T Grauby-Heywang

A V Cohen-Bouhacina

V V Zaytseva

S L Arslanov

Selektor

A V Shokurov

D N Novak

M A Grin

C Grauby-Heywang

T Cohen-Bouhacina

A V Zaytseva

V V Arslanov

S L Selektor

2-oleyl-phosphatidylcholine (POPC)

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Photodynamic therapy (PDT) of oncological diseases is one of the most promising and widely studied fields of medical chemistry. PDT is based on the penetration of a photosensitizer (PS) in cells, followed by its photo-irradiation and the formation of free-radical oxygen metabolites, able to induce irreversible photodamages and to oxidize biomolecules, and consequently to destroy tumor and its surrounding vasculature [1]. However, despite the large number of publications devoted to the development of new PDT drugs, detailed mechanism of the effect of active forms of oxygen on cell organelles and membranes is still relatively unknown.

Experimental studies in this field are carried out either on living cells [2][3][4] or on systems modelling the cell components, particularly their lipid membranes [5][6][7][8][9][10][11][12][13]. One of the key approaches used in this regard is the use of liposomes, PS being introduced during the bilayer formation process or after by insertion [7-9, 13, 14]. Afterwards, upon irradiation by light with the required wavelength, PS starts generating active oxygen forms, causing membrane damages which can be observed by various physico-chemical methods. Another approach for modelling of PDT processes is based on black lipid membranes including PS, formed at the level of a small hole drilled in a flat substrate and separating two aqueous phases [15][16][17]. In this case, changes in membrane permeability or electrical resistance can be studied.

From this standpoint, lipid bilayers formed on solid supports are especially interesting. In these systems, lipids are either tethered to the substrate [18][19][20][21][22], or stabilized on it due to strong adhesive interactions [19,[23][24][25][26]. In this regard, it can be deemed most efficient to employ the Langmuir-Blodgett (LB) and Langmuir-Schaeffer (LS) methods, which make possible to form and to study lipid bilayers, stable if they are kept in contact with an aqueous phase, with various inclusions, on virtually any solid hydrophilic supports [19,20,[27][START_REF] Girard-Egrot | Nanobiotechnology of Biomimetic Membranes[END_REF][START_REF] Gust | [END_REF][30][31][32][33][34][35].

Thus, in the present work we aim to develop a method to form model lipid bilayers from Langmuir monolayers based on 1-palmitoyl-2-oleyl-phosphatidylcholine (POPC) and to study the process of their photodestruction in the presence of pheophorbide a methyl ether (MePha) by means of combined atomic force microscopy (AFM) and wetting angle measurements. Moreover, the aims of the work include estimation of the possibility to investigate the process of bilayer photodestruction in such a way, before widening it to other PS.

RESULTS AND DISCUSSION

Lipid POPC (Fig. 1) was chosen as a model lipid for the formation of the monolayers and bilayers because it acts as a main structure-forming lipid in a large number of live cells, and also due to the large amount of available literature on its behavior in monoand bilayers. Moreover, as shown previously by AFM [30] this lipid can be oxidized, being thus a plausible target in PDT modeling.

MePha (Fig. 1) is a convenient model PS due to its previously demonstrated high efficiency and the fact that its properties are well-studied [START_REF] Grin | Anti-Cancer Agents[END_REF]37].

On a first stage, we investigated monolayers of pure MePha at the air/water interface. Reproducibility of compression isotherms (Fig. 2), coherent mean molecular areas and absorbance spectra indicates that MePha forms stable monolayers at the air-water interface, with probably negligible dissolution in aqueous subphase, formation of multilayers or micellation. This fact creates positive premises for its possible use in mixed monolayers with POPC lipid.

Mean molecular area corresponding to the start of surface pressure increase (around 130 Å 2 ) was found to be significantly lesser than the expected value taking into account the planar chemical structure of MePha, suggesting that the long axis of PS in the monolayer is oriented perpendicularly relative to the subphase surface. This assumption agrees well with the MePha monolayer absorbance spectra (Fig. 3, set of curves 2), which indicate the formation of stacking aggregates: position of all spectral bands is significantly batochromically shifted (from 30 to 50 nm) as compared to the absorbance spectrum of the forming solution in chloroform (Fig. 3, curve 1). Moreover, ratios of the intensities of individual components of these spectra also differ significantly and monolayer spectra intensity logically increases with surface pressure.

As it is well known, aggregated state of PS can inhibit generation of active oxygen forms [37][38][39].

One of the most efficient ways to inhibit aggregation is inclusion of PS into highly organized matrix [38], and ordered Langmuir lipid monolayers can be used in this way.

Thus, on the next stage of this work, we studied mixed POPC : MePha monolayers at various molar ratios at the air-water interface (Fig. 4). Mean molecular areas are calculated by POPC molecule taking into account the lipid concentration after the dilution of this solution by the PS one. The isotherm of pure POPC is in agreement with isotherms previously shown [35], showing that this lipid is in a liquid expanded phase state during compression. The presence of MePha at increasing ratio shifts the isotherm to higher mean molecular areas, indicating that MePha is inserted in the monolayer. The high shift compared to the mean molecular areas on pure MePha isotherm suggests that POPC favors the spreading of PS at the interface in a surface parallel orientation, avoiding aggregates at the same time. This hypothesis is supported by absorbance spectra (Fig. 5). Mean molecular areas of the start of surface pressure increase on isotherms for monolayers with ratios of РОРС : MePha from 20 : 1 to 4 : 1 (Fig. 4) indicate good compatibility of the components. These results allow us to assume that POPC facilitates spreading of PS on the interface and inhibits its aggregation as well.

Indeed, spectra of PS included in such mixed monolayers (Fig. 5) significantly differ from spectra of pure MePha monolayer: first, bands appear on the whole at similar wavelengths to ones in the PS solution spectrum (a bathochromic shift of 5 nm is observed, probably due to changes of polarity of the environment happening upon transition from bulk to two-dimensional air/water interface); secondly intensity ratios of separate spectral bands are analogous to those observed in PS solution. At last, compression of such monolayer does not lead to any significant spectral change. All these facts indicate inhibition of MePha aggregation in the mixed monolayer and its predominant existence in lipid matrix in monomeric form in all the surface pressure range (0-30 mN/m). However, the absence of aggregates does not explicitly indicate a homogenous distribution of PS in the lipid matrix. Thus, in order to determine the optimal composition of the monolayer, mixed monolayers with various POPC : MePha molar ratios were transferred onto quartz substrates by the Langmuir-Blodgett-Schaeffer method. Obtained lipid bilayers containing MePha were studied using fluorescence microscopy.

Typical fluorescence images are shown on Fig. 6. They demonstrate that despite the general homogeneity of the samples (at the resolution of fluorescence microscopy), increase of MePha content up to 10 mol% (POPC : MePha 10 : 1) leads to the formation of areas characterized by increased PS fluorescence intensity as compared to surrounding phase. Most probably, these areas are due to local crystallization of PS in the film. At POPC : MePha molar ratio 20 : 1 such heterogeneity is not observed. On the contrary, change of molar ratio in favor of PS (POPC : MePha 1 : 2) leads to significant enlargement of the areas with increased fluorescence intensity, which confirms the role of excessive concentration of PS in the process of formation of observed crystal-like defects.

Overall homogeneity of the macroscopic lipid bilayer structure containing 5 and 10 mol% of PS is also confirmed by contact angle measurements: contact angles are similar in the case of pure POPC bilayers and POPC : MePha (10 : 1 or 20 : 1) bilayers in the absence of irradiation (Fig. 7).

In order to study the process of photoinduced degradation of the model lipid membranes containing PS, samples were exposed to irradiation by visible light in a wavelength range of 350-800 nm during 60-150 min. Contact angle values of water droplets were measured at different steps of the exposure (Fig. 7). In the case of pure POPC bilayers, contact angle increases of only 1° after 60 min of exposure, showing that the integrity of the bilayer is maintained. On the contrary in the presence of PS, contact angles increase significantly with exposure time, showing that the surface becomes more hydrophobic. This effect is probably due to the destruction of the upper layer of the lipid bilayer and the resulting uncovering of hydrophobic alkyl chains of POPC of the lower monolayer in contact with the substrate. At last, the increase of contact angle is not Mean molecular area, Å 2 significantly different between bilayers containing 5 and 10 mol % of PS, rising to 5° and 7°, respectively. This could be due to the fact that bilayers containing 10 mol % of PS contain also more aggregates, which can inhibit generation of active oxygen forms, as previously mentioned [37][38][39]. Finally, these results support the correlation between the PS concentration and the photodestruction of POPC bilayer.

