Daniel Communal

Mr Patrick Bonnay

Frédéric Diogo Lopes

La Rizza

Josiana Afonso

Stephan Garcia

Aurelia Charrier

Srinivas Kota

Santhosh Onkaraiah

Rajasekhar Madugundo

Axay Rawal

Miral Rawal

Prof Mr Stephan Bernard Fauve

Prof Zappoli

Jerome

I take this opportunity to thank one and all, who helped me achieve my goals and aspirations in the limited time frame of 3 years of my PhD. This work would not have been possible without strong diligence and strict discipline coupled with smart thinking and constant mentoring. Firstly, I thank that ultimate source of power and knowledge, that hidden Self which pervades

Chapter 1

Introduction

The never ending quest of man to unravel the secrets of science has kept himself questioning about everything around him. Two aspects continuously drove this innate instinct of man: the necessity and the curiosity. While many of the human inventions came out of his sheer necessity to surmount various problems of the day to day life, there are many more discoveries or inventions that came into existence due to his innate curiosity. It is this intense curiosity of man that tried to answer various intriguing questions like: what is universe? How did it come into existence? Where do we come from? Can any other planetary object in space support life like Earth does? The field of space research came into existence in the process of finding answers to these questions. This led to the development of advanced space systems, most notable being the space stations, Earth and planetary observation systems, interplanetary probes, landing rovers etc. Successful development of these systems necessitated extensive experimentation (using sounding rockets, balloon experiments etc.) to test various engineering and technological hypotheses and models. Born with the first set of space station missions, parabolic flights or sounding rockets experiments, the scientific study of the physical phenomena in zero gravity has become an important element of space research carried out by the scientific community. It did not take much time for this field of research to evolve into a highly prestigious and one of the most sought after scientific fields. Microgravity experiments onboard the classical zero-g instruments are generally considered to be high profile missions and often involve cooperation between various international teams, given their high cost of experimentation. The experimental setups developed by CEA-Grenoble to cancel gravity forces in fluids of great interest to the space industry viz., Hydrogen and Oxygen, operate on the principle of magnetic levitation and have till now proven to be very efficient in creating zero-g and at the same time highly cost effective.

Weightlessness has a strong influence on the behavior of fluids in which it wipes out the effects of hydrostatic pressure, sedimentation and convection. The free surfaces of liquids in zero-g are no more horizontal, the effects of capillary forces being predominant. Zero-gravity study of fluids enables us to gain a better understanding of physical phenomena which are otherwise masked or, in some cases, deformed by the presence of Earth's gravity. Weightlessness is a great instrument to acquire knowledge about phenomena that would be otherwise impossible in the normal terrestrial conditions. An ideal example of such a phenomenon is "piston effect" which is a mechanism of heat transfer that has been discovered only because the gravity effects have been canceled.

Objective of the thesis The present thesis work is dedicated to the study of fluid phenomena subjected to zero and variable gravity conditions. It deals with mainly three problems: (i) the study of the interaction of harmonic vibration with the thermal boundary layer of a supercritical fluid under the absence of gravity, (ii) the study of the interaction of vibration with the liquid-vapor interface of a near-critical fluid under various gravity levels (Faraday and frozen wave instabilities, dynamic equilibrium of the interface) and (iii) the study of the geysering phenomenon inside a reservoir partially filled with a liquid when it is subjected to a brusque variation of gravity. The objectives of the present thesis are partly experimental and partly numerical in nature. Experiments are carried out onboard the setups HYLDE (HYdrogen Levitation DEvice) and OLGA (Oxygen Low Gravity Apparatus) using H 2 and O 2 respectively as the working fluids. The experimental investigation is carried out both in the subcritical and supercritical regions. Numerical simulations are carried out to study the stability of a thermal boundary layer in the supercritical zone using an in-house code that resolves compressible Navier-Stokes equations using a finite volume method based on the SIMPLER algorithm. Numerical simulations are also carried out to study the interface dynamics of a fluid interface subjected to brusque variation of gravity using an academic code named THETIS that is based on the VOF-PLIC algorithm.

The thesis report is organized as follows. Firstly, some basics of the near-critical fluids and the principle of magnetic levitation are presented in chapters 2 and 3, followed by a description of the experimental setups and numerical methods used to carry out the present study in chapters 4 and 5, respectively. Chapter 6 presents the numerical study of the stability aspects of a supercritical fluid subjected to vibration under zero-g. Chapters 7 and 8 deal with the stability aspects of a liquid-vapor interface of a near-critical fluid under various gravity levels and the dynamic equilibrium of a liquid-vapor interface under various gravity levels.

In chapter 9, study of geysering observed in rising large vapor bubbles is presented. Finally, chapter 10 gives the concluding remarks and defines the scope for future studies.

Introduction (Français)

La plupart des inventions et des découvertes scientifiques de l'homme ont été motivées par la nécessité de surmonter les divers problèmes de sa vie quotidienne. Cependant, certaines ne sont que la conséquence de sa curiosité innée. C'est cette intense curiosité qui le pousse sans cesse à répondre à des questions singulières, comme celles-ci: quelle est l'origine de l'univers? Quel est sa taille? Y a-t-il une vie sur d'autres planètes? Le domaine de la recherche spatiale est né dans l'idée de trouver des réponses à ces questions. Cela a conduit au développement de systèmes complexes, les plus notables étant les stations spatiales, les systèmes d'observation de la terre, les sondes interplanétaires et les rovers. Le développement de ces systèmes a nécessité une vaste expérimentation (à l'aide de fusées ou ballons sondes, vols paraboliques etc...) pour tester les diverses hypothèses technologiques et les modèles d'ingénierie. Née avec la première série de missions de la station spatiale, des vols paraboliques ou des expériences de fusées-sondes, l'étude scientifique des phénomènes physiques en apesanteur est devenue un élément important de la recherche spatiale menée par la communauté scientifique. Il n'a pas fallu beaucoup de temps pour que ce domaine de recherche évolue rapidement et devienne très prestigieux. Les expériences en microgravité à bord des instruments classiques "zéro-g" sont généralement considérées comme des missions de haut niveau et impliquent souvent une coopération entre diverses équipes internationales, compte tenu de leur coût élevé. Les dispositifs expérimentaux développés par le CEA Grenoble pour annuler les forces de gravité dans l'hydrogène et l'oxygène, deux fluides de grand intérêt pour le domaine spatial, fonctionnent sur le principe de la lévitation magnétique. Ces dispositifs permettent des expérimentations sur de petits volumes avec des coûts réduits. La gravité terrestre influence fortement le comportement des fluides. En absence de celle-ci, la pression hydrostatique devient nulle, il n'existe plus de sédimentation ni de convection naturelle. La surface libre des liquides n'est plus horizontale car l'effet des forces capillaires devient prédominant. Les études des fluides en apesanteur nous permettent d'acquérir une meilleure compréhension des phénomènes physiques qui peuvent être masqués ou, dans certains cas, déformés par la présence de la gravité terrestre. L'exemple typique d'un tel phénomène est "l'effet piston", un mécanisme de transfert de chaleur qui a été récemment découvert seulement parce que les effets de la gravité avaient été annulés. L'étude du comportement des fluides en apesanteur est également un sujet de grand intérêt pour l'industrie spatiale. Les fluides sont en effet utilisés dans différents domaines comme les moteurs cryogéniques (combustion, systèmes de pressurisation, alimentation), les systèmes optiques (systèmes de refroidissement de caméras à haute résolution), les systèmes de chauffage et de refroidissement, les réservoirs etc. . . Une bonne compréhension du comportement des fluides en apesanteur est capitale pour la conception et l'optimisation de ces systèmes.

Objective de la thèse La thèse est consacrée à l'étude des fluides lorsqu'ils sont soumis à des conditions d'apesanteur, de pesanteur partielle ou d'accélération variable. Les trois problèmes suivants sont traités: (i) Etude de l'interaction de vibrations harmoniques avec une couche limite thermique d'un fluide supercritique en absence de gravité, (ii) Etude de l'interaction de vibrations avec une interface liquide/vapeur d'un fluide sous-critique sous plusieurs niveaux de gravité, (iii) Etude du phénomène de geyser à l'intérieur d'un réservoir partiellement rempli d'oxygène lorsqu'il est soumis à une variation rapide de la gravité (ou accélération). Pour ces 3 points, la thèse comporte une partie expérimentale et une partie numérique. Des expériences ont été réalisées sur les installations HYLDE (HYdrogène Levitation DEvice) et OLGA (Oxygen Low Gravity Apparatus) du CEA Grenoble utilisant respectivement les fluides H 2 et O 2 , dans la zone sous-critique. Des simulations numériques sont réalisées pour étudier la stabilité d'une couche limite thermique dans la zone supercritique en utilisant un code numérique qui résout les équations de Navier -Stokes en utilisant l'algorithme SIMPLER. Des simulations numériques sont également effectuées pour étudier la dynamique d'une interface fluide soumise à une variation rapide de la gravité en utilisant un code académique nommé THETIS, basé sur l'algorithme VOF-PLIC. Le rapport de thèse est organisé comme suit: dans les chapitres 2 et 3 sont présentées quelques notions de base concernant les fluides proches de leur point critique et le principe de la lévitation magnétique. Dans les chapitres 4 et 5 on trouvera une description des dispositifs expérimentaux et des méthodes numériques utilisées pour réaliser la présente étude. Le chapitre 6 présente l'étude numérique des conditions de stabilité d'un fluide supercritique soumis à des vibrations en microgravité. Les chapitres 7 et 8 traitent des conditions de stabilité d'une interface liquide-vapeur d'un fluide sous-critique sous différents niveaux de gravité et soumis à des vibrations. Le chapitre 9 présente l'étude de geysers observés dans une cellule partiellement remplie d'oxygène lors d'une variation rapide de gravité. Enfin, le chapitre 10 donne les conclusions et définit des perspectives pour les études futures.

Chapter 2

Basics of near-critical fluids

Resumé (Français)

Le comportement thermique et hydrodynamique particulier des fluides proches de leur point critique est dû aux propriétés spéciales du point critique. Proche du point critique, les propriétés physiques d'un fluide peuvent être représentées par des lois de puissance universelles valables pour tous les fluides. Des propriétés comme la tension de surface d'une interface liquide-vapeur, la différence de densité entre les phases liquide-vapeur, la diffusivité thermique etc. tendent vers zéro et des propriétés comme le coefficient d'expansion thermique, la compressibilité, la chaleur spécifique etc. tendent vers l'infini. C'est cette particularité des fluides critiques ou presques critiques qui rend leur étude extrêmement intéressante. L'expérimentation avec des fluides proches de leur point critique présente des avantages. Par exemple, la diffusivité thermique qui tend vers zéro rend les phénomènes thermiques très lents et donne alors à l'expérimentateur plus de temps pour les étudier (c'est le fameux "ralentissement critique"). Aussi, comme les valeurs de la tension de surface et de la différence de densité des phases liquide-vapeur changent beaucoup quand la température du fluide est modifiée, il est possible d'étudier les phénomènes physiques pour plusieurs combinaisons de propriétés. Le but de ce chapitre est de présenter succinctement les propriétés des fluides proches de leur point critique.

The thermodynamics and hydrodynamics of a fluid close to its critical point has been for long (and still is) a very rich subject of investigation. As one approaches the critical point, all fluids behave anomalously resulting in interesting physical phenomena. The main objective of the present section is to give a brief description of this unique nature of the near-critical fluids and some remarkable thermodynamic aspects of pure substances close to the liquid-vapor critical point.

Phase diagram of the pure substances

A pure substance is one that has a homogeneous and invariable chemical composition. Pure substances exist in mainly three phases: solid, liquid and gas. The equilibrium between two or more of these phases can be completely described by specifying the three principal thermodynamic parameters: pressure (p), temperature (T ) and density (ρ) of the substance. A PVT (pressure -volumetemperature) diagram is a three dimensional diagram indicating the existence of the pure substance in any of the three phases. Figure 2.1a shows the typical phase diagram of a pure substance. It shows, in a 3D diagram, the regions where two or more phases can coexist. The projection of the PVT diagram on the PT (pressure-temperature) plane gives the PT diagram (Fig 2 .1b). Figure 2.1b shows various temperature and pressure domains in which the pure substance can exist in a single phase (solid, liquid or gas), coexist in two phases (solid-liquid, solid-gas, liquid-gas), or coexist in all the three phases (solid-liquid-gas : the triple point). Gibb's phase rule for pure substances (F=3-P) gives the number of independent variables determining the thermodynamic equilibrium between the three phases; here, F is the number of degrees of freedom, P is the number of phases. The coexistence of the three phases: solid, liquid and gas, for which P = 3, has zero degrees of freedom. This corresponds to the triple point in the PT diagram. The coexistence of two phases: solid/liquid, liquid/gas or gas/solid, corresponding to P = 2, has one degree of freedom. That is, one independent parameter (either pressure or temperature) is sufficient to determine the coexistence of the pure substance in two phases. The various pressure and temperature values for which the substance can coexist in two phases constitute the coexistence curve on the PT diagram. In the Fig 2 .1b, the lines denoted by S+G, S+L and L+G correspond to the three coexistence curves. The three coexistence curves intersect at the triple point on the phase diagram. The liquid-vapor coexistence curve, also called the liquid-vapor saturation curve is bounded on its extremities by the triple point (except for helium) and the critical point as can be seen in the Fig. 2.1b. The critical point corresponds to the thermodynamic conditions beyond which there is no distinction between a [START_REF] De La Tour | Exposé de quelques résultats obtenu par l'action combinée de la chaleur et de la compression sur certains liquides, tels que l'eau, l'alcool, l'éther sulfurique et l'éssence de pétrole rectifiée[END_REF]. While listening to discontinuities in the sound generated by a rolling flint ball in a sealed cannon filled with carbon dioxide (CO 2 ) at various temperatures, he observed the disappearance of the distinction between the liquid and the vapor phases beyond a particular temperature, which he named as the critical temperature. The critical point of a fluid is defined by its critical temperature T c , critical pressure p c and critical density ρ c . In the Fig. 2.1b beyond the critical point, the fluid exists in a single state called supercritical state. In the supercritical domain the fluid behaves like a gas with the density of a liquid. The table 2.1 lists the critical points of some pure substances.

Critical power laws and universal scaling

As one approaches the critical point, thermodynamic parameters and transport coefficients of fluids either diverge or tend to zero. For example, properties like thermal expansion coefficient, thermal conductivity, isothermal compressibility,

β p (K -1 )
1.1 × 10 -2 ε -1.24 χ T (Pa -1 ) 5.8 × 10 -8 ε -1.24 ∂p/∂T (Pa.K -1 ) 1.8 × 10 5 c v (J.kg -1 .K -1 ) 1.5 × 10 4 ε -0.11 -1.2 × 10 4 c p (J.kg -1 .K -1 )

2.1 × 10 3 ε -1.24 D T (m 2 .s -1 ) 5.1 × 10 -8 ε 0.67 ν(m 2 .s -1 )

1.5 × 10 -7 ε -0.04 Λ(W.m -1 .K -1 ) 3.3 × 10 -3 ε -0.567 specific heat capacities etc., diverge, while properties like liquid-vapor density difference, surface tension of the liquid vapor interface, thermal diffusivity etc., tend to zero. According to [START_REF] Zappoli | Supercritical fluid hydrodynamics[END_REF], the divergence or convergence of the thermodynamic parameters and the transport coefficients is due to the long range contribution from the intermolecular correlation function, called the critical contribution.

There exists also a background contribution from the molecular short range part. Thus the general behavior of the properties is obtained by fitting experimental results to a law with a constant background term and a temperature dependent critical contribution term. Thus a given quantity Y expressed in terms of the reduced thermal proximity to the critical point ε defined as:

ε = | (T -T c ) T c | (2.1)
is given as

Y (ε) = Y c (ε) + Y reg (ε)
where, Y c is the critical contribution and Y reg is the regular or background contribution. While the regular or the background part is expressed as a polynomial function according to:

Y reg (ε) = B 0 + B 1 T c + B 2 T 2 c + . . . , (2.2)
the critical contribution Y c (ε) is expressed as a scaled power law multiplied by a crossover function according to

Y c (ε) = ψ Y 0 ε ±ψ (1 + a (1)
Y ε ∆ + a (2)
Y ε 2∆ + . . .)

Here, ψ Y 0 is the critical amplitude, ψ is a universal exponent and (1 + a

Y ε ∆ + a (1)
Y ε 2∆ + . . .) is the crossover function with ∆ (≈ 0.502), a universal exponent. However in most cases the higher order terms in the crossover function are negligible. Thus in general the fluid properties can be expressed as:

Y (ε) = B 0 + ψ Y 0 ε ±ψ , (2.4)
a relation valid close to the critical point. The table 2.2 lists some important properties for n -H 2 in its supercritical state.

Critical slowing down and piston effect

Near-critical fluids, owing to their unique properties, display more interesting dynamics than compared to normal fluids. Given that the thermal diffusivity tends to zero as the critical point is approached, all heat transfer processes slowdown very close to the critical point. This phenomenon is called critical slow down, which gives a nice opportunity to the experimentalists to observe interesting phenomena evolving at a very slow rate close to the critical point. It was first believed that the effect of the critical slowing down increases as the critical point is approached. This general belief was however contradicted by Straub's [START_REF] Nitsche | Die isochore warmekapazitat am kritischen punkt unter reduzierter schwere[END_REF] sounding rocket experiments that indicated an unusually faster heat transfer close to the critical point. It was found in the experiments that the bulk temperature of the fluid followed the wall temperature of the cell without much delay. It was not until 1990 that this phenomenon was explained, by three independent teams Onuki et al. [START_REF] Onuki | Fast adiabatic equilibration in a single component fluid near the liquid-vapor critical point[END_REF], Boukari et al. [START_REF] Boukari | Critical speeding up observed[END_REF] and Zappoli et al. [START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF]. It was discovered that a 4 th mechanism of heat transfer called the piston effect is responsible for the anomalous fast heat transfer. Piston effect is the adiabatic heating of the bulk of the fluid very close to the critical point due to the hyper-compressible nature of the fluid. This mode of heat transfer is seen to be dominant close to the critical point (where the thermal diffusivity is negligibly small and isothermal compressibility is extremely large) under the absence of gravity forces (where the convective heat transfer is completely eliminated). The mechanism of the piston effect can be explained as follows: Consider a thermally insolated container filled with a supercritical fluid under zero-gravity conditions. If a heat pulse is applied on one side of the container, the fluid close to the heated side will expand without heating the bulk of the fluid, owing to its hyper-compressibility and vanishing thermal diffusivity. This expansion creates a longitudinal wave that propagates through the bulk fluid at the speed of sound.

The high density and highly compressible supercritical fluid converts some of this kinetic energy into thermal energy, heating the bulk uniformly each time the wave passes. In the absence of the gravity field, the dominant thermalization process is the conversion of the density wave into uniform heating. Since the time for a sound wave to traverse a small sample (∼ centimeters) is much shorter (∼ ms) than the time for the sample to come to equilibrium through diffusion (∼ hours or months), this process can be considered as adiabatic.

Conversely, if a wall is cooled, the fluid close to it will contract and cause adiabatic cooling throughout the bulk fluid. The mechanism of wave propagation resembles to a piston that compresses gas in basic thermodynamics, hence the term "piston effect". It can operate at time scales, orders of magnitude smaller than the classical hydrodynamic time scales when the fluid is sufficiently close to its critical point. The piston effect time scale is defined as:

t P E = L 2 (γ -1) 2 D T (2.5)
while the diffusion time scale is t D = L 2 D T , where γ is the ratio of specific heats, L is the length scale, D T is the thermal diffusivity. As one approaches critical point, γ diverges with a critical exponent of 1.13 while D T tends to zero with a critical exponent of 0.67, the net effect being a drastic decrease in the piston effect time scale thus resulting in the speeding up of the heat transfer processes.

Conclusion

It has been seen in this chapter that near-critical fluids have unique properties and that they can be described using universal scaled power laws. Working with near-critical fluids is interesting keeping in view the variability of their properties by just heating or cooling it close to the critical point. In the present thesis two near-critical regions are of interest: subcritical region where the fluid problem involves a liquid-vapor interface and supercritical region where the fluid exists in a single phase. Study of the stability aspects of a near-critical fluid is the main subject of research in the thesis work.

Chapter 3

Zero-gravity experimentation & magnetic levitation

Resumé (Français)

Les conditions d'apesanteur ou de microgravité sont un véritable instrument pour mieux comprendre les phénomènes physiques. L'absence de gravité annule en effet la convection naturelle et la stratification de densité dans les fluides. Les interfaces des fluides ne sont plus horizontales car les effets capillaires sont prépondérants en absence de gravité. C'est cette particularité de l'expérimentation en absence de gravité qui a permis de découvrir des phénomènes cachés comme l'effet piston. Le but principal de ce chapitre est de présenter les instruments classiques de la microgravité. Les bases du principe de lévitation magnétique qui est la méthode utilisée pour mener les expériences de cette thèse sont ensuite présentées.

Necessity of zero-gravity conditions

It has already been discussed in Chapters 1 and 2 that a fluid becomes more and more unstable as one goes closer and closer to the critical point. That is, one can destabilize a near-critical fluid much more easily than a normal fluid and thus carry out experimental study of these instabilities to gain a deeper understanding of them. However, experimentation close to the critical point needs zero-gravity conditions because of the hyper-compressible nature of the near-critical fluid. Close to the critical point, under normal gravity conditions, the fluid can get compressed under its own weight, even on a cm or a mm length scale, leading to strong density stratification. Zero-gravity is required to eliminate the effect of the density stratification on the dynamics of a fluid.

Weightlessness

Two bodies separated by a distance exert gravitational attractive forces on each other. The gravitational force applied by each body on the other is directly proportional to the product of the individual masses of the two bodies and is inversely proportional to the square of the distance between them. It is directed towards the other body with which the body interacts. More generally, the gravitation force acting on a body is a vector sum of all the gravitational attractive forces exerted by all the other bodies with which it interacts. A body close to a huge massive body like the Earth experiences a net gravitational force towards the center of the Earth since the contribution of the other bodies is negligible compared to the Earth's gravitational force. The term "weight" of a body close to the Earth's surface is thus popularly associated with the Earth's gravitational force. Weightlessness corresponds to the cancellation of the gravitational force by other types of forces. Since gravitational force is a body force acting on each and every particle of the body, to nullify it an equal and opposite amount of force should be applied on each and every particle of the body. There are mainly two methods of creating zero-g: inertial methods and non-inertial methods. Various classical methods of creating zero gravity conditions are presented in Fig. 3.1. All the instruments shown in Fig. 3.1 are based on the inertial method. Drop towers, parabolic flights and sounding rockets allow the test article to fall freely under the terrestrial gravity thereby applying a pseudo force on each particle of the test article equal to its weight in a non-Newtonian frame of reference fixed to it. In launch vehicles and space stations, the test article orbits around the Earth at a constant speed such that the centrifugal force due to its rotation is equal to its weight in a reference frame fixed to it. All these instruments can provide different zero-g durations and have different costs of experimentation. The typical zero-g durations for these classical modes of zero-g are ∼ 10 s (drop towers in catapult mode), ∼ 30 s (parabolic flights), ∼ 10 mins (sounding rockets), ∼ days (launch vehicles) and ∼ months (space stations). These testing methods can be exorbitantly expensive given the involved complexities. For example a typical campaign of a parabolic flight or a sounding rocket can cost some millions of Euros while experimentation onboard a launch vehicle or the space station can cost some tens of millions of euros. There are also severe safety and security constraints on the fluids utilizable in fluid dynamic problems. The non-inertial methods of creating zero gravity conditions need active application of other kinds of forces on the test article. Application of electrical or magnetic fields or a combination of both can be carried out to compensate gravity forces. CEA Grenoble has developed two instruments HYLDE and OLGA based on the principle of magnetic levitation to levitate Hydrogen (H 2 ) and Oxygen (O 2 ) respectively. These setups are used to carry out experiments on H 2 and O 2 and have practically infinite duration of zero-g. The details of the principle of magnetic levitation and its advantages and limitations are discussed in the following section.

Magnetic levitation

Magnetic induction and magnetization

When an external magnetic field H is applied to a material, the material induces its own magnetic field due to the interaction of the atoms of the material with the external magnetic field. This self-induced magnetic field is called magnetization denoted by M . The net magnetic field called magnetic induction B inside the material is a vector sum of the two fields. The relation between the three magnetic quantities is given by the relation:

B = µ 0 (H + M ) (3.1)
where, µ 0 (= 4π × 10 -7 T.m.A -1 ) is the magnetic permeability of vacuum. In SI units, B is expressed in Tesla (T) while M and H are expressed in Ampere per meter (A.m -1 ). The magnetization of the material is related to the applied magnetic field according to the relation:

M = χH (3.2)
thus resulting in the expression:

B = µ 0 (1 + χ)H (3.3)
where, χ is the magnetic susceptibility of the material. It can be positive or negative, depending on the magnetic properties of the material. The various kinds of magnetic materials are discussed in the following section.

Magnetic materials

An external magnetic field applied to a material can interact with it resulting in the induction of magnetic dipole moments inside it. The net magnetic dipole moment results in a net magnetization depending on a number of factors, such as the atomic and molecular structure of the material, and the net magnetic field associated with the atoms. The magnetic moments associated with the atoms have three origins: the electron motion, the change in motion caused by an external magnetic field, and the spin of the electrons. In most atoms, electrons occur in pairs. Electrons in a pair spin in opposite directions have zero net magnetic dipole moment. Alternately, materials with some unpaired electrons will have a net permanent magnetic dipole moment. Thus based on the interaction of a material to an applied magnetic field, most materials can be broadly classified as diamagnetic, paramagnetic or ferromagnetic. However other types of magnetic materials exist, like ferrimagnetic materials, anti-ferromagnetic materials etc. We concentrate in this section only on the three most important classes of magnetic materials. Diamagnetic materials have a negative net magnetization under an applied external magnetic field. Thus the diamagnetic materials are slightly repelled by a magnetic field. The material does not retain the magnetic properties when the external field is removed. In diamagnetic materials all the electron are paired and thus have zero permanent net magnetic moment per atom. Diamagnetic properties arise from the realignment of the electron paths under the influence of an external magnetic field. These materials are associated with a very small and negative magnetic susceptibility. Another important characteristic of these materials is that their susceptibility does not change with temperature. Most elements in the periodic table, including copper, silver, and gold, are diamagnetic. The most common fluids like hydrogen, water, carbon dioxide etc. are diamagnetic in nature.

Paramagnetic materials have a small, positive magnetic susceptibility to the applied external magnetic field. Thus these materials are slightly attracted by a magnetic field and the material does not retain the magnetic properties when the external field is removed. Paramagnetic properties are due to the presence of some unpaired electrons, and from the realignment of the electron paths caused by the external magnetic field. Magnesium, aluminum, titanium, molybdenum, etc. are some examples of paramagnetic materials. The fluid of interest in the present context, Oxygen, is paramagnetic in nature. Interestingly the magnetic susceptibility of the paramagnetic materials changes quite remarkably with temperature.

Ferromagnetic materials have a large, positive magnetic susceptibility to an external magnetic field. They exhibit a strong attraction to magnetic fields and are able to retain their magnetic properties after the external field has been removed. Ferromagnetic materials have some unpaired electrons so their atoms have a net magnetic moment. They get their strong magnetic properties due to the presence of magnetic domains. When a magnetizing force is applied, the domains align with the field to produce a strong magnetic field within the part.

Magnetic compensation of gravity

Magnetic forces can be applied on a substance by subjecting it to a strong magnetic field, exploiting the diamagnetic or paramagnetic nature of the substance. The magnitude of the magnetic force F m depends on the applied magnetic field and the magnetic susceptibility of the substance according to the relation [START_REF] Quettier | Magnetic compensation of gravity forces in liquid/gas mixtures: surpassing intrinsic limitations of a superconducting magnet by using ferromagnetic inserts[END_REF]

F m = χ 2µ 0 ∇(B 2 )V (3.4)
where, V is the volume of the substance. It is convenient to define G = ∇(B 2 ). If the applied magnetic field is strong enough this magnetic force can completely compensate for the gravitational force F g of the substance (weight). Thus the necessary condition for the total compensation of gravity is given by where, χ ρ = χ/ρ is the specific magnetic susceptibility. Thus the necessary condition for total compensation of gravity is given by

F m + F g = 0 (3.5) that is, g + χ ρ 2µ 0 G = 0 (3.6)
G 0 = - 2µ 0 χ ρ g 0 (3.7)
Here, G 0 is the value of G required for total compensation and g 0 is the gravitational acceleration. Table 3.1 shows the values of G 0 for various substances.

It can be seen from the Table 3.1 that to levitate H 2 and O 2 , G 0 values of -991.4 T 2 .m -1 and 8.156 T 2 .m -1 (at 90 K) respectively are required. The value of G 0 for total gravity compensation in Oxygen however changes with temperature (since the magnetic permeability of a paramagnetic material is temperature dependent).

