Development and improvement of the experimental techniques for fluid examination T H È S E

UNIVERSITÉ LIBRE DE BRUXELLES

ÉCOLE POLYTECHNIQUE DE BRUXELLES

T H È S E

présentée en vue de l’obtention du Grade de

Docteur en Sciences de l’Ingéneur

Présentée et soutenue par

Viktar YASNOU

Development and improvement of the

experimental techniques for fluid

examination

Thèse dirigée par:

Prof. Frank DUBOIS

Supervision scientifique:

Dr. Valentina SHEVTSOVA

Prof. Yury BOKHAN

Abstract

The aim of the thesis is the development and improvement of the experimental techniques for fluid examination. The thesis consists of two parts and both examine heat and mass transfer in liquids using the optical methods and thermal analysis. The first part deals with the measurement techniques for studying flow patterns and their stability in systems with gas/liquid interface, in particular, in a liquid bridge system. The second part is aimed at the improvement of the existing experimental techniques to study the heat/mass transfer in the mixtures with Soret effect, enclosed in a container.

Part A is motivated by preparation of the experiment JEREMI (The Japanese-European Research Experiment on Marangoni Instability) to be performed on the International Space Station (ISS). One of the objectives of the experiment is the control of the threshold of an oscillatory flow in the liquid zone by the temperature and velocity of the ambient gas. The developed set-up for a liquid bridge allows to blow gas parallel to the interface at different temperatures and investigate the effects of viscous and thermal stresses on the stability of the flow. The present study reports on isothermal experiments with moving gas and non-isothermal experiments with motionless gas when the cooling of the interface occurs due to evaporation. The discussion concerning the experimental observations is based on two sources: an interface shape measured optically and the records on thermocouples giving an indication of how temperature and frequency evolve over time.

A note of thanks

I would like to acknowledge everyone who has assisted me throughout my doctoral studies over the years.

First, I would like to express my deep gratitude to my scientific adviser, Dr. Valentina Shevtsova, for the interesting topics, professional guidance, inspiration, invaluable advice, patience, friendly supervision, constant support and encouragement. I also would like to thank Prof. Yury Bokhan, for its activity as second supervisor of my work.

My greatest appreciation and gratitude to Aliaksandr Mialdun, this thesis would not have been possible without his support and assistance! I am also grateful to other members of our group “Non-linear Phenomena in Liquids” Yuri Gaponenko, Jean-Claude Legros, Denis Melnikov, for their advice, interesting discussions and helpful suggestions.

I would like to thank Professor Frank Dubois creating favourable conditions for efficient and fruitful work in Service Chimie-Physique E.P. of ULB and for his advice and care.

Many thanks to all other colleagues of Service Chimie-Physique E.P. for their kindness, support and inexhaustible readiness to help.

A special thank to my family for their endless support throughout my studies. Without their constant assurance and assistance, completion of this work would have not been possible. I want to thank my lovely wife for her patience, support and motivation in the completion of my PhD degree.

Content

1. Part A: Measurement techniques for studying heat/mass transfer in a liquid

bridge ………...……….1

A1. Introduction ………...……….……1

A1.1. Thermocapillary convection in a liquid bridge ………...….…….1

A1.2. Deformation of interface ………....…….6

A1.3. Stability of flow under the influence of different factors ………....…….7

A1.3.1. Interfacial heat flux ……….….……….…….7

A1.3.2. Shape of interface ……….…….…….11

A1.4. Experimental methods of observation ……….……….………13

A1.4.1. Observation of flow ……….……….13

A1.4.2. Measurements of temperature field ……….……….…….14

A2. Experimental set-up ……….….……16

A2.1. General overview ……….……….16

A2.2. Fluids management ……….…….18

A2.3. Volume control ……….………….20

A2.4. Optical system ……….…….20

A2.5. Control unit ……….……….22

A3. Validation of the new set-up ……….……….….24

A4. Shape detection ……….………….30

A4.1. Preliminary shape detection with pixel-size accuracy ……….…….30

A4.2. Determination of the threshold for sub-pixel shape detection ………...…….33

A4.3. Final shape detection with sub-pixel accuracy ……….……….35

A5. Measurements of volume, evaporation and dynamic surface deformation .…..….36

A5.1. Measurements of volume ……….….….36

A5.2. Measurements of evaporation ………..……….36

A5.3. Detection of dynamic deformation ………..….41

A6. Stability of convection driven by thermocapillary and buoyant force ...45

A6.1. Variety of experimental procedures ……….……..…45

A6.1.1. Scanning by ∆T at constant volume and Tmean …………..….………47

A6.1.2. Scanning by volume at constant ∆T and Tmean ………...……52

A6.2. Analysis and comparison of the instability regimes ………….……….….58

A6.3. Stability window ……….………….……63

A6.4. Effect of the mean temperature on the flow stability ………….………..…66

2. Part B:Measurements technique for heat/mass transfer in mixture with Soret effect ……….70

B1. Introduction ……….…70

B2. Set-up ………..…73

B2.1. Initial configuration ………..73

B2.2. Cell optimization ………..……76

B2.3. Final cell design elaborated in the work ……….…….80

B2.4. Interferometer modification ………..………82

B2.5. Control unit ………..…84

B3. Image processing ……….….86

B3.1. Fringe analysis for phase-measuring interferometry ………..….86

B3.2. Subtraction of reference image ………...……90

B3.3. Experimental limitations and precision of the method ………..….92

B3.4. Beam deflection problem ………...…93

B4. Data extraction ……….….95

B4.1. Governing equations ………...………95

B4.2. Fitting equation for full path ………..……….96

B5. Results ……….…….99

B5.1. Water–Isopropanol (IPA) ………...……99

B5.1.1. Contrast factors ………...……99

B5.1.2. Summary on Soret, diffusion and thermodiffusion coefficients ….….100 B5.1.3. Region with negative Soret effect; water rich mixture, C>0.75 ….….103 B5.1.4. Intermediate concentration regime, 0.2<C<0.75 ……….104

B5.1.5. Region with low water content C<0.2 .………105

B5.1.6. Comparison with benchmark values ………105

B5.1.7. Error estimation ………....106

B5.1.8. Discussion ………107

B5.2. Binary couples in tetralin–isobutylbenzene–n-dodecane system …..…108

B5.2.1. Contrast factors ……….……108

B5.2.2. Transport coefficients ……….109

1 Part A

Measurement techniques for studying heat/mass transfer

in a liquid bridge

A1. Introduction

A1.1. Thermocapillary convection in a liquid bridge

Surface tension of liquids is usually a decreasing function of temperature, and any temperature variation along an interface generates a surface flow transporting warm fluid towards cooler regions. A fluid motion driven by surface tension differences along a liquid-gas interface is called thermocapillary convection or Marangoni convection. Such flows are very common in nature and in numerous industrial applications, in which, e.g., evaporation, melting or welding are taking place.

Figure A1.1. Floating zone technique of crystal growth.

2

developed by [1] for zone refining of Germanium, and further used for silicon by Keck and Golay [2] and by Emeis [3]. The floating zone method shortly can be described as a process when a long hard rod of semiconductor is slowly pulled through a ring heater (see Fig. A1.1) and, at first, is locally melted and then re-crystallized, while the impurities concentrate at the end of the rod. On industrial level, floating zone method was used for the first time by SIEMENS AG.