AFM was used for a further detailed study of morphological changes occurring in lipid bilayers upon irradiation. Figure 8 shows first typical AFM images obtained for control POPC bilayer during irradiation by visible light in the 0-150 min range. In the absence of exposure, height images show the presence of small "spots" regularly distributed. The contrast inversion in corresponding phase images shows that these areas are also more dissipative, and thus more "soft". Height profiles (Fig. 9a) confirm the presence of areas higher than the surrounding phase, with a step of 0.2-0.3 nm. This value, clearly lower than the POPC size, suggests that these spots are not due to areas organized in bilayer with a surrounding phase organized in monolayer. After irradiation, AFM images and height profiles demonstrate that irradiation of POPC bilayers does not lead to any significant changes of the surface (Fig. 8).

In the case of POPC : MePha bilayers in the absence of exposure (Fig. 10), AFM height images are homogenous, but phase ones reveal the presence of domains regularly distributed, some of them having larger dimensions than those previously observed in the absence of PS. After 60 min of exposure to light, these domains become observable with a "spongy" texture in height image (Fig. 9b, curve 2). Height profiles show that, these "spongy" domains are higher of around 1 nm. On corresponding phase images, domains are still visible but with an inverted contrast as compared to the previous phase images. This suggests that domains after exposure are more rigid than the surrounding lipid phase. Additionally, photodestruction of the bilayer occurs in an analogous way on the whole area of the film, manifesting through the appearance and the increase of number of small dark spots on the phase images, which confirms relatively uniform distribution of the PS in bilayer. At last, surprisingly, these domains seem to disappear after 150 min of exposure to light, height profiles showing a decrease of the steps. Moreover, they show that the degradation process has a quite complex nature. The presence of PS combined with light exposure could lead, in a first step, to the formation of degraded products of POPC, such as oxidized derivatives. Indeed, it has been shown that air oxygen is able to cause POPC oxidation into planar monolayers. The higher polarity of oxidized chains causes their reversal, raising the molecule above the plane of the monolayer and leading to a local increase of 0.8 nm [30]. This value is in agreement with the value of 1 nm observed A further exposure would lead to a re-organization of molecules in the plane of the bilayer and further molecular degradation.

CONCLUSIONS

Photodestruction of model lipid bilayer based on POPC and MePha photosensitizer by active oxygen forms was observed upon irradiation with visible light at wavelengths of 350-800 nm. The composition of mixed lipid bilayer was optimized based on surface pressure measurements, contact angle measurement and fluorescence microscopy. It was shown in particular that the value of contact angle of water droplets on the surface of these bilayers reflects well the process of their photodestruction induced by MePha PS. Changes of morphology of mixed planar bilayers upon light exposure was also studied using AFM, revealing a likely complex process involving chemical degradation and reorganization of lipids and PS.

The possibility to employ the proposed approach to study the mechanism of sensitized photodestruction of lipid membranes and comparative evaluation of the effectiveness of photosensitizers developed for photodynamic therapy is demonstrated. EXPERIMENTAL Methyl ether of pheophorbide a (MePha) is a chlorophyll a derivative obtained according to a known procedure [40].

Lipid POPC (1-Palmitoyl-2-oleoylphosphatidylcholine) was acquired from Sigma-Aldrich (≥95.5% (GC), ≥98% (TLC)).

Langmuir mini-trough from KSV (Finland) with a surface area of 273.0 cm 2 was used for surface pressure measurements. The trough is made of Teflon whereas barriers are made hydrophilic polymer polyacetal. Compression isotherms were recorded using an automated Langmuir balance and a platinum Wilhelmy plate. Before experiments Langmuir trough was cleaned by acetone, chloroform, and distilled water, and polyacetal barriers by ethanol and distilled water. Monolayers were spread on ultrapure water from solutions at 1 mM in chloroform for PS and in chloroform/ethanol (v/v) for POPC. All solvents were HPLC grade. Experiments were performed at 25°C.

Model lipid bilayers (pure POPC or POPC : PS ones) were formed by Langmuir-Blodgett-Schaeffer technique onto quartz or mica substrates. At first stage, vertical transfer according to Langmuir-Blodgett technique was carried out at a constant surface pressure of 20 mN/m, which provides orientation of hydrophobic parts of the molecules outwards from the substrate. Then, according to Langmuir-Schaef- fer method, a second layer was transferred onto horizontally oriented substrate bearing the first singlelayer film, resulting in the contact of hydrophobic parts of the lipids between the two layers. UV-vis absorbance spectra of solutions were recorded in the range of 190-900 nm using a UV 2450 PC Shimadzu (Japan) spectrophotometer.

Absorbance spectra of monolayers at the air/water interface during compression were obtained using an AvaSpec-2048 (Netherlands) fiber optic spectrometer. Our device allows to record the absorbance spectra in situ directly upon compression of the monolayer, according to the technique described previously [41].

Fluorescence images of POPC : PS planar bilayers transferred on quartz substrates were obtained using an device including a LOMO MIKMED-2 microscope equipped with mercury lamps (DRS 100, HBO 100 W/2) and light filter holders for excitation (FS-1-8, wavelength range of 335-448 nm) and emission (KS-11 wavelength range of 600-800 nm), and an Olympus XC50 camera. Integration time usually amounted from 10 to 25 s. No less than 3 images in 7 different places were obtained for each sample. Due to high numerical aperture of the objectives, resolution of the device was in the range of 0.4-0.6 μm.

Contact angle values were determined using the Contact Angles Measurement System-101 from KSV (Finland). The device is equipped with a high resolution optical camera, a high intensity red lamp and an objective stage. Droplets of distilled water were deposited onto different places of the sample and average wetting angle values were calculated using KSV-Cam100 software (4-5 measurements per sample).

Bilayers for AFM studies were onto high quality and freshly cleaved mica. These studies were conducted on a Multimode V (Veeco, USA) microscope, equipped with a Nanoscope IIIa controller and TESP brand AFM tips (standard geometry, needle height 10 μm, point radius 5 nm). All measurements were done in tapping mode. Several images through all the sample area were performed. Processing of the AFM data and height profile acquirement were done using Gwyddion 2.45 software package.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Chemical structures of (a) MePha and (b) POPC.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Absorbance spectra of (1) MePha chloroform solution and (2) its pure monolayer during compression.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Absorbance spectra of (1) MePha chloroform solution and (2) mixed POPC : MePha monolayer with a molar ratio of 10 : 1 during compression.

Fig. 7 .

 7 Fig. 7. Dependence of contact angles of water droplets deposited onto POPC : MePha bilayers upon enduring irradiation by visible light: (1) control POPC bilayer, and mixed bilayers with POPC : MePha (2) 20 : 1 and (3) 10 : 1 molar ratios.

Fig. 8 .Fig. 9 .

 89 Fig. 8. AFM images of the control POPC bilayer without PS: (a) height images, (b) phase images. From left to right: 0, 60, and 150 min of exposure to visible light, respectively.

Fig. 10 .

 10 Fig. 10. AFM images of a mixed POPC : MePha 10 : 1 bilayer. (a) Height images, (b) phase images. From left to right: 0, 60, and 150 minutes of exposure to visible light, respectively.

ACKNOWLEDGMENTS

The present work was financially supported by Russian Foundation for Basic research, (project no. 17-53-150013 CNRS_a) and by French Centre National de la Recherche Scientifique (joint research project no. 1520). Experimental studies were performed using the equipment of CKP FMI IPCE RAS.


Bruno Marchal
email: bruno@tu.ac.th

How do the representations of learners on learning culture evolve via a social network and situational problems integrate with ICT?

Keywords: ICT, situational problem, social network, social media, socio-cognitive, empowerment, self-construction

How do the representations of learners on learning culture evolve via a social network and situational problems integrate with ICT?