Strong magnetic fields are needed to create the required G 0 inside a cell of a given size. The magnetic fields can either be generated using resistive coils, which consume huge amounts of electricity, or by using superconducting coils, which require cooling to cryogenic temperatures. The simplest coil configuration that could be used to generate a strong magnetic field is a cylindrical coil. But it was shown by Quettier et al. [START_REF] Quettier | Magnetic compensation of gravity forces in liquid/gas mixtures: surpassing intrinsic limitations of a superconducting magnet by using ferromagnetic inserts[END_REF] that using a simple cylindrical configuration of a coil total compensation of gravity can be obtained only at a single point. That is, the resulting (B 2 ) field from a simple cylindrical coil is not a uniform field indicating the existence of a residual gravitational field. The non-homogeneity of non-dimensional residual gravitational field ǫ is defined as

ǫ( r) = G( r) -G 0 G 0 = g * ( r) g 0 (3.8)
where, g * is the residual gravity field and r is the spatial coordinate. Figure 3.2

shows the variation of B and B.dB/dz for a typical magnetic coil. Here, B.dB/dz represents the contribution of the magnetic field towards the compensation of the gravity (which is along the negative z direction). It can be seen that for a cylindrical coil of finite length, the magnetic field attains a maximum value at the center of the coil. It can be observed that B.dB/dz is not uniform. This indicates that the compensation of gravity is not perfect and that there exists a residual gravity field. The variation of |B.dB/dz| has two maxima denoted by the red and the blue circles. Since the diamagnetic materials need a negative B.dB/dz, to compensate the gravity forces they need to be positioned at the red circle (top portion of the coil). Similarly, since the paramagnetic materials need positive B.dB/dz, they can be levitated at the position indicated by the blue circle (bottom of the coil). The main advantage of the magnetic levitation instrument is the cost of experimentation. It is highly cost-effective and provides zero-g conditions for very long times. However these setups have some limitations: (i) total compensation of gravity is possible only at a single point and thus there exists a residual gravity field, (ii) the volume of fluid that can be levitated is limited and is a compromise with the uniformity of the gravity field sought for.

Conclusion

The chapter aims at creating a clear background to the micro-gravity research. It presents a brief description of the various zero-gravity instruments. It is seen that instruments based on magnetic levitation can be used as a cost-effective means of creating zero-g conditions and have capability to replace, partly, experiments on-board the classical experimental setups.

Chapter 4

Experimental setups

Resumé (Français)

Le Service des Basses Températures du CEA-Grenoble a développé des instruments de zérogravité basés sur le principe de la lévitation magnétique. Ces instruments s'appellent HYLDE (Hydrogen Levitation DEvice) et OLGA (Oxygen Low Gravity Apparatus). Ils permettent d'annuler les forces de gravité dans l'hydrogène et l'oxygène, respectivement. Ces instruments sont utilisés pour réaliser les expériences effectuées dans cette thèse. Le but de ce chapitre est de décrire ces instruments ainsi que les cellules expérimentales utilisées.

Introduction

The "Services des Basses Températures" (SBT) at CEA Grenoble has developed two experimental installations: HYdrogen Levitation DEvice (HYLDE), for Hydrogen and Oxygen Low Gravity Apparatus (OLGA), for Oxygen, to carry out zero gravity experimentation. Both these installations work based on the principle of magnetic levitation. To create the magnetic field required to compensate the gravity forces (10 T for H 2 and 1 T for O 2 ), these installations use superconducting coils made of Nb-Ti which need to be maintained at liquid helium temperature. The two installations are similar in construction except for some minor differences. Figure 4.1 shows the schematic diagram of a typical magnetic levitation device.

As can be seen in the Fig 4 .1, the device is essentially a cryogenic facility. It consists of a cryostat in which is mounted a superconducting coil made of Niobium-Titanium (Nb-Ti). The cryostat consists of a liquid nitrogen jacket (not shown in the figure) thermally isolated from the external and internal en-vironments using two jackets of vacuum. This serves to reduce, to minimum, radiation heat transfer to the outside environment. Liquid helium at 4.2 K is filled inside the cryostat to cool the superconducting coil below its critical temperature. The level of the liquid helium filled inside the cryostat can be monitored using a level sensor. Additional cooling system can be provided to further reduce the temperature of the liquid helium based on the working conditions and the critical temperature of the coil. The temperature of the coil can be monitored using carbon or platinum based thermometers mounted at various positions on the coil. The experimental cell is mounted inside a cylindrical vessel called the anticryostat. The anticryostat is maintained under vacuum to thermally isolate the cell. The position of the anticryostat with respect to the position of the superconducting coil can be adjusted using a motor mounted on it. Two endoscopes, one for the light source and the other for the camera are mounted inside the anticryostat with a provision to independently adjust their positions with respect to the cell. [START_REF] Cowley | The interfacial stability of a ferromagnetic fluid[END_REF]. It is well known that the surface of a ferromagnetic fluid becomes corrugated when the magnetic induction B exceeds a threshold value B c ∼ (σg∆ρ) 1/4 where, σ is the surface tension, g is the gravitational acceleration, ∆ρ is the density difference between the liquid and the vapor phases. Close to the critical point of oxygen(a paramagnetic substance with a positive magnetic susceptibility), the values of surface tension and liquid vapor density difference vanish thus reducing considerably the value of B c and leading to the formation of the Cowley-Rosenweig instability. In the presence of these instabilities it is not possible to carryout meaningful fluid dynamics experiments.

HYLDE (HYdrogen Levitation Device)

The HYLDE setup has been operational since 1995. It was initially designed to study the filling of targets with Deuterium (D 2 ) used for inertial confinement fusion (ICF). It is later recovered by CEA Grenoble to carry out scientific experiments to study the fluid dynamic and heat transfer problems under weightlessness. The details of the setup are as described below: The coil is powered using a superconducting magnet controller which can provide a DC power of 100 A and 5 V. The coil is energized at a very slow rate, 0.025 A.s -1 when going from 0 to 50 A, 0.012 A.s -1 from 50 to 60 A and 0.008 A.s -1 from 60 to 70 A. In order to protect the coil from quenching (the sudden transformation of the coil from superconducting state to resistive state), a heat discharge coil made of stainless steel is connected in parallel to the superconducting coil. Thus in the case of quenching of the superconducting coil the entire current passes through the discharge coil thus protecting the superconducting coil from melting down. A safety valve provided on the cryostat releases the pressure rise inside the cryostat that can occur during a quench due to the sudden transformation of liquid helium into vapor. Important to note is that one liter of helium at 4.2 K expands to around 700 liters of vapor at normal temperature and pressure (NTP). Two carbon temperature sensors are provided, one on the top and the other on the bottom of the coil to monitor the coil temperature.

Cryostat

The coil is housed inside a cryostat made of stainless steel. It is filled with liquid helium to cool the superconducting coil to less than 4.2 K. In order to reduce the consumption of liquid helium, it is provided with a liquid nitrogen jacket isolated from the inner helium chamber and the outside environment by two vacuum linings. The liquid nitrogen is transferred from a 200 L nitrogen Dewar using an electro-valve which automatically fills the nitrogen jacket whenever the level of the liquid nitrogen drops below a certain level. At the boiling point of helium (4.2 K), the Nb-Ti coil has a critical current of about 50 A. Thus in order to be able to pass around 70 A through the coil, the coil needs to be cooled to 2.17 K. To cool down the liquid helium from 4.2 K to 2.17 K a refrigeration system based on the Roubeau ′ s bath is used. The schematic diagram of the Roubeau ′ s bath with a Joule Thomson (JT) valve is shown in Fig. 4.3a. The liquid helium (at 4.2 K) is extracted from hotter region of the cryostat and is precooled to a lower temperature (typically around 3.5 K) before being expanded using the JT valve. The expansion of the liquid helium into the refrigeration bath produces the required refrigeration effect. The flow through the JT valve can be controlled by using a manual flow control valve.

A compromise between the refrigeration power (eventually the time required to attain 2.17 K) and the helium consumption rate is achieved by adjusting the position of the valve. The Joule Thomson valve consists of a needle valve which can be raised or lowered (to control the size of the orifice) by using a knob. The JT valve provides a refrigeration power of around 1 W to 2 W at 2.17 K. At the optimum position of the valve, the temperature of helium liquid reduces from 4.2 K to 2.17 K in roughly 3 hrs.

Roubeau ′ s bath with a capillary tube

The needle valve is extremely delicate and should not be forced upon. In one of the occasions the needle valve broke due to excessive forcing and stalled the experiment. To replace the valve, the entire valve assembly needed to be redesigned and fabricated which involved a delay of around 3-4 months. To compensate for the lack of time a make shift alternative to the Joule Thomson valve was en-visaged and designed. The alternate solution (shown in the schematic diagram Fig. 4.3b) is the use of a capillary tube to produce refrigeration. The capillary tube is a long copper tube of very small internal diameter used as a throttling device to generate refrigeration. The working of the capillary tube is as follows: when liquid helium enters the capillary tube its pressure suddenly drops due to the very small diameter of the tube. The small diameter of the capillary tube has the similar effect as an orifice. The decrease in the pressure in the capillary tube is dependent on the capillary diameter and the length of the tube. The reduction in the pressure evaporates the liquid helium thus causing it to absorb heat energy in the form of latent heat of vaporization. A capillary tube 80 mm long with an inner diameter 0.3 mm was finally chosen for the purpose. This configuration theoretically produces a refrigeration power of 3.5 W and consumes around 4.5 l.hr -1 of liquid helium. Thus liquid helium can be cooled from 4.2 K to 2.17 K, in less than 2 hrs. Figure 4.4 shows the picture of the modified Roubeau s bath with the capillary tube. The JT valve is completely eliminated and instead a copper tube is brazed in its place. The capillary tube connects the outlet of the regeneration circuit to the refrigeration bath.

It can be seen that using the capillary tube, the system becomes simpler. However there are many inconveniences with a capillary tube. First of all, the capillary tube consumes a lot of liquid helium. Once the liquid helium cools down to 2.17 K, only a very small amount of refrigeration power is needed to maintain the system at that temperature (to account for the heat leak into the system by conduction or radiation). Since it is not possible to reduce the flow rate of the system, the system consumes more liquid helium than required. Secondly, there is more risk of blocking the capillary tube due to an unintended entry of air onto the system. If the capillary tube is blocked it needs at least 3 days to heat-up the system to the room temperature and to restart the experiment. However, due to unavoidable reasons this configuration of refrigeration is retained for one set of experiments. The use of capillary tube however does not affect the quality of the results of the experiments; it only increases the liquid helium consumption and the preparation time of the experimental setup. To create zero-g conditions, a 63.5 A current is passed through the coil. Figure 4.5 shows the variation of the longitudinal components of the magnetic field intensity B and G 0 inside the coil. It can be seen that to achieve total compensation of gravity using HYLDE, we need rougly 10 T magnetic field which gives the required G z , the longitudinal component of G 0 , of -991 T 2 .m -1 . As it was shown in Fig. 3.2 B z attains a maximum value at the center of the coil while the G z has the maximum negative value at the top of the coil (corresponding to the positive z value). This is the position where H 2 will be levitated. As has already been discussed in section 3.3.3, the residual gravity field is not uniform. The uniformity ǫ of the residual gravity field inside an experimental cell depends on the size of the cell and the position of levitation with respect to the coil. Figure 4.5c shows the uniformity of the gravity field inside the cell. It can be seen that for a square cell of side 3 mm and 7 mm the uniformity of gravity obtained is around 2% and 4%.

It was shown by Clement et al. [START_REF] Lorin | Magnetic compensation of gravity by using superconducting axisymmetric coils: Spherical harmonics method[END_REF] that the homogeneity of the gravity field can be improved by using ferromagnetic inserts. When an external magnetic field is applied to a ferromagnetic material, the ferromagnetic material produces its own magnetic field called magnetization which adds up to the applied magnetic field, thus modifying the effective magnetic field of the coil. Thus, by carefully designing and positioning a ferromagnetic insert inside the coil, one can modify the field and make the resulting G more uniform. A soft iron insert of internal and external diameters 64 mm and 90 mm and of length 40 mm was used to render the gravity field more uniform for bigger experimental cells. In the presence of soft iron insert, a current of 68.5 A is required to achieve total compensation of gravity.

OLGA (Oxygen Low Gravity Apparatus)

The OLGA setup was put in service in 2004 after a successful set of experiments onboard HYLDE and keeping in view the need to levitate larger volumes of liquids for carrying out scientific experiments. The Nb-Ti coil of OLGA, originally designed in 1978 to characterize superconducting wire of the Tore Supra coil installation at Cadarache, was recovered from CEA Cadarache in 2001, keeping in view its suitability to levitate O 2 . Figure 4.8 shows an image of the OLGA station and its super conducting coil. The details of the setup are given below:

Superconducting coil

To achieve gravity compensation in oxygen, OLGA uses two concentric cylindrical coils made of Nb-Ti powered independently by using two power supplies BOUHNIK and CERN. The external coil has an inner diameter of 414 mm and an outer diameter of 650 mm and is 555 mm long. It has 3996 turns. The internal coil is smaller with dimensions: 406 mm outer diameter, 336 mm inner diameter and 570 mm long. It has 990 Nb-Ti turns. Each of the coils is independently capable of compensating gravity forces in Oxygen. This configuration of concentric cylindrical coils can be exploited to carry out experiments with sudden change in gravity. This is done by maintaining the external coil at a desired current and discharging the current in the internal coil.

Since the sudden discharge of current in the internal coil induces strong eddy currents in the external coil, the power supply (BOUHNIK) of the external coil is designed to be capable of applying powerful negative voltage (up to -400 V) to maintain its current constant.

Cryostat

The cryostat of OLGA is made up of stainless steel and has almost the same configuration as that of HYLDE. It has a liquid nitrogen LN 2 jacket separated from the inner liquid helium chamber and the external environment using vacuum linings. LN 2 is filled inside the nitrogen jacket using electro-valves and control modules whenever the liquid level of nitrogen drops below a certain level. The nitrogen cooling jacket works on the principle of thermo-syphon. However the LN 2 jacket itself is not sufficient to cool down the OLGA setup to 80 K. Additionally, a LN 2 heat exchanger coil (shown in Fig. 4.8b) is used to cool down the magnetic coil to 80 K before filling the setup with liquid helium.

Zero-g and uniformity

Zero-gravity conditions can be achieved inside the cell by using only the external solenoid (energized with a current of 239.25 A) or by simultaneous use of both the solenoids. Various combinations of currents in the coils exist but we limit the present discussion to the combination with 186.6 A on the external coil and 170 A on the inner coil. It can be seen that to achieve a total compensation of gravity using OLGA, we need rougly 1 T magnetic field which gives the required G z of 8.156 T 2 .m -1 . Figure 4.9c shows the uniformity of the gravity field inside the cell. It can be seen that for a cylindrical cell of diameter 30 mm and height of 100 mm, a uniformity of 3% can be achieved at the bottom of the cell. The uniformity is however bad at the top of the cell (close to 20%). This would however not affect the results for the reason that will be discussed in chapter 9.

Experimental cells

The experimental cells used to carry out experiments inside the setups HYLDE and OLGA need to be rigorously designed paying keen attention to the thermal, structural and magnetic aspects of the cell. The experimental cell needs to work under severe space and thermal constraints. The following are some aspects that need to be kept in mind while designing the experimental cells.

The thermal expansion coefficients of the cell and its support and various other components need to be carefully selected to avoid unintentional thermal stresses and also to avoid unintentional tilting of the experimental cell due to non-uniform contraction of various components at low temperatures. Thus, care should be taken to select material combinations with similar thermal expansion coefficients.

In order to reduce the consumption of liquid helium during the experiments the heat leak into the system should be limited by optimally choosing the cross sectional area and the length of the supports. The design of the thermal control system of the cell needs to be optimized to reduce the helium consumption keeping in mind the requirements of the experimental cell. The background temperature of the cell, usually achieved by cooling it by conduction or by convection (using helium heat exchangers) should be kept as close as possible to the required temperature of the cell. By doing this the maximum heating required to achieve the required temperature of the cell will be reduced thus optimizing the helium consumption.

Careful attention to the compatibility of various adhesives, joints and o-rings with the cryogenic temperatures needs to be taken. For example typical epoxy based adhesives lose their strength at cryogenic temperatures. STYCAST is an adhesive specially adapted for cryogenic applications. It has good electric insulation and good thermal conduction properties which can be rightly exploited in assembling the cells. Indium serves as an excellent joint at cryogenic temperatures. Given its extreme softness, it can flow under mechanical pressure and seal gaps providing very high quality joints. Mainly two types of experiments are carried out as a part of the present thesis: (i) vibration experiments involving vibration of an experimental cell filled with H 2 at a predetermined amplitude and frequency, (ii) gravity quench experiments involving sudden change in the gravity field inside a static cell filled with O 2 .

A detailed description of the experimental cells used is given in the following paragraphs.

Vibration experiments

The main objective of carrying out the vibration experiments is to study the stability of a liquid-vapor interface or a thermal boundary layer when subjected to vibration under various gravity levels. Also studied is the dynamic equilibrium of a liquid-vapor interface under horizontal vibration at various gravity levels.

In these experiments H 2 is the working fluid and the experiments are carried out using the setup HYLDE. The requirements of the experimental cell are as follows:

• Precise filling system for filling the cell at the critical density of H 2 and a provision to close the cell once H 2 is filled to the required level.

• Thermal system to cool or heat the cell to achieve temperature control within 1 mK and good uniformity of the temperature of the fluid inside the cell.

• Leak tightness of the cell.

• Experimentation at temperatures close to the critical point meaning operation at high pressures (1.5 MPa).

• Vibration system to vibrate the cell horizontally at various amplitudes (0.1 mm to 1 mm) and various frequencies (1 Hz to 50 Hz).

• Transparent cell configuration to observe the dynamics of the fluid. )) is made of sapphire and contains a cuboidal (or cubical) cavity of size 3 mm × 3 mm × 2 mm (or side 7 mm). It is mounted using plastic (i.e. non-magnetic) supports made of polyether ether ketone (PEEK) which are in turn mounted on a support bracket made of brass. The support bracket oscillates along a pivot attached to a steel support which is directly mounted on a copper flange (shown as bottom flange in the figures). Two thin strips of copper (not visible in the picture), one on each side of the cell, are used to thermally connect the sapphire cell and the brass support which is in turn connected to the bottom flange using a thermal bridge (made of strands of copper wire). Since sapphire is a very good conductor of heat (with a thermal conductivity an order of magnitude better than that of copper at 30 K), the temperature field inside the sapphire cell is quite uniform. Two resistors of identical resistance (50 Ω) are pasted directly, one each on the copper strips connecting the cell to the brass support to electrically heat the cell whenever required.

The square cavity is closed on both sides using sapphire windows of diameter 24 mm (or 30 mm) and thickness 2 mm using titanium nuts, bolts and washers. The joint between the sapphire windows and the sapphire cell is a circular strip of indium. The sapphire cell and the windows are assembled and torqued (according to experience) and are let to age for a day. During this time the indium joint flows through the gaps and seals all the leak paths. Once assembled, the cell is leak checked using a helium spectrometer, the acceptable leak being 10 -8 mbar.l.s -1 . A capillary tube of internal diameter 0.5 mm and external diameter 1 mm serves to fill the cell with H 2 . The capillary tube is fitted with a thermal switch. It is a small block of copper continuously cooled using liquid helium by conduction and heated whenever required using a resistive heater. In the absence of heating, the hydrogen inside the capillary tube close to the switch is in solid state thus closing the cell. To fill or empty the cell, the switch can be heated thus melting solid H 2 inside the capillary tube. Special thermometers, CERNOX, that work under strong magnetic fields (calibrated close to the critical temperature of H 2 ) are pasted on the top of the sapphire cell to continuously monitor the temperature of the cell. The experimental cell is vibrated using a motor driven cam mechanism. The mo-tor and cam assembly mounted on the top of the anticryostat is connected using a long shaft (not shown in the figure) to the vibration arm of the cell (shown in the Fig. 4.10). The rotational motion of the motor is converted into a rocking motion of the shaft (due to the cam assembly). The rocking motion of the shaft results in an oscillatory motion of the cell along the pivot (shown in the Fig. 4.10). The oscillation amplitude being very small, the net effect is equivalent to a horizontal vibration. The cell is oscillated along the pivot with various frequencies (10 -50 Hz) and various amplitudes (0.1 -1 mm). It is estimated that for a frequency of 50 Hz and for a maximum amplitude of 1 mm, the cell experiences an oscillation in the vertical direction of ±10µm which is negligible compared to the amplitude of the horizontal vibration. Also, the resulting centripetal acceleration compares to the vibrational acceleration as a/R p (where a is the amplitude of vibration and R p is the distance of the cell from the pivot) which comes out to be of the order of 1/60. Thus the centrifugal force is negligibly small in the experimented frequency and amplitude ranges (0-50 Hz) and (0.1-1 mm) respectively. Thus the vibration can be assumed to be in the horizontal direction. All the electrical elements (thermometers and resistors) are connected to the top of the anticryostat using electrical cables. The thermometers are connected using a 4-wire method. While the resistors are cabled using copper wires of 0.4 mm diameter, the thermometers are cabled using manganin wires of diameter 0.2 mm. The temperature control of the cell is achieved by using a standard PID control system.

Geysering experiments

The main objective of carrying out the geyser experiments is to study the dynamics of a liquid-vapor interface when subjected to a sudden change in acceleration. O 2 is the working fluid and the experiments are carried out using the setup OLGA.

The requirements of the experimental cell are as follows:

• Filling system to fill the cell to a required level with O 2 at atmospheric pressure and 90 K.

• Thermal system to cool or heat the cell to achieve temperature control. The experiment does not require very strict isothermal conditions though they are desirable.

• Leak tightness of the cell.

• Transparent cell configuration to allow observation of the dynamics of the fluid. The experimental cell is shown in the Fig. 4.11. It is a cylindrical cell made of sapphire with inner diameter 30 mm and height 100 mm. It is provided with two copper heat exchangers, in which is circulated helium gas at 4.2 K. The two heat exchangers are designed for a maximum flow rate of 1000 l.hr -1 (of equivalent helium gas at NTP) and are each capable of evacuating up to 5 W of heat energy at 90 K. The cell and the heat exchangers are mounted using long threaded rods made of brass. The joints between the heat exchangers and the cell are made of indium which flows into the gaps and seals them. Once assembled, the cell is checked for leak using helium spectrometer, the acceptable leak being 10 -8 mbar.l.s -1 .

A capillary tube with an inner diameter of 2 mm is connected through the top heat exchanger to the cell. The capillary tube communicates to the filling station and does not have a thermal switch like the one in the case of the experimental cell of vibration experiments. Thus the only way to isolate it from outside is by closing the valve at the filling station. Filling of the cell to the required level is carried out using this valve only. The initially hot O 2 gas cools down in the copper heat exchanger before entering the cell as a liquid.

Resistive heaters (200 Ω each) are pasted using STYCAST on two annular copper strips bolted to the two heat exchangers. The temperature of the cell can be changed by heating these two strips. Two CERNOX thermometers (calibrated at a temperature range around 90 K) are mounted, one on the top heat exchanger and another on the bottom heat exchanger. Temperature control of the cell is carried out using a classical PID control circuit close to 10 mK. The parameters of the PID control system need to be fixed experimentally for each experimental cell used. All the electrical elements are cabled out to the top of the anticryostat to connect them with the ground system.

Experimental methodology

Carrying out experiments onboard HYLDE and OLGA is a very tedious activity and often needs a lot of patience. The experimental campaign needs to be planned very meticulously to carry out the experiments without breaks and surprises. This is because of the long experimental preparation times and susceptibility to various problems like leaks, blockage of tubes, failure of systems, alignment problems, unplanned over-consumption of working fluids etc. A typical experimental campaign takes at least 2 weeks when everything works according to plan. It can take a couple of weeks more if some problems arise. The experimental procedure is identical for both HYLDE and OLGA. The experimental cell is first assembled and leak checked using a helium leak detector.

It is then assembled to the anticryostat and is checked for alignment by using the visualization system. Adjustment of the camera and the endoscopes is simulta-neously done according to the resolution requirements of the experiments. The anticryostat is then checked for leak using the helium leak detector. Proper working of the electrical cabling is checked at each stage to ensure correct electrical connection. The anticryostat is then ready for assembly to the cryostat. After assembling the anticryostat to the cryostat, the entire cryostat is rinsed with helium gas at least three times (in order to drive away all other gases from the cryostat). This process is extremely important for HYLDE that uses a Roubeau ′ s bath to cool down liquid helium from 4.2 K to 2.17 K. If the rinsing is not done properly, the Roubeau ′ s bath will not work and the experiments cannot be carried out.

Once the rinsing is completed, all the connections to the anticryostat are done (fluid connections, endoscopes, electrical connections, camera etc). The PID control module is checked for correct functioning. The vacuum pump is switched on and vacuum conditions are maintained inside the anticryostat as well as the cell.

Liquid nitrogen is then filled inside the cryostat and the system is allowed to cool down to 80 K over a weekend. This is important to reduce the liquid helium consumption during the experiment. Once the system attains 80 K, liquid helium can be transferred into the cell and the superconducting coil can be switched on to carry out the experiments when the temperature of the cryostat reached 4.2 K. For experimentation with HYLDE an additional step of switching on the Roubeau ′ s bath needs to be carried out before switching on the superconducting coils.

Concluding remarks

In the present chapter the experimental installations HYLDE (for H 2 ) and OLGA (for O 2 ) used to carry out the experiments are detailed. We have seen that these installations work on the principle of magnetic levitation. They use superconducting coils made of Nb-Ti and are cooled using liquid helium. The volume of fluids that can be levitated within a few %g 0 using these setups are of the order of ∼ mm 3 for HYLDE and ∼ cm 3 for OLGA. Designed to study the dynamics of fluids under zero-gravity conditions or variable gravity (oscillations, fast variations), these installations have till date carried out some extremely interesting set of experiments. Their contribution to the microgravity science is immense and they will be the instruments of great importance to the microgravity research and science.

Chapter 5

Governing equations and numerical model

Resumé (Français)

Des simulations numériques sont effectuées pour comparer les résultats expérimentaux avec la théorie. Deux problèmes sont considérés: (i) L'étude de la stabilité d'une couche limite de fluide supercritique soumise à des vibrations en absence de gravité, (ii) L'étude de la dynamique d'une interface liquide-vapeur soumise à une brusque variation de gravité ou d'accélération. Le premier problème concerne un fluide hyper-compressible monophasique refroidi brusquement par ses parois et soumis à des vibrations. Le problème est modélisé en utilisant les équations de Navier-Stokes compressibles. Une équation d'état linéarisée est utilisée. Les équations sont discrétisées sur un maillage non-uniforme décalé et ensuite résolues en utilisant l'algorithme de volumes finis SIMPLER (Version révisée de Semi-Implicit Method for Pressure Linked Equations).

Le deuxième problème concerne un écoulement incompressible diphasique soumis à une brusque variation de la gravité. Le problème est modélisé en utilisant un "one fluid model" qui est un modèle efficace pour simuler les écoulements multiphasiques pour les fluides incompressibles et immiscibles de tension interfaciale constante. Les équations sont résolues en utilisant la méthode VOF-PLIC d'un code académique THETIS développé par le laboratoire I2M-TREFLE à Bordeaux.

Dans ce chapitre sont détaillés les modèles mathématiques utilisés pour modéliser les deux problèmes ainsi que les méthodes numériques SIMPLER et VOF-PLIC utilisées pour les résoudre.

Introduction

Two problems are investigated numerically by using a finite volume method: the interaction of a thermal boundary layer with vibration under weightlessness and the interface dynamics of large rising bubbles due to a brusque change of the gravity field. The first problem deals with a hyper-compressible monophasic fluid subjected to a thermal quench simultaneously with vibration while the second problem deals with a liquid-vapor interface of two quasi-incompressible fluids subjected to a sudden gravity or acceleration change. Both the problems are entirely different and need to be analyzed using two different numerical codes. In the present chapter, the governing equations for the two problems under study are first presented and then the finite volume numerical algorithms are briefly described.

Numerical modeling of a supercritical fluid subjected to vibration

The interaction of a thermal boundary layer with vibration can produce various kinds of instabilities based on the relative direction of the vibration and the thermal boundary layer. Close to the critical point (in the supercritical zone) various non-dimensional numbers like the Rayleigh number and Prandtl number diverge meaning that a supercritical fluid is much more unstable than a normal fluid. Solving for the flow field of a supercritical fluid subjected to vibration simultaneously with sudden heating or cooling is complicated due to the hypercompressible nature of the problem.

The problem under study is a 2D square cavity of side h filled with supercritical H 2 at its critical density (ρ c ), at an initial pressure approximately equal to its critical pressure (p c , here the effect of change in pressure with a thermal quench is neglected) and temperature T i = T c + ∆T , where ∆T is the temperature difference representing the proximity to the critical temperature T c . Three thermal configurations are considered: (i) Sudden cooling of all the 4 walls (Fig. 5.1a), (ii) of the vertical walls (Fig. 5.1b) or (iii) of the horizontal walls (Fig. 5.1c). Adiabatic conditions are imposed on the remaining walls. The cell is subjected to horizontal vibration (along x direction) with an amplitude a and frequency f simultaneously with the quench. The vibration period is chosen to be small compared to the hydrodynamic time scales (high frequency vibration conditions). Zero-gravity conditions are maintained inside the cell to exclude the effects of density stratification and buoyancy-induced convection inside the fluid domain.