The floating zone is kept between two solid parts –melting and freezing ones. Being contactless to crucibles, and thus preventing potential sources of contamination, the method makes it possible to grow monocrystalline silicon with the highest purity which is important for a number of electronic and optoelectronic applications. Floating zone monocrystalline starts to grow from a high purity, small diameter seed crystal. This seed crystal is prepared in the right crystalline direction in order to grow pure silicon with no crystalline defects.

(a) (b)

Figure A1.2. Sketch of floating zone method (a) and of a liquid bridge (b). Liquid bridge, representing a bottom half of a floating zone is a model used to describe physical phenomena occurring in the real technological processing.

3

important for applications using the bulk of silicon wafers for manufacturing devices. There are two practical ways of obtaining 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. The 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, it is called ex-situ doping, and it is done in neutron irradiating reactors. The starting material for ex-situ doping is high resistivity silicon that after being irradiated with a controllable dose of neutrons changes its resistivity by transforming silicon atoms into 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. Silicon crystals are presently industrially grown with diameters up to 150 mm, weighing more than 35 kg.

One of the reasons for the non-homogeneity of the final crystals was attributed to the presence of buoyancy during the technological process. The non-steadiness of the flow generates variations of the properties of the produced materials. Among the crystal defects are voids, inclusions, distortion, strain, dislocations, inhomogeneities, striations [4].

In order to improve homogeneity, studies of convection in the melting zone, its stability and dependence on various factors were initiated. In the early days of spaceflights, it was believed that crystals of exceptional quality could be grown from the melt in the microgravity environment due to the absence of the undesired buoyant convection. However, as the ring heater creates temperature gradients along the free surface of the melt, Marangoni convection inevitably occurs in the melt independently of buoyant convection. Many microgravity experiments have been performed since the early 70s. Between the Apollo missions and the year 1995 at least 77 experiments were successfully conducted.

4

experimenting on ground, only a small liquid volume may be held between the rods. The typical radius of disks is several millimeters.

At small values of the imposed temperature difference a two-dimensional state is observed. When the temperature difference exceeds the critical value, this basic state becomes unstable and gives rise to a new spatial pattern (3D steady or oscillatory). Pioneer experiments on oscillatory thermocapillary flows were conducted in model floating zones by Chun [5]. Under microgravity conditions the first experiments were performed independently by D. Schwabe and C.H. Chun at the same missions (TEXUS3A 1980, partly successful) and (TRXUS 3b, fully successful), which showed that thermocapilalry forces provide a very strong convection [6], [7].

(a) (b)

Figure A1.3. (a) - Temperature field obtained via CFD simulations at the central cross section (top picture) and at unrolled interface of a liquid bridge. Red and blue colours show higher and lower temperature, respectively. The temperature field clearly shows a flow structure with wave number m=2. (b) – experimentally recorded temperature field in case of a well-developed travelling wave with m=1. The liquid bridge is in the centre, while the two side images are its reflections in the mirrors put behind.

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3D. For Pr >0.07, the supercritical flow is oscillatory. A hydrothermal instability in a high Pr number LB emerges through a supercritical Hopf bifurcation as either a travelling wave in the azimuthal direction or a standing wave. The magnitude of the critical temperature difference, ∆T, is related to the physical properties of the liquid, to the geometrical constraints and is sensitive to the ambient conditions. The supercritical flow is characterized by an azimuthal wave number m, which describes the azimuthal periodicity of the thermocapillary flow and may change with the aspect ratio Γ = d/R (ratio height d to radius R of LB) of the liquid bridge, the temperature difference ∆T, the gravity.

A flow in a LB has to comply with the azimuthal periodicity, consequently, m has to be an integer. The wave number m is equal to the total number of warm (or cold) spots in the temperature field (e.g., in the developed external interface or in a transverse cross- section). The azimuthal wavenumber m of a slightly supercritical flow was found to be primarily determined by the aspect ratio. The first empirical correlation between the azimuthal wave number m and the aspect ratio Γ near the critical point was suggested by Preisser et al. [8] as m∼2.2/Γ. Computer simulations in absence of buoyancy provided the same relation but with a slightly different coefficient 2.0 for Pr < 7 [9]. When including buoyancy forces, corresponding to the terrestrial conditions, it results in a different value of the coefficient [10], [11].

Furthermore, the existence of stable m = 0 mode, a hydrothermal mode running along the interface in the axial direction, was reported [12], [13], [14]. Previously, the axisymmetric oscillatory solution was predicted by [15] in an infinitely long column.

6

two classes of thermal-convective instabilities, namely, stationary longitudinal rolls and propagating hydrothermal waves (hereafter referred to as HTWs).

Riley and Neitzel [17] conducted a series of experiments in a laterally heated rectangular geometry, considering very thin liquid layers of 1cSt silicone oil (d = 0.75 − 2.5 mm; Pr = 13.9). They reported observing pure HTWs for d ≤ 1.25 mm presenting a good agreement with the results from the linear theory of Smith and Davis [16]. A definitive proof for the existence of the HTWs in a liquid bridge was presented by Kuhlmann [18].

As the distance from threshold of instability is increased, secondary instabilities may occur when a primary pattern undergoes transitions to other states, e.g., chaos [19], in a possible succession of different patterns as the system moves further from the critical point.

The thermocapillary convection cannot be totally suppressed, but the oscillatory regime can be weakened. Most of the works aimed 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 based on using magnetic field in a floating-zone of electromagnetically active melt (Dold et al. [20], and Cröll et al. [21]). Other approaches were based on creating a counter flow of the ambient gas (Dressler and Sivakumaran [22]), on imposing vibrations of the end-walls (Anilkumar et al. [23]), or on using surrounding gas at a certain temperature and pressure that decreases the surface tension (Azumi et al. [24]). Another set of attempts to weaken hydrothermal wave is based on rotating the whole system. Among evident drawbacks of these methods aimed at decreasing the effective Marangoni number is that the weakening of the basic state enhances macro-segregation of chemical compositions because of the weakening of the global mixing.

There exist more sophisticated methods consisting in heating locally the free surface by actuators and using a feedback control algorithm defined by the oscillations in the liquid bridge themselves. The inputs of the algorithm are signals from local sensors (Petrov et al. [25]).

A1.2. Deformation of interface

7

due to stationary convection (base state), and supercritical dynamic deformation due to oscillatory convection. The theoretical studies of the stability of the liquid bridge with deformed interface by Shevtsova et al [27], [28], [29], Kuhlmann et al. [30], Nienhueser et al. [31] preceded the experimental ones. The points of interest in the experimental works by Montanero et al. [32], Shevtsova et al. [33], Ferrera et al. [34], [35] were focused on measurements of the amplitudes of deformations, the fundamental frequency and their impact on Marangoni flow. These efforts were directed at both the static and the dynamic deformation of thermocapillary liquid bridges. One of the results is that the subcritical deformations grow linearly with increasing strength of the basic Marangoni flow.