How do the representations of learners on learning culture evolve via a social network and situational problems integrate with ICT?

Selection and evolution of Facebook as a platform for learning

In the 1990s sensational book was released in the United States on the decline of sociability and the disintegration of social ties. The book uses extensive data on the evolution of social relations in the United States. A survey conducted on a representative sample of 5,000 Americans followed for 20 years show that in 2003 the Americans had on average more than 4 close social contacts, 2 friends and 2 people from the family circle, while they were 6, twenty years earlier. In July 2010, 10 years later, when we 1 decided and designed the adapted screenplay for the Facebook platform, there were about 500 million users worldwide for this particular network, with over 100 million Americans and 17 million French, each with an average of 130 "friends." Four years later, in 2014, more than 1.30 billion human beings are connected on this platform. Beyond the number of 150, Robin Dunbar, professor of evolutionary anthropology at Oxford, stated that "this is the maximum number of friendships that the human mind is capable of supporting. Dunbar's number -as it is known -is applied, as the Facebook generation. "

We no longer speak of a decline in social skills, but rather the beginning of a new sociability, profoundly transformed and renewed. The paradox of the technical development of the global human communication appears to substantially reduce the lattice exchanges, but greatly increases the written exchanges as Internet and television; another person is in "contact" with a person who is at the other end of the planet. Pierre Merkle states: "The panorama of new relational practices invites rather to consider the Internet, which is a technology that accompanies, rather than cause, a number of these changes in social relationships (weakening links, transformation the concept of group, flattening and informalisation of relationships...)." For us, Facebook has imposed itself as a kind of compensatory sociability rooted in learning the language.

In addition to developing a strong network that touches so many people, we know from the data of several studies (77% of French youth aged 18 to 24 have a FB account, students 86%, with men at 38% and women at 48% students are most "connected." The platform of Facebook social network appeared to us as the most appropriate tool for a number of reasons related: its accessibility, its integrated interaction tools confirmed the potential application of teaching models.

Accessibility and mobility

The platform is free, requires no registration and all key users registering there are thus capable of promoting the construction of the social network of each. The social network on the net is a medium from which the user can create, define, and build a digital identity. Users are familiar with all the tools. It is not necessary to provide training for the use, either in-class or from a distance. The recent passage of the platform to a configuration in "Timeline" does not in itself provide a plus from the scenario as it was designed, especially as at the time of this writing, everyone has not passed to this interface. It will, however, be a necessity soon.

In addition, applets developed for all operating systems for mobile telephony allow almost unlimited access in time and space. Mobile devices developed with Android mobile devices or operating system specific Apple® offer free Facebook software. For our experiment, we can say that this software has been further improved and that all functions previously available on the application or portable computer, are now on these terminals, which provide near real-time interactions. Having practiced the learning modules on dedicated as Moodle or other platforms, we know that the recovery and monitoring of learners is paramount to maintaining presence and avoiding abandonment in online learning [START_REF] Charlier | Voyage au pays de la formation des adultes : dynamiques identitaires et trajectoires sociales[END_REF].

Integration and traceability

Built-in tools (awareness, Chat, leaving messages, creating internal groups, document sharing) raise interactions valued by Like clicks and comments added. Adding messages and tracking one's conversation can be done daily. It thereby allows traceability of interactions. Facebook offers, on creating pages or groups, statistical tools from simple (filing hours) to the arcane (number of visits per post, per day / week / month in graphs). Furthermore, a page can be opened to the entire Internet community if permission is given, allowing visibility of changes in learning who wants to connect to the Net.

Learning applications

The third reason is the one that interests teachers such as us the most: the possibility of applying recognized teaching models. B. Joyce, Weil, and E. Calhoun distinguish four main types of teaching modelssocialization, information processing, individuality and behaviorists systems. Two of them are particularly suitable to a social network platform: the socialization of a social constructivist model that offers the possibility of developing an integrative and productive interaction on the one hand and on the other hand the development of standards to support learning activities. But a social networking platform is also developing the intellect, whether by learning information retrieval, learning conceptualization, a process designed to generate hypotheses and test them, or creative thinking. We are then in a cognitive pedagogy of information processing.

Socialization

For Van Lier, the language articulates the relationship between the world and the people and therefore, context is central. The emergence of language development is through the interaction of children and learners with their spatial, social, cultural, educational environment, and creates common ground between the development of primary language, children and adults in the learning a second language and language acquisition by robots.

However, linguistic precision is important and Felix called for an approach combining social constructivism and cognitivism so that learners are exposed to a range of activities that support both automatic responses, increasingly complex engagement in independent learning, and the simultaneous development of both cognitive and metacognitive processes.

Ryberg and Christiansen also showed that social networking sites showed language acquisition by the fact signs that they offered opportunities to learn through observation before mastering an understanding. This naturally leads to increased trust and due recognition by the community members and therefore encourages learning through the community and the learner, making him become a kind of "smuggler of information and knowledge" himself.

Data processing

The pedagogy of information processing focuses on the steps to the memory using representations (concept maps, ideators, graphs) according to work by B. [START_REF] Joyce | Models of teaching[END_REF], the study of the world and society through the development of argumentation and critical thinking. Thus, the models offer inductive thinking, concept acquisition and learning (re) presentation. Based on the model ACT Anderson describes the transition from one form of knowledge to another through the working memory that acts as an activated from longterm memory area (a joint semantic knowledge and procedural knowledge). We will see below in the Scenario part that we are committed to expanding the mnemonic practices and conceptualization.

Scenario Educational progress from the initial situational problem to the final situational problem

Consisting of a set of complex situations that require concerted action to get out of these difficult situations, the learning scenario proposes motivating micro-objectives and technological tools for both language and the profile of learners. In order to be closer to the concerns of our academic profile of a learner who is twenty and follows a course of BA in French and our Internet social networking profile using Facebook late at night and mobile telephony, situational problems were scripted in what looks like a fun application (applet) micro-world. So Facebook has become not only a network for sharing photos, addresses, filing comments or messages but also a place where you can learn to play in a simulated world in real time where we must. Collaboration and exchanges build collective knowledge from knowledge of each. There have been numerous research studies of situational problems; to summarize their purpose and interest, we can say that a situational problem is a concrete task under conditions that assume that people cross a number of unavoidable obstacles to get there. The situational problem is always a fiction under control and it is part of a pedagogy based on self-building knowledge, know-how, and skills. The barriers are directly related to lack of learners: lack of knowledge, know-how or lack of conduct and proper attitude, the conditions necessary for the work of the task. They can be physical or human resources, the various constraints that will determine the learning of learners and make them indispensable. Situational problems can also be built with cascading sub-problems to solve situations at each stage to advance in the overall process.

Analysis and Results of the Research

Our research aims to develop learning strategies among young students. We have seen progress in three categories of strategies: metacognitive, cognitive and socio-emotional. These analyses are mainly based on the indicators proposed by Paul Cyr and Claude Germain in "Learning Strategies" (Clé International, 1998).

At a metacognitive level, the observed progresses are in skills anticipation, planning, attention, self-management and self-regulation.

What is particularly interesting is that the Facebook group may dispose of an awareness tool, that is to say the ability to see group members present -if they want to show themself -and speak to them directly. All educational platforms do not supply this tool that dramatically facilitates exchanges between members of a group. And this is, in our opinion, the advantage of a social platform like Facebook. Imagine the following scene: a student goes home at the end of the day. He realizes he is late in his group work and decided to turn on the computer. But he is like any modern young person who has a Facebook account: unable to start his work session without taking a look at notifications and the news of his personal page thread. But just arrived on Facebook, his attention is arrested by a fellow who is present on the platform, who calls for help because he is too late in his work and needs help. Then the two are drawn into a purely educational conversation because that's where the misunderstanding, reformulations and explanation, signs of empowerment among learners. This conversation is all the more rewarding if the language is the target language, as was the case in our group. As guardian-observer training, we had fun to hide from us every evening on the platform to witness this kind of educational exchange between students, particularly abundant before the end of the period of a task.

The second category is about cognitive strategies like practicing the language (the students must take every opportunity offered to them to communicate in the target language); taking notes and classifying ideas, especially through the meeting Chats; and performing information retrieval; translation and comparison; paraphrase, preparation and summary.

