Since the early 60s, it is known that **bubble** dynamics and mass transfer rate can be drastically influenced by the solutal Marangoni effect resulting from surface active contaminants accumulation at gas liquid **bubble** surface. For a clean system with no impurities or surfactant at the **bubble** surface, the terminal rise velocity of a spherical **bubble** can be accurately predicted by using the relation of Mei et al. [13] . For a fully contaminated surface, the terminal rise velocity of a spherical **bubble** can be determined with the correla tion established by Schiller and Nauman [14] for solid particle. As for mass transfer, with a clean interface system, mass transfer rate is usually based on the Higbie’s penetration theory [15] with a con tact time defined as the ratio of the **bubble** diameter to the **bubble** rise velocity which is also known as the Boussinesq solution for a **single** spherical **bubble** [16] . Numerical simulations [17,18] have shown that this analytical solution appears to be very accurate to describe interfacial mass transfer for a **single** clean spherical bub ble rising in a still liquid, at large **bubble** Reynolds and Péclet num bers. Some corrections based on results for a **single** **bubble** have been introduced by Winnikow [19] , Takemura and Yabe [17] or Colombet et al. [20] to account for the effect of a finite Reynolds number. For a fully contaminated **bubble**, the mass transfer rate can be described using the mass transfer relation established for a solid spherical particle [21 23] . For partially contaminated bub

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icate glass melt at 1150 ◦ C. Nine glass samples are synthesized and investigated,
utilizing three different amounts of Ce 2 O 3 and three different redox ratios (Ce-
(III)/Ce total ). Employing in-situ observation, the **single**-**bubble** behavior is recorded
with a camera. For each glass melt, five experiments are performed with different initial **bubble** radii. The shrinkage rate (𝑑𝑎∕𝑑𝑡) depends strongly on the cerium con- tent as well as the redox ratio. Numerical calculations are also conducted to support the understanding of the **bubble** shrinkage mechanism in the given cases. The model adequately estimates the experimental data for several cases, and an explanation is proposed for the cases, in which it does not. Moreover, we demonstrate, physically and mathematically, the influence of the initial radius of the **bubble** on the mass transfer between the rising **bubble** and the melt. We confirm the utilization of the “modified Péclet number”, which is a dimensionless number that takes into consid- eration the influence of multivalent elements on mass transfer. Finally, we master the **bubble** shrinkage behavior by normalizing the experimental data employing a characteristic time for the mass transfer (𝜏).

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To cite this version : Filella, Audrey and Ern, Patricia and Roig, Véronique Motion of a **single** **bubble** rising in a countercurrent flow in a Hele-Shaw cell. In: European Two-Phase Flow Group Meeting ETPFGM2013, 13 May 2013 - 15 May 2013 (Lyon, France). (Unpublished)

rather striking, since the latter results where obtained with a laser **bubble** collapsing near an infinite solid boundary.
Finally, the **bubble** disappearance in such experiments can be easily correlated with an abrupt drop of the mean harmonic con- tent of the microphone signal. However the latter event was found to occur about 4000 acoustic cycles after the **bubble** actually disappeared. This delay may be attributed to the extinction time of the levitation cell modes excited by the **bubble** harmonics. For practical applications, these results demonstrates that despite acoustic monitoring is a simple and cheap method to detect non- predictable events occurring near the **single** **bubble**, it is thousands of acoustic cycles late in issuing the diagnostic. This phenomenon must be anticipated if one wish to use this method to get pre-trigger frames of a camera, for example by allowing for an oversized camera memory. Aside from this restriction, this opens the way to the easy detection of uncontrollable foreign objects, such as a micron-sized crystal, allowing its visualization as early as possible during its growth.

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generation of at least one nucleus are plotted on Fig. 5 , along with the line corresponding to inertial cavitation threshold. The latter was obtained by using the dynamic equation of **bubble** wall motion (part 1, [4] ) and the cavitation criterion of **bubble** radius expanding at least twice its initial value. Each contour curve has an upper and lower branch and the zones of possible nucleation lie within these branches. This figure shows even more clearly the existence of an optimal pressure range centered on 225 kPa, especially for low undercooling. There is also an optimal **bubble** size range centered on 8.5 l m. Nucleation can be already achieved at !5 !C over a large size domain (radius from 3 to 13 l m) providing the appropriate acoustic pressure. The influence of the initial radius on nucleation threshold reflects the nonlinear evolution of the **bubble** dynamics with the initial radius. Even if it is not as critical as the applied acoustic pressure, the control of initial **bubble** size may be helpful for achieving nucleation at a desired temperature.