Another source of dynamic fluctuations of interface is the gas flow around the LB. Herrada et al. [36] performed calculations of an isothermal liquid bridge with a straight cylindrical interface in a coaxial gas flow. The conditions of a fully developed flow were prescribed for gas at both ends of a liquid zone. They reported that the maximum magnitude of the free surface deformation depends almost linearly on the gas velocity. A similar study was performed by Gaponenko et al. [37]. They were interested in deformations of an isothermal interface caused by a co-axial gas flow entering from the top. Matsunaga et al. [38] measured on ground the dynamic fluctuations of an interface in an isothermal LB of 3 mm in radius caused by a shear-driven flow. The experiments were conducted with 5cSt silicone oil and nitrogen as the outer fluid. The gas was blown parallel to the interface. They obtained that the magnitude of the dynamic deformation varies from 1 to 15 microns, that is between 0.03 and 0.5% of the radius of the liquid bridge, depending on the gas velocity and the volume ratio. The dynamic deformations are larger when gas enters from the top, from the hot side.

A1.3. Stability of flow under the influence of different factors A1.3.1. Interfacial heat flux

The important role of the heat transport through the liquid-gas interface on the stability of the thermocapillary flow in LB has been reported since the 80s. Heat flux modelled by Newton’s law of cooling relates the rate of the change of the temperature to the difference between the temperature of the medium T and the ambient temperature Tamb:

λ

𝑑𝑑

𝑑𝑑 = −ℎ(𝑑 − 𝑑

),

(A1.1)

8

The local rate of heat transfer through the interface is characterized by the Biot number Bi:

𝐵𝐵 =

ℎ𝑅

_λ

𝑙

.

(A1.2)

The value of the heat transfer depends on a number of factors. The most important ones are convective mechanism of heat transfer, evaporation and thermal radiation: h=hc +

he + hr. While the radiative flux is difficult to estimate, the others can be estimated.

Melnikov et al. [39] have shown how to estimate the convective heat transfer coefficient hc due to the presence of a moving gas surrounding the interface of a liquid

bridge. Their approach is based on a work by Kays et al. [40]:

ℎ

= 0.664𝑅𝑅

𝑃𝑑

λ

𝑑 ,

(A1.3)

where ν is the kinematic viscosity, Re=Vid/νgas is the Reynolds number in gas phase, the

subscript ‘gas’ denotes the ambient gas. Vi is the velocity of the liquid at the interface. It

enters into the definition of Re as it is assumed that the ambient gas is entrained by the moving interface, and thus its characteristic velocity equals to that of the liquid. The Reynolds number of the moving gas is calculated via computer modelling for the experimental conditions. Evaporation is characterized by the mass loss dm/dt through the interface, and consequently, by the related loss of energy. Knowing the evaporation rate and neglecting the thermal diffusivity and kinetic energy of the surrounding air, one obtains from the energy balance:

λ

𝜕𝑑

𝜕𝑑 =

𝐿

𝑆

𝑑𝑑

𝑑𝑑 ,

(A1.4)

where Lv is the latent heat of evaporation, S=2πRd is the area of the evaporating surface.

Hence,

ℎ

= −

_𝑆(𝑑

𝐿

− 𝑑

)

𝑑𝑑

𝑑𝑑 ,

(A1.5)

where (Ti-Tgas) is the difference in temperature between the interface and surrounding gas.

The mass loss may be experimentally evaluated by monitoring the liquid volume change with time.

9

drag opposing the Marangoni shear at the free surface. An average reduction of 66% of the Marangoni velocities was successfully obtained during their experiments.

Velten et al. [41] carried out ground experiments on liquid bridges formed by sodium nitrate (Pr = 9), potassium chloride (Pr = 1,) and tetracosane (Pr = 49). They studied both positive and negative temperature gradients. The authors somewhat confirmed the observations of [22] that the air motion around the liquid column, mainly caused by buoyancy due to the heating and cooling arrangement of the experiment, has a strong effect on the onset of an oscillatory flow. This work reported that values of ∆Tcr were

higher in the heated-from-below cases. The difference was attributed to alterations in the flows of gas surrounding the zone, which was confined by a larger quartz cylinder playing the role of shielding. In the heating-from-above case, this gas flow exhibits a pair of counter-rotating tori which, in turn, modify the radial heat transfer.

Kamotani and co-workers [42], [43] and [44] experimentally measured the effect of surface heat loss/gain for 2 and 5 cSt silicone oils with high Prandtl numbers (Pr = 24 - 49). They performed ground-based experiments on LB using 2 and 3mm diameter rods and reaching Γ= d/R = 0.8 - 1.4. A wide range of ambient temperatures was studied by placing the experimental set-up into an oven. The authors estimated the Biot number to be about unity or smaller. It was obtained that even at such moderate values of Bi the enhanced heat loss from the surface significantly destabilized the Marangoni flow. Changing the air temperature alters the critical temperature difference by a factor between two and three. The airflow analysis showed that even though the heat loss was relatively small, the critical temperature difference was affected appreciably. Analyzing a very high sensitivity of the critical conditions to the heat transfer through the free surface reported in [43], the authors made as a possible conclusion that this effect could be due to some changes of dynamic free surface deformation [44], which, in turn, is small and is itself strongly affected by the heat loss.

10

Schwabe [47] performed microgravity experiments onboard the sounding rocket MAXUS-4 in 2cSt silicone oil (Pr=28) LB of very large aspect ratio Γ= 5 (the length d = 15.0 mm and the radius R = 3.0 mm). The opposite direction of the hydrothermal wave with respect to the theoretical predictions of [15] was attributed to strong surface cooling, as the estimated Biot number was Bi = 3.5. The authors stated that the high heat loss at the interface promotes instability waves travelling counter to the surface flow.

In the last several decades, many numerical studies of the thermal convection in liquid bridges with interfacial heat transfer have been performed. Xu et al. [15] considered an infinitely long liquid bridge under weightlessness. Having kept constant the ambient gas temperature and chosen only Bi = 0 and 1 they predicted an increase of the critical Marangoni number when increasing the Biot number.

Sim and Zebib [48] studied the effect of free surface heat loss on the critical conditions by means of direct numerical simulations in LB with unity aspect ratio and Pr = 30. For that study, a uniform temperature in the gas was chosen equal to the temperature of the cold disk. It was obtained that cooling the free surface stabilizes the flow. Increasing the Biot number from 0 up to 20 led to a monotonous increase of ∆Tcr.

A liquid bridge with high Prandtl number fluid, Pr=18, surrounded by an ambient gas of constant temperature was studied by [49] for a large aspect ratio, Γ=1.8. They reported that increasing the Biot number from zero up to 1.8 does decrease the critical Marangoni number by 33%.

Recently, the effect of interfacial heat exchange on thermocapillary flow in a cylindrical liquid bridge of 1 cSt silicone oil (with Prandtl number about 16) with aspect ratio 1.8 has been investigated in absence of gravity by a linear stability analysis [50]. With both constant and linearly distributed ambient temperature, the computed results predicted that the stability curve for the thermocapillary flow exhibits a roughly convex trend with variation of the Biot number.

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The influence of heat loss (or gain) on the stability of the thermocapillary flow in LB is strongly influenced not only by the Biot number and the ambient temperature in the gas, but also by the parameters governing the liquid flow, i.e., the Prandtl number, the aspect ratio, gravity, and the flow in the gaseous phase [45], [51]. The recent experimental benchmark [52] demonstrated large scattering (~15-20%) of the results for the critical parameters between different experimental groups working with the same liquid. Any comparison between different results could be made only if they were obtained under identical conditions. In some theoretical works, the Biot number was varied with large increments, and thus, possible local changes in the dynamics of the flow were disregarded. One should perform the study carefully by varying the Biot number by small steps, especially between 0 and 1, as at high heat loss the behaviour of the pattern is more or less the same because the cooling prevails any other factor.