Governing equations

The problem under study is mathematically modeled using compressible Navier-Stokes and energy conservation equations. A linearized equation of state, to describe the relation between density, pressure and temperature inside the fluid domain, is understood to be sufficiently accurate when the relative thermal quench considered ( δT ∆T ) is small. The governing equations with standard notations are modeled as:

∂ρ ∂t + ∇.(ρ u) = 0 ρ[ ∂ u ∂t + ( u.∇) u] = -∇p + µ[∆ u + 1 3 ∇(∇. u)] + ρaω 2 sin (ωt) i ρc p [ ∂T ∂t + ( u.∇)T ] = β p T dp dt + λ∇.(∇T ) + µφ visc ρ = ρ 0 + χ T (p -p 0 ) -β p (T -T 0 )                    (5.1)
where, φ visc is the dissipation rate defined as φ visc = u i,j u j,i + u i,j u i,j -2 3 u i,i u j,j (using the Einstein summation convention of repeated indices), c p is the specific heat capacity at constant pressure, χ T is the isothermal compressibility, β P is the thermal expansion coefficient, µ is the dynamic viscosity, λ is the thermal conductivity, ρ 0 , p 0 and T 0 are the initial density, pressure and temperature fields. The equations are non-dimensionalized using

ρ ′ = ρ ρc , T ′ = T Tc , p ′ = p ρcc 2 , u ′ = u c
for density, temperature, pressure and velocity respectively and x ′ = x h , y ′ = y h and t ′ = t ta for space and time with t a = h c with c the sound velocity is the corresponding acoustic time. The resulting dimensionless equations are as follows:

∂ρ ′ ∂t + ∇.(ρ ′ u ′ ) = 0 (5.2) ρ ′ [ ∂ u ′ ∂t ′ + ( u ′ .∇) u ′ ] = -∇p ′ + 1 Re [∆ u ′ + 1 3 ∇(∇. u ′ )] + 1 F r 2 v ρ ′ sin (ω ′ t ′ ) i (5.3) ρ ′ [ ∂T ′ ∂t ′ + ( u ′ .∇)T ′ ] = β ′ p E c T ′ dp ′ dt ′ + 1 Re.P r ∇.(∇T ′ ) + E c Re φ ′ visc (5.4) ρ ′ = ρ ′ 0 + χ ′ T (p ′ -p ′ 0 ) -β ′ p (T ′ -T ′ 0 ) (5.5)
Here φ ′ visc is the non-dimensional dissipation rate defined as

φ ′ visc = u ′ i,j u ′ j,i + u ′ i,j u ′ i,j -2 3 u ′ i,i u ′ j,j , ν is the kinematic viscosity, D T is the thermal diffusivity. The non-dimensional numbers are: isothermal compressibility χ ′ T = χ T ρ c c 2 , Reynolds number Re = hc ν , vibrational Froude number F r v = c √ h(aω 2 )
, thermal expansion

coefficient β ′ P = β P T c , Eckert number E c = c 2 cpTc
, Prandtl number P r = ν D T and angular frequency ω ′ = ωt a . The relative temperature proximity to the critical point is defined as ε = T -Tc Tc and is used to calculate all the physical parameters and the derived quantities (see Table 2.2). These properties are assumed to be constant for a small temperature quench δT , which is typically fixed as ∆T 10 . The initial conditions are set as:

ρ ′ 0 = 1, p ′ 0 = pc ρcc 2 , T ′ 0 = 1 + ε i , where ε i = T i -Tc Tc .
No slip velocity boundary conditions are applied on the walls. Boundary conditions on the temperature field are defined in Fig. 5.1. The problem under study is a multi-scale hyper compressible problem involving various time scales viz., the thermal diffusion time scale

t D = h 2 D T , viscous diffusion time scale t v = h 2
ν , piston effect time scale t P E = h 2 γ 2 D T , vibration time scale

t vib = 1
f and the acoustic time scale t a = h c , where γ is the ratio of specific heats. The piston effect plays an important role in the dynamics of the fluid. Its timescale tends to zero as ε 1.6 when nearing T c . Nevertheless, a rigorous analysis [START_REF] Zappoli | Acoustic saturation of the critical speeding up[END_REF] has shown that thermal equilibration is governed only by acoustic phenomena, showing that the piston effect mechanism cannot be faster than the acoustic propagation. This mechanism is called "acoustic saturation of the critical speeding up". The governing equations comprising of the compressible Navier-Stokes equations coupled with the energy and the equation of state are discretized spatially using a staggered non-uniform mesh with higher resolution near the walls and are solved using the finite volume algorithm SIMPLER (revised version of the Semi-Implicit Method for Pressure Linked Equations) [START_REF] Accary | A 3d finite volume method for the prediction of a supercritical fluid buoyant flow in a differentially heated cavity[END_REF][START_REF] Patankar | Thermal Vibrational Convection[END_REF] which will be described in the section 5.2.4.

Figure 5.2: Staggered mesh

By using a staggered mesh the mass flow rates across the control volume faces can be calculated without any interpolation for the velocity components, thus not compromising on the accuracy of the results. The difficulties associated with solving for the pressure terms that appear inside the source term of the momentum equation and the application of the continuity condition are completely removed by using a staggered grid. Figure 5.2 shows a staggered mesh. In the mesh, the solid lines represent the primary grid for the scalar quantities T , p and ρ and the dashed lines represent the secondary grid indicating the control volumes for the two components of the velocity field. In the staggered grid, the velocities are calculated on face centers of the control volume (denoted by the red (for x component) and green arrows (for y component)). All other variables are defined on the primary grid points denoted by the circle symbols. The SIMPLER algorithm used to solve the problem is described as follows. Firstly, the governing equations are discretized using the staggered grid. The discretization equations for the momentum and energy equations have similar form. Discretization equations can thus be derived for a generalized equation, for a generalized variable φ, which can be either temperature (T ) or velocities (u and v). This generalized equation can be written as:

Generalized discretization equation

∂ ∂t (ρφ) + ∂ ∂x (J x ) + ∂ ∂y (J y ) = S C + S P φ P (5.6)
where, J x (= ρuφ -Γ( ∂φ ∂x )) and J y (= ρvφ -Γ( ∂φ ∂y )) are the total fluxes (convection + diffusion) and S = S C + S P φ P is the linearized source term. The discretization equations are derived for a control volume as shown in Fig. 5.3. The control volume is indicated in dashed lines inside the grid with grid size ∆x and ∆y; δx and δy are the grid size of the primary grid; P is the grid point under focus while E, W , N and S are the neighboring grid points; e, w, n and s are the faces of the control volume. Integration of the generalized convection diffusion equation over the control volume shown in Fig. 5.3 and using a first order Euler discretization in time gives:

(ρ P φ P -ρ 0 P φ 0 P ) ∆x∆y ∆t + J e -J w + J n -J s = (S C + S P φ P )∆x∆y (5.7)
and integration of the continuity equation 5.2 over the control volume gives:

(ρ P -ρ 0 P ) (∆x∆y) ∆t + F e -F w + F n -F s = 0 (5.8)
where J e , J w , J n and J s are the total fluxes through the control volume faces (shown in Fig. 5.3), F e = (ρu) e ∆y, F w = (ρu) w ∆y, F n = (ρv) n ∆x and F s = (ρv) s ∆x are the mass flow rates through the control volume faces. The superscript '0 ′ indicates the values from the previous time step. Manipulation of eq. 5.7 gives:

(φ P -φ 0 P )

ρ 0 P ∆x∆y ∆t + (J e -F e φ P ) -(J w -F w φ P ) + (J n -F n φ P ) -(J s -F s φ P ) = (S C + S P φ P )∆x∆y (5.9)
Though the convection and the diffusion fluxes are grouped into a single term, their resolution is not straight forward. Convection flux involves a velocity com-ponent while the diffusion term involves a velocity gradient. Choosing a velocity gradient is quite straightforward but choosing the convection velocity (which is a value defined in between two grid points) is not simple. We need a model to calculate the right velocity that can correctly satisfy the continuity equation.

Various schemes exist, viz., upwind scheme, exponential scheme, hybrid scheme, power-law scheme etc. to determine the value of the velocity based on the relative strength of the convection and diffusion fluxes defined by the Peclet number (P e) based on the grid size (the details of which will not be presented here and can be found in [START_REF] Patankar | Thermal Vibrational Convection[END_REF]).

The generalized form of the discretization equations can be written as:

a P φ P = a E φ E + a W φ W + a N φ N + a S φ S + b (5.10)
where, a P , a E , a W , a N and a S are the discretization coefficients that involve the diffusion conductances D e , D w , D n and D s , flow rates F e , F w , F n and F s and Peclet numbers P e , P w , P n and P s , defined as below.

a E = D e A(|P e |) + M AX(-F e , 0) a W = D w A(|P w |) + M AX(-F w , 0) a N = D n A(|P n |) + M AX(-F n , 0) a S = D s A(|P s |) + M AX(-F s , 0) a 0 P = ρ 0 P ∆x∆y ∆t b = S C (∆x∆y) + a 0 P φ 0 P a P = a E + a W + a N + a S -S P ∆x∆y                                  (5.11
)

D e = Γ e ∆y (δx) e P e = F e D e D w = Γ w ∆y (δx) w P w = F w D w D n = Γ n ∆x (δy) n P n = F n D n D s = Γ s ∆x (δy) s P s = F s D s                          (5.12) Table 5.1: Function A(|P |) for different schemes Scheme Formula for A(|P |) Central difference 1 -0.5|P | Upwind 1 Hybrid M AX(0, (1 -0.5|P |)) Power law M AX(0, (1 -0.1|P |) 5 ) Exponential (exact) |P |/[exp(|P |) -1]
The function A(|P |) can be defined using various schemes. A list of various popular schemes is given in Table 5.1. The power law scheme was used in our study.

Discretization equations for NS equations

a e u e = Σa nb u nb + b + (p P -p E )A e a n v n = Σa nb v nb + b + (p P -p N )A n (5.13)
where, a nb , u nb and v nb are the neighboring coefficients and neighboring velocities, A e and A n are the areas of the faces e and n. The above equations can further be rewritten in terms of the pseudo velocities û and v as:

u e = ûe + A e a e (p P -p E ) v n = vn + A n a n (p P -p N )        (5.14) where ûe = Σa nb u nb + b a e vn = Σa nb v nb + b a n        (5.15)
Pressure equation: The discretization equation for the pressure is obtained from the continuity equation and can be written with usual notation as:

a P p P = a E p E + a W p W + a N p N + a S p S + b (5.16)
where b is given by b

= (ρ 0 p -ρ p )∆x∆y ∆t + [(ρû) w -(ρû) e ]∆y + [(ρv) s -(ρv) n ]∆x (5.17)
Energy equation: The general discretized equation 5.10 can be directly used to solve for the energy equation just by replacing φ with T .

a P T P = a E T E + a W T W + a N T N + a S T S + b (5.18) b = S C (∆x∆y) + a 0 P T 0 P a P = a E + a W + a N + a S -S P ∆x∆y (5.19)

The SIMPLER algorithm

The discretization equations are linear algebraic equations and can be solved using classical methods like TDMA. The classical compressible Navier-Stokes equations can be numerically solved using the SIMPLER (revised version of the Semi-Implicit Method for Pressure Linked Equations) algorithm. The sequence of operations in the SIMPLER algorithm are as follows.

a. Start with a guessed velocity field (usually taken as the velocity field from the previous time step.

b. Calculate the coefficients of the momentum equations and then calculate û and v from Eqns 5.15. c. Using the pseudo velocities û and v, calculate the coefficients for the pressure equation 5.16 and solve for p.

d. Treating the obtained pressure field as p * , calculate for u * and v * (the new velocity fields) using the discretization equations 5.13 e. Calculate the source term b using Eq. 5.17 by using u * and v * instead of û and v and hence solve for the pressure correction p corr . f. Correct the velocity fields using eq. 5.14 (with u * and v * instead of û and v and p corr instead of p) but not the pressure field.

g. Solve for the temperature using the discretization equation 5.18 by replacing φ with T .

h. Update the density values using the linearized state equation i. Repeat from b till convergence. The convergence is characterized by the residue at iteration "k". The condition for convergence is expressed as (i,j) are the nodes in the x and y directions. In the present study, the largest residue was obtained by the pressure equation and we have fixed eps = 10 -10 .

RES = M AX i,j | (φ k ij -φ k-1 ij ) φ k ij | < eps,

Numerical model used for the geysering phenomenon

The problem under study consists of an oxygen vapor bubble suspended inside liquid oxygen, subjected to a brusque variation of gravity. The objective of the study is to analyze the complex interface dynamics resulting in the formation of a geyser at the liquid-vapor interface. To carry out the numerical simulations a (2D) finite volume code based on VOF-PLIC method of the academic multipurpose software "THETIS" developed by the I2M-TREFLE laboratory in Bordeaux is used. The numerical modeling of the two-phase flow involving separated phases is achieved with a one-Fluid model [START_REF] Kataoka | Local instant formulation of two-phase flow[END_REF]. This model is very efficient in handling multi-phase fluid flows involving immiscible incompressible and isothermal fluids with constant surface tension. The exact details of the multiphase code used in THETIS is beyond the scope of the present thesis. Some superficial details of the method are however presented below. The numerical modeling of two-phase flows involving separated phases can be achieved with a one-Fluid model [START_REF] Kataoka | Local instant formulation of two-phase flow[END_REF]. The interfacial jump conditions, valid at the interface, are integrated into the Navier-Stokes equations according to the CSF model of Brackbill et al. [START_REF] Brackbill | A continuum method for modelling surface tension[END_REF]. The resulting one-Fluid model is then similar to the single phase Navier-Stokes equations (see below) to which extra-terms have been added so that local modifications of equations are included through local viscosities, densities and surface tension forces.

(ρ( ∂ u ∂t + ( u • ∇) u) = ρg -∇p + ∇ • (µ(∇ u + ∇ t u)) + F st (5.20) ∇ • u = 0 (5.21) ∂C ∂t + u • ∇C = 0 (5.22)
where u is the local velocity field, t is the time, p is the pressure, g is the gravity and F st is the surface tension force defined as F st = σκn i δ i , σ is the surface tension, κ is the local curvature of the interface and n is the normal to the interface. C is the phase indicator function which describes the evolution of the interface with time using an material advection equation. For each fluid k the phase function C k is supposed to behave like a Heaviside function:

C k (x, y, t) =
1 if the spatial coordinate (x,y) has the phase k 0 otherwise (5.23) The interface between the phase k and the other phases is defined by the isosurface C k = 0.5. Using the phase function, the local quantities like the density (ρ) and the dynamic viscosity (µ) are defined according to:

ρ = Σ N -1 k=0 C k ρ k (5.24) µ = Σ N -1 k=0 C k µ k (5.25)
where, N is the number of phases in the fluid system, ρ k and µ k are the density and viscosity of the k th fluid. The interfacial jump conditions are expressed as:

( u 1 -u 2 ). n i = [ u]. n i = 0 (5.26) pn k -µ(∇u + ∇ t u) • n = σκ (5.27)
For the calculation of the interface curvature and the interface normal, a VOF-PLIC function is built. Finite volume discretization on fixed Cartesian grids is applied to approximate the system of conservation equations 5.20, 5.21 and 5.22. Concerning the resolution of the pressure-velocity coupling (Eqs. 5.20 and 5.21) the method of projection [START_REF] Goda | A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows[END_REF] was used. The space discretization uses a first order upwind scheme and a first order Euler scheme is implemented for the discretization of the time dependent terms. The resulting algebraic systems are inverted by a direct solver called MUMPS (MUltifrontal Massively Parallel sparse direct Solver) [START_REF] Amestoy | Multifrontal parallel distributed symmetric and unsymmetric solvers[END_REF].

Conclusion

In the present chapter the governing equations and the numerical algorithms used to solve the problems are detailed. The results of the numerical simulations will be presented in the chapters 6 and 9 for the supercritical fluid problem and the geysering problems respectively.

Chapter 6

Stability under vibration of a supercritical thermal boundary layer

Resumé (Français)

En absence de gravité, l'interaction de vibrations avec une couche limite thermique peut entrainer des dynamiques remarquables dans un fluide supercritique (SCF

Introduction

It is known that vibrations can induce mean flows inside a fluid that exhibits density non-homogeneities. For example, under Earth ′ s gravity, a two phase fluid subjected to vibration can display stationary or propagating waves on the inter-face or, in special cases, a stationary relief phenomenon called "frozen waves" [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF][START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF][START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF] for horizontal vibration (vibration direction tangential to the interface) and Faraday waves [START_REF] Faraday | On a peculiar class of acoustical figures and on certain forms assumed by groups of particles upon vibrating elastic surfaces[END_REF] for vertical vibration (vibration direction perpendicular to the interface) in the case of immiscible fluids. Miscible fluids (with vanishing interfacial tension) also can respond to vertical vibration leading to Faraday waves [START_REF] Zoueshtiagh | Experimental and numerical study of miscible faraday instability[END_REF] and to horizontal vibration, giving rise to frozen waves [START_REF] Legendre | Instabilités à l'interface entre fluides miscibles par forçage oscillant horizontal[END_REF]. Supercritical fluids have densities close to those of liquids and viscosities close to those of gases. Due to the high thermal expansion coefficient of a supercritical fluid, a small amount of non-uniform heating/cooling inside the domain can create strong density gradients. These density gradients act as non-homogeneities inside the fluid domain and remarkable dynamics can be excited [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF][START_REF] Beysens | The effect of vibrations on heterogeneous fluids: Some studies in weightlessness[END_REF] by subjecting them to vibration. However working with supercritical fluids needs zero-gravity conditions to eliminate the undesirable density stratification due to the hypercompressible nature of the fluid. The present chapter studies various kinds of instabilities, i.e., corner, Rayleigh vibrational and parametric instabilities, observed when an advancing thermal boundary layer interacts with vibration. Both numerical and experimental investigations carried out with supercritical H 2 will be presented. Experiments carried out by various research groups on supercritical CO 2 and H 2 under zero-gravity conditions have given evidence for thermal boundary layer instabilities under vibration. Experiments onboard HYLDE involving vibration with a sudden thermal quench close to the critical point of Hydrogen showed boundary layer fingers propagating in a direction perpendicular to the direction of vibration [START_REF] Beysens | The effect of vibrations on heterogeneous fluids: Some studies in weightlessness[END_REF]. Similar kind of fingers were observed in experiments involving supercritical CO 2 onboard Maxus-7 sounding rocket [START_REF] Beysens | Phase transition under forced vibrations in critical co2[END_REF]. These finger-like structures are a hall mark of the Rayleigh vibrational instability originating in the temperature gradient of a thermal boundary layer. This instability was theoretically investigated by Gershuni and Lyubimov [START_REF] Gershuni | Thermal Vibrational Convection[END_REF]. They carried out a linear stability analysis of a semi-infinite layer of fluid in an averaging approach using incompressible governing equations with Boussinesq approximation under the high frequency assumption. Two-dimensional numerical analysis by Amiroudine and Beysens [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF] using a finite volume based numerical code to solve compressible Navier-Stokes equations with a linearized equation of state successfully predicted this type of instability. The critical Rayleigh vibrational number was plotted as a function of the temperature proximity to the critical point and the results showed a strong increase of this number when approaching the critical point. Lyubimov et al. [START_REF] Lyubimov | Stability of a thermal boundary layer in the presence of vibration in weightlessness environment[END_REF] solved a linear stability problem involving averaged equations for a semi-infinite layer using incompressible Navier-Stokes equations with Boussinesq approximation, for the cases of low amplitude (with respect to the sample size) and high frequency (with respect to the inverse fluid typical times) vibrations. The stability curve and predicted values of the critical Rayleigh vibrational number for various temperatures matched with those of Amiroudine and Beysens [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF] for temperatures sufficiently far away from the critical point.

In the present work the stability aspects of supercritical Hydrogen involving the interaction of a thermal boundary layer with vibration is studied. This chapter is organized as follows: Firstly a brief description of the problem being studied will be presented followed by a description of the general results obtained. Subsequently, a study of the various kinds of instabilities observed and their analysis are presented.

Numerical investigation

The problem under study is a 2D square cavity of side h filled with supercritical H 2 at its critical density (ρ c ), at an initial pressure approximately equal to its critical pressure (p c ) and temperature T i = T c +∆T , where ∆T is the temperature difference representing the proximity to the critical temperature T c . Three thermal configurations are considered: (i) Sudden cooling of all the 4 walls (Fig. 6.1a), (ii) on the vertical walls (Fig. 6.1b) or (iii) on the horizontal walls (Fig. 6.1c). Adiabatic conditions are imposed on the remaining walls. The cell is subjected to horizontal vibration (along x direction) with an amplitude a and frequency f simultaneously with the quench. The vibration period is chosen to be small compared to the hydrodynamic time scales (high frequency vibration conditions). Zero-gravity conditions are maintained inside the cell to exclude the effects of density stratification and buoyancy-induced convection inside the fluid domain. The problem is modeled using compressible Navier-Stokes equations with a linearized state equation. The governing equations and the numerical method used to solve the problem are presented in the section 5.2. The various time scales involved are: thermal diffusion time scale

t D = h 2 D T , viscous diffusion time scale t v = h 2
ν , piston effect time scale t P E = h 2 γ 2 D T , vibration time scale t vib = 1 f and the acoustic time scale t a = h c , where γ is the ratio of specific heats. The simulation time step should be smaller than all the characteristic time scales including the piston effect time scale t P E . This is the reason why the computational times can be very high for small values of ε. It can be seen in Table 2.2 that for all the cases of frequencies used in this study, the hydrodynamic times t D and t v are large compared to the vibration time t vib . Since the physical phenomenon at the acoustic timescale is not the primary interest, low-Mach number based calculations can be used [START_REF] Accary | A 3d finite volume method for the prediction of a supercritical fluid buoyant flow in a differentially heated cavity[END_REF][START_REF] Amiroudine | Direct numerical simulation of instabilities in a two-dimensional near-critical fluid layer heated from below[END_REF]. They filter (or average out) the acoustic waves at time steps larger than the acoustic timescales. A linearized equation of state (as in Ref. [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF]) and the finite volume method, which integrates (and thus filters) the partial differential equations are used in the present study. Fixed time steps (10 -3 s) much larger than the acoustic timescales (of the order of 20 µs) were used without making the calculations to diverge. The residue at iteration "k" can be expressed as:

RES = M AX i,j | (φ k ij -φ k-1 ij ) φ k ij |,
where φ is the field variable (components of the velocity field, pressure or temperature) and (i,j) are the nodes in the x and y directions. The governing equations comprising of the compressible Navier-Stokes equations coupled with the energy and the equation of state are discretized spatially using a staggered non-uniform mesh with higher resolution near the walls and are solved using the finite volume algorithm SIMPLER (revised version of the Semi-Implicit Method for Pressure Linked Equations) [START_REF] Patankar | Thermal Vibrational Convection[END_REF][START_REF] Amiroudine | Direct numerical simulation of instabilities in a two-dimensional near-critical fluid layer heated from below[END_REF] (see 5.2. A first-order Euler discretization is used for time. A square cell of size h = 7 mm is used to carry out the study. The thickness of the viscous boundary layer is calculated as δν = 2ν ω which ranges from 49 µm (for f = 25 Hz) to 150 µm (for f = 2.77 Hz). A spatial discretization of 80 × 80 cells using a geometrical power factor of 0.2 (with the first point of the mesh at a distance of about 10 µm from the origin) is sufficient to ensure that at least 5 to 10 cells are present inside the boundary layer. The field parameters (pressure, temperature, density and velocity) are recorded once every half vibrational time period. The condition for convergence is given by RES < eps, where eps is a small number chosen as eps = 10 -10 . The largest residue obtained by the pressure correction equation was as small as 10 -10 , which ensures the convergence of the present method with time steps as large as 10 -3 s.

Results

The simulations are carried out for temperature proximities ∆T = 10 mK, 100 mK and 1 K, with the corresponding Prandtl numbers ranging from 950 at ∆T = 10 mK to 35 at ∆T = 1 K. The vibration frequencies range between 2.77 Hz and 25 Hz and vibration amplitudes between 1 mm and 30 mm. The calculations for each case are carried out for at least 15 s capturing the scalar and vector fields data once every half period of vibration. The thermal boundary layer (TBL), defined as 2π √ D T t, is not stationary and is moving with time. The instability occurs when a threshold value of the TBL thickness is crossed. The application of the vibration after waiting for some diffusion times less than the critical time does not seem to change the results. The threshold is thus not modified by the time when the vibration is applied. The observation of the onset of the fluid instabilities is carried out by plotting the streamlines of the fluid flow with time. An instability is said to have onset when the streamlines deform giving rise to vortex structures inside the flow domain. This method is similar to the classical method in which vorticity function is examined for observing the instability onset. In the simulations with isothermal conditions on all the 4 walls, three types of instabilities are observed as shown in Fig. 6.2.

• Corner instability forming fingers in the thermal field at the corners of the 2D cell. They appear at the beginning of the process with a frequency equal to the frequency of vibration of the cell.

• Parametric instability from the vertical walls oscillating at half the frequency of vibration.

• Rayleigh-vibrational fingers appearing from the horizontal walls.

The oscillation frequency of all these instabilities are verified by recording the data at various time intervals. It was observed in all our numerical simulations that for low vibrational accelerations the Rayleigh vibrational instability is favored, while the parametric instability is seen only for high vibrational accelerations.

For a fixed frequency it was observed that as the amplitude of vibration is increased (from 1 mm to 30 mm), Rayleigh vibrational instability appears at lower amplitudes, a coexistence of the Rayleigh-vibrational and parametric instabilities happens for intermediate amplitudes and at higher amplitudes parametric instability predominates.

A strong interaction of these three types of instabilities is observed from the simulations. For higher amplitudes of vibration the parametric instability tends to suppress the apparition of the Rayleigh vibrational instability whereas for moderate amplitudes the Rayleigh-vibrational instability wins over the parametric instability. The corner instability that forms due to the interaction of the fluid with the corners of the domain also has a significant influence on the parametric instability.

In order to study each one of these instabilities, one needs to isolate the influence of one over another. For isolating the effect of the corner instability on the parametric instability, adiabatic conditions are imposed on the horizontal walls and simulations are carried out. The configuration is the one shown in Fig. 6.1b. Similarly, simulations are carried out with adiabatic vertical walls (Fig. 6.1c to isolate the effect of parametric instability on the corner instability. The following sections present the results of such an analysis.

Corner instability

When the cell is thermally quenched, the SCF develops a thermal boundary layer that results in strong density gradients close to the walls by virtue of its high thermal expansion coefficient. These density non-homogeneities, when subjected to vibration, produce strong velocity gradients near the walls, which lead to the formation of two counter-rotating vortices in the top and bottom halves of the cell. The interaction of these vortices with the vertical walls leads to the formation of the corner instability. They appear near a wall only during the half period of vibration when the direction of acceleration coincides with the direction of propagation of the boundary layer. Figure 6.3 shows the evolution of the corner instability for f = 2.78 Hz, a = 20 mm, ∆T = 1 K with a temperature quench of 100 mK. Figures 6.3a -6.3e show the temperature fields of the domain while Figs. 6.3f -6.3j show the corresponding streamlines for every period of vibration.

As can be seen, vortices (primary vortices indicated as PV in the figure) are formed in the top and bottom half of the cell due to the density gradient in the fluid. The interaction of these vortices with the vertical walls of the cell deforms the streamlines. In subsequent cycles of vibration this deformation evolves into secondary vortices (SV) as seen in the Figs. [START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF].3i -6.3j. The counter rotating vortex pairs at the corners squeeze-in cold dense fluid from the corners into the bulk of the fluid thus appearing as fingering structures in the thermal field. Fingers on a particular wall appear during a favorable half cycle of vibration while fingers on the opposite wall recede. The figures 6.3a -6.3j show the thermal and streamline patterns for the extreme right position of the cavity when the fingers on the right wall attain their maximum while the fingers on the left wall are at their maximum recession. The corner instability is associated with threshold amplitudes. Figure 6.4 gives the stability curve for this instability for two temperature proximities ∆T = 10 mK and 100 mK. It is observed that the stability domain reduces as the critical point is approached. The critical amplitude a cr decreases as the frequency f increases.

Parametric instability

Vibration perpendicular to a two-phase fluid interface (miscible or immiscible) is known to exhibit parametric instability [START_REF] Zoueshtiagh | Experimental and numerical study of miscible faraday instability[END_REF][START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF]. Another example of parametric instability is the excitation of convection in fluid subjected to modulated gravity field and vertical temperature gradient [START_REF] Gershuni | Thermal Vibrational Convection[END_REF]. In this case, if the static gravity is absent, only parametric instability occurs [START_REF] Gershuni | Thermal Vibrational Convection[END_REF]. A thermal boundary layer of a supercritical fluid close to its critical point can exhibit similar type of instability when subjected to vibration. The configuration with vertical isothermal walls subjected to horizontal vibration (Fig. 6.1b) shows evidence for this. For sufficiently high amplitudes of vibration, the thermal boundary layer of the vertical wall becomes parametrically unstable forming fingers in the thermal field oscillating with a frequency equal to half the frequency of vibration.

The results for a typical vibration case are presented in Fig. 6.5. The top row (Figs. 6.5a -6.5f) shows the thermal field depicting the finger pattern for 6 consecutive half vibration time periods for the case ∆T = 100 mK, f = 2.78 Hz and a = 20 mm. The corresponding streamline patterns are shown in the bottom row (Figs. 6.5g -6.5l). The phenomenon involved can be understood as follows.

When the vibrational acceleration direction is anti-parallel to the temperature gradient a Rayleigh-Taylor like configuration occurs, which tends to destabilize the boundary layer. In the subsequent vibration half-cycle the vibrational acceleration direction is parallel to the temperature gradient, which has a stabilizing effect on the boundary layer. The stabilizing and destabilizing vibration halfcycles continue while the thickness of the boundary layer increases with time.

When a threshold boundary layer thickness is attained, counter rotating vortex pairs are formed during a destabilizing half cycle. It is at this point that fingerlike structures start forming. The number of the counter rotating vortex pairs near the wall corresponds to the number of fingers seen in the thermal field. A rocking motion of the vortices in the lateral direction eventually ensues, due to the vibration, leading to a hit and rebound process between the adjacent vortices.

