walls (Anilkumar et al. [3]) or rotate the whole System;

or simply to use surrounding gas of certain température and pressure that decreases the surface tension (Azumi et al. [5]). More sophisticated methods consist for example in local heating the free surface by actuators using feedback control algorithm defined by the oscillations in the liquid bridge themselves. The input of the algorithm is signais from the local sensors (Petrov et al. [91]).

2.2 LIQUID BRIDGE (LB)

The study of convection in liquid bridges is related to the floating zone crystal-growth process.

What are the model used for theoretical description of the real phenomenon and its origin?

The heater is situated in the middle of the melting zone held by rigid rods at the top and the

bottom (Fig. 2.1), where the température profile has its minimal values. Thus, the température

reaches is maximum at the mid-height of the liquid-gas interface. Spreading from the hot to the cold areas along the free surface, two sets of convective rolls (above and below the heater) are developing. Hence, to simplify the theoretical considération of the problem, only a half-zone can be focused on (Fig. 2.2). A model used for analyzing the processes taking place in melt during the floating zone procedure of crystal growth is called liquid bridge.

Figure 2.2: Liquid bridge (half zone) model

In the half-zone model (liquid bridge) a small volume of liquid is held between two coaxial

circular disks, which are kept at different températures yielding a température différence AF =

Thot — Tcoid- As the applied température gradient is parallel to the interface, motion from the

hot to the cold région appears for any non-zero value of AT. When the température différence

between the disks exceeds the critical value, AT > AT

, unique for a given set of parameters,

the flow is three-dimensional and/or unsteady. Thus, the définition of the critical température

différence for the liquid bridge problem is the minimal AT when the thermocapillary flow gets

a three-dimensional structure and not uniform in azimuthal direction any more. The transition

from steady to oscillatory flow, as illustrated in Fig. 2.3, is an important feature of Marangoni

convection in the liquid bridge configuration. Generally, two hydrothermal waves propagating

in opposite directions bifurcate from two-dimensional state at the critical point. They resuit in

standing (SW) or traveling (TW) wave depending on the ratio of their amplitudes [34, 152,

122, 76]. For high Pr liquid bridges a mechanism of hydrothermal instability was proposed

by Wanschura et al. [152] is identical with that suggested by Smith and Davis [132] and

Smith [133] (velocity - température coupling in case of an infinité layer subjected to horizontal

température gradient). It is also internai velocity - surface température coupling that could

be responsible for oscillatory instability. With high Pr fluids, as the flow becomes faster, the

température field gets more distorted resulting in big gradients on the cold corner. Thus, more

hot fluid is transported by the surface flow and the return flow becomes cooler. In return, this

température change by the flow changes the thermocapillary driving force on the interface, and

the flow rate is changing. Kamotani and Ostrach [47] suggested that the free surface dynamical

deformations could play an important rôle in the development of the instability. Their argument is that one could not neglect the interface deformations since the thermal boundary layer is very thin, and thus its thickness is altered. This is not the case with small Pr liquids {Pr w O(10~^)) which hâve a feature of establishing almost linear liquid température gradients. In this case, the type of instability was identified by Levenstam and Amberg [64] as a purely hydrodynamic instability similar to the one of a vortex ring.

Type of the wave which will onset in the System is an open question. There is no concensus reached on this point. It is clear that the stable type of the hydrothermal wave dépends upon the parameters of the System (Prandtl number, température différence, gravity level, liquid volume, ambient conditions etc.). Different experimental and numerical works, performed even for the same parameters, report different results. Having mentioned that their results being in agreement with microgravity experiment [75], Savino and Monti [103] showed that the instability appears as SW. They performed calculations for Pr = 30, F = 0.5, l.For Pr = 4, 7 and aspect ratio r € (0.5,1.3), Leypoldt et al. [68] obtained the TW the only stable solution. In annular configuration, for ly — 2cSt and a wide range of aspect ratios, Kamotani et al. [48] reported that at the onset of oscillations TW is stable. This resuit was confirmed by calculations of Sim and Zebib [130] made for Pr = 17. But calculations by Lavalley et al. [63] gave SW as a stable solution. Carrying out laboratory experiments with silicone oil liquid bridge, i/ = l — 5cSt, Pr « 18 — 90 (as they used température of the cold rod of —20°C), and varying aspect ratio in a wide range T € (0.3, 2.0), Ueno et al. [148] observed only SW at the onset of oscillatory régime.

The calculations performed by Melnikov et al. [74] for ly = lc5t, T = 1 case confirmed the experimental results.

The hydrothermal waves are characterized by (azimuthal) wave number m which reflects the spatial symmetry of the flow. In oscillatory régime, one may see in température distribution a set of m hot and m cold patterns that rotate (TW) or pulsate (SW). Wave number dépends upon the Prandtl number, the aspect ratio, gravity level and may be upon the température différence AT. The azimuthal wave number at the threshold of instability is called critical mode. The flow structure in liquid bridge in the supercritical parameters’ région is very similar to that observed for a flow in infinité liquid column (P = oo) by Xu and Davis = [156]. However, the system’s spatial limitation changes the critical mode. For P = 1, Gr = 0, it is not m = 1 but m = 2.

The first empirical corrélation for the détermination of the azimuthal wave number, rricT ~ 2.2/r, has been suggested by Preisser et al. [94] by analyzing the experimental data for a fluid of Pr = 8.9. Here T = d/R is the aspect ratio (see Fig. 2.2). The slightly different corrélation, rUcr ~ 2.0/F, has been obtained numerically for Pr < 7 assuming pure Marangoni convection by Wanschura et al. [152], Leypoldt et al. [68].

By analogy to low Pr instability and that of a thin vortex ring [64], it follows that the azimuthal wave number may satisfy the relation m = 2.5/F. In reality the coefficient of the proportionality for liquid bridges was found smaller than ’2.5’. The numerical calculations by Wanschura et al. [152] showed that the relation mer ~ 2.0/F remains also valid for small Pr number, although the mechanism of the instability has different origin for relatively high, 0.5 < Pr <7, and low Prandtl liquids.

The discussed above empirical relation mer ~ 2.0/P does not hold for high Prandtl number

liquid bridges Pr > 30. Indeed, it has been theoretically obtained by Xu and Davis [156] for high

Prandtl numbers fluids that the critical azimuthal wave number mer is equal to one for infinitely

long cylindrical jet. Experiments [12], [125], [77] with different silicone oils demonstrate that

the critical azimuthal mode corresponds to mer — 1 for Pr > 30 for the unit aspect ratio and

aspect ratio close to one. The 3D numerical calculations by Shevtsova et al. [122] hâve also

confirmed that this empirical relation is not valid for Pr = 35.