Experimenting with a liquid bridge of 10 cSt silicone oil, Shevtsova et al. [53] experimentally demonstrated that the convective mode may change when the temperature of the cold rod of the liquid bridge varied while keeping its volume constant. This discovery has a direct connection with the influence of heat transfer on a thermocapillary flow. Changing the cold temperature (the environmental thermal conditions stay the same) one manipulates the heat flux through the interface. Along with the mode change, the two-dimensional flow destabilizes (∆Tcr drops by almost 25%) while the temperature at the

cold rod increases from 10 to 30°C. A1.3.2. Shape of interface

Another important factor with respect to the stability of a two-dimensional flow is the volume of the liquid (denoted as Vol) contained in the bridge and, correspondingly, the static deformation of interface. The magnitude of the deformation is determined by the ratio of the hydrostatic to the capillary pressure, i.e., by the so-called static Bond number

𝐵𝐵 =

ρ

𝑔𝑑

_σ

,

(A1.6)

where ρ is the liquid density, g is the gravity acceleration, σ is the surface tension. At zero-gravity, the Bond number is zero, and the interface of a liquid bridge with volume of the corresponding cylinder equal to πR2_{d is straight. On the contrary, on ground Bo≠0 and}

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Both experimental and theoretical results on the influence of the liquid volume on the stability of a thermocapillary flow are usually presented in the form of a curve ∆Tcr

versus volume ratio Vr, which is the ratio of the volume of the liquid bridge to the volume

of the corresponding cylinder, i.e.,

𝑉𝑑 =

_π

𝑉𝐵𝑉

_𝑅

_𝑑.

(A1.7)

One of the first experimental evidences is that the critical temperature difference depends strongly on the liquid volume. It was obtained by Cao et al. [54], Hu et al. [55], Masud et al. [56] and Hirata et al. [57]. It was also found experimentally, that the wave number m at the onset of instability changes with the volume of the liquid bridge. A non-monotonic response of the stability curve to increasing the liquid volume was obtained for high-Prandtl liquids. Shevtsova et al. [58] showed for high Prandtl numbers under terrestrial conditions that the stability diagram (ΔTcr vs. Volume) consists of two branches

with different oscillatory modes. In a LB of 10 cSt silicone oil of aspect ratio Γ = 4/3 they identified two branches corresponding to different azimuthal wave numbers. In slim liquid bridges with a concave interface the stable mode corresponds to m = 1, but it is m = 2 when the interface is convex. In the region around Vr=1 the stability curve has a local maximum, meaning a strong stabilization of the flow. These results were confirmed later in [59].

Performing experiments with n-Decane in a wide range of Vr∈[0.7;1.05] Melnikov et al. [39] measured the critical temperature difference and confirmed the existence of two branches with a gap at around Vr=0.9. The numerical results have shown that these two branches of ∆Tcr are not associated with a change of mode.

When the interface is not straight, the relation between the mode m and the aspect ratio should be revised. Considering a LB with small Prandtl numbers, Pr = 0.01, no buoyancy, Lappa et al. [60] suggested a modified empirical relation, which re-defines the aspect ratio via the radius of the liquid column at the mid-height:

𝑑

≈

2

Γ

� ,

Γ

� =

𝑑

ℎ(𝑧 = 𝑑2)

(A1.8)

13

With respect to the stability of deformable liquid bridges, the theoretical developments have not progressed too much. The reason is mainly that modelling a flow in non-straight geometries becomes technically difficult and CPU time-consuming. The first calculations on steady thermocapillary convection in a floating zone with a deformable free surface were performed in the limit of small deformations by Kozhoukharova et al. [61]. It is worth mentioning a few numerical simulations of two-dimensional thermocapillary flows in liquid bridges with strong deformation of the free surface performed by Shevtsova et al. [27], [28], Sumner et al. [62], Tang et al. [63]. They studied the influence of g-jitter on thermocapillary convection in an axisymmetric LB with deformation of the free surface. Lappa [64] made a study on the influence of liquid volume and gravity on the flow instability in a floating zone.

Being aware of the difficulties related to modelling a three-dimensional thermocapillary flow in a liquid bridge with a non-cylindrical interface a benchmark was performed, where 9 research groups were participating. A goal was to analyze the validity of the models of the flow and to predict the onset of hydrothermal instability in a liquid bridge with Pr=0.02 [29].

A1.4. Experimental methods of observation A1.4.1. Observation of flow

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A1.4.2. Measurements of temperature field

The temperature field is measured by two main methods: contact thermometry and radiation (contactless) thermometry. Contact thermometry is carried out with a sensor, e.g., a thermocouple, or a Platinum resistance thermometer, which always remains in contact with the device under test. The principle of any contact thermometer is that it is designed to have some physical parameter changing in a well-known way with temperature. A contactless thermometer is put into a “contact” with the tested object by means of electromagnetic radiation, e.g., infrared radiometer.

Radiation thermometry measures the radiation of the device under test without contact, by means of an infrared sensor. The thermal radiation emitted by an object is not only defined by the temperature of the object, but also by the emissivity ε of its surface. The power output of an object with surface temperature Ts surrounded by a medium with

temperature Tamb is given by the Stefan–Boltzmann law:

𝑃 =

_{𝜎𝑆(𝑑}

_{− 𝑑}

),

(A1.9)

where S is the radiating surface area, σ= 5.670373×10−8 W m−2 K−4 is the Stefan– Boltzmann constant.

The emissivity deserves special attention. It is not a universal constant, but a dimensionless number between 0 (for a perfect reflector) and 1 (for a perfect emitter). Non-metallic and non-transparent objects are generally good radiators with an emissivity larger than 0.8. The emissivity of metals can vary between 0.05 and 0.9. Shiny, highly reflective metal surfaces will have lower emissivities. The emissivity of a surface depends not only on the material but also on the nature of the surface, its temperature, wavelength and angle. For example, a clean and polished metal surface will have a low emissivity, whereas a roughened and oxidised metal surface will have a high emissivity. Radiation thermometers are generally calibrated using blackbody reference sources that have an emissivity close to 1. Unfortunately, the emissivity of a material surface is often very difficult even to estimate. It must either be measured or modified in some way, for example, by coating the surface with high emissivity black paint, to provide a known emissivity value. When working with liquid surfaces, this approach is obviously not possible. Thus, one needs to calibrate the thermometer before using it for reliable experimental temperature records.

laser-15

based methods (e.g., laser-absorption radiation thermometry or Rayleigh-scattering thermometry).

Contactless thermometry has both advantages and disadvantages as compared to contact thermometry. One does not need to wait for establishing thermal equilibrium between the object and the thermometer. It is, thus, better suited for the measurement of fast temperature changes. It is always better to use the radiation thermometry as any direct contact with a flow inevitably generates disturbances. IR cameras allow high spatial resolutions of surface temperatures. Their response to temperature changes is immediate. On the other hand, records of a thermocouple are much more accurate than those from a reasonably priced IR camera. Temperature measurements that are taken with thermocouples, however, have a very limited spatial resolution. Another drawback of the radiation method of temperature measurements is that one cannot access the interior of the tested domain. The signal to be measured by contactless thermometry may be perturbed on its way from the object to the thermometer, or the signal may merely correspond to an average temperature (averaged over the line of sight, for example). To determine the mode of the supercritical flow, one needs to use either an IR camera or a set of thermocouples.