The resonance between the rocking motion and the vibration leads to the parametric excitation of the vortices. Figures 6.5a The wavelength Λ(= λ h ) is plotted versus the reduced acceleration Γ (= aω 2 g ) for the parametric instability in Fig. 6.7a for ∆T = 10 mK, 100 mK and 1000 mK. It reflects the fact that the wavelength of the instability decreases as the acceleration of vibration increases. A power law curve fits well with the data as λ ∼ (aω 2 ) b , with b ∼ -0.35 ± 0.02 (uncertainty: one standard deviation). It can be observed that the slope of the curve in the log-log plot is approximately the same for all the three critical point proximities. Let us compare the above case with typical Faraday instabilities in immiscible fluids [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF]. The dispersion relation for Faraday waves in the absence of viscosity (see below) is given by:

ω 2 = [ (ρ 1 -ρ 2 )kg 0 + σk 3 ρ 1 + ρ 2 ] (6.1)
where ω is the frequency of vibration, ρ 1 and ρ 2 are the densities of the two fluids under study, k is the wave number, g 0 is the acceleration due to gravity and σ is the surface tension. According to Ref. [START_REF] Fauve | Parametric instability of a liquid-vapour interface close to the critical point[END_REF], the dispersion relation for the Faraday waves in immiscible fluids is not valid for T c -T < 20 mK in CO 2 .

The same relative distance (1 -T Tc ) in H 2 corresponds to about 2 mK. Since the temperature proximities dealt with in this paper are far from this limit (10 mK to 1 K), we remain in the domain of validity of the relation. Reu ′2 term (taking into consideration the order of magnitude of u ′ ) is found to be of the order of 4 × 10 -24 , 8 × 10 -23 and 2 × 10 -21 respectively whereas the thermal diffusion 1 Re.P r term is of the order of 10 -10 , 6 × 10 -10 and 2 × 10 -9 respectively. The same order of magnitude analysis in the momentum equation gives, for the smallest amplitude a=1 mm and smallest frequency f = 2.77 Hz: u ′ Re ∼ 1.3 × 10 -13 , 10 -13 , 7 × 10 -14 and 1 F rv2 ∼ 4 × 10 -8 , 3 × 10 -8 , 1.7 × 10 -8 for ∆T = 0.01 K, 0.1 K and 1 K respectively. The above order of magnitude analysis therefore shows that the cases we deal with in our analysis are near-inviscid in nature.

In the equation 6.1 the first term is related to the gravitational waves and the second term to the capillary waves. For miscible fluids in the presence of gravity, the second term is negligible with respect to the gravity-induced term because the surface tension is very weak. Thus k is proportional to ω 2 g and the exponent for the power law dependence of λ with (aω) 2 is equal to b = -1 exactly as was observed in Ref [START_REF] Zoueshtiagh | Experimental and numerical study of miscible faraday instability[END_REF]. For the case of immiscible fluids, in zero-gravity, the first term in the dispersion relation being strictly zero, k is proportional to (ω 2 ) 1 3 and the power law dependence of λ with (aω) 2 gives b ∼ -0.33. The latter case compares well with our case of parametric instability in supercritical fluids (b ∼ -0.35 ± 5%). It can be understood by the influence of the non-zero weak surface tension associated with the boundary layer density gradients [START_REF] Lacaze | Transient surface tension in miscible liquids[END_REF], which thus dominates the behavior under zero-g. The slope of aω 2 (ρ 1 + ρ 2 ) vs k 3 plot gives the value of surface tension (as also suggested in Ref. [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] for a method of measurement of the surface tension of sub-critical fluids). Figure 6.8 shows ω 2 (ρ 1 + ρ 2 ) vs k 3 plots for various temperature proximities. The slopes of the curves are found out to be 1.28 × 10 The stability domains depicting the critical vibration amplitude (a cr ) vs frequency (f) and the critical dimensionless vibrational acceleration (Γ cr ) vs frequency (f ) are plotted in Figs. [START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF].9a and 6.10a for ∆T = 10 mK, 100 mK with quenches of 1 mK, 10 mK, respectively. The fluid is stable below the curves and unstable above them. It can be seen from Fig. 6.9a that the critical amplitude a cr decreases as the frequency f increases and saturates at higher frequencies. The curves for ∆T = 10 mK, 100 mK seem to converge for smaller frequencies. The critical acceleration plot Γ cr vs f (Fig. 6.10a) seems to approximately follow the power law Γ c = k 0 + k 1 f 2 . Extraction of data from the similar curve in the case of Faraday instabilities in the immiscible fluids [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] and fitting according to the above power law also gave 2 as exponent value. Simulations were carried out for a fixed critical point proximity (example ∆T = 100 mK) and fixed vibration parameters (f = 8.33 Hz and a = 15 mm) but for various quenches (δT = 10 mK, 8 mK, 6 mK, 5 mK and 1 mK) to see if the wavelengths of the parametric instabilities are modified by the thermal quenches applied. It was observed that the wavelength remains constant but the stability domain of the parametric instabilities is modified. Figures 6.9b and 6.10b show the stability domains for ∆T = 100 mK, and quenches of 10 mK and 5 mK. The behavior of the curves is as explained in the last paragraph. It is important to note that the amplitude of the quench has a great influence on the onset of the instabilities. Also important is to note that the (a cr ) vs (f ) curves (Fig. 6.9b) seem to diverge as the frequency of excitation is decreased, in contrast to what was observed in the cases studied in Fig. 6.9a.

Rayleigh-Vibrational instability

Vibration of a thermal boundary layer in a direction perpendicular to the thermal gradient results in a Rayleigh-vibrational instability. It occurs due to shear between the dense fluid inside the boundary layer, which has a higher momentum due to its higher density, and the less dense bulk fluid. The instability thus has a Kelvin-Helmholtz type origin. The term "Kelvin-Helmoltz instability" is however usually used for the case of two immiscible fluids with different velocities [START_REF] Talib | The influence of viscosity on the frozen wave instability: Theory and experiment[END_REF][START_REF] Talib | Instability of a viscous interface under horizontal oscillation[END_REF]. When only a single fluid phase is concerned, as it is the case in the present paper, we use rather the term "Rayleigh-vibrational instability". The Rayleigh vibrational instability is triggered by a Bernoulli-type pressure difference proportional to the velocity difference between the fluid element and its surroundings [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF]. The order of magnitude of the Bernoulli-type pressure difference ∆p, resulting from the relative velocity ∆v with the surrounding fluid can be expressed as ∆p ∼ ρ c ∆v 2 = ρ c ( ∆ρ ρc ) 2 a 2 ω 2 resulting in a driving force r 2 0 ∆p oriented perpendicular to the direction of the vibration (here r 0 is the dimension of the fluid element considered). The density difference is related to the thermal expansion coefficient according to the relation ∆T ρc = β p δT .In the above-mentioned lines of reasoning a non-dimensional Rayleigh-vibrational number can be defined as:

Ra v = (aωβ p ∆T δ T BL ) 2 2νD T (6.2)
When this number crosses a threshold value or in other sense, when a critical boundary layer thickness δ T BLc is crossed (which occurs at a "critical" time t cr ), the thermal boundary layer becomes unstable and leads to the formation of the Rayleigh vibrational instability. Amiroudine and Beysens [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF] observed in their simulations with CO 2 that the critical Rayleigh-vibrational number Ravc was strongly dependent on ∆T for fixed frequency and amplitude. It was seen to increase when approaching T c with power laws with two different exponents (-0.86 close to T c and -0.31 away from T c ). Their analysis was limited to high frequency (with respect to the inverse hydrodynamic time scales) and low amplitude conditions (a < 0.1h). In the present study, we widen the amplitude range (∼ 0.05h to 5h) for the high frequency case.

The value of Ravc is obtained from the simulations by extracting the time t cr at which the first signs of the instability are observed. This time is then used to calculate the critical boundary layer thickness δ T BLc = 2π √ D T t cr . This value is in turn used in the formula for the Rayleigh vibrational number (see Eq. 6.2) to obtain its critical value. Simulations are carried out for various amplitudes and frequencies of vibration. The results plotted in Fig. 6.11 show that Ravc depends strongly on a and f . Fig. 6.11a gives the evolution of Ravc with respect to f for a = 10 mm at two different temperatures (∆T = 0.1 K and 1K). Fig. 6.11b shows a similar plot but with respect to a for f = 5.56 Hz and for the same temperatures to the critical point (∆T = 0.1 K and 1 K). It can be noted that Ravc increases as the critical point is neared (shown by the curve for ∆T = 0.1 K being higher than that for ∆T =1 K). The explanation of the variation of Ravc as a function of temperature variation was given in [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF] and can be summarized as follows. The critical value Ravc (= 2129), as determined by the linear stability analysis by Gershuni and Lyubimov [START_REF] Gershuni | Thermal Vibrational Convection[END_REF], corresponds to the case of a layer in contact on both sides with infinite walls of thermal conductivity much larger than the fluid. In the present problem, the fluid is in contact with a finite solid wall on one side and with the bulk fluid on the other. This configuration corresponds to the non-stationary states in Ref. [START_REF] Gershuni | Thermal Vibrational Convection[END_REF] when the Prandtl number takes large values. The instability here develops in the TBL during the transient. High critical values of Ravc have been found also with CO 2 near its critical point [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF][START_REF] Lyubimov | Stability of a thermal boundary layer in the presence of vibration in weightlessness environment[END_REF]. The fact that Ravc increases with both acceleration and frequency can be explained as follows. The linear stability analysis carried out in Refs. [START_REF] Legendre | Instabilités à l'interface entre fluides miscibles par forçage oscillant horizontal[END_REF][START_REF] Lyubimov | Stability of a thermal boundary layer in the presence of vibration in weightlessness environment[END_REF] uses a small amplitude assumption (aβ p δT ≪ h) enabling the application of averaging method. In this method nonlinearities in the pulsating field are neglected and kept only in the average field. It thus results in a Ravc value that does not vary with a or f . This simplification is no longer valid for the cases considered in the present study where aβ p δT can be of the same order or even larger than h as can be observed in Table 6.2 for various proximities to the critical point and for a vibration amplitude of 30 mm. Fig. 6.12 shows the evolution of Ravc vs ∆T for three frequencies (f = 2.78 Hz, 5.56 Hz and 8.33 Hz) and for a constant vibration amplitude a = 10 mm. The solid lines indicate the two power law curves with exponents -0.83 and -0.31 according to the data published for CO 2 by Amiroudine and Beysens [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF] and recalculated for the same T -Tc Tc . It can be seen that the data points follow approximately the same power law as for CO 2 . It can also be observed that, as f increases, Ravc also increases and the power law curves displace vertically along the y axis. Figures 6.13a and 6.13b shows the stability curves for the Rayleigh-vibrational instability depicting critical vibration amplitude (a cr ) vs frequency (f ) and critical vibrational velocity (af ) cr vs frequency f for two critical point proximities 10 mK and 100 mK. It can be seen that the critical amplitude (Fig. 6.13a) decreases with an increase in the frequency of vibration. The curves seem to converge as the frequency decreases similar to the case of the parametric instabilities. When the critical velocity (af ) cr is plotted with respect to f (Fig. 6.13b) a linear curve with a non-zero intercept is obtained. This non-zero intercept shows that when f → 0, a cr → ∞. It simply reflects the fact that the critical Rayleigh vibrational number is proportional to the critical velocity of vibration Ravc ∼ (af ) cr .

Conclusion

A 2D finite volume numerical modeling of thermo-vibrational instabilities has been performed in a square cell filled with supercritical H 2 . Three kinds of instabilities have been put in evidence and analyzed: corner, Rayleigh-vibrational and parametric instabilities. The latter is well documented in immiscible [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] and miscible [START_REF] Zoueshtiagh | Experimental and numerical study of miscible faraday instability[END_REF] fluids. It has, however, not yet been evidenced in one-phase thermal boundary layers. Figures 6.14a and 6.14b show the combined stability diagrams of the three types of instabilities for ∆T = 0.1 K and 0.01 K. It can be seen from the plots that there are zones in the stability diagram where all the instabilities are simultaneously observable. For example, the region shown by point 1 in Fig. 6.14a indicates the region in which corner, Rayleigh vibrational and parametric instabilities are simultaneously observable while the region shown by 2 indicates simultaneous presence of corner instability as well as Rayleigh vibrational instability. The regions 3 and 4 indicate the presence of only Rayleigh vibrational instability or only corner instability respectively. A crossover between the Rayleigh-vibrational instabilities and the corner instabilities can be observed. The behavior of the three stability curves does not seem to change much with the change in the critical point proximity as can be observed in Fig. 6.14b. Comparing these three kinds of instabilities, it is seen that for a given frequency of vibration, the parametric instability needs substantially higher amplitudes to appear than the two other types of instabilities. Simulations with all four walls isothermal subjected to a thermal quench showed that the parametric instability has a stabilizing effect on the Rayleigh vibrational instability. For the corner instability the stability domain reduces as one approaches the critical point. Similar kind of behavior is observed for the Rayleigh-vibrational and parametric instabilities. It was observed that the wavelength of the parametric instability varies according to a power law with an exponent of approximately -0.35, which is the same as the exponent for classical Faraday instabilities in immiscible fluids under zero-gravity. The values of the effective surface tension were calculated and it was observed that the surface tension decreases as the critical point is approached. The critical vibration acceleration vs angular velocity plot for the parametric instabilities showed an increase in the vibration acceleration with the square of the angular velocity. This is in conjunction with the results of Ref. [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] for the onset of Faraday waves in immiscible fluids subjected to vertical vibration (however in the presence of gravity). It can thus be concluded that the parametric instabilities in supercritical fluids are similar to the Faraday waves in immiscible fluids. For the Rayleigh-vibrational instability it was observed that the critical Rayleigh-vibrational number varies with the amplitude and frequency of vibration as against a constant value (2129) calculated by Gershuni and Lyubimov [START_REF] Gershuni | Thermal Vibrational Convection[END_REF]. The latter considered a semi-infinite layer with amplitudes of vibration much smaller than the distance between the two walls, thus, authorizing the application of averaging methods involving linearization of the pulsating field. In contrast, the configuration studied here involves vibration amplitudes that can be larger than the cell size, thus inducing non linearities responsible of the variation of the critical Rayleigh vibrational number. The 2D simulations presented in this paper are assumed to be sufficient to accurately predict the three kinds of instabilities. The Rayleigh-vibrational instabilities observed earlier in experiments onboard sounding rockets and in experiments at CEA Grenoble [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF][START_REF] Beysens | The effect of vibrations on heterogeneous fluids: Some studies in weightlessness[END_REF][START_REF] Beysens | Phase transition under forced vibrations in critical co2[END_REF][START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF] in cylindrical as well as in cubical cells, were successfully reproduced by 2D simulations [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF]. It is possible to complement the available very few experimental results with these 2D simulations to gain more understanding. However, they need to be validated using suitable experiments.

Chapter 7

Stability of a liquid-vapor interface subjected to vibration Resumé (Français) L'interaction d'une interface liquide-vapeur avec des vibrations peut donner plusieurs types d'instabilités selon la direction relative de l'interface et de la vibration. L'instabilité de type Faraday est obtenue quand l'interface est vibrée perpendiculairement. Elle est produite à cause de l'excitation paramétrique de l'interface. Les ondes gelées sont des instabilités de type Kelvin-Helmholtz et elles sont obtenues quand l'interface est tangentielle par rapport à la direction des vibrations. Les ondes de Faraday apparaissent quand la différence des densités entre les deux phases est grande alors que les ondes gelées apparaissent quand la différence des densités est faible. Dans ce chapitre, deux types d'instabilités sont étudiés expérimentalement dans l'hydrogène proche du point critique: l'instabilité de Faraday en absence de gravité et les ondes gelées pour différents niveaux de gravité. Des résultats intéressants ont été obtenus. Le but de ce chapitre est de présenter ces résultats et aussi d'expliquer un phénomène étrange observé en zéro-g: la disparition des ondes Faraday suivie par l'apparition de bandes liquide-vapeur perpendiculaires à la direction de la vibration.

Introduction

Vibration can induce average motion inside a fluid with density non-homogeneities. The density non-homogeneities can result from the fact that the fluid domain is composed of various fluid phases with different densities or due to local heating inside the fluid. Based on the relative direction of the gravity field and the vibration, the fluid can be either stabilized or destabilized depending on vibrational parameters (amplitude and frequency of vibration) and physical properties (density contrast, interfacial tension etc). For example, a Rayleigh-Taylor configuration consisting of a heavier fluid initially placed over a lighter fluid under non-zero gravitational field (directed downwards) can be stabilized by applying a strong vertical vibration [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF]. Similarly, various types of instabilities can be provoked by vibration. Under vibrations normal to the interface, a parametric instability in the form of Faraday waves can develop. Under vibrations tangential to the interface, a Kelvin-Helmholtz (K-H) type of instability [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF][START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF][START_REF] Khenner | Stability of plane-parallel vibrational flow in a two-layer system[END_REF] can appear. Working with a fluid system involving a liquid layer surrounded by its own vapor has many advantages. As has earlier been discussed in the chapter 2, close to the critical point, various fluid properties such as thermal expansion coefficient, isothermal compressibility, thermal conductivity etc., diverge while properties like surface tension, liquid-vapor density difference and coefficient of thermal diffusivity etc., vanish. So just by heating or cooling the fluid (or in other words, by approaching or moving away from the liquid-vapor critical point) one can modify the properties of the liquid and vapor phases and thus study the stability of the interface for a variety of fluid conditions without having to change the fluid combination. Thus using a near-critical fluid, a single experiment is sufficient to study the effect of various factors on the stability aspects of a fluid interface. The present chapter deals with Faraday wave instability at zero-g and frozen wave instability [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF][START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF] under various gravity levels. When a vapor bubble is subjected to vibration under zero-g conditions, some parts of its surface turn out to be nearly perpendicular to the vibration direction and the other parts are parallel to the vibration direction. In this situation both Faraday and K-H instability could arise at different parts of the bubble surface. However, the conditions for the excitation of these two types of instability are different; they would thus develop in different parameter ranges. Vibration experiments close to the critical point are able to show both types of instabilities. In the present chapter, section 7.2 presents a detailed experimental study of the Faraday wave instability under weightlessness. Section 7.3 presents the experimental investigation of the frozen wave instability (a type of Kelvin-Helmholtz instability) for various gravity levels. In Section 7.4 the coexistence of the Faraday instability and the Kelvin-Helmholtz instability is discussed with some experimental evidence.

Faraday wave instability 7.2.1 Introduction

When a vessel containing a liquid is vibrated perpendicular to its surface a pattern of standing waves is often observed. These waves occur due to the parametric excitation of the interface under vibration. These waves were first observed by Faraday [START_REF] Faraday | On a peculiar class of acoustical figures and on certain forms assumed by groups of particles upon vibrating elastic surfaces[END_REF] and are widely known as Faraday waves. They oscillate at a frequency equal to half the frequency of vibration. Study of Faraday waves has since then been widely carried out theoretically as well as experimentally. Most of the Faraday wave experiments involve a liquid layer separated by air, but very few experiments involve a liquid separated by its own vapor. The only known study of the dynamics of a vibrated liquid-vapor interface is from Fauve et al. [START_REF] Fauve | Parametric instability of a liquid-vapour interface close to the critical point[END_REF]. They have carried out experiments on a liquid-vapor (L-V) interface of Carbon dioxide (CO 2 ) under terrestrial gravity and close to its critical point. They reported a detailed study of the Faraday wave instability close to the critical point. They demonstrated that the measurement of instability wavelength and the critical acceleration at the instability onset can help to determine important characteristics of the L-V interface like the capillary length and the relative density difference. They reported that the instability wavelength saturates to a finite value very close to the critical point. Also, they observed that at such a proximity to the critical point, the most preferred instability pattern is a "one-dimensional standing wave pattern" rather than the usual square pattern. They observed that the dispersion curve obtained by Benjamin and Ursell [START_REF] Benjamin | The stability of the plane free surface of a liquid in vertical periodic motion[END_REF] which neglects viscous effects is insufficient to describe the behavior of the Faraday waves sufficiently close to the critical point where the line pattern is the preferred pattern. In this section we present the results of the experiments carried out on near-critical Hydrogen (H 2 ) under zero-g conditions. Working under zero gravity conditions has two major advantages: (a) it removes the effect of density stratification of the fluid, due to its hyper-compressibility, on the instabilities, (b) the behavior can be put on universal, master curves, valid for all fluids.

Experimental cell

The experiments are carried out with hydrogen as the working fluid using the facility HYLDE, described in the sections 4.2. The experimental cell used is a cuboidal cell of side 3 mm as shown in the Fig. 4.10a. The experiments were carried out with a ferromagnetic insert and as was explained in section 4.2.4, a uniformity close to 1% can be achieved inside the cell. To carry out the experiments the cell is filled with H 2 . The experiments are performed within 24 hours after filling the cell with H 2 at room temperature, thus the working fluid is n -H 2 (a 3:1 composition of the spin isomers ortho and para hydrogen, stable at 300 K). The critical point of n -H 2 is defined by: T c = 33.19 K, p c = 1.315 M.Pa, ρ c = 30.11 kg.m 3 [START_REF] Zappoli | Supercritical fluid hydrodynamics[END_REF]. The filling is carried out at the critical density of H 2 . To determine the critical density, the cell is filled up to half its height at a temperature very close to the critical point (say 50 mK) and the meniscus is monitored for small temperature increase or decrease of the cell (50 mK on either side of the filling temperature). If the level of the meniscus does not change with temperature, it means that the cell is filled at its critical density. This method of filling the cell at the critical density of the fluid is quite precise and is close to 0.2% of the critical density ρ c . Zero-gravity conditions are then established inside the cell by switching on the superconducting coil and passing 68.5 Amps current through it. Figure 7.1 a shows the H 2 bubble in zero-g when the total compensation of gravity is achieved at the center of the cell. In the picture we can see that the vapor bubble is spherical and that the liquid surrounds it, sticking to the walls of the cell. The thick dark circular portion separating the liquid and the vapor phases is the meniscus of the liquid vapor interface, appearing thick due to the partial refraction of light across it. The perfect spherical shape of the bubble inside the cell shows that the bubble is in zero-g. Figure 7.1b shows the shape of the bubble as T c -T is decreased from 38 mK to 11.5 mK. It can be seen that close to the critical point the interface becomes flatter due to the decrease in the capillary length. The latter varies with temperature according to:

l c = σ ∆ρg ∼ ε 0.47 (7.1)
where, g is the gravity level, in this case, the residual gravity (of the order of 0.01g 0 ), σ corresponds to the surface tension, ∆ρ is the density difference between liquid and vapor phases and ε = Tc-T Tc . The surface tension and the liquid-vapor density difference obeys the scaling laws σ = σ 0 ε 1.26 (7.2) and ∆ρ = 2Bρ c ε 0.325 (7.3) respectively, where σ 0 = 0.00542 N.m -1 and B = 1.62 for n -H 2 close to the critical point [START_REF] Zappoli | Supercritical fluid hydrodynamics[END_REF]. The interface deforms close to the walls when the critical point is approached due to the remaining g levels, since the uniformity achieved inside the cell is of the order of 0.01g 0 .

Determination of the fluid temperature

It is important to mention that the experiments have a serious limitation with respect to the measurement of the temperature. Since the thermometers fixed to the cell vibrate along with the cell inside an intense magnetic field, eddy currents are induced inside them and their electric cabling, provoking unwanted oscillations in the temperature electric signal. This renders the values of the temperature during the vibration experiments useless. These limitations can however be surmounted by estimating the temperature with the measure of the thickness of the meniscus of the liquid. The thickness of the meniscus (t m ) is directly dependent on the capillary length (l c ) of the interface and the liquid-vapor refractive index difference (∆n) i.e., t m ∼ ∆nl c . The capillary length of a liquid-vapor interface varies with temperature as ε 0.47 (see Eq. 7.1) and the refractive index between vapor and liquid phases varies as the density difference between the two phases [START_REF] Zappoli | Supercritical fluid hydrodynamics[END_REF], that is:

∆n ∼ ∆ρ ∼ ε 0.325 (7.4)
We thus can expect the thickness of the meniscus to vary with temperature as t m ∼ ε 0.79 . The perfectly spherical shape of the bubble can be explained as follows: far from the critical point, the thickness of the meniscus is large so that the non-uniformity of the gravity field near the liquid layer (close to the walls) cannot be felt by the vapor bubble. But closer to the critical point (see Fig 7 .1b), the shape is no more perfectly symmetrical as the thickness of the meniscus has increased and the vapor bubble now feels the non-uniform gravity close to the walls. The proportionality constant of the above relation can be estimated by fitting it to the theoretically estimated values of the temperature for the onset of the Faraday wave instabilities as will be seen subsequently. Figure 2b shows the variation of the thickness of the meniscus and the temperature is calculated from the above correlation. Fig 7.1c shows the scaling of the meniscus thickness with respect to the temperature. The solid line shows the scaling according to the law t m ∼ ε 0.79 . The meniscus thickness is measured from the pictures of Fig 7 .1b and the theoretical curve (solid line) is used to determine the corresponding value of the temperature which is given in Fig 7 .1b. This method is precise and temperatures measured using this method match well with those of the expected values from the theoretical dispersion relation (see below).

Results

To carry out the experiments the cell is initially maintained at a temperature slightly lower than the critical temperature, T c . The amplitude of vibration, a, is set to a fixed value (by adjusting the eccentricity of the cam) and the cell is set to vibration while simultaneously heating it at a very slow rate (∼ 20 mK.min -1 ) towards the critical point. The experiment is started with a small value of frequency (f ) and is repeated by increasing the value of the frequency. For each value of the pre-set amplitude of vibration, critical values of the frequency (f c ) corresponding to various wavelengths of the Faraday wave instability are obtained. Given that the video camera used to carry out the experiments is of 25 fps, the fact that the frequency of oscillation of the fingers is equal to half the frequency of vibration was verified for vibrational frequency upto 25 Hz. The figures show various rows of fingers when viewed from a lateral direction for a typical case with five fingers inside the domain. Figure 7.3a corresponds to the case where the rows of fingers are staggered and this kind of an arrangement is called a square pattern. Figure 7.3b shows a case in which the rows of fingers are aligned; this arrangement is called the line pattern. Fig. 7.2 shows that below a particular value of T c -T , the Faraday fingers align giving an impression that the fingers penetrate deeply into the domain. In the figures, A and B are two rows of Faraday fingers in the domain. The two rows are not in the same plane but are in two parallel planes in the lateral direction. It can be seen from these figures that at 11.5 mK and 10.5 mK the two rows of fingers A and B have a phase difference of π radians whereas at 7.5 mK and 7 mK the two rows of fingers are perfectly aligned. This phenomenon is exactly what has been described in Fauve et al. [START_REF] Fauve | Parametric instability of a liquid-vapour interface close to the critical point[END_REF], that is, a transition from a square pattern (figures corresponding to 11.5 mK, 10.5 mK) to a line pattern (figures corresponding to 7.5 mK and 7 mK). The image corresponding to 9 mK indicates an intermediate phase in the transition from square pattern to line pattern. Series of experiments consistently showed this transition. It is also observed that this transition occurs for higher values of vibrational velocity (aω) very close to the critical point. It is suggested in [START_REF] Fauve | Parametric instability of a liquid-vapour interface close to the critical point[END_REF] that the above transition is due to strong viscous dissipation. The viscous dissipation per unit volume between two adjacent fingers (with wavelength λ) can be written to be proportional to µu 2 f λ 2 , where, µ is the fluid viscosity (equal for both phases close to the critical point) and u is the velocity of the fingers (of the order of aω). Thus the viscous dissipation term is of the order of µ(aω) 2 f λ 2 . It can be seen that the viscous dissipation increases as aω is increased or λ is decreased, µ being almost independent of the temperature near the critical point. The viscous dissipation term increases as the critical point is approached. The vibrational energy term which is proportional to ∆ρ(aω) 2 decreases with a power law exponent of 0.325 as T c is approached. We can define a non-dimensional number V as the ratio of the viscous dissipation energy to the vibrational energy:

V = µ f λ 2 ∆ρ (7.5)
A mapping of the viscous dissipation versus the vibrational energy for a= 0.55 mm and various frequencies is shown in Fig. 7.4a. The solid part of the curve indicates the conditions under which the square pattern is observed whereas the dashed line indicates the conditions suitable for the line pattern formation. The arrow marks show the direction of temperature increase along the iso-frequency curves. It can be seen that as the temperature approaches the critical point, the viscous dissipation increases while the vibrational energy decreases. Figure 7.5 shows the plots of the temperature (T c -T ) and wavelength for the square pattern-line pattern transition over a range of vibrational velocities (aω). Above the curves Faraday waves have square pattern while, below the curves, line pattern is observed. It is observed that the critical transition temperature (T c -T ) c from square to line pattern (Fig. 7.5a) increases as aω is increased and saturates to a constant value for higher values of aω. The critical wavelength also increases as aω is increased and subsequently saturates to a constant value for higher values of aω (Fig. 7.5b). The transition from square to line pattern happens when the viscous dissipation energy becomes more than about 10% of the vibrational energy.