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a b s t r a c t
This paper deals with the inertial cavitation of a **single** gas **bubble** in a liquid submitted to an ultrasonic wave. The aim was to calculate accurately the pressure and temperature at the **bubble** wall and in the liquid adjacent to the wall just before and just after the collapse. Two different approaches were proposed for modeling the heat transfer between the ambient liquid and the gas: the simplified approach (A) with liquid acting as perfect heat sink, the rigorous approach (B) with liquid acting as a normal heat conduct- ing medium. The time profiles of the **bubble** radius, gas temperature, interface temperature and pressure corresponding to the above models were compared and important differences were observed excepted for the **bubble** size. The exact pressure and temperature distributions in the liquid corresponding to the second model (B) were also presented. These profiles are necessary for the prediction of any physical phenomena occurring around the cavitation **bubble**, with possible applications to sono-crystallization.

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Two numerical models for studying the dynamics of formation and rise of single bubbles in high‐viscosity ionic liquids were implemented using the level‐set method. The models describe [r]

The radial motion of an inertial cavitation bub- ble is sufficiently violent so that one may reason- ably think that pressure diffusion produces notice- able effects. Indeed, calculations of the **bubble** dy- namics predict accelerations of the liquid near the **bubble** wall of the order of 10 12 g [2]. In fact, pres- sure diffusion has been considered by Storey & Sz- eri [3], along with other secondary mass-diffusion process, to study the segregation of gases inside a sonoluminescing **single** **bubble**. Here we investigate the effect of pressure diffusion on the segregation of a binary mixture surrounding a radially oscillating **bubble**. We report in this paper theoretical predic- tions from an analytical solution of the transport problem obtained in an earlier paper [4], and we propose a qualitative microscopical mechanism for sono-crystallization, consistent with our theoretical predictions.

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Many irreversibly adsorbed systems have been used to stabilise foams, which have indeed proven to stop coars- ening, with a surprisingly good agreement with the Gibbs criterion [15, 16, 14, 17, 18]. However, many questions remain as to how the behaviour of a **single** **bubble** can be related to that of a complex foam which contains bub- bles of different sizes since some of them will shrink and others will grow. Moreoever, the coarsening process can be slowed down or arrested for other reasons, the **bubble** surfaces might become impermeable to gas ar- resting the diffusion process or the shear viscoelasticity of the bubbles might stop rearrangements which would

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The motion of a **bubble** sliding over an inclined wall from moderate to high **bubble** Reynolds number is studied experimentally for a wide range of liquid properties and bubbles sizes, considering wall inclination angles from nearly horizontal to nearly vertical. All experiments are restricted to sliding behavior, below the transition to steady bouncing motion. We study both the shape of the **bubble** and its drag coefficient. For small angles, the **bubble** shape is dominated by gravitational effects resulting in a flattened shape against the wall; for large angles, the **bubble** remains in constant contact with the wall but adopts a shape that is aligned perpendicularly to the wall, closer to that observed for an inertia- dominated free rising **bubble**. We model this transition of shape considering balances among surface tension, gravitational, and inertial forces; we observe good agreement with experiments. We found that the drag coefficient is strongly influenced by the shape that the **bubble** adopts as it slides over the wall. By considering the flow in the film and around the **bubble**, we propose a correlation to predict the drag coefficient for each regime of **bubble** shape. In the regime dominated by viscous effects, the drag of a **single** **bubble** is increased due to the mirror effect with the wall and by the friction in the film formed between the wall; conversely, for the case dominated by inertial effects, the drag coefficient is constant. The behavior for a **single** **bubble** is changed: no significant increase due to deformation. In both shape regimes the proposed expression agrees well with the experimental measurements.

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or identified. In the figure, the prediction from Eq. ( 26 ) is shown for two values of the capillary number. Clearly, for large Re w the prediction asymptotes to a similar value.
IV. CONCLUSION
In this paper we studied the motion of bubbles sliding steadily over a flat inclined wall. Surprisingly, previous studies had not proposed a generalized correlation for the drag coefficient of the **bubble** in this configuration. More importantly, the alignment of the **bubble** with respect to the wall had not been identified as a relevant feature to determine such a coefficient. Thanks to the wide range of experimental parameters that were covered in the study, we were able to, first, identify the conditions for the transition between **bubble** alignment with the wall or perpendicular to it. Consequently, the experimental data were divided into two groups. In this manner, drag correlations were proposed for each case, taking into account only the relevant physical mechanisms involved in each condition. What our results show is that the transition from spherical to oblate shape that leads for a **single** **bubble** to a significant increase of the drag coefficient due to deformation (see Fig. 5

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There are two ways to initiate cavitation in water ac- cording to its phase diagram [20]: by lowering the pres- sure or by raising the temperature. In practice, pressure- based mechanisms are widely used due to their relatively simple setups [21], however with these methods it is dif- ficult to precisely control the cavitation bubble’s loca- tion. As an alternative method, superheated cavitation bubbles are generated by laser pulses [22] or by electric sparks [23, 24], which is more convenient to control and study a **single** **bubble** and its effects [25–27]. We have chosen the spark-induced approach in this study. The electric spark is generated by the discharge of capacitors (the equivalent capacitance of the circuit is 23.5 mF) that can be charged up to 50 V. Two tinned copper electrodes linked to the circuit, approximately 0.17 mm in diameter, are touched together at the desired location of the nucle- ation. A trigger initiates the discharge of the capacitors, creating a short-circuit and thereby a spark that nucle- ates a cavitation **bubble**. Experiments are performed in a plexiglas tank filled with filtered water at room temper- ature. The **bubble** is nucleated far enough from the tank walls and from the air-water surface to neglect the effects of these boundaries. Three different voltages are used to charge the capacitors: 40, 45 and 50 V. Below 40 V no significant movement of the particles has been observed. This nucleates bubbles with respective maximal radii of R b,max = 2.6 ±0.1, 3.3±0.2, 3.9±0.2 mm, growing times