The point that seems most interesting as a teacher of French as a second language is the language practice. Among the various ways to "seize" the opportunity to communicate in training, the more convincing are the Chats. Indeed, the first element is at the invitation to participate in a Chat: if the person decides to come and be present, this shows an intention to communicate, at least of reading comprehension; our sample group showed 64% attendance to all Chats that we have collected. If the person is invited, but did not come to apologize, he or she will normally contact the tutor by email or private messaging FB to explain his or her absence. In cases where the learner does not come and apologizes before or after the meeting, this clearly indicates a lack of willingness to communicate.

In contrast, in the copies of Chats we can count the number of times each student participated as well as check their quality by checking whether they wrote sentences or simply by assents, interjections, or emoticons.

In the three Chats we analyzed (early training SP1 -SP4 medium training -and end SP8 training), we find a frequency of interventions and a increased learner speaking as the training advances, even though the number of times the tutor on line intervenes is important because he answers many questions. Interjections even if they do not have a qualitative nature show proof of presence and monitoring the progress of the Chat.

We now turn to the socio-affective strategies, which consist in skills: questioning for clarification and verification; cooperation to interact with peers to accomplish the task; management of emotions aware of the affective dimension; techniques to increase self-confidence and motivation.

In distance learning, the geographical distance, cultural distances as well as time differences do not allow instant understanding of emotions expressed; silence or absence may indicate a lot. One can also say that the emotions are often expressed by hyperbole; there is a whole range to express: emoticons, punctuation, rehearsals, games typography. Often the walls are allowed in addition to clicking "like" comments in which one can read the emotions: the joy of success; the desire to win; relief; contentment. In contrast, we read discontent, pressure, guilt, depression, fear of timeout... One might add that the relational learninglearning, learning -tutor was not only rewarding for the time after training but also because trade occurred, and continued.

Motivation is crucial: the training is not mandatory and does not fit into any curriculum; registrations are made on a voluntary basis (as well as the attachment to the professor who invited them to participate). Despite all the technical problems, few have had the urge to stop if they do, it is often for reasons beyond their control. One can even say that some have resumed training when their personal problems have been resolved.

Conclusion

On the educational level, the learning experience on a social network platform that matches the natural environment of the learner is different from the proposed landscaped environment on campus. The distance learner is faced with a learning experience that moves away from the traditional socio-academic background, realistic and known, and engages with cognitive and affective solitudes. It should then promote the development of learner autonomy, which will naturally have the effect of the student actively engaging in its approach, to motivate him.

The autonomist media model learning from Deschenes and Lebel (1994) support the broad outlines of this approach. For them, the autonomous model:

-Is part of a major paradigm shift that goes from behaviorism andcognitivism to constructivism, and so from objectivism and observable performance to a subjectivism of knowledge and focus on the mental activity of the individual, often unobservable. -Wants to integrate all aspects of the learning process: cognitive, metacognitive and affective.

-Based on a conception of learning that gives a key role in the interaction between the learner and the learning object. -Assume that the learner must be active in their learning process by assuming control himself.

This model, as can be seen, relies on a broad responsibility on the part of the learner, particularly in regard to the awareness of mental activity. In doing so, it would promote the development of autonomy described as the ability to manage their learning process (Deschenes, 1991) or, in other words, the ability to identify ways to implement to achieve its objectives.

However, and this goes to the point of view of Tardif mentioned above, all learners do not begin the learning process with the same level of autonomy:

Autonomy is not a simple but a higher quality integrated line (a métaconduite) and, for most individuals' mode, this conduct does not form a natural part of their repertoire; it must be learned. If distance education wants to support this autonomy, which determines its existence, it must help learners develop the measure; it must be learned (Linard, 2006, p 2). Thus, for Linard, it is not enough to empower the learner but he must be able to fulfill the responsibilities imposed on it; in other words, teach him to learn more effectively through systematic reflexive analysis of his own way of learning.

Metacognition is somehow the ultimate measure of the development of autonomy, which could also be called an empowerment process. It is for the learner to self-evaluate on an ongoing basis the learning process as it rises above itself to hear oneself think, act, and be aware of the mental process that engages in managing this process. This exercise, when used in the context of a real process of knowledge construction, will act positively on the planning, execution, control and regulation of the learning process (Lafortune, Jacob and Hebert, 2000).

This way of interacting is driven by the ability of the learner to understand the mechanisms of this interaction and adjust. It is therefore not accessible to all age levels but is intended for university students. It could be adapted to ongoing training but it requires a considerable investment in time (3 hours / week). Moreover, considering the strong consensus culture in Asian countries playing positively in reports and socialcognitive conflicts, we hope to offer this model to integrate into university curricula, alongside more traditional platforms in distance learning.

Marchal B. & Do Q.H., (2010)


Daniel Fuster

Stéphane Popinet

An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension

Keywords: All-Mach formulation, bubble dynamics, multiphase flows, compressible flows, volume-of-fluid method

This paper presents a generalization of an all-Mach formulation for multiphase flows accounting for surface tension and viscous forces. The proposed numerical method is based on the consistent advection of conservative quantities and the advection of the color function used in the Volume of Fluid method avoiding any numerical diffusion of mass, momentum and energy across the interface during the advection step. The influence of surface tension and liquid compressibility on the dynamic response of the bubbles is discussed by comparing the full 3D solution with the predictions provided by the Rayleigh-Plesset equation for two relevant problems related to the dynamic response of bubbles to pressure disturbances: The linear oscillation of a single bubble in an acoustic field and the Rayleigh collapse problem. Finally, the problem of the collapse of a bubble close to a wall is compared with experimental results showing the robustness of the method to simulate the collapse of air bubbles in liquids in problems where bubbles generate a high velocity liquid jet.

Introduction

The simulation of bubbles accounting for compressible effects is a topic of interest in various applications related to underwater explosions [1, 2], engineering applications [START_REF] Fourest | Analysis of bubbles dynamics created by hydrodynamic ram in confined geometries using the rayleigh-plesset equation[END_REF][START_REF] Wang | Three dimensional microbubble dynamics near a wall subject to high intensity ultrasound[END_REF] or medical treatment such as HIFU or lithotripsy [START_REF] Coralic | Finite-volume weno scheme for viscous compressible multicomponent flows[END_REF][START_REF] Maeda | Modeling and experimental analysis of acoustic cavitation bubble clouds for burst-wave lithotripsy[END_REF]. In all these processes it is crucial to understand the dynamic response of bubbles to pressure waves motivating the development of numerical methods for the Direct Numerical Simulation of bubbles in slightly compressible liquids [START_REF] Hejazialhosseini | High order finite volume methods on wavelet-adapted grids with local timestepping on multicore architectures for the simulation of shock-bubble interactions[END_REF]. Despite the development of numerical techniques to solve the Navier-Stokes equations in the presence of multiple phases [START_REF] Shyue | An eulerian interface sharpening algorithm for compressible two-phase flow: the algebraic thinc approach[END_REF][START_REF] Coralic | Finite-volume weno scheme for viscous compressible multicomponent flows[END_REF] there are still numerical difficulties when coupling compressibility, viscous and surface tension effects [START_REF] Wesseling | Principles of computational fluid dynamics[END_REF]. For instance classical solvers for compressible flows based on Riemann solvers rarely take into account surface tension [START_REF] Garrick | A finite-volume hllc-based scheme for compressible interfacial flows with surface tension[END_REF]. This is due to the intrinsic difficulties related to the treatment of sharp interfaces and also how to deal with the jump of properties across the interface ensuring that surface tension terms satisfy basic relations such as the balance between pressure and capillary forces in an equilibrium state [START_REF] Abu-Al-Saud | A conservative and well-balanced surface tension model[END_REF].