Stability domain of the Faraday wave instability

The Faraday wave instability under weightlessness can be modeled as two layers of immiscible and incompressible viscous fluids subjected to vibration perpendicular to the interface in the absence of gravity. The stability problem derived from the Navier-Stokes equations was treated by Kumar and Tuckerman [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] using a Floquet analysis. They carried out numerical calculations based on the linearized Navier-Stokes equations and analytical calculations by introducing the viscosity in a phenomenological way into Mathieu equation for low viscosity case. The dispersion relation for zero-gravity conditions in low viscosity case is given by: where, ρ l and ρ v , µ l and µ v are the densities and the dynamic viscosities of the liquid and the vapor phases respectively. Figure 7.6a shows the variation of the wavelength at the onset of Faraday waves (critical wavelength λ c ) with respect to the frequency of vibration for various values of T c -T ranging from 6 mK to 12 mK. The symbols are the experimental values while the solid lines correspond to the values predicted by the dispersion equation 7.6. It is observed that the experimental values agree well with those predicted by the dispersion equation for T c -T more than 7.5 mK. However for T c -T = 6 mK we can see that the experimental values do not fall exactly on the theoretical curve. Errors in the measurement of temperature using the method of meniscus thickness cannot be completely excluded given that the accuracy is strongly dependent on the resolution of the image. For small values of T c -T these errors can be non-negligible. It can be observed from the plots that for a particular proximity to the critical point the value of the wavelength at the onset decreases as the frequency of vibration is increased. It can also be seen that a particular wavelength of instability onset is obtained farther from the critical point for higher frequencies of vibration. It is true that the effects of compressibility cannot be neglected. But interestingly, the good match of the experimental values with theory indicate negligibly small effect of the compressibility for the amplitudes and frequencies of vibration considered.

(ω/2) 2 ≈ σk 3 ρ l + ρ v -4k 4 ( µ l + µ v ρ l + ρ v ) 2 (7.6)
It is because the mach numbers associated are very small. Figure 7.6b shows the values of ω 2 /k 3 versus the wave number k for T c -T = 7.5 mK. The dots are the experimental values while the solid line is a linear fit.

It can be seen that ω 2 /k 3 varies, as expected, linearly with respect to k with a negative slope. The y-intercept of the curve corresponds to the contribution of surface tension and the slope of the curve gives the contribution of the viscous term (see details in Eq. 7.6). The surface tension calculated from the y-intercept is 1.1 × 10 7 N m -1 as compared to the analytical value 1.3 × 10 -7 N m -1 from Eq. 7.2. This agreement is a cross-check of the method of determination of the temperature values from the thickness of the meniscus. The stability domain of the Faraday instability plotting the critical amplitude of vibration versus critical frequency of vibration is shown in Fig. 7.7a. The stability curves for constant critical point proximities are plotted as solid lines whereas the stability curves for constant wavenumbers are plotted in dotted or dashed lines. It can be seen that for a constant temperature, the critical vibration amplitude decreases with an increase in vibration frequency. Also for a constant wavenumber, the critical amplitude decreases with an increase in the frequency of vibration. The non-dimensional critical vibrational acceleration A c /g(= acω 2 c g ) against frequency for various values of T c -T is shown in Fig. 7.7b. It can be seen that the critical acceleration increases with an increase in the frequency of vibration according to:

A c /g = k 1 + k 2 ω 2 (7.7)
in agreement with the results of Kumar et al [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF].

Conclusion

The experiments on near-critical Hydrogen under magnetic compensation of gravity and under vibration and weightlessness have put in evidence Faraday instability in zero-g. The results seem to be in good agreement with the dispersion relation derived by Kumar and Tuckerman [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF]. Stability diagrams for various temperatures and various wavenumbers are plotted. The critical amplitude vs frequency plots seem to follow a quadratic law, in conjunction with Kumar and Tuckerman [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF]. It was observed that the Faraday waves chose a square pattern away from the critical point. For temperatures very close to the critical point, where the influence of viscosity dissipation is predominant, a transition from square pattern to line pattern is observed, similar to the results of Fauve et al. A [START_REF] Fauve | Parametric instability of a liquid-vapour interface close to the critical point[END_REF].

7.3 Frozen wave instability and the effect of gravity

Introduction

Horizontal vibration applied to a vessel containing two immiscible fluids destabilizes the initially flat interface and induces a wavy profile on the interface. This wavy profile is stationary and does not change phase with time, in a reference frame fixed to the cell and thus is called frozen wave instability.

Wolf [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF] reported waves with a "saw-tooth structure", when subjecting a fluid interface to horizontal vibration. These waves have a waveform unchanging (or frozen) with time in a reference frame fixed to the experimental cell. The name "frozen waves" for this type of waves has been coined in Wunenburger et al. [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF]. Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] carried out a linear stability analysis of the frozen wave instability for two incompressible immiscible fluids. Their analysis was based on the fact that the frozen waves do not change with time and are thus associated with averaging effect. They accordingly carried out a stability analysis using the "high frequency" approximation where the vibration period is small compared to the typical hydrodynamic time scales and the vibration amplitude is small compared to the sample length scale. The method permitted to split the general problem, using incompressible Navier-Stokes equations, into fluctuating and averaged parts. The equations for the fluctuating part, which are linear under accepted approximation, are solved analytically and then the equations for average components are solved using linear stability analysis and weakly non-linear calculations. They explained that the instability mechanism for the frozen waves is similar to what is found in the classical case of the Kelvin-Helmholtz instability. They showed that the frozen wave instability is associated with a threshold. The bifurcation curve derived by them clearly demonstrated that the frozen wave instabilities are possible only for fluids with comparable densities but not for a free surface. They also showed that at the onset the most preferred wavelength of the instability is proportional to the capillary length of the fluid interface. Lyubimov and Lyubimova [START_REF] Lyubimov | Modelling in Mechanics[END_REF] numerically investigated the onset and non-linear dynamics of frozen waves in the framework of the high-frequency approach. The numerical results on critical conditions for instability and bifurcation type are found to be in good agreement with Ref. [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. It is shown that with an increase in the vibration velocity above some critical value the wave height grows until it reaches the boundaries. Thus at high values of the vibrational velocities, when vibrations play dominating role and gravity role is negligible, the system is split into "strata", i.e. domains occupied by the two different fluids with interfaces perpendicular to the vibration direction. Wunenburger et al. [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF] and Beysens et al. [START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF] reported experimental results of the frozen wave instability in liquid CO 2 in equilibrium with its own vapor close to its critical point under 1-g (terrestrial gravity conditions). They carried out experiments by varying the temperature of the cell as a result of which the density difference between the liquid and vapor phases and the surface tension of the interface could be varied. Their results show a good agreement with the results of Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] and Khenner et al. [START_REF] Khenner | Stability of plane-parallel vibrational flow in a two-layer system[END_REF] carried out a linear stability analysis of the problem of two immiscible fluids subjected to horizontal vibration. They explained that the instability mechanism for the frozen waves is similar to the Kelvin-Helmholtz instability where the velocity grows and the pressure falls over the elevation of the interface. Interface deformations can thus grow beyond a threshold. They investigated the effect of viscosity on the instability with finite frequency and amplitude considering the case of equal kinematic viscosities of two fluids. Ivanova et al. [START_REF] Ivanova | Interface dynamics of immiscible fluids under horizontal vibration[END_REF] investigated the stability aspects of the frozen wave problem at an interface between two immiscible fluids with a small surface tension. They carried out a quantitative comparison of experiments with a theoretical analysis of the interface stability in the purview of the high frequency assumption. They showed that, as the surface tension vanishes, the stability of the interface depends on only the ratio of the velocities of the interface and gravity waves. Legendre et al. [START_REF] Legendre | Instabilités à l'interface entre fluides miscibles par forçage oscillant horizontal[END_REF] reported a study of the frozen wave pattern in miscible fluids. They showed a good agreement of their results with those of Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. They were also successful in estimating the effective surface tension of the transient interface between two miscible fluids. Talib et al. [START_REF] Talib | The influence of viscosity on the frozen wave instability: Theory and experiment[END_REF][START_REF] Talib | Instability of a viscous interface under horizontal oscillation[END_REF] and Jalikop and Juel [START_REF] Jalikop | Steep capillary gravity waves in oscillatory shear driven flows[END_REF] carried out an experimental in-vestigation and a linear stability analysis of two immiscible fluids superimposed over each other to study the effect of viscosity contrast on the frozen wave instability onset. Their analyzed the effect of the viscosity contrast N 1 (= ν 1 /ν 2 ) on the threshold of the frozen wave instability for N 1 in the range of 1 to 10 4 and identified four regions of N 1 in which the instability thresholds display different behaviors. An experimental study of the nonlinear growth of the frozen wave instability was performed by Jalikop and Juel [START_REF] Goda | A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows[END_REF] for a few combinations of viscosity contrasts. They varied the vibrational Froude number (W ) and identified a critical value (W c ) at which the wave undergoes a transformation from a weakly nonlinear state to a strongly nonlinear state. This transition apparently corresponds to a qualitative change in the variation of the wave heights and the wavelength. The studies by the same group of authors [START_REF] Talib | The influence of viscosity on the frozen wave instability: Theory and experiment[END_REF][START_REF] Talib | Instability of a viscous interface under horizontal oscillation[END_REF][START_REF] Jalikop | Steep capillary gravity waves in oscillatory shear driven flows[END_REF] found that for finite frequencies of vibration, the model of Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] underestimates the threshold for fluids of equal viscosities, but overestimates the threshold for fluids with large viscosity contrast. In both cases, however, with an increase in the dimensionless frequency, the threshold values found in [START_REF] Talib | The influence of viscosity on the frozen wave instability: Theory and experiment[END_REF][START_REF] Talib | Instability of a viscous interface under horizontal oscillation[END_REF][START_REF] Jalikop | Steep capillary gravity waves in oscillatory shear driven flows[END_REF] approach the values obtained in the high frequency approximation in [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. Yoshikava and Wesfreid [START_REF] Yoshikawa | Oscillatory kelvin helmholtz instability -part1[END_REF][START_REF] Yoshikawa | Oscillatory kelvin helmholtz instability -part2[END_REF] studied the frozen wave phenomenon using harmonically oscillated vessel filled with a fluid. They investigated the effects of viscosity contrast and the ratio of the amplitude of vibration with respect to the wavelength of the instability on the onset of the frozen wave instability. In [START_REF] Yoshikawa | Oscillatory kelvin helmholtz instability -part1[END_REF], they carried out a linear stability analysis of two thick viscous fluid layers of oscillatory flows for the asymptotic case of small K (ratio of the amplitude of vibration with respect to the wavelength of the instability). They showed that the asymptotic theory predicts a significant influence of the oscillation frequency and the viscosity contrast on the stability threshold. Comparison of the results of their theoretical predictions with that of Talib et al. [START_REF] Talib | The influence of viscosity on the frozen wave instability: Theory and experiment[END_REF][START_REF] Talib | Instability of a viscous interface under horizontal oscillation[END_REF] showed good agreement. In [START_REF] Yoshikawa | Oscillatory kelvin helmholtz instability -part2[END_REF], Yoshikava and Wesfreid carried out experiments with high viscosity contrast fluids and compared the results with the low amplitude linear stability theory of [START_REF] Yoshikawa | Oscillatory kelvin helmholtz instability -part1[END_REF]. The stability problem solved numerically for a high amplitude case was shown to agree well with the experiments. In this section, results of frozen wave experiments carried out on near-critical H 2 at various g levels are presented. A brief description of the experimental cell used to carry out the experiments is first given followed by the results of the experiments and some general conclusions at the end of the section.

Experimental procedure

The experiments are carried out with H 2 as the working fluid using the cryogenic facility HYLDE, described in the section 4. The experiments are carried out with a soft iron insert. It was earlier seen that with an insert we obtain a uniformity of 3% inside a cell of 7 mm side. To carry out the experiments the cell is first maintained at a temperature slightly lower than the critical temperature T c (say T c -50 mK) and then horizontal vibration is applied. The temperature of the cell is increased very slowly (20 mK/min) so that the fluid can be assumed to be under thermodynamic equilibrium at all temperatures. Experiments are conducted for various combinations of amplitudes (ranging from 0.2 mm to 0.8 mm) and frequencies (between 10 Hz and 35 Hz).

Various gravity levels: 0.18g 0 , 0.1g 0 , 0.05g 0 and 0g 0 (g 0 is the earth gravitational acceleration) are obtained by adjusting the current intensity in the coil.

Determination of the fluid temperature

It is important to mention that the experiments have a serious limitation with respect to the measurement of the temperature. Since the thermometers fixed to the cell vibrate along with the cell inside an intense magnetic field, eddy currents are induced inside them and their electric cabling, provoking unwanted oscillations in the temperature electric signal. This renders the values of the temperature during the vibration experiments useless. Thus an indirect method of estimating the cell temperature needs to be used which is described as below.

According to [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF], frozen wave instability can be characterized by 4 parameters: the effective gravity (g), the vibrational velocity (aω), the surface tension (σ), the liquid-vapor density difference (∆ρ = ρ l -ρ v ). As the behavior of these quantities is universal near the critical point, the results of the frozen wave experiments carried out by Wunenburger et al. [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF] and D.Beysens et al. [START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF] on another fluid (near-critical CO 2 ) can be used to determine specific behavior. In other words, the behavior of the wavelength of the frozen waves for super-critical values of aω (i.e., aω > (aω) c , the critical vibrational velocity) in the framework of the linear stability analysis of by Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] is used to estimate the temperature of the cell in our experiments. The analysis in [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] is valid when the dimensionless vibration frequency is large (Ω = ωh 2 ν ≫ 1) and the dimensionless vibration amplitude is small (a/h ≪ 1, h is the container size). The first assumption allows the fluctuating flow to be considered as inviscid since the thickness of the Stokes boundary layer is small in comparison with the container size. The second assumption permits to neglect the non-linear terms in the equation of the fluctuating flow. In our experiments the dimensionless vibration frequency Ω is varied in the range of 3000 to 15000 and the dimensionless vibration amplitude in the range of 0.01 to 0.1. The applicability of the inviscid theory of the linear stability analysis of Lyubimov and Cherapanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] can then be assessed for the present results. The stability problem considered by [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] is concerned with two immiscible incom-pressible fluids of different densities, the lighter fluid over the heavier one, filled in a 2D vessel of height H and vibrated under a gravitation field with frequency ω and amplitude a. The onset of the instability is determined by performing a linear stability analysis considering the viscous terms as negligibly small and using a high frequency approximation. A perturbation of wavelength ω becomes unstable when:

(aω) 2 ≥ 1 2 (aω) 2 c ( λ λ c + λ c λ ) tanh ( πH λ ) (7.8)
where, λ c is the critical wavelength defined as λ c = 2πl c , l c being the capillary length, (aω) c is the critical vibrational velocity defined as:

(aω) 2 c = (ρ l + ρ v ) 3 ρ v ρ l (ρ l -ρ v ) ( σg ρ l -ρ v ) 1 2
(7.9)

The method used to estimate the temperature values is described as follows. The wavelength of the frozen waves (λ) is assumed to be related to aω, (aω) c and λ c according to the relation:

λ = l c ( aω (aω) c ) m (7.10)
where, m is an exponent to be determined. This exponent m can be obtained by fitting the λ vs T c -T curves of [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF] and [START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF] to the above equation (note that ( aω (aω)c ) depends on (T c -T )), which is found out to be equal to 1.2. We make an assumption that this exponent is universal and thus can be applied to H 2 . It is interesting to express the capillary length l c in terms of aω, ∆T (= T c -T ) and g. The quantities σ and ∆ρ obey the scaling laws as described in eqs. 7.2, 7.3 respectively. From equations 7.1 -7.3 and 7.8 -7.10, the λ vs aω relation can empirically be written as: λ = 0.075 ∆T 0.38 g 0.8 (aω) 1.2 (7.11) Equation 7.11 can thus be used to estimate the temperature of the cell corresponding to the wavelength of the frozen wave pattern. However, for temperatures very close to the critical point, where the wavelength of the instability is very small, there exists a strong viscous dissipation due to stronger velocity gradients. This effect is not taken into consideration for the temperature estimation. Thus minor deviations in the temperature values at temperatures very close to the critical point are unavoidable.

Results

Stability diagram for the frozen wave instability

It is possible to obtain experimentally the values of the critical amplitude a c and critical frequency f c of the frozen wave instability for various gravity levels. For instance, the value of the critical frequency of onset for a given gravity level and for a particular amplitude is obtained by carrying out experiments by varying frequency each time until the instability onsets. This is done to obtain the critical values of frequencies for 0.18g 0 , 0.1g 0 and 0.05g 0 for various values of amplitudes. The determination of the critical values does not involve any measurement of temperature and thus are obtained without taking into account the Eq. 7.11. The theoretical values can be derived from Eqs. 7.8-7.10. Since the temperatures at which the onset happens in the experiments are not known, we make an assumption that, if the instability does not occur even for T c -T = 1 mK, then the couple (a,ω) lies in the stable zone of the stability diagram. This temperature value of 1 mK corresponds to the value beyond which the effects of shear flow and viscous dissipation completely blur the liquid-vapor interface, rendering the frozen waves unobservable. The value of the critical velocity (aω) c corresponds to the velocity calculated at T c -1 mK using the Eqs. 7.3, 7.2. Vapor density data (ρ v ) from NIST [42] is used to calculate the value of (ρ l ) using Eq. 7.3. The comparison of the critical curves obtained experimentally and theoretically is carried out in the Fig. 7.9. We can see a good agreement between the experimental and the theoretical curves. This validates the applicability of the inviscid linear stability dispersion equation 7.8 to our experiments. The minor deviation in the values is due to the fact that the critical frequencies measured in the experiments have an uncertainty of ± 2.5 Hz (as shown by the error bars in Fig. 7.9). This is due to the fact that the frequency is changed in steps of 5 Hz to find out the critical frequencies corresponding to each vibrational amplitude.

Amplitude measurements of the frozen wave instability

.10 shows the frozen wave pattern observed with the experimental cell of side 7 mm for the vibration case a = 0.83 mm and f = 25 Hz and for three different gravity levels g = 0.18g 0 , 0.1 g 0 and 0.05 g 0 as the cell is slowly heated towards the critical point. It was observed that the wave pattern does not change phase with time, clearly showing that the pattern corresponds indeed to the frozen wave instability. The waveform of the instability is triangular in shape far from the critical point where the wavelength of the instability is large enough. The waveform appears more and more diffused as the temperature approaches the critical point. Close to the critical point, when the wavelength of the frozen wave is small enough, viscous dissipation becomes predominant. The triangular shape of the waveform may be due to the fact that the surface tension is very small. Small temperature fluctuations, of the order of some mK can appear leading to phase transition from liquid phase to vapor phase and vice-versa and thus make the interface blurred. Shear flow ultimately "mixes" the interface, i.e., lowers the critical point as is well established in [START_REF] Onuki | Phase Transition Dynamics[END_REF][START_REF] Beysens | New developments in the study of binary fluids under shear flow[END_REF]. It can be seen that the wavelength of the instability decreases as the critical point is approached. From Eq. 7.11 it can be seen that the wavelength decreases as ∆T decreases with an exponent of 0.38. This is because as the critical point is approached the surface tension of the liquid-vapor interface and the liquid vapor density difference vanish according to Eqs. 7.3 and 7.2, making the interface more and more flexible. It can generally be observed that the amplitude of the frozen wave pattern is more or less constant and is not dependent on the temperature proximity (which will be seen, in subsequent discussion, to be not true very close to T c ). For given T c -T and aω, it can be seen from Eq. 7.11 that λ increases with a decrease in gravitational acceleration. This can be observed from the images corresponding to around 13 mK for 0.18g 0 , 0.1g 0 and 0.05g 0 in the Fig. 7.10. The waveform of the instability is triangular in shape far from the critical point where the wavelength of the instability is large enough. The waveform appears more and more diffused as the temperature approaches the critical point. To study the evolution of the shape of the wave, one can define the vibrational Froude number W based on the ratio of vibrational velocity to the capillary wave [START_REF] Jalikop | Steep capillary gravity waves in oscillatory shear driven flows[END_REF] as:

W = aω gl c (7.12)
As has been explained in [START_REF] Jalikop | Steep capillary gravity waves in oscillatory shear driven flows[END_REF], W is a determining factor for the shape of the waveform. There exists a critical vibrational Froude number W c below which the frozen wave pattern has a near-sinusoidal shape. For W > W c , the crests and troughs of the wave "broaden" looking like triangular shapes. For the cases dealt with in [START_REF] Jalikop | Steep capillary gravity waves in oscillatory shear driven flows[END_REF], with the upper and lower layers viscosities of the order of 10 -4 and 10 -6 m 2 .s -1 respectively, the value of W c was found to be around 4.

For the experiments considered in the present paper, the value of W lies between 5 and 40. Thus it is generally expected that the frozen waves be triangular in shape. The fact that surface tension vanishes close to the critical point makes the tips of the crests and troughs sharp and pointed. However, the experimental results contradict with this fact very near to the critical point. As can be seen in Fig. 7.10 the waves have a sharp triangular shape far from the critical point while they attain a somewhat rounded shape very close to the critical point. The reason is the fact that there is a limit to the radius of curvature of the wave, due to increasing viscous dissipation. Then, as the wavelength decreases near the critical point, and the radius of curvature is almost constant at the tip, the waves become rounded. Note also that shear-induced phenomena near the critical temperature have been well established in [START_REF] Onuki | Phase Transition Dynamics[END_REF][START_REF] Beysens | New developments in the study of binary fluids under shear flow[END_REF]; shear flow "mixes" the phases at the interface and thus make them blurred. The evolution of the interface shape can be studied by analyzing the harmonic content of the wave shape. The waveform is fitted using the truncated Fourier series:

y = a 0 + a 1 cos(k(x -b)) + a 2 cos(2k(x -b)) + a 3 cos(3k(x -b)) (7.13)
where a 1 , a 2 , and a 3 are the amplitudes of the first, second and third harmonic terms, k is the wavenumber, a 0 and b serve to fix the reference frame at the mean position of the wave profile. More than 50 profiles of the frozen waves are fitted using the least square method to analyze their evolution with W . It was seen that three harmonics are sufficient to approximately represent the wave profile.

While the amplitude a 1 contributes to the height of the wave, the amplitudes a 2 , and a 3 contribute to the overall shape of the wave. The higher the values of a 2 , and a 3 , the more pointed is the waveform. Figure 7.11 shows the original (solid dots or squares) and fitted (dashed line) wave forms of the frozen wave pattern for the case a = 0.83 mm, f = 25 Hz, g = 0.05g 0 and for two temperatures T c -T = 13 mK and T c -T = 0.5 mK. It can be seen that the truncated Fourier series formulation with three harmonics is indeed sufficient to fit the profiles. For the 13 mK case, however, many more harmonics are needed to get the sharp crest tip. One sees that the shape of the tip is nearly invariant as due to the limiting effect of viscosity. Figure 7.12 shows the variation of the three amplitudes with W . The plots are made for two temperature ranges, T c -T > 2.5 mK where the frozen wave wavelength is large enough to make visible the saw tooth shape and T c -T < 2.5 mK where high viscous dissipation due to very small wavelengths of the frozen wave make the wave rounded. It can be observed that among the three amplitudes, the amplitude of the first harmonic is the leading term and is at least an order of magnitude higher than the other two amplitudes. This term directly contributes to the wave height of the frozen wave. It can be seen from Fig. 7.12a that for both the temperature ranges the amplitude a 1 increases with an increase in W . It can be seen that the data points corresponding to T c -T > 2.5 mK are the x direction with almost negligible velocity in the y direction. Given the fact that, close to the critical point, the viscosity contrast between the two phases is close to 1, the viscous dissipation term can be expressed as µ f ( ∂u ∂x) 2 which is of the order of µ(aω) 2 f λ 2 . The vibrational energy term is given by ρ(aω) 2 . The relative strength of the viscous and vibrational energy terms can be expressed as the following non-dimensional number V :

V = ν f λ 2 (7.14)
It can be seen that V is the inverse of the Reynold's number as defined at the scale of the wavelength of the frozen waves. A plot of the non-dimensional number V with respect to the vibrational Froude number W for various experimental cases is shown in Fig. 7.13 for temperature ranges T c -T < 2.5 mK and T c -T > 2.5 mK. It is interesting to see that for temperatures close to the critical point, the value of V is around an order of magnitude larger than that far from it, indicating a stronger viscous dissipation. Interestingly, the trend of the coefficients of the second and third harmonics, a 2 and a 3 , as shown in Figs. [START_REF] Quettier | Magnetic compensation of gravity forces in liquid/gas mixtures: surpassing intrinsic limitations of a superconducting magnet by using ferromagnetic inserts[END_REF].12b and 7.12c is similar to that of Fig. 7.13. This establishes the role of viscous dissipation on the rounding of the waveform close to the critical point. Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] carried out a non-linear analysis of the solutions near stability threshold and derived an expression for wave amplitude. If A is the wave height of the frozen wave defined as twice the amplitude of the frozen wave, then, according to [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]:

A = λ c 4(ρ l + ρ v ) 2 ) πρ v ρ l (-11(ρ l /ρ v ) 2 + 42(ρ l /ρ v ) -11) 1/2 ∆(aω) * (7.15)
where

∆(aω) * = (aω) 2 -(aω) 2 c (aω) c (7.16)
Going in the similar line of reasoning as in [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF][START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF], we define a non-dimensional density difference δρ * = ∆ρ/(2ρ c ). Since for the temperature range being investigated ǫ < 1.5 × 10 -3 ≪ 1 and δρ * ≪ 1, the wave height A can be developed as (see [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF][START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF]): Figure 7.14 shows the variation of the non-dimensional wave height A h with ε for a = 0.83 mm for various frequencies of vibration and various gravity levels.

A = 2λ c ( 64 20π 2 ) 1 2 ∆(aω) * (1 + ∆ρ * + O(∆ρ * 2 )) (7.17)
It can be seen that far from the critical point the amplitude of the frozen wave increases weakly as ε is reduced. Interesting to note is that for a given gravity level, the A h vs ε plots are parallel for various frequencies of vibration, well in agreement with Eq. 7.17. However, much closer to the critical point (for T c -T typically less than 5 mK corresponding to ε less than approximately 0.0002), a decrease in the wave height with ε is observed (see Fig. 7.14b). We attribute this result to the enhanced effect of viscous dissipation and shear mixing when the wavelength becomes very small. This phenomenon is less remarkable for gravity levels 0.1g 0 and 0.18g 0 even for vanishing wavelengths since A h is quite small. To study the effect of g on the wave height it is convenient to define from Eq. 7.17 the non-dimensional wave height A * = A/(λ c (1+δρ * )). Figure 7.15 shows a plot of A * vs ∆(aω) * for various g levels. The solid line is plotted according to the equation A * = 2( 64 20π 2 1 2 ∆(aω) * ). It can be seen that, irrespective of the gravity level, the general trend of A * vs ∆(aω) * curve is in good agreement with the theoretical curve for smaller values of ∆(aω) * . For larger values of ∆(aω) * , the data points deviate from the theoretical curve. In other words, the larger values of ∆(aω) * depart from the weakly non-linear model of Lyubimov and Cherapanov [START_REF] De La Tour | Exposé de quelques résultats obtenu par l'action combinée de la chaleur et de la compression sur certains liquides, tels que l'eau, l'alcool, l'éther sulfurique et l'éssence de pétrole rectifiée[END_REF].The dispersion in the data points may be due to measurement errors or may come from the fitted equation 7.11 for the temperature measurement. The agreement of the experimental results with the theoretical correlation seems to be better than with CO 2 in 1-g conditions ( [START_REF] Wunenburger | Frozen wave induced by high frequency horizontal vibrations on a co2 liquid-gas interface near the critical point[END_REF][START_REF] Beysens | Effect of oscillatory accelerations on two-phase fluids[END_REF]). This shows that Eq. 7.17 is valid for any gravity level (except for the zero-g case, where λ c and A -→ ∞).

Figure 7.16: Non-dimensional wave height A h (twice the amplitude of the instability) vs non-dimensional gravity g/g 0 .(Log-log plot) Figure 7.16 plots the wave height for the frozen wave pattern with respect to g observed in the experiments with the 7 mm cell for a = 0.83 mm and for various T c -T and various frequencies. It can be seen that for a particular value of T c -T and f , the amplitude of the frozen waves increases as the gravity is reduced. The symbols in the Fig. 7.16 which correspond to experimental results can be fitted to a power law (represented by the solid line). The data points show good agreement with power law fits according to the law A = A 1 g -0.7 , where A 1 is a constant. The theoretical prediction eq. 7.17 gives a power law exponent of -0.75 which is not far from the exponent of the fit. Thus from the equation A = A 1 g -0.7 , as g -→ 0, A -→ ∞. That is, the amplitude of the frozen wave instability diverges as the gravity is reduced to zero.

Conclusion

Vibration experiments carried out using the experimental setup HYLDE on the liquid-vapor interface of H 2 close to its critical point have successfully put in evidence the frozen wave instability under non-zero gravity fields. Experiments are conducted for various gravity levels varying from 0.18g 0 to 0.05g 0 primarily to study the effect of gravity on the wave height of the instability. The linear stability plots obtained from the experiments agreed well with the inviscid linear stability model by Lyubimov and Cherepanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] for all the three gravity levels experimented. The theoretical correlation for the amplitude of the frozen wave pattern obtained using a non-linear analysis by [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] compares well with the experimental results irrespective of the gravity level. The experiments clearly show that the wave height of the frozen waves increases with a decrease in the gravity level according to a power law with an exponent of 0.7. It can be concluded that as the gravity level is decreased to zero, the wave height diverges and becomes of the order of the cell size.

