List of Figures

List of Figures

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.). M a

means 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 1D mesh. . . . 22

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

3.3 Numerical algorithm of solving governing equations. . . . 35

4.1 P r = 4, Re = 1300, Γ = 1, ν = const. Temperature distribution (a) and disturbance flow (b) at the cross section z=0.5. . . . 47

4.2 P r = 4, Re = 1300, Γ = 1, ν = const. Temperature distribution on the free surface. . . . . 48

4.3 P r = 4, Re = 1300, Γ = 1, ν = const. Phase shift between temperature and velocity for the traveling wave m = 2. . . . . 49

4.4 Mixed mode, P r = 30, Re = 1000, Γ = 1. (a) The temperature signals from two thermocouples at different azimuthal positions (ϕ = 0 and ϕ = π) and (b) power spectrum for the beginning of the process (t ≤ 4.0) confirming existence of the mixed mode. . . . . 50

4.5 P r = 30, Re = 1000, Γ = 1, ν = const. Temperature distribution on the free surface. . . . . 51

4.6 P r = 30, Re = 1000, Γ = 2, ν = const. Temperature distribution (a) and disturbance flow (b) at the cross section z=1.0. . . . 52

4.7 P r = 30, Re = 1000, Γ = 2, ν = const. Temperature distribution on the free surface. . . . . 53

4.8 P r = 30, Re = 1000, Γ = 2, ν = 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 temperature in two different azimuthal positions. P r = 4, Re = 1300, Gr = 0, Γ = 1 . . . . 56

4.11 Steady-state distribution of temperature and isolines of stream functions when the reference temperature is (a) the temperature of the cold disk T

= T

, (b) the mean temperature in the system T

= (T

+ T

)/2. Full and dashed lines correspond to constant and variable viscosity respectively. P r = 4, Re = 1000, Γ = 1. . . . 57

v

4.12 Dependence of (a) Re

and (b) ω

upon viscosity variation R

for different Prandtl numbers, Pr=3, 4 and 5. . . . 59 4.13 Surface temperature distribution for different Prandtl numbers near onset of in-

stability (a) P r = 4, m = 2 and (b) P r = 35, m = 1. . . . 60 4.14 Temperature disturbance field in a z = 0.5 horizontal cross section for P r = 4,

Re = 3000, Γ = 1. (a)constant viscosity, R

= 0 and (b) temperature-dependent viscosity, R

= − 0.5. The axisymmetric part is subtracted from the total tem- perature distribution. . . . . 61 4.15 Dependence of (a) temperature amplitude upon √

and (b) net azimuthal flow upon , Pr=4. . . . 62 4.16 Isolines of mean azimuthal velocity, V

, for (a) P r = 4, Re = 1030, Gr =

0, R

= 0 (lines) and R

= − 0.9 (shadows), and (b) P r = 35, Re = 370, Gr = 491, R

= 0 . . . . 64 4.17 The time-periodic temperature profiles and their power spectra for P r = 35, Bo =

1.227.(a)Re = 345, (b)Re = 400, (c)Re = 488 . . . . 67 4.18 Obliquity of TW on the free surface. The axial phase difference φ at Re = 400

and 488 calculated by using maxima of oscillatory component of the temperature on the free surface. . . . 68 4.19 P r = 35, Re = 345, Bo = 1.227, R

= −0.083. Standing wave, m=1. Tem-

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

ϕ = 0, ϕ = π and (b) power spectrum when P r = 35, Re = 620, R

= − 0.21, Bo = 0. 71 4.21 P r = 35, Re = 620, Bo = 0.0, R

= − 0.21. Traveling wave, m=2. Temperature

distributions in horizontal cross-section z=0.5 and the surfaces of equal temperature. 73 5.1 Phase plane far beyond the oscillatory bifurcation: P r = 4, m = 2, Re =

4800. Closed, thin but deformed trajectory indicates presence of one fundamental frequency and harmonics 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 temperature profiles correspond to Re = 700, P r = 4, Γ = 1. 80 5.3 Temperature disturbance fields in a z = 0.5 horizontal cross section(upper part)

and on the free surface (lower part) for P r = 4, Re = 1500, Γ = 1, R

= −0.5.

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

terns in three-dimensional representation for P r = 4, Re = 1500, Γ = 1, R

=

− 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 have one fundamental frequency and harmonics. No broadband noise is generated. . . 83 5.6 Ratios of the amplitudes of harmonics to the fundamental frequency in the appro-

priate powers (a) and global entropy (b), Eq.(6.2), as functions of the Reynolds

number for m = 2 traveling wave. A

, i = 1, 2 etc., means the amplitude of the

i-th harmonic, A

is the fundamental frequency. The rhombs correspond to the

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

5.7 The evolution of the maxima of axial velocity signals as the Reynolds number increases for m=2 oscillatory solution. One-maximum oscillations undergo tran- sition to two-maxima ones at Re ≈ 1000. . . . 85 5.8 The dependence of the logarithms of the first harmonic 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 Θ and V the three different regimes are seen for the m = 2 solution. . . . . 86 5.9 Evolution of spectra of temperature 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

ω

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 function 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 Temperature record and its power spectrum for m = 3 solution in the quasi-

periodic regime, Re = 4000. ω

= 65.19, ω

= 32.21. . . . 93 5.16 Temperature temporal power spectrum for m = 3 solution in the aperiodic regime,