One family of numerical methods capturing compressibility effects resort to the generalization of numerical schemes developed for incompressible fluids [START_REF] Yoon | The unified simulation for incompressible and compressible flow by the predictor-corrector scheme based on the cip method[END_REF][START_REF] Xiao | Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow[END_REF][START_REF] Kwatra | A method for avoiding the acoustic time step restriction in compressible flow[END_REF][START_REF] Xiao | Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows[END_REF][START_REF] Jemison | Compressible, multiphase semiimplicit method with moment of fluid interface representation[END_REF][START_REF] Caltagirone | A multiphase compressible model for the simulation of multiphase flows[END_REF]. Among these methods, all-Mach semi-implicit formulations are especially appealing as they have shown to provide accurate results in different situations ranging from pure incompressible flows to compressible problems involving the presence of shock waves. One of the advantage of these methods is that they allow for a natural extension of the numerical strategies developed for incompressible multiphase flows, which have been extensively developed and tested in problems where surface tension forces are relevant. In addition, these methodologies avoid the classical time step restriction defined by the speed of sound of the least-compressible fluid, which is convenient for the simulation of subsonic flows that combine substances with very different compressibility. This is typically the case in bubbly flows where the gas compressibility is much larger than that of the liquid, sometimes considered as an incompressible substance.

In this work we present an extension of the all-Mach semi-implicit solver using the Volume-Of-Fluid (VOF) method including surface tension and viscous forces. The main advantages of VOF methods is that they keep the representation of the interface sharp, they enforce volume conservation in the incompressible limit and handle naturally topological changes of the interface. The method takes into account viscous and surface tension effects by generalizing the classical formulation used in previous incompressible solvers [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF]. The extended formulation accounts for fluids with arbitrary values of the thermal dilatation coefficient, which is typically not the case in previous works. We chose the Basilisk solver [START_REF] Popinet | A quadtree-adaptive multigrid solver for the "Serre-Green-Naghdi" equations[END_REF] as a platform to implement the new numerical scheme. The methods implemented in the code have been well validated for the simulation of multiphase flows in various problems involving interfaces [START_REF] Fuster | Numerical simulation of droplets, bubbles and waves: state of the art[END_REF]. Moreover, the method has been designed to reduce the appearance of parasitic currents induced by surface tension forces [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF], which is an appealing characteristic of the numerical scheme if one wants to reduce any artificial energy dissipation/generation in the system.

The manuscript is structured as follows: The basic equations are presented first. Then we describe the numerical methods used to solve the equations. Two numerical tests for single phase flows are introduced to discuss the properties of the method in two limiting situations: the linear propagation of pressure waves and the propagation of shock waves. Then, we quantify the numerical oscillations induced by surface tension forces in an equilibrium state and the velocity and pressure oscillations generated by the advection of an interface before discussing various tests proposed to assess the capability of the method to capture the dynamic response of bubbles to pressure changes. Finally the problem of the collapse near a wall is used to show the robustness of the solver before presenting the conclusions.

Basic equations

In this work we solve for the Navier-Stokes equations for a mixture of two immiscible substances where the position is defined by a Heaviside function H that takes the value of 1 in the reference phase [START_REF] Tryggvason | Direct numerical simulations of gas-liquid multiphase flows[END_REF]. The interface position is then given by the solution of an advection equation for H

∂H ∂t + u • ∇H = 0. ( 1
)
The continuity and momentum equation for the ith phase are

∂ρ i ∂t + ∇ • (ρ i u i ) = 0, ( 2
)
∂ρ i u i ∂t + ∇ • (ρ i u i u i ) = -∇p i + ∇ • τ i , (3)
where ρ is the density, u is the fluid velocity vector, p is the fluid pressure and τ = µ ∇u + (∇u) T -2 3 µ∇ • uδ is the viscous stress tensor depending on the dynamic viscosity µ.

The equations between the two phases can be weighted by the respective volume fraction and then added to obtain evolution equations for the various averaged quantities. To do this it is important to account for the relation between the velocity and pressure established by mass and momentum balances across a differentially thin interface located at x I . In the absence of mass transfer the velocity is continuous u 1 (x I ) = u 2 (x I ) and therefore it is natural to work with a single continuous averaged velocity field u. The Laplace equation gives the pressure jump at the interface as

p 1 -p 2 = σκ ( 4
)
where σ is the surface tension coefficient and κ is the curvature of the interface. Note that following classical formulations for incompressible multiphase flows we have neglected the viscous stress jump across the interface. Applying these conditions for the cells containing an interface we can write the momentum equation for the averaged quantities as [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] ∂ρu ∂t

+ ∇ • (ρuu) = -∇p + ∇ • τ i + σκ∇c. (5)
When accounting for compressibility we aditionally solve an equation for the energy. In this paper we restrict ourselves to situations where thermal diffusion and mass transfer effects are not relevant (e.g. high frequency oscillations [START_REF] Bergamasco | Oscillation regimes of gas/vapor bubbles[END_REF]), so that the total energy equation is

∂ρ i e i + 1/2ρ i u 2 i ∂t + ∇ • (ρ i e i u i + 1/2ρ i u 2 i ) = -∇ • (u i p i ) + ∇ • (τ i u i ) , (6)
where e denotes the internal energy. Alternative forms of the energy equation include the internal energy equation, which in the absence of thermal diffusion takes the form

ρ i c p,i DT i Dt = β T,i T i Dp i Dt + Φ v,i , (7)
where T is temperature, c p,i is the specific heat measured at constant pressure, β T,i is the thermal dilatation coefficient and Φ v,i = ∇u : τ is the viscous dissipation function. To obtain an evolution equation for pressure we follow the procedure proposed in [START_REF] Fuster | Liquid compressibility effects during the collapse of a single cavitating bubble[END_REF], where the internal energy (Eq. 7) and the continuity equation (Eq. 2) are combined to express the divergence of the velocity as a function of pressure. To do that we obtain the differences between two thermodynamic states as a sum of the changes of density in an isothermal process plus the density change in an isobaric process

dρ = ∂ρ ∂p T dp + ∂ρ ∂T p dT, (8)
which can be written using the definitions of the speed of sound, polytropic coefficient and thermal dilatation coefficient as

dρ = γ c 2 dp -ρβdT. (9)
Using Eqs. 2, 7, 9 we obtain

1 ρc 2 eff Dp Dt - β T Φ v ρc p = -∇ • u, (10)
where

1 ρc 2 eff = γ ρc 2 - β 2 T T ρc p .
For pure substances such as ideal gases (β T ≈ 1/T ) and liquids like water (γ ≈ 1, β T ≈ 0) the classical approximation 1 ρc 2 eff ≈ 1 ρc 2 is justified if we neglect viscous dissipation. In problems involving multiple phases, we average Eq. 10 implying that the mixture compressibility is given by the harmonic mean of the compressibilities

1 ρc 2 eff = C ρ 1 c 2 eff,1 + 1 -C ρ 2 c 2 eff,2 . (11)
The system of equations solved is closed by adding an equation of state (EOS) for each phase that establishes the relation between the various thermodynamic variables EOS i (p i , T i , ρ i ) = 0. Note that from the state equation we can obtain various quantities such as the speed of sound, the thermal dilatation coefficient, etc. In this work we use the general EOS written in the Mie-Gruneisen form (see for example [START_REF] Saurel | A multiphase "Godunov" method for compressible multifluid and multiphase flows[END_REF][START_REF] Johnsen | Implementation of weno schemes in compressible multicomponent flow problems[END_REF])

ρ i e i = p i + Γ i Π i Γ i -1 (12)
which defines the sound speed as

c 2 i = Γ i p i + Π i ρ i , (13)
but in principle any other choice of the EOS is possible.

Numerical scheme

In this section we present the numerical scheme used to solve the equations detailed in the previous section using a cell centered formulation for octree grids [START_REF] Popinet | A quadtree-adaptive multigrid solver for the "Serre-Green-Naghdi" equations[END_REF].