Transition of the Faraday wave instability to frozen wave instability in zero-g

Vibration experiments carried out with various fluids under zero-g conditions put in evidence an interesting phenomenon: transition from horizontal Faraday fingers to vertical bands of alternate liquid and vapor phases. Vibration experiments carried out using the experimental cell used in 7.2 under zero-g conditions demonstrated this strange phenomenon. Figure 7.17 shows the results of zero-g vibration experiments carried out on H 2 filled inside the cell of side 3 mm for a = 0.29 mm and f = 40 Hz. The experiments involve vibration of the cell filled with H 2 at its critical density while heating it at a very slow rate in order to study the dynamics as one approaches the critical point. An initially spherical vapor bubble (Fig. 7.17a) surrounded by its liquid under zero-g conditions, when vibrated, first showed Faraday waves (Fig. 7.17b, 20 mK -7.5 mK) which oscillate at a frequency equal to half the frequency of vibration (as already seen in section 7.2). As the temperature of the bulk approaches the critical point, at a particular proximity to the critical point the Faraday waves disappear completely and a reorganization as alternate layers (band pattern) of liquid and vapor phases aligned perpendicular to the direction of vibration occurs (Fig. 7.17b, 6.5 mK -3 mK). This phenomenon was earlier observed in experiments conducted on CO 2 [START_REF] Beysens | The effect of vibrations on heterogeneous fluids: Some studies in weightlessness[END_REF][START_REF] Beysens | Phase transition under forced vibrations in critical co2[END_REF] and also on H 2 [START_REF] Beysens | The effect of vibrations on heterogeneous fluids: Some studies in weightlessness[END_REF]. While one mechanism is predominant far from the critical point, the other mechanism is predominant close to it. This is reflected by the fact that there exists a small transition region in which the Faraday waves start weakening out while the band pattern gain importance. Figure 7.18 clearly shows this transition from the Faraday waves to the band pattern for the case a = 0.41 mm and f = 32.5 Hz. It can be seen that while Faraday waves start to fade away as the temperature is increased, thin line-like structures (denoted by B) start forming. Then, the Faraday waves completely disappear and the band pattern becomes predominant. Figure 7.19 plots the critical point proximity T c -T against the vibrational velocity (aω) for the coexistence of the Faraday and band pattern. The solid line indicates the temperature proximity at which the band pattern first appears when the cell is heated from a temperature below T c towards T c . Similarly, the dashed line gives the temperature proximity at which the Faraday wave instability ceases to exist, again when the cell is heated from a temperature below T c towards T c . It can be seen that at higher vibrational velocities Faraday waves disappear or band pattern first appear at temperature closer to the critical point. There exists a crossover between the two lines for values of aω smaller than around 0.04 ms -1 , the value above which there exists a narrow band of temperature range where the two patterns (Faraday and band patterns) coexist. The three images of Fig. The vertical band like patterning of liquid and vapor phases emerges from the horizontal walls. This indicates that the band pattern is caused by the destabilization of the horizontal part of the interface when it is subjected to vibration.

Thus it is likely that the band pattern is related to the Kelvin-Helmholtz instability. The detailed analysis of the band pattern instability is itself part of another detailed experimental campaign and is beyond the scope of the present paper.

Similarities between band pattern and frozen wave instability

The band pattern shares some similarities with frozen waves under non-zero gravity fields. Similar to the frozen wave pattern, band pattern cannot occur for higher liquid-vapor density differences. Both frozen waves and band pattern occur at temperatures very close to the critical point. In both types of phenomena the waves are frozen with time. Thus there is a probability that the band pattern we observe in zero-g experiments very close to the critical point are actually frozen waves in zero-g. This hypothesis can be defended using the following arguments. Figure 7.20 shows the wave pattern of the frozen wave instability for gravity varying from 0.18g 0 to zero-g. The results are shown for two vibration amplitudes a = 0.53 mm and 0.83 mm and frequency 25 Hz. The wave pattern is evidently not for the same temperature, which does not matter for the present comparison. It can be seen that the amplitude of the frozen waves increases quite remarkably as the gravity is reduced, to reach the sample size at some weak g-value. We presume that this latter configuration corresponds to the zero-g case where the observed band pattern would correspond to the frozen wave pattern at finite g. It is important to note that the frozen wave instability is associated with a threshold (with respect to the vibration parameters a and ω) while the band pattern have no threshold. Figure 7.9 shows that as the gravity level is reduced the stability domain of the frozen wave instability decreases. It is thus likely that when the gravity is reduced to zero, the problem becomes completely unstable, with zero threshold. It follows from the equation for the bifurcation curve obtained in [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] (Eqs. 7.8-7.9) that with the lowering of the gravity level the critical vibration velocity amplitude tends to zero, so there is zero instability threshold for zero gravity conditions. The selection of wavenumber should be only cell size dependent. It is indeed another instability whose study is beyond the scope of

Conclusion

Zero-g experiments involving vibration very close to the critical point put in evidence a coexistence of two types of instabilities: Faraday instability and band pattern. While the Faraday wave instability is favored when the liquid-vapor density difference is large enough (far from the critical point), periodic layering (band pattern) predominates for small difference in the densities of the liquid and vapor phases (close to the critical point). Experimental study of frozen wave instability for various gravity levels has indicated that the band pattern and frozen waves share quite some similarities. It is thus believed that the band pattern is of Kelvin-Helmholtz origin, a statement to be verified by further studies. A particular investigation of the periodic layering at variable gravity is envisaged and should give more insight into this interface instability.

Chapter 8

Dynamic equilibrium of a liquidvapor interface of H 2 under vibration at various g-levels

Introduction

A fluid interface subjected to vibration shares quite some similarities with that of simple mechanical systems under vibration. For example, a simple pendulum can be stabilized in an upside down position by vertically vibrating its support at a frequency much larger than the natural frequency of the pendulum, i.e., when the condition aω ≥ √ 2gl 0 [START_REF] Landau | Mechanics. MIR[END_REF] is satisfied (a is the amplitude of vibration, ω is the angular frequency of vibration and l 0 is the length of the pendulum). Sim-ilarly, when the support of the pendulum is vibrated horizontally at frequencies much larger than the natural frequency of the pendulum and vibrational velocity amplitudes aω higher than a threshold value (aω ≥ √ 2gl 0 ), the equilibrium position of the pendulum is no more vertical and makes an angle α defined by sin α = 2gl 0 a 2 ω 2 [START_REF] Landau | Mechanics. MIR[END_REF] with respect to the horizontal. Vibration of a fluid interface can demonstrate similar phenomena. Using vibration it is possible to stabilize two-fluid configurations generally unstable when the vibration is absent. For example, a heavier fluid floating over a lighter fluid under terrestrial gravity field is unstable (Rayleigh-Taylor configuration) under normal conditions. Application of strong vertical vibration can dynamically stabilize the above configuration [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF]. When subjected to strong horizontal vibration, the interface of an initially horizontal fluid interface can attain a dynamic equilibrium at an angle to the horizontal plane [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF]. The present chapter deals with the dynamic equilibration of a fluid interface when subjected to horizontal vibration. The problem is relatively less investigated till now and only two research articles related to this phenomenon could be found. Wolf [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF] first reported the phenomenon when experimenting with aqueous solution of potassium iodide and oil. He observed that for large amplitude-frequency cases the interface tilts towards a wall at an angle α to the vertical. Assuming that the width of the vessel is much smaller than the depth of the two fluids and neglecting the effects of viscosity and capillarity, he estimated that for small angles of α (large values of aω), the following relation should hold for a rectangular cavity:

sin α = ( 2gl πa 2 ω 2 )( ρ l + ρ v ρ l -ρ v ) (8.1)
where, l is the width of the cavity, ρ l and ρ v are the liquid and vapour densities respectively. It is worth noting here that for some values of vibration parameters and density or gravity values the expression 8.1 can give values larger than unity. It simply means that sin α saturates to unity. Lyubimov et al. [START_REF] Lyubimov | Modelling in Mechanics[END_REF] carried out the analysis for a free surface, where the density of the lesser dense phase is much smaller than the density of the denser phase, using high frequency approach.

For the case of weak gravity level (gl ≪ (aω) 2 ), they obtained an analytical expression for the interface shape from which it follows that sin α is proportional to gl (aω) 2 similar to the expression by Wolf [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF]. As it is also shown in [START_REF] Lyubimov | Dynamics of fluid interfaces in vibrational fields[END_REF], at small enough values of the vibration intensity (aω) the equilibrium position of interface is nearly horizontal and symmetric with respect to vertical plane (i.e. sinα=1). With the increase of (aω) spontaneous symmetry breaking takes place at some critical value (aω) and the interface takes a position inclined to the horizontal. The experiments were carried out for a fixed value of aspect ratio ( h l = 1). In [START_REF] Lyubimov | Dynamics of fluid interfaces in vibrational fields[END_REF] the variational principle was formulated for the determination of the free surface shape of a fluid subjected to the high frequency horizontal vibrations. Numerical minimization of the corresponding functional has shown that the results obtained for different aspect ratios are qualitatively the same and that for the same vibration velocity amplitude (aω) the inclination with respect to the vertical is larger for the values of h l much larger than 1. The objective of the present chapter is to investigate the above phenomenon when the density difference between liquid and vapour phases and the g level vary. It has been seen earlier that just by heating or cooling the fluid (or in other words, by approaching or moving away from the liquid-vapor critical point) one can modify the properties of the liquid and vapor phases in a scaled way and thus study the interface for a variety of fluid conditions without having to change the fluid combination. The effect of gravity is studied by changing the gravity level inside a small cell by using a magnetic levitator. Also, it is interesting to note that working close to the critical point decreases the density difference between the two fluids which, according to Eq. 8.1 leads to a magnification in the angle of the tilted interface under vibration. The present chapter is organized as follows: The experimental setup used to carry out the experiments is given in section 8.2 followed by the results of the experiments and comparison with theory in section 8.3. Finally some concluding remarks are made in section 8.4.

Experimental procedure

The experiments are carried out with H 2 as the working fluid using the cryogenic facility HYLDE (HYdrogen Levitation DEvice), described in section 4.2. The experimental cell used is a cubical cavity of side 7 mm (shown in the Fig. 4.10b), made of sapphire. The experimental cell and the conditions under which the experiments are realized are identical to that for the frozen wave experiments described in section 7.3.

Results and discussion

To carry out the experiments the cell is first maintained at a temperature slightly lower than the critical temperature T c (say T c -100 mK) and then horizontal vibration is applied. The temperature of the cell is increased very slowly (20 mK.min -1 ) so that the fluid can be assumed to be under thermodynamic equilibrium at all temperatures. Experiments are conducted for various combinations of amplitudes (ranging from 0.2 mm to 0.8 mm) and frequencies (between 10 Hz and 35 Hz). Various gravity levels: 0.18g 0 , 0.1g 0 , 0.05g 0 , 0.01g 0 and 0g 0 are obtained by adjusting the current intensity in the coil. The experiments have been conducted for various amplitudes, frequencies and gravity levels and the results show no signs of hysteresis.

Determination of the fluid temperature

It is important to mention that the experiments have a serious limitation with respect to the measurement of the temperature. Since the thermometers fixed to the cell vibrate along with the cell inside an intense magnetic field, eddy currents are induced inside them and their electric cabling, provoking unwanted oscillations in the temperature electric signal. This renders the values of the temperature during the vibration experiments useless. Thus an indirect method needs to be used to estimate the temperature of the bulk fluid. Figure 8.1 shows the evolution of the interface as the cell is heated from a subcritical temperature (T c -45 mK) towards T c at a very slow rate (20 mK.min -1 ). Figure 8.1a shows the interface attaining a dynamic equilibrium, tilting towards the right wall. In our experiments the interface always seemed to tilt towards the right wall. This non-random behavior is presumably due to a slight initial tilt in the cell with respect to the vibrational direction. In the absence of this initial tilt, the interface should randomly choose a particular wall based on the experimental perturbations. It is not supposed to flip to the other wall once it found its equilibrium position.As the temperature of the cell is further increased, the interface deforms giving rise to the frozen wave instability. Experiments and subsequent analysis of the frozen wave instability (section 7.3) have shown that the wavelength (λ) of the frozen wave instability depends on the thermal proximity to the critical point ∆T (= T c -T ) according to the empirical relation 7.11. The temperature of the bulk fluid can be approximately estimated by first calculating the temperature for a particular wavelength of the frozen wave instability and then extrapolating it using the fact that the heating of the cell is carried out at approximately 20 mK.min -1 . This method is extremely approximate and is sufficient for the present analysis.

Phenomenological understanding of the tilting of the interface

It is well known that when a liquid filled in a reservoir is subjected to a constant acceleration a 0 , the interface of the liquid tilts at an angle α given by tan(α) = (g + a 0y )/a 0x with respect to the vertical. Though this phenomenon is a little bit different from the phenomenon observed in our experiments, they share the same root as will be demonstrated below. The tilting of the liquid-vapor interface in our experiments is caused by the non-linear interaction of vibration with the interface. This happens when a threshold vibrational acceleration is exceeded. Lyubimov and Cherapanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] analytically treated the frozen wave problem involving two incompressible inviscid fluids subjected to horizontal vibration. Their approach involved, under high-frequency approximation (Ω = ωh 2 ν ≫ 1), the splitting of the equations of motion into pulsating and average parts. A supplementary assumption of low amplitude ( a h ≪ 1) allows linearizing the pulsating fields and solving them separately. For the present experiments, Ω lies between 3000 and 15000 while a h lies between 0.1 and 0.01 showing that we are in the validity range of the Lyubimov and Cherapanov theory [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. The resulting averaged equations using the standard notation are reproduced from [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF] as below:

du β dt = - ∇p β ρ β + ν β ∆u β -(g γ + ∇((aω) 2 V 2 β )) (8.2)
where, u β is the averaged velocity field, V β is the relative amplitude (with respect to aω) of the pulsating velocity field and the subscript β refers to the two fluids, γ is the unit vector in the direction of the gravitational acceleration. The Eq. 8.2 along with the relevant boundary conditions can be solved to obtain the average flow field of the problem. Such an analysis is, however, out of the scope of the present paper. Our interest in discussing the above equation is merely to explain the phenomenology of the tilted interface. As can be seen from Eq. 8.2, the pulsating velocity fields induce an extra acceleration term ∇((aω) 2 V 2 β ) in the average flow field of the problem. The averaged equations indicated by Eq. 8.2 are phenomenologically similar to a fluid filled inside a square cavity subjected to a constant acceleration equal to a 0 = ∇((aω) 2 V 2 β . The nature of this acceleration term needs to be determined by appropriate analytical methods to compare it with the Eq.8.1 which is not taken up in the present paper.

Effect of vibrational parameters on the angle of the interface

From the temperature of the bulk fluid estimated according to the procedure described in the previous section, theoretical values of sin α can be calculated using the Eq. 8.1. The results of the experiments are compared with the theory in the Fig. 8.2 for 0.05g 0 and for various values of amplitudes and frequencies. The solid dots in the plot are the results of the experiments for various values of a 2 ω 2 . The solid line is derived from Eq. 8.1 for corresponding values of a 2 ω 2 . The following observations can be made. When the vibrational velocity is increased, the angle of the interface with respect to the vertical decreases. For smaller values of aω, the interface remains relatively horizontal (sin α ∼ 1). It can be seen that Eq. 8.1 compares well with the experiments for smaller values of α. The values sin α > 1 are mathematically undefined (indicated by the dashed line in Fig. 8.2) and indicate the region of a 2 ω 2 where the dynamic equilibrium position of the interface is horizontal. The dispersion in the experimental values is attributed to errors in the calculation of the temperature values and errors in the measurement of the angle of the interface. However, an exact agreement with Eq. 8.1 cannot be expected due to the approximation made in the equation. The experiments carried out using the gravity level 0.01g 0 however do not match very well with the Eq. 8.1 due to the fact that the uniformity of the gravity field itself is of the order of 0.03g 0 which is much higher than the gravity sought for. It is important to note that the observed tilted interface cannot be an incomplete frozen wave with a wavelength larger than the size of the liquid-vapor interface. Independently from the fact that its wavelength should be a multiple of the cell length, when the conditions corresponding to the instability threshold are met, frozen waves are seen to grow from the tilted interface (Fig. 8.1b). In addition, the wavelength of a frozen wave instability (see 7.11) increases with aω, corresponding to an angular tilt of the interface that increases with respect to vertical. This is opposite to what is observed in Fig. 8.2 where the interface angle α decreases with aω.

Effect of critical point proximity on the dynamic equilibrium of the interface

As has been mentioned earlier, as one approaches the critical point, various properties of a near-critical fluid vary with the proximity to the critical point according to universal scaled power laws as explained in the 2 [START_REF] Zappoli | Supercritical fluid hydrodynamics[END_REF]. Thus studying the effect of the temperature on the angle of the interface is an important aspect of the problem. Figure 8.3 presents the results of experiments for the case a = 0.83 mm, f = 35 Hz and g = 0.05g 0 . In the pictures presented, the temperature of the cell changes from T c -62 mK to T c -20 mK. It can be seen that the angle of the interface does not change much between T c -62 mK and T c -25 mK. A small bump is already evident on the interface (shown as A in Fig. 8. 8.1 describes the dependence of the angle of the interface on the relative density ratio (ρ l + ρ v )/(ρ l -ρ v ). The liquid-vapor density difference ∆ρ(= ρ l -ρ v ) varies with ε = (T c -T )/T c , the relative critical point proximity, according to Eq. 7.3 [START_REF] Zappoli | Supercritical fluid hydrodynamics[END_REF] The values of (ρ l +ρ v )/(ρ l -ρ v ) are calculated for various temperatures using data from NIST [42] and from Eq. 7.3 and are then used to calculate the theoretical values of sin α from Eq. 8.1. Figure 8.4 shows the effect of critical point proximity (T c -T ) on the angle of the interface and compares the results of the experiments with theory. It can be observed from the plot that as the fluid approaches the critical point, the angle of the interface increases. In the range of the temperatures experimented, for a drastic change in the angle of the interface is not seen because the values of (ρ l + ρ v )/(ρ l -ρ v ) change only by ±7% with respect to the median temperature. Comparison of the results of the experiments and theory show good agreement. It can be seen that the trend of the plots is the same. It can be seen that the general trend of the plots is the same. The point corresponding to 0.01g 0 deviates considerably from the line. This is due to the fact that the uniformity of the gravity levels inside the cell is of the order of 0.03g 0 , greater than the gravity level sought.

Effect of gravity on the angle of interface

Conclusion

Vibration experiments were carried on a liquid-vapor interface of H 2 to study the dynamic equilibrium of the interface. The study was performed for various combinations of amplitude and frequency of vibration and for various gravity levels, thanks to the magnetic gravity compensation instrument HYLDE. Experiments showed that under harmonic vibrations the equilibrium position of an interface can considerably deviate from its normally horizontal position and attain large enough angles. When compared with the theoretical correlation derived by Wolf [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF], the data compared well with theory, although this latter suffers from substantial approximation. It is interesting to note that under vanishing gravity, vibration acts on an interface in the same way as an artificial gravity. Further theoretical work seems to be desirable to deepen our understanding of this phenomenon.

Chapter 9

Interface dynamics and geysering during an abrupt change in acceleration

Resumé (Français)

Ce chapitre est consacré à l'étude de la réorientation d'un liquide dans un réservoir quand celui-ci est soumis à une accélération. Les missions spatiales complexes et avancées ont besoin de rallumages multiples des moteurs. En absence de gravité, la position exacte des propergols dans les réservoirs est inconnue. Pour être sûr que le propergol liquide se positionne à l'entrée du moteur, une petite accélération doit être appliquée. Sous l'effet de cette accélération, la phase liquide s'écoule vers le bas du réservoir et ensuite rebondit vers le haut. Si l'accélération est trop forte, le rebondissement du liquide peut entrainer la formation d'un geyser dans le réservoir. La bonne compréhension de ce phénomène est très importante pour optimiser les besoins des missions spatiales. Des expériences sur l'installation OLGA ont été effectuées et ont démontré la capacité de cette installation à mener des études sur la réorientation d'un liquide dans les réservoirs spatiaux. Des expériences avec plusieurs niveaux de remplissage d'oxygène dans une cellule de 30 mm de diamètre et de 100 mm de hauteur ont été réalisées. Des simulations numériques ont été effectuées pour comparer les résultats expérimentaux avec la théorie. Enfin, une étude numérique a été effectuée pour développer des corrélations empiriques prenant en compte l'effet de la paroi sur les vitesses de la bulle et du geyser.

Introduction

Understanding the physics of the interaction of different phases present in a multi-phase fluid is important in view of various interesting applications involved. For example, the presence of vapor bubbles inside a heat exchanger has a non-negligible effect on the overall performance of the system. Another interesting problem is cavitation where a liquid phase can give rise to vapor bubbles at low pressures, which subsequently can implode at locations where the local fluid pressure is high enough. The rising of a bubble inside a vertical column of liquid under buoyancy forces is another problem with interesting industrial applications. For example in nucleate boiling under normal gravity, small bubbles form at suitable nucleation sites and rise vertically inside the vessel while interacting with the surrounding liquid phase. Interaction of vapor bubbles with a liquid phase is seen in various practical applications such as rising of steam inside a boiler or rising of gas bubbles in oil wells. In space vehicles, bubbles of various sizes may form inside the liquid systems due to various reasons like boiling at heat sources or due to maneuvering induced sloshing resulting in turbulent mixing of liquid-gas phases. This has a non-negligible effect on the overall performance of the system. Another interesting problem related to the interaction of bubbles with liquid phase is that of a coupled structure and propulsion system instability called "pogo effect" that can have a detrimental effect on the performance of a liquid propulsion rocket. Bubbles of nitrogen gas are injected at a controlled rate into the two propellant inlets to dampen the unwanted oscillations. Another important problem is the geyser phenomenon (impingement of liquid propellant into a huge gas bubble) inside a propellant reservoir of a spacecraft when reorientation acceleration is applied. An understanding of the maximum acceleration required to achieve a good collection rate of the liquid propellant near the engine inlet without the formation of geysers is a very important mission requirement. The motion of gas/vapor bubbles rising inside a column of liquid is the problem of interest in the present chapter. The problem has historically been studied under two major streams. (a) rising vapor bubbles and (b) reorientation studies, which studies the reorientation of liquids inside reservoirs when a small acceleration is applied. The problem of rising vapor bubbles is restricted to cases where the size of the bubbles (R b ) is much lesser than the size (R) of the container (R b /R < 0.1) whereas the vapor bubbles dealt in the reorientation studies occupy typically around 20-80% of the container volume. The bubble shape deformation in the case of rising vapor bubbles depends on the variation of pressure along the interface and the surface tension. In contrast, it is the inertia of the liquid phase and its eventual rebounding, which determines the bubble shape deformation in the case of reorientation studies. To optimize the performance of a liquid propulsion rocket system that is destined to launch multiple satellites in orbits around earth, multiple restarts with intermediate phases of coasting are required. At the end of each coasting phase it is required to ensure that the liquid propellant is available near the inlet of the engine. The position of the liquid propellant inside the rocket tank at the end of a coasting phase is however unknown, and is dictated by various factors like the orbit maneuvers, orbital perturbations, start and shut-down transients etc. It is thus important to apply a small acceleration (called reorientation acceleration) to collect the propellant at the inlet of the engine. This acceleration, if large enough, can cause the liquid propellant to rebound from the bottom end of the reservoir and reorient itself to form a geyser, thus limiting the amount of liquid collected near the engine inlet pipe. Extensive experimental research has been carried out till date to study the reorientation phenomenon in scaled rocket reservoir models as well as simple reservoir configurations. Notable among these studies is the work by [START_REF] Masica | Motion of liquid-vapor interface in response to imposed acceleration[END_REF][START_REF] Salzman | Experimental investigation of liquidpropellant reorientation[END_REF][START_REF] Salzman | Low gravity reorientation in a scaled model centaur liquid-hydrogen tank[END_REF][START_REF] Haggard | Motion of single bubbles under low gravitational conditions[END_REF] in which experiments with a fill ratio inside the reservoir of around 50% were performed. Tanks with various shapes and sizes were used to carry out the experiments using liquids with near-0 • contact angles with air. Reorientation accelerations in the range of 0.01g 0 to 0.1g 0 were applied to study the dynamics of the liquid-vapor interface. The experimental findings of [START_REF] Masica | Motion of liquid-vapor interface in response to imposed acceleration[END_REF][START_REF] Salzman | Experimental investigation of liquidpropellant reorientation[END_REF][START_REF] Salzman | Low gravity reorientation in a scaled model centaur liquid-hydrogen tank[END_REF][START_REF] Haggard | Motion of single bubbles under low gravitational conditions[END_REF] can be summarized as follows. The stability of a liquid-vapor interface depends on the reorientation Bond number (ratio of buoyancy forces to the surface tension forces):

Bo = ∆ρg * R 2 b σ (9.1)
where ∆ρ is the density difference, g * is the reorientation acceleration and σ is the surface tension of the interface. The interface becomes unstable for Bo > 0.842. In other words geysering phenomenon is possible only for Bo > 0.842. Under the application of a reorientation acceleration to a partially filled reservoir initially under zero-g conditions, the liquid propellant accelerates towards the bottom of the tank. The liquid rebounds once it reaches the bottom of the reservoir and if the leading edge velocity of the liquid front is sufficiently large, it forms a geyser. The geyser under severe acceleration conditions can progress along the longitudinal axis of the tank and reach the fore-end of the tank. The geysering phenomenon involves the following stages: leading-edge impingement, initiation of geyser, geyser formation, geyser progression, geyser impingement and finally recirculation. For sufficiently large accelerations the geyser tip attains a constant velocity indicating that the effect of acceleration, surface tension, viscosity etc., becomes negligible and that the inertia of the developing geyser becomes dominant. The magnitude of the instantaneous leading-edge velocity immediately prior to impingement V L is an important parameter determining the severity of a geyser. The velocity V L can be calculated using the formula [START_REF] Salzman | Experimental investigation of liquidpropellant reorientation[END_REF]:

V L = (2a L R b ) 1/2 (9.2)
where a L = (3.8V 2 0 )/R b , is the acceleration rate of the leading edge, V 0 is the vapor penetration rate or the ullage velocity given by the relation (for Bo > 12):

V 0 = 0.48(g * R b ) 1/2 (9.3)
Surface tension has a stabilizing effect on the geysering phenomenon as is seen in the expression for the Bond number, whereas the inertial forces are destabilizing in nature. The Weber number defined as

W e = (ρV 2 L R b )/σ (9.4)
is a measure of the relative effect of these two forces and is an indicator of the severity of a geyser. It was reported that geysering mechanism can occur only when W e > 4 [START_REF] Salzman | Experimental investigation of liquidpropellant reorientation[END_REF]. This is when the inertial effects are predominant and the surface tension forces are not sufficient to stop the geysering phenomenon. For 4 < W e < 10 the geyser is seen to form but the axial propagation stops; for 6 < W e < 60 axial progression is decelerating and impingement happens on the end of the reservoir and for 10 < W e < 200 the geyser forms, progresses at constant velocity to the end of the reservoir. These results were found to be independent of the tank shape, fill ratio and reorientation accelerations. Baumbach et al. [START_REF] Baumbach | The transient behaviour of a large bubble in a vertical tube[END_REF] studied the transient motion of a large spherical cap bubble in a vertical cylindrical tube and compared with a theoretical model. Pichavant et al. [START_REF] Pichavant | Using superconducting magnet to reproduce quick variations of gravity in liquid oxygen[END_REF] presented a method of achieving variable gravity fields in a cell filled with oxygen using a setup based on the principle of magnetic levitation. They demonstrated that this method can be used to carry out experiments on reorientation studies. Walters et al. [START_REF] Walters | The initial motion of a gas bubble formed in an inviscid liquid. part 1. the two-dimensional bubble[END_REF] presented a 2D analysis of a bubble rising from rest inside a cylinder. They predicted that the bubble rises with the acceleration of gravity over a distance of at least the initial bubble radius and that a tongue of liquid should be projected up from the base of the bubble into its interior. They carried out a 3D analysis [START_REF] Walters | The initial motion of a gas bubble formed in an inviscid liquid. part 2. the three-dimensional bubble and the toroidal bubble[END_REF] of the same problem as in [START_REF] Walters | The initial motion of a gas bubble formed in an inviscid liquid. part 1. the two-dimensional bubble[END_REF] and showed that the three-dimensional bubble initially rises with an acceleration equal to twice that of gravity distorting eventually into the form of a mushroom. This distortion ultimately breaks up the bubble. They reported that while the 2D bubble always detaches two small bubbles at its rear, the three-dimensional bubble breaks up into a small spherical-cap bubble with a large toroid below.

In the present chapter, we consider the dynamics of geyser formation in large vapor bubbles for which 0.1 < R b R < 1. Experiments as well as numerical simulations are carried out to study the bubble rising velocity and geysering. The other aspect of the present chapter is that we deal with Oxygen, a fluid of great interest in the field of space technology. Oxygen is chosen due to the possibility of carrying out experiments onboard the setup OLGA (Oxygen Low Gravity Apparatus, see Section 4.3), which has the capability of realizing experiments with gravity quench (a very fast variation of gravity), from zero-g to 0.4g 0 inside a cell filled with Oxygen. Moreover, the very low surface tension of its liquid-vapor interface allows us to attain high values of Weber and Bond numbers for relatively low gravity quenches. The experimental results will be compared with simulations carried out with a 2D Finite Volume code THETIS based on VOF-PLIC [START_REF] Youngs | Numerical methods for fluid dynamics[END_REF][START_REF] Pianet | Simulating compressible gas bubbles with a smooth volume tracking 1-fluid method[END_REF][START_REF] Hua | Numerical simulation of bubble rising in viscous liquid[END_REF][START_REF] Gopala | Volume of fluid methods for immiscible fluid and free-surface flows[END_REF][START_REF] Van Der | Bi-cgstab: A fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems[END_REF]. The present chapter is organized as follows: Section 4.4.2 describes the experimental setup used to carry out the experiments. The 2D numerical model used to carry out the simulations is explained in Section 5.3. The results of the experi-ments as well as the numerical simulations are presented in Section 9.4.1 followed by a discussion and a comparison with the theoretical results in Section 9.4.3. In Section 9.5 concluding remarks are given.