16

A2. Experimental set-up

A2.1. General overview

Considering the numerous requirements to be met by the set-up, its design appears to be quite complex (Yasnou et al. 2012) [65]. First, it has to allow blowing gas axially in the duct around the liquid bridge in both directions with the possibility to control the temperature of the gas. Second, independent and precise temperature control of both top and bottom metal rods and their precise alignment are needed. The temperature inside the liquid has to be measured at different points. As there is no direct access to the liquid as it is surrounded by a glass tube of the air gas channel, all sensors and supplying pipes have to be driven through the supporting rods. The general view of the set-up is shown in Fig. A2.1.

(a) (b)

Figure A2.1. a) View of the set-up, b) Simplified sketch of the set-up

1 - lower plate, 2 - lower clamp, 3 - upper plate , 4 - upper clamp, 5 - middle plate, 6 - cylinder guides, 7 - screw pairs, 8 - toothed belt, 9 - liquid bridge position

The set-up is a structure consisting of a lower plate with a clamp for holding the lower rod, an upper plate with an auto-centering clamp for holding the upper rod and a moving middle plate on which the optical system is integrated. The middle plate moves vertically along cylindrical guides. This displacement is driven by screw pairs synchronized by a toothed belt.

The clamp of the lower rod has the possibility of precise two-axis positioning as well as precise control of verticality. The exact adjustment of the liquid bridge height is performed by the upper rod driven by a micro-screw.

17

The geometry of the liquid bridge is shown in Fig. A2.2. The working parts of the rods 3mm in radius are made of brass. The minimal distance between them is hmin=0.5 mm, which is conditioned by the space necessary for the thermocouples installed

in the upper rod. The maximum distance between the rods is determined by the hydrostatic stability of the liquid bridge at normal gravity and is approximately hmax=5-7 mm. The

rods are placed inside a glass tube with inner radius Rext=5mm. Gas can be injected

between the internal walls of the glass tube and the rods. For producing a laminar gas flow after the gas inlet, the rods have a system of six azimuthally symmetric wide ducts (marked 9 in Fig. A2.2). To additionally homogenize the gas flow, a relatively long distance of L=25 mm is left between the ducts, where the gas is injected, and the liquid bridge. According to direct numerical simulation of a gas flow in such geometry [66], one can expect formation of a pure parabolic profile in the gas channel. The gas supplying tube can be easily linked either to the upper or to the lower rod depending on the gas flow direction which is chosen for the investigation.

Figure A2.2. Rods structure

1 - planting cylinders, 2 - cavity for heat-carrying liquid, 3 - tubes, 4 - thermistors, 5 - thermocouples, 6 - capillary for injecting working liquid, 7 - heat-insulating tubes, 8 - pneumoconnectors, 9 - ducts, 10 - side windows, 11 - connectors, 12 - plastic disk.

To maintain the required temperature of the end parts of the rods which are in contact with the test liquid, the cavity is equipped with tubes through which heat-carrying liquid is pumped from the heat exchanger. To reduce the heat exchange between the working parts of the rods and the bases of the rods, the former are fixed through heat-insulating and mechanically rigid composite plastic tubes. Low thermal conductivity of the

18

plastic (0.2 W/m/K) also reduces heat exchange between the ends of the rods and the blowing gas.

The temperature of the surfaces which are in contact with the test liquid (the tips of the rods) is controlled by means of thermistors placed at a distance of 0.3 mm from them. Three thermocouples (with wires’ diameters of 24 µm) for measuring the temperature of the liquid at different positions of the bridge are installed on the upper rod; their junctions are located at a distance of 0.3 mm from the surface. The injection of liquid to establish the liquid bridge is performed through a capillary placed in the lower rod.

A2.2. Fluids management

The temperatures of the rod tips are maintained by pumping heat-carrying liquid through the cavities by means of a hydraulic system. The principle of the operation of this system is shown in Fig. A2.3.

Figure A2.3. Hydraulic system

1 - heat exchangers, 2 - filters, 3 - dampers, 4 - peristaltic pump, 5 - copper blocks, 6 - Peltier elements, 7 - thermal bath, 8 - heat-insulating casing, 9 - PID-controller

The hydraulic system consists of heat exchangers, filters, dampers and a peristaltic pump. The heat exchangers are a set of three copper blocks between which Peltier elements are placed. Heat-carrying liquid passes through the central copper block, and

19

liquid coming from the thermal bath is pumped through two external blocks. The whole assembly is placed in a casing in order to establish a temperature isolating barrier. Such a structure is necessary for maintaining a stable and precise temperature. The peristaltic pump is connected between two dampers smoothing pressure pulsations and eliminating vibrations transmission to the liquid bridge. Due to the fact, that liquid is supplied to the working part of the rods through thin capillaries, filters are installed to prevent plugging of the latter. The tubes supplying liquid to the rods are heat-insulated. Temperature is controlled by a PID-controller. The temperature of the surface test liquid is read out from the PID-controller and kept in the computer. The temperature stability of the rods ensured by the system is better than 0.1 K (rms value).

Figure A2.4. Pneumatic system

1 - gas bottle, 2 - mass flow controller, 3 - heat-exchanger, 4 - aluminium blocks, 5 - Peltier elements,6 – thermoregulated bath, 7 - heat-insulating casing, 8 - PID-controller

The gas flow and its temperature are managed by the pneumatic system shown in Fig. A2.4. A heat exchanger produced on the same principle as in the hydraulic system is used in the pneumatic system. It differs in that it is made of aluminum blocks of a much larger area and has a turbulator installed in the centre of the gas channel. Working gas flows from the bottle into the heat exchanger through a pressure regulator and a computer-driven mass flow controller, which allows maintaining the gas injection rate with a great precision. As in the hydraulic system, the temperature is controlled by a PID-controller. The data on the gas temperature are read out from the PID-controller and stored in the computer.

6

4 3 5 7 8 2 1

Thermal bath ath

20 A2.3. Volume control

To establish a liquid bridge and to keep its volume constant, the liquid evaporation is compensated, as shown in Fig. A2.5.

Figure A2.5. System for compensation of liquid evaporation

1 - tank for discharging liquid, 2 - tap, 3 - syringe pump, 4 - computer, 5 - thermal insulation

To establish a liquid bridge, the liquid is injected between two rods by means of a syringe pump. The shape of the liquid bridge is controlled by an optical system. The volume of the liquid is calculated in real-time, a command is sent to the syringe pump to inject or pump out a calculated volume of liquid. The volume stabilization is done according to PID law, which allows maintaining the volume with a precision of 0.1%. To minimize the effect of ambient air temperature change (thermal expansion of liquid) on the liquid bridge volume, the syringe pump and supplying capillaries are insulated.

A2.4. Optical system

The optical system assembled on the middle plate consists of two identical optical couples (illuminator – camera) located at a right angle to each other. Each couple operates at its own and different wavelength (using corresponding LEDs and filters) to avoid cross-reflections. We use green 520 nm and red 655 nm LEDs in combination with long pass and short pass filters with a cut-off wavelength of 600 nm. These filters are mounted between the lens and the camera.