For a two-component mixture, we obtain the evolution of the interface position (Eq. 1) using the Volume-Of-Fluid (VOF) method that solves the advection equation for the color function C representing the volume of the reference phase with respect to the total volume of the grid. This equation is solved numerically following the split advection method proposed by Weymouth & Yue [START_REF] Weymouth | Conservative volume-of-fluid method for free-surface simulations on cartesian-grids[END_REF] where the advection equation for the volume fraction is written as

∂C ∂t + ∇ • (uC) = C∇ • u. ( 14
)
This form of the equation is valid for both compressible and incompressible substances and has been used in previous schemes based on Riemann solvers for multiphase flow simulations [START_REF] Johnsen | Implementation of weno schemes in compressible multicomponent flow problems[END_REF]. In a direction-split advection method the discretized advection equation for C takes the form

C n+1 = C n - f F C (u f ∆t/∆x, C) + C * f ∇ f u∆t. ( 15
)
where f denotes the cell face, F C = C adv u f is related to the flux of C across the face during ∆t, and C adv is the fraction of the reference phase crossing the face during ∆t evaluated geometrically. The right-hand-side of the equation represents the volume change of the reference phase during ∆t due to compressibility effects and as explained in [START_REF] Weymouth | Conservative volume-of-fluid method for free-surface simulations on cartesian-grids[END_REF], C * must be held constant for all sweep directions in order to ensure exact volume (e.g. mass) conservation in the incompressible limit.

In the time-split scheme the set of conservative equations is solved in various steps. The advection step solves for the advection of mass, momentum and energy in each phase separately to obtain provisional values of the conservative quantities associated with the ith component as

(C i Y i ) * = (C i Y i ) n - f F C i Y i (∆tu f /∆x, C i , Y i ) ( 16
)
where

Y i = (ρ i , ρ i u, ρ i e i + 1/2ρ i |u| 2 )
T and F Y i represents the amount of mass, momentum or energy transfered across a given computational face during the time step. Because the transport of the color function C and the conservative quantity C i Y i are obviously linked, the calculation of the flux terms is expressed as a function of the flux of C obtained geometrically to solve Eq. 15

F C i Y i (∆tu f /∆x, C i , Y i ) = F C (u f ∆t/∆x, C i )Y i,adv , (17)
where Y i,adv represents the averaged value of the conservative quantity related to the ith component within the advected volume crossing the face during the timestep. It is important to remark that by recycling the value of F C we can freely chose the method to obtain Y i,adv while minimizing the error coming from inconsistencies between the advection of the color function and the conservative quantity C i Y i in mixed cells during the overall timestep. To see this, let's consider a problem where we know the amount of c crossing a cell face during a given ∆t

F c 1 = f c 1 (t)u f (t)dt. ( 18
)
In this case the corresponding flux of c 1 y 1

F y 1 c 1 = f y 1 (t)c 1 (t)u f (t)dt (19)
could be computed using an approximation of the full integral

g(t) = y 1 (t)c 1 (t)u f (t)
by the Taylor expansion

g(t) = g(t 0 ) + ∂g ∂t t 0 (t -t 0 ) + 1 2 ∂ 2 g ∂t 2 t 0 (t -t 0 ) 2 + O(∆t 3 ). ( 20
)
However this approximation does not converge when, as it is the case, g(t) is a discontinuous function. The problem can be solved if we define a smooth approximation of y 1 at the nth order as y 1 ≈ y 1 + O(∆t n ). In this case we can rewrite Eq. 19 as

F y 1 c 1 = F c 1 (y 1 + O(∆t n )) (21)
which is consistent with the calculation of the flux of C i obtained from Eq. 18 at the order of the approximation used to estimate y i . Following this procedure, the problem of the transport of a discontinuous function depending on C is reduced to the problem of obtaining the volume fraction flux (Eq. 18) naturally handled by the VOF method. For practical purposes we evaluate Y i,adv using the Bell-Collela-Glaz second-order upwind scheme [START_REF] Bell | A second-order projection method for the incompressible navier-stokes equations[END_REF],

Y i,adv = Y i + 1 2 sign(u f ) min(1, 1 -sign(u f ) u f ∆t ∆x ) ∂Y i ∂x ∆x, if C i > 0 0, otherwise, (22)
where the values of C i and Y i are upwind cell centered values. The scheme is adapted to define a smooth function of Y i,adv in regions close to the interface by obtaining the derivative of Y i in the jth cell as

∂Y i ∂x j =            1 ∆x f L (Y k-1 i , Y k i , Y k+1 i ), if C k i > 0 k = j -1, j, j + 1 Y k i -Y k-1 i ∆x , if C k+1 i = 0 ∧ C k-1 i > 0 Y k+1 i -Y k i ∆x , if C k-1 i = 0 ∧ C k+1 i > 0 0, otherwise, (23)
where

f L (Y k-1 i , Y k i , Y k+1 i
) is a function that provides an estimation of the increment of a variable in a given direction. In this work we use the generalized minmod slope limiter

f L (y -1 , y 0 , y 1 ) =      min(θ(y 0 -y -1 ), y 1 -y -1 2 , θ(y 1 -y 0 )), if y -1 < y 0 < y 1 max(θ(y 0 -y -1 ), y 1 -y -1 2 , θ(y 1 -y 0 )), if y -1 > y 0 > y 1 0, otherwise, (24)
where θ is a freely adjustable parameter that allows to recover various classical slope limiters such as superbee (θ = 2) and minmod (θ = 1). Note that we use slope limiting rather than flux limiting. The numerical calculations included in this manuscript show a negligible dependence on the parameter θ.

The advection scheme proposed presents some interesting properties. First, by construction, the advection step is fully conservative in the discrete framework. Second, the fluxes obtained for the color function and the conservative quantities are consistent, significantly reducing errors appearing during the advection of a discontinuity. Finally, because the advection step is taken for each phase separately, we avoid any artificial numerical diffusion between the two immiscible phases. This is important in the case of large density differences since methods ensuring the conservation of momentum and energy are known to reduce the appearance of numerical instabilities in the simulation of multiphase incompressible flow problems [START_REF] Rudman | A volume-tracking method for incompressible multifluid flows with large density variations[END_REF][START_REF] Le Chenadec | A monotonicity preserving conservative sharp interface flow solver for high density ratio two-phase flows[END_REF][START_REF] Vaudor | A consistent mass and momentum flux computation method for two phase flows. application to atomization process[END_REF].

After the advection step the provisional momentum is corrected and viscosity or surface tension effects are taken into account as

(ρu) * * -(ρu) * ∆t = ∇ • µ ∇u + (∇u) T - 2 3 ∇ • (µ(∇ • u)I) , (25)
(ρu) * * * -(ρu) * * ∆t = σκ∇C. (26)
The calculation of the surface tension forces is critical for accurate results. This step is similar to that already described in Popinet [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF][START_REF] Abu-Al-Saud | A conservative and well-balanced surface tension model[END_REF] and therefore the reader is referred to these publications for further details.

Finally, in the classical projection step we take the divergence of the remaining terms in the momentum equation

u n+1 -u * * * = - ∆t ρ ∇p. ( 27
)
The divergence of the velocity at the end of the time step is expressed as a function of pressure by splitting Eq. 10 in time as

p * -p n ∆t + u • ∇p = 0, ( 28
)
1 ρc 2 e p n+1 -p * ∆t = -∇ • u n+1 + β T Φ v ρc p . ( 29
)
Replacing Eq. 29 into Eq. 27 we find the following Poisson-Helmhotz equation for pressure

1 ρc 2 eff p n+1 -p * ∆t + ∇ • u * * * - β T Φ v ρc p = ∇ • ∆t ρ ∇p , (30)
where the divergence terms are computed by linear interpolation of the cell centered values for u * * * , which defines a provisional staggered velocity field.

For the calculations reported in this manuscript we use

1 ρc 2 eff ≈ 1 γ-1 p 1 γ-1 + 1 + Πγ γ-1 (31)
where

1 γ -1 = C γ 1 -1 + 1 -C γ 2 -1 (32) Πγ γ -1 = C Π 1 γ 1 γ 1 -1 + (1. -C) Π 2 γ 2 γ 2 -1 (33)
The method followed to solve for Eq. 28 to obtain p * significantly influences the performance of the method. Among the various alternatives proposed in the literature [START_REF] Xiao | Unified formulation for compressible and incompressible flows by using multi-integrated moments I: one-dimensional inviscid compressible flow[END_REF][START_REF] Kwatra | A method for avoiding the acoustic time step restriction in compressible flow[END_REF][START_REF] Xiao | Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows[END_REF][START_REF] Jemison | Compressible, multiphase semiimplicit method with moment of fluid interface representation[END_REF] we chose the solution proposed by Xiao et al. [START_REF] Xiao | Unified formulation for compressible and incompressible flows by using multi-integrated moments II: Multi-dimensional version for compressible and incompressible flows[END_REF] (also used in [START_REF] Jemison | Compressible, multiphase semiimplicit method with moment of fluid interface representation[END_REF]) where pressure is computed from the total energy obtained after the advection step. Thus, although Eq. 10 is sufficient to close the system of equations because it implicitly contains the energy equation, we explicitly solve the conservative form of the energy equation in order to compute the provisional pressure p * as

p * = (ρE) * -1 2 ρ|u| 2 -Πγ γ-1 1 γ-1 , (34)
where (ρE) * = cρ 1 e 1 + (1 -c)ρ 2 e 2 + 1 2 ρ|u| 2 is the total energy computed with the provisional values after the advection step. This step guarantees that the energy is conserved at the discrete level in the absence of viscous dissipation and surface tension.