Experimental procedure

Experiments are carried out on oxygen using the facility OLGA which uses strong magnetic fields to levitate oxygen inside an experimental cell. The description of the experimental setup OLGA and the experimental cell used to carry out the experiments are given in sections 4.3 and 4.4.2. The uniformity of the gravity field achieved inside the cell is described in section 4.3.3. It can be seen that while a good uniformity can be obtained in the radial direction of the cell, the uniformity of the gravity field in the longitudinal direction is not very good (with 1% at the bottom to 20% at the top of the cell). This however does not affect the results due to the fact that the field of observation of the cell is limited to z = 0 mm to 30 mm where the uniformity is around 1%. To carry out the experiments, Oxygen is filled inside the cell maintained at 90.1 K and 1 bar. Zero-g conditions are setup inside the cell by energizing the solenoid. The bubble is levitated initially with its center at a distance of around 20 mm from the bottom of the cell. The gravity quench is performed by suddenly discharging the inner solenoid to 0 A that takes place with a time constant of around 340 ms (as shown in the Fig. 9.1). The effective gravity inside the cell is ideally supposed to change from 0g 0 to 0.4g 0 with a time constant of 320 ms. However, the presence of the copper heat exchangers, whose electrical conductivity is large at cryogenic temperatures (3.57 × 10 8 S.m -1 [START_REF] Matula | Electrical resistivity of copper, gold, palladium and silver[END_REF]), modifies the variation of magnetic field at the bubble location. When the inner coil of the electromagnet is discharged, the sudden change in the current in the coil induces eddy currents inside the copper heat exchangers, located one above the cell and the other below. The eddy currents induced inside the copper heat exchangers act to oppose the change in the magnetic field induced by the rapidly changing current inside the coil. 2D axisymmetric simulations have put in evidence the effect of the copper heat exchangers on the time variation of the gravity field inside the cell. Figure 9.1 presents the temporal variation of gravity field inside the cell at different longitudinal positions (z) along the axis of the cell. It can be seen that gravity quench occurs instantaneously at the bottom of the cell while at the top of the cell over-compensation of gravity occurs for the initial several tens of milliseconds. The bubble displacement corresponds to heights between z = 25 and 35 mm, corresponding to an acceleration relaxation of about 150 ms, which is appreciably lower than the coil current time constant (340 ms). The geyser formation in the experiments takes place between the heights z = 10 mm and 20 mm. The resulting dynamics of the liquid-vapor interface is captured using a high speed camera at 500 fps. The initial shape of the bubble is dictated by the size of the bubble, the surface tension of the interface and the residual gravity field. The experiments are conducted for a fill ratio of 6% and 15%. For the given size of the bubbles dealt with in the present study, they are ellipsoidal in shape. The details of the bubbles used in the experiments are given in Table 9.1 (here a is the dimension of the semi-minor axis and b is the semi-major axis. approach is used where a local phase volume fraction is introduced to capture the interface. In the present study, we have used the improved method of VOF-PLIC (piecewise linear interface calculation, see [START_REF] Youngs | Numerical methods for fluid dynamics[END_REF]) which is a volume-preserving scheme with geometric reconstruction of the interface. The numerical scheme used to carry out the simulations is described in the section 5.3. The bubble is modeled as an ellipse with the same semi major and minor axes as in experiments. The fluid domain used to carry out the simulations is shown in Fig. 9.2. The domain denoted by V is the vapor phase while that denoted by L is the liquid phase. Convergence criteria in terms of mesh size and time step have been carefully checked. The grid is constituted by an optimum of 240 × 800 mesh points in the cell of size d = 30 mm and h = 100 mm. In all calculations, a constant time step of 5 × 10 -5 s is used and corresponds to a time much lower than all the characteristic timescales of the problem considered in this study. The results of the experiments for the Bubble 1 (fill factor 6%) and Bubble 2 (fill factor 15%) are shown in Figs. 9.3 and 9.4. Initially at t 0 an ellipsoidal bubble (distorted from a spherical shape due to the residual gravity field inside the cell) is suspended inside the liquid column by virtue of zero-g. The application of the acceleration of 0.4g 0 on the system causes the liquid inside the cell to descend (since it is heavier) and the vapor bubble to rise (due to buoyancy) as can be observed from the pictures. The downward moving liquid drop gathers sufficient inertia to rebound and eventually impinges into the vapor phase resulting in the formation of a geyser (as seen in the pictures t 0 + 0.045 s to t 0 + 0.09 s in Fig. 9.3 and t 0 + 0.078 s to t 0 + 0.09 s in Fig. 9.4).

Numerical simulations set-up

Results

Figure 9.4: Left: Vapor bubble at t 0 (L is the liquid phase, V is the vapor phase, W is the white patch due to reflection of light from the center of the bubble). Right: Evolution of the shape of Bubble 2 with fill factor 15% for a gravity variation from 0g 0 to 0.4g 0 at times (a) t 0 + 0.078 s, (b) t 0 + 0.082 s, (c) t 0 + 0.086 s and (d) t 0 + 0.080 s. The numbers 1, 2, 3 and 4 show how the geyser evolves over time.

Comparison of the numerical results with experiments

Following the discussions in the previous section, a simple numerical simulation with gravity variation from 0g 0 to 0.4g 0 with a zero time constant of the variation of gravity is carried out to compare the experiments with the numerical simula-tions. A zero relaxation time can be considered as an oversimplifying description of the acceleration relaxation. However, as discussed in Section 9.2, an exact description of the relaxation would be extremely complex. As the simulation is already simplified to 2D we thus chose to carry out the simulation with this simplification that will not be accurate only at the beginning of the relaxation.

The results of the simulations for Bubble 1 are presented in Fig. 9.5. Comparison of the geysers for the experiments and the simulations shows that the trailing edge of the bubble in the experiments rises faster in the beginning than in the simulations. This is due to the reason that in the beginning the trailing edge of the bubble is closer to the bottom wall of the cell where the gravity change is much more severe than, say, at 10 mm (see Fig. 9.1). However the distortions of the bubble shape seem to synchronize quite well. The geyser formation happens at almost the same time. This indicates that the effect of the copper heat exchangers on the effective gravity field is indeed to almost nullify the relaxation time.

Figure 9.5: Bubble shapes for experiment and simulations at times (t 0 , t 0 +0.045 s, t 0 +0.06 s, t 0 +0.065 s, t 0 +0.075 s, t 0 +0.08 s and t 0 +0.09 s respectively from left to right) for the Bubble 1.

Two velocities of interest, the bubble rising velocity and the geyser edge velocity are measured for the experiments and the simulations at the positions shown in The geyser formed due to the rebounding of the liquid phase into the vapor phase rises against the applied acceleration. The velocity of the geyser edge is thus expected to attain a maximum value before it starts decreasing. The geyser edge velocity as a function of time is plotted in Fig. 9.6c for the experiments and the simulations for Bubble 1. It can be seen for both the experiments and the simulations that it attains a maximum value and then starts decreasing. It can be seen that the geyser edge moves slower in the simulations, in the very beginning, than in the experiments as is observed in the Fig. 9.5 also. This is, as has already been mentioned, due to the complex nature of the gravity field inside the cell. It is however interesting to note that the general trend of the velocities for the simulations and the experiments is the same and maximum velocity observed for both the experiments and the simulations are comparable. The qualitative agreement in the results of experiments and simulations allows us to go further into the numerical study of the bubble and the geyser dynamics. Figures 9.7a and 9.7b show the plots of the bubble and geyser edge velocities for the two bubbles. These figures are the continuation of Figs. 9.6b and 9.6c, for Bubbles 1 and 2. They reveal the phenomenon which could not be observed in the experiments due to the constraints in the visualization system. It can be observed for both the bubbles that the rising velocity of the bubble saturates to a constant value after an initial transient. The geyser edge however does not show the same behavior. After an initial transient, the geyser edge velocity reaches a maximum value and then starts decreasing under the opposing reorientation acceleration field. The progressing geyser inside the bubble deforms it to such an extent that the bubble breaks-up. Figure 9.8 shows Bubble 1 with a geyser progressing from its trailing edge. We see small bubbles detaching from the trailing edge at around 0.235 s. It is very interesting to note that the break-up of the bubble happens at almost the same time (see Fig. 9.9). Similar observations were made for Bubble 2. It is seen that the geyser edge velocity attains its minimum at the time of breakup as is indicated in Figs. 9.7a and 9.7b. The velocity plotted in these figures after the break-up corresponds to the velocity of the trailing edge of the primary bubble. It can be seen that this velocity quickly reaches the rising velocity of the bubble. Oscillations in this velocity as seen in Fig. 9.7b for bubble 2 are created during the fragmentation and at this point the trailing edge has a corrugated, irregular shape like that of a spherical cap bubble. The two characteristic velocities of a rising bubble depend on both the size of the bubble and the size of the container. In order to quantify the wall effect on bubble and geyser dynamics, a parametric analysis of the bubble rising problem is carried out. Simulations are performed to study the effect of the size of the (spherical) bubble on the velocities of the rising oxygen bubbles. Bubbles of sizes varying from 4 mm to 10 mm are set inside a container with a constant relative size ( R R b ) of 2. The plots of bubble displacement vs. time and bubble velocity vs. time are shown in Figs. 9.10a and 9.10b for bubble sizes 4 mm to 10 mm. It can be clearly seen that, for all bubble sizes, the bubble velocity reaches a constant value (terminal velocity) after an initial transient. The variation of the terminal velocity of the bubble V 0 with bubble size is plotted in Fig. 9.10c; the hollow circles are the data points and the solid line is a fit according to the law (for V 0 expressed in ms -1 and R b expressed in m):

V 0 = 0.5 g * R b (9.5)
The plots of geyser edge displacement vs. time and geyser edge velocity vs. time are shown in the Figs. 9.11a and 9.11b for bubble sizes 4 mm -10 mm. It can be clearly seen that the geyser edge velocity first increases, then attains a maximum value and finally starts dropping. The variation of the maximum velocity of the geyser edge V L with bubble size is plotted in Fig. 9.11c; the hollow circles are the data points and the solid line is a fit according to the law (for V L expressed in ms -1 and R b expressed in m):

V L = 1.35 g * R b (9.6)
Simulations of the study of the effect of the wall proximity on the terminal velocity and the geyser edge velocity of the bubble are carried out by arbitrarily choosing a spherical bubble of size 5 mm and varying the cell diameter from 12 mm to 30 mm. The variation of the terminal velocity of the rising bubble (V 0 ) and the geyser edge velocity (V L ) with respect to the ratio of the cell diameter to the bubble diameter (R/R b ) are plotted in the Fig. 9.12a and 9.12b respectively. Very interestingly, the data points fit well according to the new fitted laws:

V 0 = 0.5 g * R b (1 -tanh( R b R - 1
)) (9.7) and

V L = 1.35 g * R b (1 -tanh( R b R - 1
)) (9.8)

Important to note is that the coefficients 0.

Comparison of the numerical and experimental results with theory

The bubble and geyser edge velocities are compared for the above empirical relations (Eqs. 9.7 and 9.8), with the experiments and the simulations in Table 9.2. The values of Bond and Weber numbers for the geyser in the two bubbles from theory (Eqs. 9.1 and 9.4) experiments and simulations are compared in Table 9.3. It can be seen that the theoretical correlation and the simulations match very well while the values of the experiments seem to be little bit higher. It is interesting to note that irrespective of the complex nature of variation of the gravity field inside the cell and the involved difficulty to precisely calculate the dynamics of the problem using simplified numerical simulations or theoretical correlations, the Weber and Bond numbers of the experiments match quite well with simulations and the theoretical correlations. This shows that the so-thought over-simplification of the experimental results using numerical simulations with zero gravity-relaxation time worked out really well for both the bubbles.

Conclusion

Experiments have been carried out on oxygen to study the reorientation phenomenon inside a long cylinder for two fill ratios (6% and 15%) using a new technique of generating variable gravity fields by magnetic fields. The eddy current in the cell copper flasks has the advantage of accelerating the gravity relaxation. These are the only experiments carried out till date with liquid oxygen, a liquid used in spacecraft engines. The experiments evidence a geyser formation on the trailing edge of the rising bubble. Two-dimensional numerical simulations with zero relaxation time of gravity quench compare well with the experiments. Further numerical simulations helped to adjust the expressions for the terminal velocity of bubble rise and the maximum velocity of the geyser to adapt them to a wide range of problems. The applicability of the expressions is then checked for arbitrary cases and a good agreement in the values is obtained. This good agreement in the results demonstrates the capacity of the method of magnetic levitation to recreate the reorientation phenomenon on earth for low gravities. This method can thus be used for further more focused studies with oxygen liquid for various gravity quenches, fill ratios and pressures inside the cell. Custom designed copper-made components positioned at some pre-defined positions could probably help to totally eliminate the relaxation time and provide a step variation of the gravity field. Eventually, the experiments demonstrate the capability of the instrument OLGA to provide an ideal platform to carry out reorientation studies using oxygen.

Chapter 10

Conclusions and perspectives Conclusions

In the present thesis various fluid dynamic problems involving near-critical fluids are studied under zero-gravity or variable gravity conditions. Complex experiments carried out using the magnetic levitation devices, HYLDE and OLGA, are complemented by numerical and theoretical models to study the stability aspects of near-critical fluids. Also studied are the dynamics of fluid interfaces subjected to vibration or sudden change in gravity/acceleration. The principal conclusions that could be drawn from the study are presented in the following paragraphs.

Fluid instabilities -Supercritical hydrogen: Study of the effect of vibration on supercritical hydrogen filled inside a 2D square cavity when subjected to a sudden thermal quench was carried out using a 2D finite volumes code. The study was realized for fluid temperatures close to the critical point and under zero-g conditions. It was observed that the thermal boundary layer (TBL), formed due to the sudden cooling, propagated inside the fluid domain and destabilized when its thickness crossed a particular threshold value, producing various instabilities depending on the relative direction of vibration and the thermal boundary layer. Three types of instabilities were observed: corner instability at the corners of the cavity, parametric instability when the direction of vibration is perpendicular to the TBL, Rayleigh-vibrational instability when the direction of vibration is tangential to the TBL. The stability diagrams of the three kinds of instabilities were plotted and it was observed that it is possible to observe all the three instabilities simultaneously under certain conditions of vibration and critical point proximity. The parametric instability in a single phase fluid was observed for the first time. It was seen from the analysis that the instability follows a dispersion law similar to that of the Faraday instability in immiscible fluids, suggesting that a weak surface tension is associated with the boundary layer density gradients. The values of the surface tension are estimated from the results, which were found to be of the order of 10 -8 N.m -1 .

Fluid instabilities -Subcritical hydrogen: Vibration experiments on a liquid-vapor interface of hydrogen, realized using the HYLDE setup, have demonstrated various instabilities depending on the relative orientation of the interface with respect to the vibration direction: Faraday instability when the interface is perpendicular to vibration, frozen wave instability when the interface is parallel to vibration. The study was focused on the Faraday wave phenomenon in zero-gravity conditions and frozen wave instability for various gravity levels.

The following observations were made. Under the absence of gravity forces a liquid-vapor interface takes a spherical shape. When subjected to vibration, this spherical interface can develop (i) Faraday waves on the part of the interface perpendicular to the vibration and (ii) bands of alternate liquid and vapor phases developing from the part of the interface tangential to the vibration. It was observed that the Faraday wave instability follows the dispersion law derived by Kumar and Tuckermann [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] for the zero-gravity conditions. Far enough from the critical point, the Faraday waves choose a square pattern while close to the critical point they align to form a line pattern. The pattern wavelength decreases when nearing the critical point. Thus the transition from the square to line pattern is due to the enhanced viscous dissipation effects when the wavelength becomes small close to the critical point. The experiments successfully plotted the critical temperature of transition from square to line pattern for various vibrational accelerations.

Frozen wave experiments were realized with the gravity levels varying from 0.18g 0 to 0.05g 0 to study the effect of gravity on the amplitude of the frozen wave instability close to the critical point. It was seen that the frozen wave instability agreed well with the inviscid linear stability analysis of Lyubimov and Chera-panov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. The experimental study indicated that for a particular vibration parameters, far from the critical point, the amplitude increases weakly with an increase in the temperature of the fluid. However very close to the critical point, the amplitude decreases quite remarkably with an increase in the temperature due to strong viscous dissipation effects. The reduced amplitude versus reduced vibrational velocity plot showed a good agreement with the weakly non-linear amplitude correlation of Lyubimov and Cherapanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. It was seen that for a given temperature, the amplitude of the frozen wave instability changes with gravity according to a power law with exponent -0.7. It was observed from the frozen wave experiments that as the gravity tends to zero, the amplitude of the frozen wave may become more than that of the height of the cell. It thus appears as parallel bands of liquid and vapor phases. It is thus concluded that the band pattern observed in the zero-g experiments share the same kind of origin as that of the frozen wave instability.

Dynamic equilibrium of a liquid-vapor interface under vibration: Vibration experiments carried out on near-critical hydrogen successfully demonstrated a tilted dynamic equilibrium position of the interface under vibration.

The experiments demonstrated a good agreement with the theoretical correlation derived by Wolf [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF] for various gravity levels and for various vibrational parameters. This is the first time this phenomena is experimentally investigated, thanks to the magnetic levitation device HYLDE.

Effect of brusque change in gravity/acceleration on the dynamics of an interface: Rapid gravity variation experiments on large elliptical oxygen vapor bubbles carried out using the magnetic levitation device OLGA put in evidence the rebounding of the liquid phase and the eventual formation of a geyser from the aft part of the interface. Bubble aft and fore end velocity measurements were carried out using image processing softwares. Two-dimensional numerical simulations carried out for a simpler case of gravity change match well with the experiments. Further numerical simulations carried out with spherical bubbles helped to fine tune the existing theoretical correlations to take into account the effect of walls on the geyser and bubble velocities. The new theoretical correlations seemed to match well with the experiments.

Perspectives

The research carried out in the present thesis deepens our knowledge of interfacial phenomena and thermal boundary layer instabilities. The study provides interesting insight into various kinds of instabilities in the supercritical as well as subcritical region of a fluid under vibration. Numerical as well as experimental study of the effect of a sudden variation of gravity on the dynamics of an interface helped fine tune theoretical correlations of the geyser phenomenon. Most of the above studies were performed with fluids near (above and below) their critical point, where they are more unstable and emphasize the mechanical disturbances. The results obtained so far, however, should also be observed with fluids and/or liquid mixtures far from their critical conditions. There is further scope for research in the field.

Stability of a thermal boundary layer under vibration:

The present thesis work successfully demonstrated, through numerical simulations, the existence of parametric instability in supercritical fluids when they are subjected to vibration. It also extended the study of Rayleigh-vibrational instabilities by Amiroudine et al. [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF]. There are however some more aspects that still need some attention. There is a need to demonstrate experimentally the existence of the parametric instability in supercritical fluids when they are subjected to vibration. This work could not be carried out using the existing experimental setup due to constraints with respect to the maximum achievable amplitude of vibration. The experimental investigation of the Rayleigh-vibrational instabilities and comparison with the existing numerical results is needed to validate the results from the numerical simulations. This could not be carried out with the present setup due to a difficulty in imposing a sudden cooling with the existing cooling system.

Stability of a liquid-vapor interface under vibration: The present work carried out a comprehensive study of the Faraday wave instability in zero-g and frozen wave instability under various gravity levels. It successfully explained the root cause of the transition of Faraday wave instability into band pattern, suggesting that the band pattern phenomenon is closely related to the frozen wave instability. However a more extensive study of the band pattern is pending.

Geyser phenomenon: Experiments on the geyser phenomenon compared well with numerical simulations for a simplified case. They also helped to fine tune theoretical correlations to account for the wall effect. There is however uncertainty regarding the variation of the gravity field inside the cell. It is understood that careful design of the copper flanges can improve the gravity variation (temporal as well as spatial) inside the cell and thus can improve the reliability of the experimental results.

Conclusions et perspectives (Français)

Conclusions L'objectif de la thèse concerne l'étude de la dynamique des fluides proches de leur point critique sous un champ de gravité variable. Des expériences ont été effectuées en utilisant les installations de lévitation magnétique HYLDE ("HYdrogen Levitation DEvice") et OLGA ("Oxygen Low Gravitation Apparatus"). Des théories et des modélisations numériques ont été utilisées et comparées avec résultats issus des expériences sur la dynamique d'une interface liquide-vapeur soumise à une variation rapide de l'accélération de gravité et/ou à un "quench" de température. Les conclusions principales de ces études sont résumées ci-dessous.

Instabilités hydrodynamiques -l'hydrogène supercritique: L'effet de vibrations sur l'hydrogène supercritique contenu dans une cavité carrée 2D et soumise à un brusque refroidissement par les parois a été étudié avec un code numérique en volumes fini et l'algorithme SIMPLER. L'étude est réalisée proche du point critique et en absence de gravité. Il a été observé que la couche limite thermique issue du brusque refroidissement se propage à l'intérieur du domaine fluide et se déstabilise lorsque son épaisseur franchit un seuil, produisant des instabilités dépendant de la direction relative de la vibration et de la couche limite thermique. Trois types d'instabilités ont été observées: l'instabilité de "coin" au niveau des coins de la cavité, l'instabilité paramétrique quand la direction des vibrations est perpendiculaire à la couche limite thermique et l'instabilité Rayleigh-vibrationnelle lorsque la direction des vibrations est tangentielle à la couche limite thermique. Les diagrammes de stabilité des trois types d'instabilités ont été tracés et il a été observé qu'il est possible d'obtenir les trois types d'instabilités simultanément sous certaines conditions de vibrations (amplitude et fréquence) et proximités du point critique. L'instabilité paramétrique dans un fluide monophasique a été observée pour la première fois. Il a été vu que cette instabilité suit une loi de dispersion de même type que celle de l'instabilité de Faraday dans des liquides immiscibles, indiquant une tension de surface associée au gradient de densité de la couche limite. Les valeurs de la tension de surface estimées à partir des résultats sont de l'ordre de 10 -8 N.m -1 .

Instabilités hydrodynamiques -l'hydrogène sous-critique: Les expériences de vibrations sur une interface liquide-vapeur de l'hydrogène, réalisées en utilisant l'installation HYLDE, ont démontré différents types d'instabilités dépendant de l'orientation relative de l'interface par rapport à la direction des vibrations: l'instabilité de Faraday lorsque l'interface est perpendiculaire aux vibrations et l'instabilité sous forme d'ondes gelées quand l'interface est parallèle aux vibrations. L'étude des ondes de Faraday a été faite dans des conditions d'apesanteur et celle des ondes gelées pour différents niveaux de gravité.

Les observations suivantes donnent un aperçu de ces études. En absence de gravité, l'interface liquide-vapeur prend une forme sphérique. Lorsqu'elle est soumise à des vibrations, cette interface sphérique peut se déstabiliser donnant: (i) des ondes de Faraday sur la partie de l'interface perpendiculaire aux vibrations et (ii) des bandes alternées des phases liquide et vapeur sur la partie de l'interface tangentielle aux vibrations. Il a été observé, lors de ces expériences, que l'instabilité de Faraday suit la loi de dispersion obtenue par Kumar et Tuckermann [START_REF] Kumar | Parametric instability of the interface between two fluids[END_REF] en apesanteur. Assez loin du point critique, les ondes de Faraday s'organisent suivant une structuration carrée tandis que près du point critique elles s'alignent pour donner des lignes verticales. La longueur d'onde diminue quand on s'approche du point critique. La transition de la structuration carrée à la forme des lignes est due à l'augmentation des effets de dissipation visqueuse lorsque la longueur d'onde devient petite proche du point critique (les gradients de vitesse augmentant). La température critique de transition de la structuration carrée à la forme des lignes a été tracée pour différentes accélérations vibrationnelles.

Les expériences ont été réalisées pour différents niveaux de gravité variant de 0,18g 0 à 0,05g 0 afin d'étudier l'effet de la gravité sur l'amplitude de l'instabilité de l'onde gelée près du point critique. On a vu que l'instabilité de l'onde gelée est en bon accord avec l'analyse de stabilité linéaire basée sur une hypothèse de fluide non-visqueux Lyubimov et Cherapanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. L'étude expérimentale a indiqué que, pour des paramètres de vibration donnés, loin du point critique, l'amplitude augmente faiblement avec une augmentation de la température du fluide. Mais très proche du point critique, l'amplitude décroît assez remarquablement avec une augmentation de la température due aux effets de dissipation visqueuse. La comparaison de l'amplitude en fonction de la vitesse vibrationnelle a montré un très bon accord entre les résultats issus de nos expériences et la loi de dispersion de Lyubimov et Cherapanov [START_REF] Lyubimov | On the development of steady relief on fluid interface in a vibrational field[END_REF]. Il a été observé que pour une température donnée, l'amplitude de l'instabilité change avec la gravité selon une loi de puissance avec un exposant de -0.7. Il a aussi été observé dans les expériences d'ondes gelées que quand la gravité tend vers zéro, l'amplitude de l'onde gelée peut devenir supérieure à celle de la hauteur de la cellule. Il apparaît alors, suivant toute la hauteur de la cellule, des bandes parallèles de phases liquide et vapeur. On a donc conclu que les bandes liquide-vapeur observées dans les expériences en apesanteur proviennent de l'instabilité d'ondes gelées.

Equilibre dynamique d'une interface liquide-vapeur sous vibrations: Les expériences réalisées sur l'hydrogène proche du point critique ont démontré avec succès la position d'équilibre dynamique de l'interface inclinée par rapport à l'horizontale sous vibrations. Les expériences ont montré un bon accord avec la corrélation théorique de Wolf [START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF] pour différents niveaux de gravité et paramètres vibratoires (amplitude et fréquence). C'est la première fois que ce phénomène a été étudié expérimentalement, grâce au dispositif HYLDE de lévitation magnétique.

Effet de la variation brusque de gravité / accélération sur la dynamique d'une interface: Les expériences de variation rapide de la gravité sur des grosses bulles elliptiques d'oxygène, effectuées en utilisant le dispositif de lévitation magnétique OLGA, ont mis en évidence le rebondissement de la phase liquide sur le fond de la cellule et la formation éventuelle d'un geyser.

Les mesures de vitesse des bulles ont été effectuées en utilisant des logiciels de traitement d'images. Des simulations numériques bidimensionnelles effectuées pour un cas simple de changement de gravité correspondent bien avec les observations expérimentales. Des simulations numériques réalisées avec des bulles sphériques ont contribué à affiner les corrélations théoriques existantes pour prendre en compte l'effet des parois sur le geyser vis-à-vis des vitesses de bulles. Les nouvelles corrélations théoriques sont en très bon accord avec les résultats issus des expériences.

Perspectives

La recherche effectuée dans cette thèse a approfondi notre connaissance des phénomènes d'interface et des instabilités de couche limite thermique. Cette thèse donne un aperçu intéressant sur les différents types d'instabilités dans la région supercritique ainsi que la région sous-critique d'un fluide sous vibrations. Les études numériques et expérimentales de l'effet de variations rapides de la gravité sur la dynamique d'une interface ont permis d'affiner les corrélations théoriques du phénomène de geyser. La plupart de ces études ont été réalisées avec des liquides près de leur point critique, où ils sont beaucoup plus instables. Les résultats obtenus devraient également être observés avec des liquides et / ou mélanges de liquides loin de leurs conditions critiques mais avec des valeurs des différents paramètres nettement plus importants.

Stabilité d'une couche limite thermique sous vibrations: Le travail de thèse a démontré avec succès, grâce à des simulations numériques, l'existence d'instabilité paramétrique dans les fluides supercritiques quand ils sont soumis à des vibrations. Il a également étendu l'étude des instabilités Rayleigh-vibrationnelles par Amiroudine et Beysens [START_REF] Amiroudine | Thermovibrational instability in supercritical fluids under weightlessness[END_REF]. Il y a cependant quelques aspects qui nécessitent encore une attention particulière. Il est par exemple nécessaire de démontrer expérimentalement l'existence de l'instabilité paramétrique dans les fluides supercritiques quand ils sont soumis à des vibrations. Ce travail n'a pas pu être effectué car le dispositif expérimental existant avait des limitations quant à l'amplitude maximale possible des vibrations. L'étude expérimentale des instabilités de Rayleigh-vibrationnel sont également nécessaires afin de valider les résultats issus des simulations numériques. Cela n'a pas non plus pu être effectué avec l'installation actuelle en raison de la difficulté à imposer un refroidissement brusque avec le système de refroidissement existant.

Stabilité d'une interface liquide-vapeur sous vibrations: Le travail de thèse a permis d'effectuer une étude approfondie de l'instabilité des ondes de Faraday en apesanteur et de l'instabilité des ondes gelées sous différents niveaux de gravité. Il a permis d'expliquer la cause de la transition d'une instabilité de Faraday à une instabilité de type bandes alternées des phases liquide vapeur associée à l'instabilité de l'onde gelée. Cependant, la compréhension du passage des bandes horizontales de forme rectangulaire à la formation de lignes verticales alternées est actuellement en cours.