1 5

PC

4

21

Figure A2.6. Optical system

1 - collimated light source, 2 - glass tube, 3 - cylindrical lens, 4 - liquid bridge, 5 - compensating cylindrical lens, 6 - objective lens, 7 - camera, 8 - filter

The structure of one of the optical modules is shown in Fig. A2.6. A collimated light source is used for obtaining a sharp image of the liquid bridge. Since the glass tube surrounding the liquid bridge is acting like a diverging cylindrical lens (with an estimated focal length of -65 mm), a pair of compensating converging cylindrical lenses (of 150 mm focal length) is introduced into the optical path (#3 in Fig. A2.6)to make the beam lighting the liquid bridge perfectly parallel. Another (#5 in Fig. A2.6) compensating cylindrical lens, necessary for compensating bridge geometry distortion, is placed between the liquid bridge and the camera. Calibration of the overall optical system has been done by use of a square grid of 0.1 mm step mounted inside the same glass tube in the meridional plane of the liquid bridge. The procedure was basically the same as the one applied by Gaponenko et al. [66] and Matsunaga et al. [38].

The high-speed cameras can record the changes in the shape of the liquid bridge surface with a frequency of 500 frames per second. Using two cameras with perpendicular views enables to record four surface profiles simultaneously, which allows obtaining the tomographic scanning of the liquid bridge surface.

22 A2.5. Control unit.

The control unit diagram of the set-up is shown in Fig. A2.7.

Figure A2.7. Control unit diagram of the set-up

The set-up is operated by a computer by means of a control unit. The volume of the liquid bridge, the volume of the injected liquid, the area of the interface, the temperature of

PSU 0-13V

PID-CONTROLLER

USB-RS232

LED DRIVERS PSU 5V 5A

23

the rods, the consumption and temperature of gas as well as the temperature of the liquid in three points of the liquid bridge are recorded by the computer. Two other computers are used for capturing images from the high-speed cameras.

The control unit is a shielded modular structure designed and produced in our laboratory which contains the main electronic blocks of the set-up. The photo of the control unit is shown in Fig. A2.8.

Figure A2.8. Photo of the control unit.

24

A3. Validation of the new set-up

Professor D. Schwabe from Giessen University offered to our laboratory a liquid bridge set-up, which he had used for many years to investigate instabilities in liquid bridges and particle accumulation phenomena (PAS). This set-up is shown in Fig. A3.1.

Figure A3.1. Photo of Schwabe's set-up.

To validate the new set-up, we have performed comparable experiments for determination of the onset of instability using both set-ups. The data on critical ∆T obtained on Schwabe’s set-up [67] are in a favorable agreement with the series of experimental data obtained with our new set-up. The experiments on both set-ups were conducted for different volumes of the liquid bridge with n-Decane as the working liquid. The results of the experiments are summarized in Fig. A3.2.

Figure A3.2. Stability map: values of critical ∆T as a function of relative liquid bridge volume measured on our new set-up and Schwabe's set-up. V0 is the volume of the straight cylinder.

25

As a general trend, the critical ∆T in Schwabe’s set-up is lower than in the new one. A detailed analysis has been carried out to understand why there is some discrepancy between data from different set-ups. One possible justification for this might be that in the new set-up the mean temperature of the test liquid is constant, Tmean=25K, while in

Schwabe’s set-up it varies with ∆T. Another reason is attributed to various thermal conditions around liquid bridges because they have different structural features. The new set-up was initially designed for studying the effect of blowing gas at different flow rates on hydrodynamic behaviour of the fluids. The photos of both set-ups are presented in Fig. A3.3.

Photo of Schwabe’s set-up Photo of the new set-up

Sketch of Schwabe’s set-up Sketch of the new set-up

26

In spite of the fact, that the principal geometry and dimensions of the liquid bridge are identical (the rods’ radius is 3 mm, the aspect ratio is Γ=H/R0 =1), the materials used to

27

Figure A3.4. Computed temperature (a) and flow (b) fields in the two set-ups at ∆T=7°C in two phases: n-Decane and air. The left part of the plots presents distribution in Schwabe’s set-up and the right part presents fields in the new set-up. The geometry of calculations is presented in correct scale: the radius of rods is R0=3mm, the external radius of the new and Schwabe’s

set-ups are Rext=15mm. The total heights of chambers filled with air are 21mm and 9mm for the new and Schwabe’s set-ups, respectively.

28

Indeed, one of the driving forces, i.e., the thermocapillary force which is proportional to the temperature gradient ~dT/dz, acts on the liquid/gas interface. The temperature profiles at the interface are shown in Fig. A3.5 (a) in both cases. The temperature in the new set-up is fraction higher and, more importantly, is almost constant at the major part of the interface. This provides a smaller effective temperature gradient than in Schwabe’s set-up. Consequently, the interface velocity (see Fig. A3.5 (b)) is larger in Schwabe’s set-up and it may lead to the destabilization of the flow as observed in Fig. A3.2.

(a) (b)

Figure A3.5. Computed temperature (a) and axial velocity (b) along the interface filled with n-Decane at ∆T=7°C Geometries of the liquid bridges correspond to set-ups’ design.

In the presence of evaporation, one of the key parameters for the flow instability is the heat flux through the liquid/gas interface. The local heat flux q(z) through the free surface area is determined as:

𝑞(𝑧) = −𝑘

_�

𝜕𝑑

𝜕𝑑�

(A3.1)

where k is the thermal conductivity of the liquid, R0 is the radius of the liquid bridge rods.

29

shows that the heat flux is positive and the interface loses heat over the entire length. The heat flux in the central part of the interface grows almost linearly towards the cold side in both set-ups, but in the new set-up the slope is smaller. Furthermore, the heat lost in the new set-up is smaller than in Schwabe’s set-up. Consequently, the interface in Schwabe’s set-up is cooling faster and non-uniformly and it may contribute to the increase of the interface velocity and, consequently, to the destabilization of convective flow.

Figure A3.6. Computed distribution of the local heat flux q along the interface n-Decane/air (see Eq.A3.1 for definition). The geometry of calculations corresponds to the geometry of the set-ups.

Table A3.1. Physical properties of n-Decane [70], [71]

Density, ρ kg/m3 730

Specific heat capacity, cp J/(kg*K) 2190

Thermal conductivity, λ w/(mK) 0.1351

Dynamic viscosity, μ Pa*s 9.05E-04

Kinematic viscosity, ν m2/s 1.24E-06

Surface tension, σ N/m 2.39E-02

Variation of surface tension with temperature, dσ/dT N/(m*K) 1.18E-04

Thermal expansion coefficient, β×103 1/K −1.06

Latent heat of evaporation Ws/kg 3.503E+05

30

A4. Shape detection

A4.1. Preliminary shape detection with pixel-size accuracy

The preliminary shape detection is done by a very robust algorithm of edge detection, the so-called Canny method. The robustness of this algorithm is conditioned by the fact that it utilizes a double threshold and finds both strong and weak edges and connects them, if necessary. Therefore, the processing can hardly be affected by noise. Previously, the somewhat similar technique was used in joint research by MRC, ULB and University Extremadura, Spain [32], [33], [34], [35].