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Mancas and Rosu ( 2016 )].
Indeed, according to Ohl ( 2002 ), **bubble**-**bubble** interac- tions, including the collapse of microscopic bubbles at the early stage of cavitation development, should be considered for accurate modelling of cavitation inception. However, although such a phenomenon was clearly demonstrated by experimen- tal studies, most of the current models do not include these effects. Typically, the equation proposed by Gilmore ( 1952 ) provides the evolution of a unique **bubble** submitted to a pressure variation. Such a **single**-**bubble** model enables to pre- dict correctly the evolution of one or several bubbles that survive during the first stage of cavitation expansion. How- ever, it does not take into account the vapour volume tran- siently contained in all the bubbles that have disappeared during that time. Most of the studies devoted to the cloud of bubbles ( Chahine , 1982 ; D’Agostino and Brennen , 1989 ; and Kubota et al. , 1992 ) make the hypothesis of a constant number of bubbles. This assumption was also used by Kubota

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Abstract: Central to market fundamentals are three ideas: (1) Nominal money (2) Dividend (3) Existing stock. In connection with the cumulative dividend stream criterion of fundamental and noise movement, the conception of sequentially stable Markov process is grounded on the theory of bubbles. This paper firstly embodies the origin of speculative **bubble** burst with overconfidence. Then, unique equilibrium with inertia is re-illuminated by the overconfidence. Keywords: externalities, speculative bubbles, heterogeneous beliefs, overconfidence, speculative **bubble** burst, equilibrium with inertia. JEL: D01; D52; D62; D84

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In this paper, we study **bubble** generation around a ship model submitted to waves and motions in order to understand the me- chanisms of air entrainment at real scale despite the similarity issues discussed in the ﬁrst part of this study. A quantiﬁcation of these mechanisms is performed on a 1/30 model of the Pourquoi pas?, corresponding to a ship model of 3.13 m length between perpendiculars (Lpp), 0.67 m beam and 0.18 m draft, in a wave and current tank. Delacroix et al. (2014) describe the experimental set- up developed to reproduce the real conditions of **bubble** sweep- down in a circulating tank (presented in Fig. 1 ). Acquisition sys- tems and ﬁrst observations of **bubble** generation are also detailed. This study was facilitated by the use of a hexapod allowing to consider the four base con ﬁgurations: (1) with current only, (2) with current and waves, (3) with current and motions and (4) with current, waves and motions. Two phenomena of air en- trainment have been observed: air entrainment by vortex shed- ding or by the breaking bow waves, for which a schematic de- scription is given in Fig. 2 . The distinct **bubble** clouds (as opposed to **single** bubbles) frequency are recalled in Fig. 3 .

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[0003] The anti-**bubble** has many potential applica-
tions: air nets formed by anti-**bubble** foams can filter the air, the gas-liquid surface areas of the anti-**bubble** foams are very large and can be used for chemical cleaning, or for dissolving gas in liquid, and a specific liquid may be contained inside the anti-**bubble** for substance transport, and so on. However, practical applications of the anti- **bubble** are rare, and one of the main reasons is that gen- eration of the anti-**bubble** is very difficult, generation con- ditions are strict and the yield is small.

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Let us initiate a discussion of the separation between asset pricing bubbles and speculative bubbles. For example, during the German hyperinflation, Flood and Garber (1980) had immense appeal for hypothesis of no speculative bubbles. How- ever, theoretically, financial market setting emphasizes on finite wealth, the finite horizon and short-selling behavior of risk-averse finitely lived agents. Hence, the **bubble** condition cannot be paralleled with transversality or boundary conditions. The methodological part of speculative **bubble** has come to debate on a Markov process and an equilibrium. The article “Sunspots and Cycles” (Azariadis and Guesnerie, 1986) is devoted to the detailed technique with sequentially stable Markov process in accordance with perfect foresight. Unstable beliefs in the form of the transition probability matrix are intended to yield a regular stationary sunspot equi- librium (SSE).

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4.4 Kinematics of radiation shells and the domain wall There will be a domain wall that separates the bubble spacetimes unless the bubbles were each in the same classic[r]

to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr
This is an author’s version published in: http://oatao.univ-toulouse.fr/20504
To cite this version:
Saboni, Abdellah and Alexandrova, S. and Karsheva, M. and Gourdon, Christophe Mass transfer into a spherical **bubble**. (2016) Chemical Engineering Science, 152. 109-115. ISSN 0009-2509