Le phénomène de Geyser: Les expériences sur le phénomène de geyser concordent bien avec les résultats des simulations numériques pour un cas simplifié. Les résultats issus des simulations numériques ont également aidé à affiner les corrélations théoriques pour tenir compte de l'effet de paroi. Il y a cependant, une incertitude due à la variation complexe du champ de gravité à l'intérieur de la cellule. Une conception précise des brides en cuivre peut améliorer la variation de la gravité (dans le temps ainsi que dans l'espace) à l'intérieur de la cellule et peut éventuellement améliorer la fiabilité des résultats expérimentaux.

8 9

 9 Dynamic equilibrium of a liquid-vapor interface of H 2 under vibration at various g-levels 119 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 121 8.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 122 8.3.1 Determination of the fluid temperature . . . . . . . . . . . 122 8.3.2 Phenomenological understanding of the tilting of the interface123 8.3.3 Effect of vibrational parameters on the angle of the interface125 8.3.4 Effect of critical point proximity on the dynamic equilibrium of the interface . . . . . . . . . . . . . . . . . . . . . 126 8.3.5 Effect of gravity on the angle of interface . . . . . . . . . . 128 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 vii CONTENTS Interface dynamics and geysering during an abrupt change in acceleration 130 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 135 9.3 Numerical simulations set-up . . . . . . . . . . . . . . . . . . . . 137 9.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.4.1 Comparison of the numerical results with experiments . . . 140 9.4.2 Effect of walls on the bubble dynamics . . . . . . . . . . . 145 9.4.3 Comparison of the numerical and experimental results with theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 10 Conclusions and perspectives 151 Nomenclature 166

Figure 2 . 1 :

 21 Figure 2.1: (a) PVT Diagram and (b) PT diagram of a pure substance

Figure 3 . 1 :

 31 Figure 3.1: Various classical methods of creating zero-g (a) Drop towers, (b) Parabolic flights, (c) Sounding rocket, (d) launch vehicles and (e) Space stations.

Figure 3 . 2 :

 32 Figure 3.2: Variation of B and B.dB/dz inside a typical magnetic coil similar to that of HYLDE. The red and blue circles indicate the points for levitation of diamagnetic and paramagnetic materials.

Figure 4 . 1 :

 41 Figure 4.1: Schematic diagram of a magnetic levitation device

4. 2 . 1

 21 Superconducting coil HYLDE (shown in Fig 4.2) uses a cylindrical superconducting coil (shown in Fig. 4.2b) of inner diameter 90 mm, external diameter 186 mm and height 200 mm and contains 32500 turns of Nb-Ti cable. The inductance of the coil is 52 H.

Figure 4 . 2 :

 42 Figure 4.2: (a) picture of HYLDE station (b) magnetic coil of HYLDE

Figure 4 . 3 :

 43 Figure 4.3: Schematic diagram of the Roubeau's bath with (a) Joule Thomson valve and (b) capillary tube.

Figure 4 . 4 :

 44 Figure 4.4: Roubeau's bath with capillary tube

Figure 4 . 5 :

 45 Figure 4.5: HYLDE: Variation of (a) longitudinal component of magnetic field intensity B z and (b) longitudinal component G z with the height z inside the coil. (c) shows the uniformity ǫ of the gravity field inside a cell window of 8 mm × 8 mm.

Figure 4 . 6 :

 46 Figure 4.6: HYLDE: Variation of the longitudinal component of magnetic field intensity B z with height z (a) without and (b) with a soft iron insert.

Figure 4 .

 4 6 compares G z inside the coil for the cases without (Fig.4.6a) and with (Fig.4.6b) insert. It can be observed that the presence of the insert deforms G z and makes it uniform at the top of the coil thus improving the uniformity of the g-field.

  Figure 4.7 compares the uniformity of the gravity field inside a square cell for the cases without (Fig. 4.7a) and with (Fig. 4.7b) an insert. It can be seen that the gravity uniformity inside the cell of sides 3 mm and 7 mm are improved to 1% and 3% respectively.

Figure 4 . 7 :

 47 Figure 4.7: HYLDE: Uniformity of the gravity field inside a cell of size 8 mm (a) without and (b) with soft iron insert. The red square shows a 3 mm cell while the blue square shows a 7 mm cell.

Figure 4 . 8 :

 48 Figure 4.8: (a) Picture of the OLGA station, (b) magnetic coil of OLGA

Figures 4 .

 4 9a,b show the variation of the longitudinal components of the magnetic field intensity B and G inside the coil.

Figure 4 . 9 :

 49 Figure 4.9: OLGA: Variation of (a) B z and (b) G z with the height z inside the coil. (c) shows the uniformity of the gravity field inside a cell window of 30 mm × 100 mm in % of g 0 .

Figure 4 .

 4 Figure 4.10: (a) 3 mm vibration cell, (b) 7 mm vibration cell

Figure 4 . 11 :

 411 Figure 4.11: Experimental cell with copper heat exchanger for geyser experiments.

Figure 5 . 1 :

 51 Figure 5.1: Three different computational configurations: (a) isothermal conditions on all 4 walls, (b) isothermal conditions on vertical walls and adiabatic conditions on horizontal walls, (c) isothermal conditions on horizontal walls and adiabatic conditions on vertical walls.

Figure 5 . 3 :

 53 Figure 5.3: Control volume

Figure 5 . 4 :

 54 Figure 5.4: Control volume

Figure 6 . 1 :

 61 Figure 6.1: Three different computational configurations: (a) isothermal conditions on all 4 walls, (b) isothermal conditions on vertical walls and adiabatic conditions on horizontal walls, (c) isothermal conditions on horizontal walls and adiabatic conditions on vertical walls.

Figure 6 . 2 :

 62 Figure 6.2: Evolution of the thermal field for the configuration of Fig. 6.1a (4 isothermal walls) with f = 5.56 Hz, a = 20 mm, ∆T = 1 K and δT = 100 mK. Different types of instabilities are observed (1): corner instability; (2)-(6): Parametric instability on the vertical walls; (4)-(6): Rayleigh vibrational instability on the horizontal walls.

Figure 6 . 3 :

 63 Figure 6.3: Evolution of the corner instability at each period of vibration. Configuration of Fig. 6.1c (adiabatic vertical walls, isothermal horizontal walls): f = 2.78 Hz (period 0.36 s), a = 20 mm, ∆T = 1 K with a temperature quench of 100 mK. (a,f) 1.44 s, (b,g) 1.8 s, (c,h) 2.16 s, (d,i) 2.52 s and (e,j) 2.88 s respectively. Top row: temperature fields; bottom row: corresponding stream lines. PV: primary Vortex, SV: secondary Vortex.

Figure 6 . 4 :

 64 Figure 6.4: Stability curves for the corner instability for ∆T = 10 mK with a quench of 1 mK and for ∆T = 100 mK with a quench of 10 mK. The fluid is unstable above the curve and stable below it.

, 6 .

 6 5g correspond to the initiation of the fingers on the left wall.Similar to the classical Faraday instabilities, fingers on a particular wall can be seen only during a destabilizing half-cycle (for example on the left wall in Figs.[START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF].5a, 6.5c, 6.5e and on right wall as in Figs. 6.5b, 6.5d, 6.5f) and a smoothening of the boundary layer is seen during a stabilizing half cycle (as on the left wall in Figs.[START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF].5b, 6.5d, 6.5f) and on the right wall in Figs.[START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF].6a, 6.6c, 6.6e). The size of the fingers (or vortices) increases with time as can be observed on the left wall in Figs.[START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF].5a, 6.5c, 6.5e.

Figure 6 . 5 :

 65 Figure 6.5: Parametric instability in the Fig. 6.1b configuration (adiabatic horizontal walls, isothermal vertical walls) for the case f = 2.78 Hz (period 0.36 s), a = 20 mm, ∆T = 100 mK and δT = 10 mK in 6 consecutive half periods of vibration. (a,g) 14.58 s, (b,h) 14.76 s, (c,i) 14.94 s, (d,j) 15.12 s, (e,k) 15.3 s and (f,l) 15.48 s Top row: temperature fields; Bottom row: corresponding streamlines.

Figure 6 . 6 :

 66 Figure 6.6: (a-d) Temperature field of the parametric instability in the Fig. 6.1b configuration (adiabatic horizontal walls, isothermal vertical walls) for the case f = 2.78 Hz (period 0.36 s),a = 20 mm, ∆T = 100 mK and δT = 10 mK in 4 consecutive time periods at (a) 14.4 s, (b) 14.76 s, (c) 15.12 s and (d) 15.48 s respectively plotted for the same vibration phase. (e-h) Temperature field of the parametric instability for a = 20 mm, ∆T = 100 mK, δT = 10 mK for frequencies: (e) 2.78 Hz, (f) 5.56 Hz, (g) 8.33 Hz and (h) 16.66 Hz.

Figure 6 . 7 :

 67 Figure 6.7: (a) Reduced wavelength (Λ) of the parametric instability versus reduced acceleration (Γ) for the cases ∆T = 10 mK, 100 mK and 1000 mK with corresponding quenches of δT = 1 mK, 10 mK and 100 mK respectively (log-log plot). Line: best fit (see text). (b) Reduced wavelength (Λ) of the parametric instability versus critical point proximity (∆T ) for f = 2.78 Hz, 5.56 Hz and 8.33 Hz and a = 15 mm (log-log plot).

  -8 N.m -1 , 1.72 × 10 -8 N.m -1 and 4.03 × 10 -8 N.m -1 for ∆T = 10 mK, 100 mK and 1000 mK respectively. It is interesting to note that the effective surface tension of the fluid decreases as one approaches the critical point. The values of surface tension of the liquid-vapor interface of H 2 for the three proximities to the critical point ∆T = 10 mK, 100 mK and 1000 mK are 1.98 × 10 -7 N.m -1 , 3.61 × 10 -6 N.m -1 and 6.58 × 10 -5 N.m -1 respectively. It can be seen that the estimated values of the transient surface tension are very small in comparison with those of a liquid-vapor interface. Figure6.7 shows the plot of wavelength Λ(= λ h ) versus critical point proximity ∆T for the parametric instability for the cases f = 2.78 Hz, 5.56 Hz and 8.33 Hz and a = 15 mm. It is seen that the curves in the log-log plot for various frequencies are parallel to each other. The wavelength varies with the critical point proximity according to a power law with exponent 0.075 ± 0.05.

Figure 6 . 8 :

 68 Figure 6.8: ω 2 (ρ 1 + ρ 2 ) vs k 3 plot for ∆T = 10 mK, 100 mK and 1000 mK.

Figure 6 . 9 :

 69 Figure 6.9: Stability curves for the parametric instability showing the critical amplitude (a cr ) vs frequency f for (a) ∆T = 100 mK and 10 mK with quenches of δT = 10 mK and 1 mK respectively and (b) ∆T = 100 mK with quenches of δT = 10 mK and 5 mK.

Figure 6 . 10 :

 610 Figure 6.10: Critical vibrational acceleration (Γ cr ) vs frequency (f ) of vibration for the parametric instability for (a) ∆T = 100 mK and 10 mK with quenches of δT = 10 mK and 1 mK respectively and (b) ∆T = 100 mK with quenches of δT = 10 mK and 5 mK.

Figure 6 . 11 :

 611 Figure 6.11: Critical Rayleigh Vibrational number Ravc for two proximities to the critical point: (a) vs frequency, (b) vs amplitude.

Figure 6 . 12 :

 612 Figure 6.12: Critical Rayleigh vibrational number Ravc vs critical point proximity ∆T for f = 2.78 Hz, 5.56 Hz and 8.33 Hz and a = 10 mm. The solid lines are the corresponding variations in CO 2 from Ref. [23] with slopes -0.83 for ∆T < 0.1 K and -0.31 for ∆T > 0.1 K.

Figure 6 .

 6 Figure 6.13: (a) Stability curve (amplitude a cr vs frequency f ) for the Rayleigh vibrational instability for ∆T = 10 mK and 100 mK with quenches δT = 1 mK and 10 mK respectively, (b) Critical velocity (af ) cr vs frequency plot for ∆T = 10 mK and 100 mK with quenches δT = 1 mK and 10 mK respectively.

Figure 6 . 14 :

 614 Figure 6.14: Stability domain for the three types of instabilities: corner (solid line with dots), parametric instability (solid line with boxes) and Rayleigh vibrational instability (solid line with circles) for (a) ∆T = 0.1 K and (b) ∆T = 0.01 K

Figure 7 . 1 :

 71 Figure 7.1: (a) Hydrogen bubble in zero-gravity. (b) Variation of the thickness of the liquid-vapor interface with critical point proximity T c -T in a cell of size 3 mm filled with H 2 . (c) Log-log plot of the meniscus thickness t m vs T c -T in a cell of size 3 mm filled with H 2 showing a power law with exponent 0.79 (see text).

Figure 7 . 2 :

 72 Figure 7.2: Variation of the wavelength of the Faraday wave instability with T c -T (from 3 fingers at 20 mK to 9 fingers at 7 mK) for a = 0.29 mm and f = 40 Hz. The letters A and B stand for the two rows of Faraday fingers in two different planes.

Figure 7 .

 7 Figure 7.2 shows the results of the Faraday wave experiments for a = 0.29 mm and f = 40 Hz. As one approaches the critical point, the surface tension of the liquid-vapor interface and the liquid-vapor density difference decrease according to Eqs. 7.2 and 7.3. The decrease in the surface tension of the interface renders it more flexible, resulting in the decrease in the wavelength of the instability. This can be observed very clearly in Fig. 7.2 where the number of Faraday fingers increases from 3 at 20 mK to 9 fingers at 7 mK. In accordance with the work of Fauve et al. [30], the wavelength does not decrease indefinitely due to the viscous dissipation and saturates to a finite value close to the critical point.

  Figure 7.4b shows the variation of the non-dimensional number V with the relative critical point proximity ε for the case a = 0.55 mm and frequencies ranging between 13 and 30 Hz. It can be seen that the viscous dissipation increases with a decrease in ε showing the strong viscous dissipation effects close to the critical point. The role of the viscous dissipation on the square to line pattern transition is thus evident. It is seen that the threshold value of V for the transition is around 0.1.

Figure 7 . 4 :

 74 Figure 7.4: (a) Viscous dissipation µ(aω) 2 f λ 2 vs vibrational energy δρ(aω) 2 (b) non-dimensional viscous dissipation µ f λ 2 ∆ρ for a = 0.55 mm and various frequencies showing the transition from square to line pattern.

Figure 7 . 5 :

 75 Figure 7.5: (a) Border between square and line pattern (b) Wavelength at the square-line transition for a = 0.551 mm and f varying between 10 and 35 Hz. The solid triangles are the experimental results. Solid curves are smoothing of the data

Figure 7 . 6 :

 76 Figure 7.6: (a) Critical wavelength λ c at instability onset as a function of frequency f for different values of the proximity to the critical point T c -T , (b) Plot of σ 2 /k 3 vs k for T c -T = 7.5 mK. The solid lines are theoretical values and the symbols are the experimental results.

Figure 7 . 7 :

 77 Figure 7.7: (a) Stability diagram for various critical point proximities T c -T and various wavenumbers k, (b) Critical vibrational acceleration A c /g = (a c ω 2 c )/g versus frequency f = ω/2π for different values of the proximity to the critical point T c -T .

2 . 8 Figure 7 . 8 :

 2878 Figure 7.8: Experimental configuration

Figure 7 . 9 :

 79 Figure 7.9: Stability diagram for the frozen wave instability in H 2 for three gravity levels, comparison between experiment and theory. The solid lines correspond to the theoretical values at T c -1 mK and the symbols are the experimental values (see text).

Figure 7 . 10 :

 710 Figure 7.10: Frozen wave instability for the vibration case a = 0.83 mm and f = 25 Hz for gravity levels: (a) 0.18g 0 , (b) 0.1g 0 and (c) 0.05g 0 . (g 0 is the Earth acceleration constant).

Figure 7 . 11 :Figure 7 . 12 :

 711712 Figure 7.11: The waveform of frozen wave instability at T c -T = 13 mK and 0.5 mK fitted using truncated Fourier series as in Eq. 7.13 for a = 0.83 mm, f = 25 Hz, g = 0.05g0.

Figure 7 . 13 :

 713 Figure 7.13: Variation of the parameter V (= ν f λ 2 ) with W .

Figure 7 .

 7 Figure 7.14: (a) Variation of the the non-dimensional wave height A h as a function of ε for various frequencies and gravity levels for a = 0.83 mm. (b) Zoomed view for g = 0.05 g 0 .

Figure 7 . 15 :

 715 Figure 7.15: Reduced wave height (A * = A/(λ 0 (1 + δρ * ))) vs ∆(aω) * for various gravity levels 0.18g 0 , 0.1g 0 and 0.05g 0 . The solid line corresponds to the theoretical correlation. The symbols are the experimental results.

Figure 7 . 17 :

 717 Figure 7.17: Experiments with 3 mm cell. (a) Shape of the Hydrogen bubble under zero-g. (b) Transition from Faraday waves to band pattern as temperature T c -T is increased from 20 mK to 3 mK.

Figure 7 . 18 :

 718 Figure 7.18: Transition from Faraday wave instability to band pattern showing the coexistence of the two kinds of instabilities; The letter B in the figure shows the initiation of the line marks inside the fluid domain. a = 0.41 mm, f = 32.5 Hz.

  7.18 (a = 0.41 mm and f = 32.5 Hz) are plotted on Fig. 7.19. The images corresponding to the thermal proximities 10 mK, 9 mK and 6 mK are shown as the points (I), (II) and (III) on the transition diagram. The point (I) denotes the Faraday instabilities, point (II) indicates a coexistence of the Faraday waves and the band pattern and (III) is the thermal proximity where the Faraday waves are completely eliminated and only the band pattern exist.

Figure 7 . 19 :

 719 Figure 7.19: Transition from Faraday wave instability to band pattern.

Figure 7 . 20 :

 720 Figure 7.20: Frozen wave amplitude for various gravity levels (a) a = 0.53 mm, f = 25 Hz (b) a = 0.83 mm, f = 25 Hz. (Temperature is not reported as the amplitude is only weakly T c -T dependent).

  Figure 8.1a shows the interface attaining a dynamic equilibrium, tilting towards the right wall. In our experiments the interface always seemed to tilt towards the right wall. This non-random behavior is presumably due to a slight initial tilt in the cell with respect to the vibrational direction. In the absence of this initial tilt, the interface should randomly choose a particular wall based on the experimental perturbations. It is not supposed to flip to the other wall once it found its equilibrium position.As the temperature of the cell is further increased, the interface deforms giving rise to the frozen wave instability. Figure8.1b shows the transition region when the tilted interface destabilizes giving birth to the frozen waves (Figs. 8.1c and 8.1d).

Figure 8 . 1 :

 81 Figure 8.1: Evolution of the dynamics at the interface for the vibration case a = 0.83 mm and f = 35 Hz and gravity level 0.05g 0 as the temperature is increased from approximately (a) T c -45 mK to (d) T c -15 mK.

Figure 8 . 2 :

 82 Figure 8.2: Comparison of the experimental results with theory (g = 0.05g 0 ).

  3) at T c -25 mK which indicates the nascence of the frozen waves. They develop much more at T c -20 mK (shown as B in Fig. 8.3).

Figure 8 . 3 :

 83 Figure 8.3: Effect of T c -T on the angle of the interface when the temperature of the cell is changed from T c -62 mK to T c -20 mK for the vibration case a = 0.83 mm and f = 35 Hz and gravity level 0.05g 0 . A and B indicate the start of frozen waves instabilities (see text).

Figure 8 . 4 :

 84 Figure 8.4: Effect of T c -T on the angle of the interface (comparison with theory).

Figure 8 .

 8 Figure 8.5 shows the images of the interface corresponding to the vibration case a = 0.83 mm and f = 35 Hz for various gravity levels 0.1g 0 , 0.05g 0 and 0.01g 0 .It can be seen that as the gravity is reduced, the angle of the interface reduces, well in agreement with Eq. 8.1.

Figure 8 . 5 :

 85 Figure 8.5: Effect of g on the angle of the interface for a = 0.83 mm and f = 35 Hz.

Figure 8 .

 8 Figure 8.6 compares the results of the experiments with theory. The solid dots are the experimental results while the solid line traces the Eq. 8.1.It can be seen that the general trend of the plots is the same. The point corresponding to 0.01g 0 deviates considerably from the line. This is due to the fact that the uniformity of the gravity levels inside the cell is of the order of 0.03g 0 , greater than the gravity level sought.

Figure 8 . 6 :

 86 Figure 8.6: Effect of g on the angle of the interface for a = 0.83 mm and f = 35 Hz.

Figure 9 . 1 :

 91 Figure 9.1: Spatio-temporal variation of the gravity field inside the cell and variation of the current (I A ) in the internal coil. Geyser phenomenon is observed between the heights z = 5 and z = 10 mm.

Figure 9 . 2 :

 92 Figure 9.2: Computational domain for 2D numerical simulations

Figure 9 . 3 :

 93 Figure 9.3: (a) and (b) Describe the various parts of the image: Vapor bubble at t0 (L is the liquid phase, V is the vapor phase, W is the white patch due to reflection of light from the center of the bubble). (c) -(f): Evolution of the shape of Bubble 1 (fill factor 6%) for a gravity variation from 0g 0 to 0.4g 0 at times (c) t 0 + 0.045 s, (d) t 0 + 0.065 s, (e) t 0 + 0.075 s and (f) t 0 + 0.080 s. The numbers 1, 2, 3 and 4 show how the geyser evolves over time.

Fig. 9 .

 9 Fig.9.6a. The velocity measured at point A, the leading edge of the bubble, also called the ullage velocity or the vapor penetration rate, corresponds to the rising velocity of the bubble whereas the velocity measured at point B is the velocity of the geyser edge. The velocity of the vapor bubble as a function of time is plotted in the Fig.9.6b for the Bubble 1 both for experiments and simulations. The rising velocity of the bubble matches quite well with that of the experiments. It can be seen that the bubble rising velocity saturates to a constant value (called the terminal velocity V 0 ) after a certain time. The value of V 0 seems to agree well for the experiments and the simulations.

Figure 9 . 6 :

 96 Figure 9.6: (a) Measurement of ullage velocity (measured at A) and geyser leading edge velocity (measured at B) for Bubble 1; (b) variation of the ullage velocity with time; (c) variation of the geyser leading edge velocity with time.

Figure 9 . 7 :

 97 Figure 9.7: Plot of numerical simulations of bubble velocity and geyser edge velocity w.r.to time for (a) Bubble 1 and (b) Bubble 2.

Figure 9 . 8 :

 98 Figure 9.8: Dynamics of the geyser showing the events before the bubble fragmentation at 0.09 s, 0.12 s, 0.18 s, 0.21 s, 0.225 s, 0.23 s and 0.235 s for Bubble 1.

Figure 9 . 9 :

 99 Figure 9.9: Dynamics of the geyser showing the bubble fragmentation in experiments (a) at 0.225 s and in simulations (b) at 0.235 s for Bubble 1.

9. 4 . 2

 42 Effect of walls on the bubble dynamics

Figure 9 .

 9 Figure 9.10: (a) Bubble displacement vs time and (b) bubble velocity vs time for 4 mm, 6 mm, 8 mm and 10 mm spherical bubbles for R R b = 2. (c) Variation of the bubble terminal velocity V 0 with R R b (data points in circles and solid line fitted according to 0.5 √ g * R b ).

5 √

 5 g * R b and 1.35 √ g * R b in the expressions for V 0 and V L are for R b /R = 1/2.

Figure 9 .

 9 Figure 9.11: (a) Geyser edge displacement vs time and (b) geyser edge velocity vs time for 4 mm, 6 mm, 8 mm and 10 mm spherical bubbles for R R b = 2. (c) Variation of the maximum geyser edge velocity V L with R R b (data points are in squares and the solid line corresponds to fitted values (see text)).

Figure 9 . 12 :

 912 Figure 9.12: Variation of (a) V 0 and (b) V L with R/R b for a bubble of size R b = 5 mm (data points are in squares and solid lines are the fitted values (see text)).

  Diffusion conductance through control volume face e F e Mass flow rate through control volume face e J e Total flux through control volume face e k Wavenumber (m -1 ) P e Peclet number control volume face e R Radius of the container (m) R b Bubble radius (m)

Table 2 .

 2 1: Critical points of some fluids Substance T c (K) p c (MPa) ρ c (kg.m -3 )

	n -H 2 O 2	33.19 154.58	1.315 5.043	30.11 436.2
	CO 2	304.13	7.378	467.8
	H 2 O	647.1	22.06	322.2
	SF 6	318.63	3.761	742
	Table 2.2: Physical properties of supercritical n -H 2 Property n -H 2 T c (K) 33.19
	ρ c (kg.m -3 )	30.11
	p c (MPa)	1.315

Table 3 .

 3

	1: G 0 for total compensation of gravity for various substances
	Substance Magnetic nature G 0 (T 2 .m -1 )
	H 2	diamagnetic	-991.4
	H 2 O	diamagnetic	-2717
	CO 2	diamagnetic	-4316
	N 2	diamagnetic	-4578
	O 2 (90 K)	paramagnetic	8.156

  While the HYLDE setup can be used to compensate gravity forces in both H 2 and O 2 , OLGA can create zero-g only in O 2 . However for a 1% uniformity of zero-g, OLGA can levitate relatively large volumes of O 2 (up to 0.5 ml) while HYLDE can levitate only about 0.1 ml of H 2 or O 2 for the same gravity uniformity. One has to note that the OLGA setup cannot be used to carry out meaningful experiments close to the critical point due to the magneto-capillary instability called the Cowley-Rosenweig instability

Table 6 .

 6

		1: Orders of magnitude of various time scales involved in the problem for
	two typical critical point proximities
	f (Hz)		∆T = 10mK			∆T = 100mK
		t v ib(s) t v (s)	t D (s) t P E(s)	t v ib(s) t v (s)	t D (s) t P E(s)
	2.77 5.55 8.33 16.66 25	0.36 0.18 0.12 0.06 0.04	236 219 × 10 3 236 219 × 10 3 236 219 × 10 3 236 219 × 10 3 236 219 × 10 3	0.059 0.059 0.059 0.059 0.059	0.36 0.18 0.12 0.06 0.04	259 47 × 10 3 259 47 × 10 3 259 47 × 10 3 259 47 × 10 3 259 47 × 10 3	1.7 1.7 1.7 1.7 1.7

Table 6 . 2 :

 62 Comparison of the size of the cavity with the length aβ p δT for a = 30

	mm
	∆T (K) δT (K) β p (K -1 ) h (m) aβ p δT (m)
	0.01 0.1 1.0	0.001 0.01 0.1	245 0.007 14 0.007 0.82 0.007	0.0074 0.0042 0.0024

Table 9 .

 9

	1: Comparison of bubble and geyser edge velocity for theory, experiments
	and simulations.
	Bubble a (mm) b (mm) Bubble volume Fill ratio
	Bubble 1	13.5	8.5	4100	5.9
	Bubble 2	16.5	12.5	10750	15.3

Table 9 .

 9

	2: Comparison of bubble and geyser edge velocity for theory, experiments
	and simulations.
	Bubble	Ullage velocity V 0 (m.s -1 )	Leading edge velocity V L (m.s -1 )
	Theory Experiment Simulation Theory Experiment Simulation
	Bubble1 0.083 Bubble2 0.072	0.093 -	0.088 0.077	0.24 0.195	0.29 0.198	0.27 0.192
	Table 9.3: Comparison of Bond and Weber numbers for theory, experiments and
	simulations.
	Bubble Bond number (Bo)	Weber number (We)
				Theory Experiment Simulation
	Bubble 1 Bubble 2	31.5 63.5	41.9 42.1	61.5 42.7	55.3 41.4

  Thermal diffusion time scale tD = h 2 D T (s) t v Viscous diffusion time scale t v = h 2 ν (s) t P E Piston effect time scale t P E = h 2 γ 2 D T (s)

	S	Source term per unit volume
	t a	Acoustic time scale t a = h c (s)
	t D
	t vib	Vibration time scale t vib = 1 f (s)
	RES Residue
	PHYSICAL PARAMETERS
	β

p Thermal expansion coefficient (K -1 )

L'application de vibrations peut stabiliser des systèmes mécaniques normalement instables. Par exemple, un pendule simple peut être stabilisé dans sa position inverse sous l'effet de vibrations verticales de forte intensité. Un pendule simple peut être stabilisé dans une position nonverticale sous l'effet de vibrations horizontales de forte intensité. Ce même type de phénomène peut être observé dans les fluides. L'application de vibrations verticales peut stabiliser une instabilité de Rayleigh-Taylor des fluides. De la même façon, sous vibrations horizontales une interface liquide-vapeur peut se retrouver dans une position d'équilibre inclinée par rapport à l'horizontale. Des expériences avec l'hydrogène ont mis en évidence le phénomène d'inclination de l'interface sous vibrations horizontales. Le but de ce chapitre est de présenter les résultats de ces expériences et de les comparer avec un modèle analytique développé par Wolf[START_REF] Wolf | The dynamic stabilization of the rayleigh-taylor instability and the corresponding dynamic equilibrium[END_REF].

Acknowledgements

, my father and my mother, who have been my "Gurus" right from my childhood and who have been a source of inspiration throughout my life. I thank my loving wife without whose constant support and love this work would not have been possible. I thank my brother, my sister and my brother-in-law who helped me in taking vital career decisions at the right time. I earnestly thank CEA and CNES for providing financial support to carry out my research work at CEA Grenoble. I thank my host laboratory CEA Grenoble for giving me a wonderful opportunity to carry out this research work. It is with great amount of reverence that I thank the director of my thesis, Prof. Daniel Beysens, for his constant inspiration and mentoring and the co-director of my thesis, Prof.

Nondimensional wave height (=