Once the pressure at the end of the timestep has been obtained we update u at the faces using Eq. 27 and the total momentum defined at the cell center as

(ρu) n+1 -(ρu) * * = -∆t∇ c p, (35)
where ∇ c p is obtained as a simple average between the face values of the pressure gradient. Finally, the total energy for each phase is updated as

(ρ i e i + 1/2ρ i u 2 ) n+1 = (ρ i e i + 1/2ρ i u 2 ) * + ∆t (-∇ • (up i ) + ∇ • (τ i u)) . ( 36
)
For the calculation of the divergence terms we use the face velocities already obtained and the pressure of the ith component at the face. For faces in contact with two single-component cells we define the pressure at the face as the average between the two neighboring cells. When the interface is present we account for the fact that Eq. 30 is indeed solved for the averaged pressure p = Cp 1 + (1 -C)p 2 , from which we compute the pressure of the ith face using the Laplace relation

p 1 = p + (1 -C)σκ (37) p 2 = p -Cσκ. ( 38
)
The face pressure is finally taken as the cell-centered pressure of the cell that contains the largest volume fraction between the two neighboring cells. The full description of the algorithm is resumed in the flow chart 1.

Numerical results

4.1. Single phase problems 4.1.1. Propagation of a linear pressure disturbance in a fluid In order to investigate the numerical properties of the method in the linear regime we measure the numerical dispersion of a small, one-dimensional pressure disturbance in an inviscid fluid. The initial condition is given by the pressure field p(x, t = 0) = p 0 [1 + 2∆p exp (-4x 2 /χ 2 )] in a fluid at rest (zero velocity) where the domain of size L 0 = 10 is discretized with a regular cartesian grid of size ∆x/L 0 = 1/2 10 . The liquid density and speed of sound is one and the pressure perturbation is obtained using ∆p = 5 × 10 -3 and χ = 0.1.

In the acoustic limit, ∆p 1, nonlinear terms are negligible and the advection step does not impact the results. The pressure pulse splits and generates two waves propagating in opposite directions. In the absence of any physical dissipation the frequency content remains unchanged as the wave propagates at the reference sound speed c 0 . We obtain the spatial distribution of pressure at four different instants ( c 0 t χ = 4, 12, 20, 28) and we perform a Fast Fourier Transform (FFT) to decompose the signal into spatial Fourier modes. For each wavelength we fit the decay rate in time of the absolute module of the FFT to an exponential decay of the type C 0 e -At , Algorithm 1 Summary of the full algorithm Initialization of where A is directly related to the numerical dissipation at a given length scale. Figure 1 shows the results obtained with CFL ac = c 0 ∆t ∆x = 0.5. The pulse peak propagates at the correct speed but it is dissipated as a consequence of the properties of the numerical method. The measurement of the effective attenuation as a function of wavelength, which is a measure of the discretization error, converges to second order. It is important to remark that the numerical dissipation of the current method is larger than that measured using a classical second-order Riemann solver. To obtain comparable results in terms of accuracy between the two methods, one needs to double the resolution of the all-Mach formulation. Figure 2 shows the impact of the CFL number. As we can see the method allows for stable solutions for CFL ac larger than 0.5 at the expense of increasing the effective attenuation of the wave and increasing the cut-off distance beyond which attenuation saturates. Defining the critical scale λ c as the minimum length beyond which we do no longer observe convergence, we find that λc ∆x = 40 × CFL ac , which can be also written as λc c 0 ∆t = 40. We can interpret this as the fact that the proposed scheme neglects compressibility effects for wavelengths λ < λ c , with λ c tending to infinity as we approach the incompressible limit.

c 1 ρ 1 , c 2 ρ 2 , ρu, c 1 ρ 1 E 1 , c 2 ρ 2 E 2 while t < t end do Set ∆t Obtain q i = c i ρ i ρu c 1 ρ 1 +c 2 ρ 2 Obtain C * = c > 0.

Att. (present) FFT(p) Riemann

Propagation of a shock wave in an ideal gas

In this problem we solve for a 1D shock wave propagating in an ideal gas with γ = 1.4 in order to test the scheme in a situation where nonlinear advection terms are relevant. The Rankine-Hugoniot conditions are used to initialize the solution defining the right (pre-shocked) and left (shocked) state as p L = 10, p R = 0.1, ρ R = 1. The theoretical shock speed is

s c 0 = γ+1 2 γ p L p R -1 + 1 , (39)
where

c 0 = γp R ρ R .
Figure 3 shows the shock wave structure in the frame of reference of the theoretical front solution, x front = st. The method is shown to be able to capture the correct shock speed. The thickness of the shockwave is approximately three cells. It should be noted that the numerical oscillation behaves in a way different from the conventional density-based shock capturing schemes. The upwind method minimizes the spurious oscillations at the expense of a larger error for a given resolution relative to the BCG 2ndorder advection schemes, which shows an almost negligible influence of the parameter θ on the solution. Remarkably the overall error computed based on the L 1 norm is indeed smaller in the case of the BCG advection scheme and first-order convergence is obtained for all cases as expected for a problem including shocks.

Two phase problems 4.2.1. Isolated interface problem

In this section we test the full solver for the problem of advection of an interface between two different ideal gases at uniform velocity and pressure. This test proposed in [START_REF] Johnsen | Implementation of weno schemes in compressible multicomponent flow problems[END_REF] quantifies the amplitude of spurious pressure and velocity oscillations induced by the method when advecting an interface with different material properties. The interface is initially placed at the middle of as (ρ, u, p, γ) T L = (1, 0.5, 1/1.4, 1.2) T (ρ, u, p, γ) T R = (10, 0.5, 1/1.4, 1.4) T By construction the method is designed to consistently transport the color function and all the conservative quantities during the advection step leading to a uniform provisional pressure and velocity that does not need to be corrected during the projection step. The results summarised in Figure 4 demonstrate that the present method is able to advect the interface between the two fluids at the theoretical speed without inducing any significant oscillation on the pressure and velocity fields keeping the interface sharp. Discrete conservation is also observed within round-off error.

Spurious currents in a static configuration

One common test for multiphase incompressible methods dealing with surface tension is the measurement of the parasitic currents induced by the discretization errors on the initialization [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF][START_REF] Abu-Al-Saud | A conservative and well-balanced surface tension model[END_REF]. In this section we report the results obtained for a two-dimensional bubble of radius R = 0.4 containing an ideal gas (Γ g = γ g = 1.4, Π g = 0) in a slightly compressible liquid (Γ l = 7.14, Π l = 300). The ambient liquid pressure is p 0 = 1 and the liquid and gas density is set to an initial constant value ρ l,0 = ρ b,0 = 1. The initial gas pressure is given by the Laplace pressure p g,0 = p 0 + σ R 0 , where the surface tension is set to σ = 1. We test various values of the fluid viscosity (equal for both substances) in order to modify the Laplace number La = ρD 0 σ µ 2 that controls the dissipation rate of the fluctuations induced by the initial discretization error. The grid size is kept constant and equal to R/∆x = 12.8.