This preliminary detection is done by the ‘edge’ function built in the Image Processing Toolbox of Matlab. Since the function finds any edge in the image, the result has necessarily to be sorted for ‘useful’ and ‘useless’ edges (see Fig. A4.1).

(a) (b)

Figure A4.1. Raw image (a) and the result of preliminary edge detection (b).

31

Figure A4.2. Selection of true edges from the total edge detection result.

After such a selection, only the coordinates of the sidewalls of the LB and supporting rods are left, as shown in Fig. A4.3.

Figure A4.3. Result of sidewall edge pixels detection.

After detecting the sidewalls, the necessary step is to cut off the edges of the supporting rods, since only the region of liquid is of interest for the present study.

To do so, the bottom of the groove cut around the end of each supporting rod is chosen as a reference point. The exact vertical distance between the bottom of the groove

32

and the end of the rod is carefully measured using a snapshot of an ‘empty’ liquid bridge (supporting rods without any liquid). Then, the measured distance in pixels is calculated on an individual snapshot to find the exact location of the liquid interface. Figure A4.4 illustrates this procedure.

Figure A4.4. Selection of the region of liquid.

Although the result of liquid interface detection looks rather smooth in this low-resolution figure, plotting it with magnification immediately visualizes all pixel steps, as shown in Fig. A4.5.

33

So, this result of interface detection with pixel accuracy is acceptable for the raw detection of the surface position and some estimation of the liquid bridge volume, but is definitely not acceptable for precise detection of small static and dynamic deformations of the free surface. It necessitates improvement of the above procedure by adding a sub-pixel detection step.

A4.2. Determination of the threshold for sub-pixel shape detection

All techniques for edge detection typically utilize a threshold approach. To find the necessary threshold, a method based on the analysis of an image histogram was used. It is possible to use the histogram of a complete image, but very often an image is ‘wasted’ by smooth black-to-white transitions and is essentially diffused. To avoid this problem, the histogram was calculated for an artificially created auxiliary image. This image consists of pixels of two stripes cut out of the original image. The shape of these stripes follows the preliminary determined edges with margins of ±30 pixels to the left and to the right of these edges, as shown in Fig. A4.6.

34

A typical horizontal intensity profile within this stripe is shown in Fig. A4.7.

Figure A4.7. Intensity profile over the region selected for histogram calculation.

It is evident, that the auxiliary image created in such a way will show a histogram with two sharp peaks corresponding to the dark and bright regions. With such an improved histogram, it becomes very easy to detect the above-mentioned intensity peaks and subsequently to determine the threshold. A typical histogram is plotted in Fig. A4.8.

35

A4.3. Final shape detection with sub-pixel accuracy

To find the exact shape of the liquid bridge with sub-pixel accuracy, one needs to combine and use all the data obtained in the previous steps, namely, the threshold value, the coordinates of edge pixels and the original image itself.

To correct the edge position, a part of horizontal intensity profile with margins of ±10 pixels around the edge pixel detected by Canny method is taken from the original image row by row. Then, the corrected position of the edge is found for this profile using the predetermined threshold value by a simple linear interpolation (see Fig. A4.9). The so-called global threshold is used for sub-pixel detection; it means that the threshold value is the same for all the edges in each particular image.

Figure A4.9. Way of sub-pixel detection.

The result of sub-pixel detection is much smoother (see Fig. A4.10), which allows evaluating the free surface deformations with a sensitivity of a micron.

36

A5. Measurements of volume, evaporation and dynamic

surface deformation

A5.1. Measurements of volume

Knowing the exact profile R(z) of the surface, the volume of liquid can be precisely determined as:

𝑉 = 𝜋 � 𝑅

(𝑧)𝑑𝑧

(A5.1)

and the surface of the interface is:

𝑆 = 2𝜋 � 𝑅

(𝑧)�1 + [𝑅′(𝑧)]

_𝑑𝑧

_(A5.2)

Here, R(z) is the dependence of LB radius upon vertical coordinate and R´(z) is its derivative.

A5.2. Measurements of evaporation

37

Figure A5.1. Decrease of LB volume with time due to evaporation for ethanol and a few shape snapshots taken at times (a) 0 min, (b) 3 min and (c) 6 min. Experimental conditions: the radius is R0=3mm, the height is d=3mm and Tamb=22°C.

The images were acquired at a frequency of 1 frame per second during 14 minutes. This final time is close to the critical moment of the thinning and breaking of the bridge for the most volatile liquid, i.e., ethanol. The curves of volume variations of an ethanol liquid bridge with time are shown in Fig. A5.1 for different gas velocities. Plots (a), (b), (c) illustrate the evolution of the liquid bridge shape with time.

Figure A5.2. LB volume change with time for different liquids; experiments are performed when evaporation is enhanced by a gas stream parallel to the interface with velocity Ug=47 cm/s.

a

b

c

38

Regardless of the presence of a forced gas stream, the evaporation rate depends on the physical properties of liquids and gas. The evolution of the relative volume of the liquid bridge with time at gas flow rate Q=24 ml/s, Ug=47 cm/s is shown in Fig. A5.2 for

three different liquids: silicone oil 5 cSt, n-Decane and ethanol. Figure A5.2 shows that the liquid bridge filled with silicone oil 5 cSt displays a slower evaporation and the volume is approximately constant during the characteristic time of flow stabilization (10-15 min). The situation is a little worse in the case of n-Decane. During the experiment the relative volume decreased from 1 to ≈0.85. Note that the decrease of LB volume with time is almost linear for n-Decane.

For a highly volatile liquid such as ethanol, the curve exhibits a nonlinear behaviour and, consequently, the volume loss rate is not constant during the experiment. One of the explanations of the non-linear behaviour may be related to reduction of the surface area (S) of the liquid bridge with a decrease in volume. Furthermore, our experiments have shown that mass evaporation rate per units of the surface area defined as

𝐸𝑅 =

𝜌

_𝑆

𝑑𝑉

_𝑑𝑑

(A5.3)

provides a value independent of time and can be used for comparison of different experiments. Here ρ is the density of liquid.

Figure A5.3. Mass evaporation rate defined by Eq. A5.3 as a function of gas velocity for three different liquids.

39

comparison with ethanol. The evaporation rate ER of ethanol increases almost linearly at small gas velocities and then achieves saturation at Vg~60cm/s.

Figure A5.4. Comparison of the evaporation rates of n-Decane with and without gas blow at room temperature.

Our current experiments on the stability of a liquid bridge use n-Decane as a test liquid and, correspondingly, we have studied this liquid in more detail. The comparison of the evaporation rate of n-Decane liquid bridge with and without gas blow shown in Fig. A5.4 demonstrates an essential increase of liquid loss when a gas flow is applied. These experimental results showed the necessity of dynamic adjustment of the liquid volume during the experiment with n-Decane. Thus, we have to impose experimental

requirement to compensate the volume of liquid even for moderately volatile liquid when gas is blown around.

40

Various types of non-isothermal experiments to be described below were performed during the study. Consequently, to analyze the dynamics of evaporation, in addition to the isothermal (∆T=0) experiments described above we have conducted non-isothermal experiments (∆T≠0) with n-Decane. Non-isothermal experiments were performed at different mean temperatures, Tmean=22°C and Tmean=25°C and various ∆T.