Re = 4500. Two characteristic frequencies ω

= 70.56 and ω

= 32.36. . . . 93 5.17 Ratios of the amplitudes of harmonics to the fundamental frequency in the appro-

priate powers (a) and global entropy (b), Eq.(6.2), as functions of the Reynolds number for m = 3 traveling wave. A

, i = 1, 2 etc., means the amplitude of the i-th harmonic, A

is the fundamental frequency. The rhombs correspond to the calculated points, and the solid line is the result of spline interpolation. . . . . . 94 5.18 The evolution of the maxima of axial velocity signals as the Reynolds number in-

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

=

0.5(Θ

− Θ

). Solid line and asterics represent m = 2, while the dashed line with rhombs denotes m = 3 mode. . . . 96 5.20 Dependence of the fundamental frequency upon the Reynolds number. Solid line

and asterics represent m = 2, while the dashed line with rhombs denote m = 3 mode. . . . 97 5.21 Net azimuthal flow, defined by Eq.(5.2), vs. the Reynolds number. Solid line

and asterics represent m = 2 solution, while rhombs denote m = 3 solution and

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

periodic. . . . 98

5.22 Schematic graphs of the dynamics of the various stable solutions with the m = 2 and 3 wave–numbers as the Reynolds number increases. m = 2 traveling wave is 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 denote: 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 temperature field

disturbances in z = 0.5 transversal section (upper drawing) and on the free surface (lower drawing). (a) Traveling wave at ∆T = 18.0K (Re = 1904): m = 1 mode;

(b) mixed standing wave at ∆T = 18.5K (Re = 1957): m = 1 + 2 mode. . . 106 6.2 Evolution of spatial power spectrum with the increase of ∆T . Spectra of the

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

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

with the increasing ∆T . The dimensional frequency in Hz can be calculated as f = ω

/2πτ

. . . 110 6.5 Evolution of temperature oscillations with the increase ∆T : (a) ∆T ≈

8.03K (Re ≈ 850), (b) ∆T ≈ 18.30K (Re = 1936) and ∆T ≈ 20.00K (Re = 2116).112 6.6 Evolution of spectra of temperature oscillations with the increase of ∆T (values

of ∆T are shown in upper right corners). . . 113 6.7 Splitting of the maxima of the temperature oscillations. Following rhombs one

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

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

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

temperature difference. (a) - plotted for the whole range of ∆T ∈ [6, 40], (b) - shown for ∆T ∈ [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 regime, 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 present. . . 121 7.1 Experimental set-up used for study of thermocapillary convection in liquid bridge

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

liquid bridge. . . 124

7.2 Temperature distribution along the free surface. 3D calculations when Pr=107, Re = 110, R

= − 0.34969, Bi = 0. . . 127 7.3 Surface temperature distribution for different Biot numbers when T

= T

.

3-D calculations for P r = 108, Γ = 1.2. . . 129 7.4 Disturbance surface temperature distributions for different profiles in the ambient

gas T

= T

. 3D calculations for P r = 108, Γ = 1.2. Steady solution is subtracted. . . 130 7.5 The dependence of temperature upon time. 3-D calculations when Pr=108, Re =

120, R

= − 0.38148, Bi = 0.5 (a) T

= T

(b) T

= T

. . . 131 7.6 Sketch of the gas circulation in the case of shielding. Two convective vortexes

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

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

0.9, Γ = 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 result for Bi = 5.0. . . . 134 8.1 Experimental dependence of the critical temperature difference ∆T

upon liquid

bridge volume, obtained by Mojahed and Shevtsova for Γ = 1.2, P r = 105, Bo

= 2.3. The branches have 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 Temperature field disturbances in mid-cross section and on the free surface for

P r = 0.017, Γ = 1, Re = 3500, Bo = 0, α

= 60

. . . 144 8.5 Temperature time-series for P r = 0.017, Γ = 1, Re = 3500, Bo = 0, α

= 60

. . 145 8.6 Distributions of v

along the free surface for different contact angles. P r =

0.017, Γ = 1, Re = 2500, Bo = 0. . . 145 8.7 Distributions of v

along the free surface for different contact angles. P r =

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

pure buoyant flow when Ra

= 32.97, Ra

= 86.6, Ra

= 0, (∆T /∆x = 3 K/mm).154 10.3 Velocity field (a) and isolines of the deviation of the temperature from linear

profile (b) in XZ-cross section for pure buoyant flow when Ra

= 32.97, Ra

= 86.6, Ra

= 0. The particular cross section is shown in the small cell. . . 155 10.4 Velocity field (a) and temperature isotherms (b) in YZ-cross section for pure

buoyant flow when Ra

= 32.97, Ra

= 86.6, Ra

= 0. The particular cross section is shown in the small cell. . . 156 10.5 Temperature 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 flow. (b) Mixed buoyant and Marangoni convection. . . 157

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

10.7 Combined thermocapillary and buoyant convection. Large bubble case, 1.2mm, M a = 74 000. Temperature fields in mid-crossection Z = 25mm and on the liquid-gas interface where the bubble is situated. . . 160 10.8 Combined thermocapillary and buoyant convection. Small bubble case, 5.5mm,

M a = 74 000. Temperature 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.

Temperature fields in mid-crossection Z = 25mm and on the liquid-gas interface where the bubble is situated. . . 161 10.10Combined thermocapillary and buoyant convection. Small bubble case, 5.5mm,

M a = 32 300. Temperature fields in mid-crossection Z = 25mm and on the

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