The experimental points are presented in Fig. A5.5 by different symbols. We found that the most efficient way to present the evaporation rate for isothermal and non-isothermal experiments together is a function of the surface temperature. The importance of the surface temperature during evaporation has been recently discussed for the geometry of droplets [72], [73]. The non-linear simulations of Navier-Stokes equation in two phases without and with a small evaporation rate demonstrate that the interface temperature can be approximated as

𝑑

≈ 𝑑

+ 0.7∆𝑑

(A5.4)

To illustrate this, we presented the temperature profile on the interface in a dimensionless form Θ=(T-Tcold)/∆T in Fig. A5.6, which supports the suggestion that the

temperature at the major part of the interface can be described by relation (A5.4).

Figure A5.6. Results of numerical simulations; variation of the dimensionless temperature Θ=(T-Tcold)/∆T along the height of a liquid bridge.

Furthermore, all the experimental results which presented evaporation as a function of the surface temperature follow a similar trend, which is shown by the dashed line in Fig. A5.5. This trend can be described by polynomial of 5th order:

|R| = 9·10-10 Ts5 - 1·10-7 Ts4 + 5·10-6 Ts3 - 0.0001·Ts2 + 0.0013· Ts - 0.006

41

symbols). We would like to draw attention that Fig. A5.5 comprises data from the experiments with and without Marangoni/buoyant flow. This plot presents experimental evidence of the general trend unifying experiments with and without convective flows. It gives hint that convective flow (or moving interface) does not provide dominant contribution to the evaporation rate.

A5.3. Detection of dynamic deformation

In addition to bulk flows, a gas blow causes mechanical disturbances and dynamic deformations of the free surface. These deformations do not have azimuthal symmetry, and a thorough analysis requires the analysis of the deformations on the entire free surface.

Tests of the optical system developed for 3D mapping of liquid bridge surface deformation caused by the mechanical effect of a gas flow were conducted with the following parameters: relative bridge volume 1.0, gas velocity 260 cm/s with an upward flow direction, i.e., against gravity. The working fluids were 5 cSt silicone oil and nitrogen. These experiments were conducted at room temperature.

Two perpendicular views of the liquid bridge, as shown in Fig. A5.7, were recorded simultaneously by two cameras. The acquisition frequencies of both cameras are identical and equal to 100 fps and exposure time 5 ms, thus, giving a time step of 10 𝜇s between the consecutive images. Each camera provides a set of 818 images.

Figure A5.7. Schematics of observation.

42

The profiles were filtered to remove optical noise of high spatial frequencies and then rescaled to have the same 101 points over the liquid bridge height for all of them.

Each profile from two sides and two views was averaged over the 818 shapes and the dynamic deformation was calculated as a deviation of the profile at a particular time instant from its average state. The four profiles that came from the equally numerated images of the front and side cameras are shown in Fig. A5.8. They point out that for such a large gas velocity, Ug=260m/s at the entrance, the symmetry between two profiles at the

same view is broken and all profiles are different. Creative fusion of four profiles at the same time instant provides a complete reconstruction of dynamic surface deformation at any cross-section.

Figure A5.8. Snapshot of dynamic surface deformation (deviation from the mean value) at different azimuthal positions of the LB when Ug=260cm/s.

Figure A5.9 demonstrates a pattern of the dynamic surface deformation in a horizontal cross-section of the liquid bridge at the mid-height. The deformation pattern is radially and azimuthally non-symmetric, it is oblate from the lateral side at φ=0 and φ=180 but oblateness is larger at the side φ=180.

43

To provide a smooth view of dynamic deformations over the full surface, an interpolation was done at 101 height levels of the LB. The initial four points with a step of π/2 radians were transformed into 40 points with a step of π/20 radians by spline interpolation. Then, the map of surface deformation was built for each time instant as shown in Fig. A5.10 (a).

(a)

(b)

(c)

(d)

44

A series of consecutive snapshots of the reconstructed map of deformations made with a time step of 10-3 s is shown in Fig. A5.10. The acquisition frequency in the given experiment was not sufficient for producing a continuous time sweep of the process but allowed obtaining a series of snapshots of the bridge gas-liquid interface. An estimation of the natural frequency of surface waves was done with the assumption that the liquid bridge length (height) can accommodate from half to one surface wave. According to this estimation, the used acquisition frequency can allow getting from 1 to 4 images per period of the wave, which is surely not enough. Due to this fact, it is too early to draw definite conclusions about the types of vibrational modes caused by the gas flow. To increase the acquisition frequency up to 500 snapshots per second (the maximum capacity of the used cameras), more powerful light sources have to be used. In addition, experiments with a smaller gas velocity may provide a smaller frequency of surface waves.

These tests have shown that the developed optical technique allows reliable identification of the liquid bridge surface deformation with a magnitude of less than 1 micron (1/10 of pixel size). The maximum deformations of the bridge in the presented experiment are achieving ± 10 microns. In the previous experiments, sub-micron deformations of the free surface by a system with high magnification and a small field of view were measured by Ferrera et al. [34]. That system needed multiple scanning in the vertical direction and matching of images to cover one complete profile of the liquid shape. The present set-up allows instant tracking of four profiles over the full LB height with a minor reduction of accuracy.

To check the consistency of interface tracking, the volume of the liquid bridge was integrated over four reconstructed profiles at every time instant. The calculated volume demonstrates some minor frame-to-frame oscillations of 0.04% magnitude, which roughly corresponds to uncertainty of shape detection.

45

A6. Stability of convection driven by thermocapillary and

buoyant forces

A6.1. Variety of experimental procedures

In ground experiments with non-uniformly heated interface the convective flow is driven by the combined effects of buoyancy and thermocapillary forces. Experimental evidence that the critical conditions for the onset of oscillatory flows in ground conditions strongly depend on the interface shape was suggested by Hu et al. [55] who carried out experiments with 10 cSt silicone oil (Pr≈108). The stability diagram (∆Tcr vs Volume)

consists of two branches which formally can be assigned to small and large volumes or slender and fat liquid bridges. For a liquid volume roughly corresponding to a cylindrical interface the critical ∆T has a peaked maximum indicating a very stable flow. In experiments this "peak" is transformed into a "gap", the width of which depends on how large values of ∆T can be achieved in the experiments. For silicone oil with viscosity 5cSt and higher, the peak occurs above ∆T~60K [59], [74] and due to experimental constraints the "peak" is transformed into an open "gap". The experiments with silicone oils of 1-2 cSt [74] showed that the peak maximum between branches drops down to ∆T≈10K or less with decreasing of the viscosity. However, these silicone oils are volatile and the previous experiments did not compensate the loss of a liquid volume. The evaporation of the liquid from the interface, even weak, may modify the instability threshold.

The purpose of this section is to study the stability of weakly evaporating n-Decane under strict control of liquid volume change due to evaporation. The physical properties of n-Decane are listed in Table A3.1. To do so, the images of the liquid bridge were captured every second and the volume of the liquid bridge was also calculated every second. These data were used by the program of PID-control for compensation of the volume of evaporating liquid (stabilization of the liquid bridge volume). This program is commanding the syringe pump regulating the volume of liquid. If not stated otherwise the mean temperature of the liquid was kept at Tmean=25°C. Each experiment consisted of the

following steps:

• The rods are carefully cleaned with fresh working liquid and, if necessary, coated with anti-wetting agent.