As shown in Figure 5, the method reduces the amplitude of the spurious currents on a time scale proportional to the viscous scale, t visc ∝ D 2 /µ for all the Laplace numbers tested here. Comparing the decaying rate of the fluctuations with those obtained in [START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF] in the incompressible limit, significant differences are only observed for very large Laplace numbers where the incompressible version tends to dissipate the spurious currents faster than the current version of the solver including compressibility effects. Note that at large Laplace numbers the viscous boundary layer is most likely not captured by the grid resolution and therefore the observed dissipation rate is mainly controlled by numerical scheme properties which are different in the incompressible and compressible framework. In particular the incompressible formulation does not explicitly include any form of energy equation whereas in the compressible formulation the total energy dissipation is directly controlled by the explicit calculation of the viscous dissipation and the discretization errors on the representation of the surface energy included through the pressure jump across the interface.

Another important difference between the compressible and incompressible formulation is the appearance of high frequency fluctuations that occur in a time scale much shorter than the viscous time scale. Figure 6 reveals that indeed these fluctuations occur due to the resonance properties of the gas bubble whose characteristic time is t res = 1 f 0 ∝ R 0 ρ l γgp 0 . We can then conclude that the discretization errors on the surface tension acts as an ini- tial impulsive force that makes the bubble oscillate mainly at its resonance frequency.

Linear oscillation of a single bubble

In this section we investigate the three-dimensional problem of the linear bubble oscillation induced by a sinusoidal acoustic wave of given frequency ω where surface tension and viscous effects are neglected. Properties are made dimensionless using the reference pressure and liquid density, p 0 = 1, ρ l,0 = 1. We initialize a gas bubble (Γ g = γ g = 1.4 Π g = 0) of initial radius R 0 = 1 and density ρ g,0 /ρ l,0 = 10 -3 at the center of a cubic domain of length L 0 = 3λ, where λ = c 0 /f is the wavelength of the wave induced in the system. The EOS of the liquid is defined by Γ l = 7.14 and Π l p 0 = 3000 and the initial velocity field u = (U 0 , 0, 0) is given by

u(x, 0) = U 0 sin (k(x + 2)) -λ < x + 2 < 0 0 otherwise, (40)
where U 0 = 10 -3 and k = ω c 0 is the wavenumber that will be varied. The initial pressure field is given by the linear approximation p(x, 0) = p 0 + ρ l,0 c 0 u(x). Each simulation is run for tf = 1. An adaptive grid is used to refine the region near the bubble but grid is progressively coarsened far from the domain center.

In the linear regime and in absence of viscous and surface tension effects the bubble response solely depends on the ratio between the excitation frequency, ω, and the bubble resonance frequency,

ω res = 1 R 0 3p 0 γg ρ l = √ 4.2.
Because in the linear regime non-spherical bubble deformation effects are high order corrections of the exact solution and therefore negligible, the evolution of the averaged bubble radius obtained from the solution of the Rayleigh-Plesset (RP) equation is expected to provide an accurate estimation of the exact solution

RR + 3 2 Ṙ = 1 ρ l p b,0 R 0 R 3γ -4µ l Ṙ R -p ∞ (t) , (41)
where p ∞ (t) = p 0 + ρ l c 0 U 0 sin(ωt).

Figure 7 shows the temporal evolution of the bubble volume V as a function of the grid size for an excitation frequency above and below the bubble resonance frequency, ω ωres = 2, and ω ωres = 0.1 respectively. For frequencies above the bubble natural frequency the method converges accurately to the solution predicted by the Rayleigh-Plesset model. For frequencies much below the resonance frequency the problem becomes challenging due to the very different scales between the wavelength and the bubble radius. We observe the appearance of fluctuations of small amplitude at the bubble resonance frequency during the initial transient state that are only partially captured due to the limited resolution of the full three-dimensional simulation and the sensitivity of these fluctuations to the characteristics of the incoming pressure pulse. These transient, high-frequency fluctuations are indeed damped by adding a small amount of viscosity to the liquid in the Rayleigh-Plesset model. From the results of Figure 7 we can then conclude that coarsening the grid increases the effective damping of the solution obtained which is an expected behavior of the numerical scheme when the resolution is low. In any case the agreement between numerical and theoretical solutions is satisfactory both, above and below the bubble resonance frequency.

V/V 0 -1 t f RP model R/∆x=5 R/∆x=10 R/∆x=20 R/∆x=40

Weakly non-linear collapse of a single bubble in a viscous fluid

In order to test nonlinear terms in a multiphase flow problem, we consider the so-called Rayleigh collapse problem [START_REF] Brennen | Cavitation and bubble dynamics[END_REF] where a single bubble with initial pressure p b,0 is suddenly exposed to a pressure increase in the surrounding liquid p l,0 p b,0 > 1. An initially spherical gas bubble (Γ g = γ g = 1.4, Π g = 0) of radius R 0 = 1 at uniform pressure p b,0 = 100 and density r = ρ b,0 ρ l,0 , collapses in a liquid with density ρ l,0 = 1 where the parameters of the EOS for the liquid are Γ l = 7.14 and Π l p b,0 = 300. Because even the compressible versions of the Rayleigh-Plesset equation (Keller-Miksis, Gilmore...) are only small modifications of the solution in the incompressible limit, it is important to correctly initialize the pressure field around the bubble to reproduce the conditions modelled in Rayleigh-Plesset models [START_REF] Fuster | Liquid compressibility effects during the collapse of a single cavitating bubble[END_REF]. Given the pressure far from the bubble p l,0 the analytical pressure field for the Rayleigh collapse problem in the limit of an incompressible liquid is p(r, 0) = p l,0 + p I l,0 -p l,0

R 0 r , (42)
where p I l,0 = p l,0 -2σ R 0 is the initial liquid pressure at the interface given by Laplace's equation (Eq. 4). The cubic domain length is set to L 0 /R 0 = 100 and the initial fluid velocity is set to zero everywhere.

Assuming spherical symmetry the Rayleigh collapse time is t R = 0.915R 0 ρ l ∆p 0 , where ∆p 0 = p l,0 -p b,0 . In this work we use this time to define the characteristic velocity of the bubble collapse process, U c = 0.915R 0 /t R and to define the main non-dimensional quantities on which the problem depends: the pressure ratio Table 1. Figure 8 shows the grid convergence of the temporal evolution of the effective radius R = V (t) 4/3π 1/3 including as reference the solution obtained with a Rayleigh-Plesset model and the corrected solution provided by the Keller-Miksis equation [START_REF] Keller | Bubble oscillations of large amplitude[END_REF]. Consistent with previous numerical results [START_REF] Tiwari | A diffuse interface model with immiscibility preservation[END_REF][START_REF] Koch | Numerical modeling of laser generated cavitation bubbles with the finite volume and volume of fluid method, using openfoam[END_REF], the numerical simulations tend to converge to a solution close to the prediction of the Keller-Miksis model. The collapse time matches well RP predictions showing very little sensitivity to the grid resolution. On the contrary, the amplitude of the rebound is significantly influenced by the grid resolution. Convergence is reached for a grid resolution of 40 points per bubble radius not only for the averaged bubble radius, but also for the amplitude of the non-spherical distortion computed from the standard deviation of the radial distance to the bubble center x c of the centroid of all cells containing an interface (Figure 9). As predicted by the Rayleigh-Taylor instability models, the growth of non-spherical modes is promoted during the compression stage and also during the instants of possitive acceleration R > 0. These modes are significantly damped during the expansion stage when deceleration occurs R < 0, when the bubble recovers a nearly spherical shape at the end of the expansion after the rebound.

Strongly non-linear collapse of a single bubble in an inviscid fluid

In this section we test the solver for a strong collapse characterized by large values of the Weber and Reynolds numbers. The set of all nondimensional parameters is included in Table 2. Figure 10 presents the evolution of the effective bubble radius compared to the Keller-Miksis equation. In this case viscous effects play a negligible role and liquid compressibility is the main mechanism responsible for damping and the resulting reduction of the amplitude of the collapse. The development of non-spherical instabilities at the interface during the collapse of the bubble is clearly seen in Figure 10 showing a local minimum at the moment of maximum radius during the rebound. The agreement with the Keller-Miksis solution is satisfactory during the overall process except on the amplitude of the rebound. We note that the Keller-Miksis solution is only an approximation of the full problem and previous numerical simulations solving the full compressible Navier-Stokes equations including compressibility effects and imposing spherical symmetry also show that the Keller-Miksis equation tends to overestimate the amplitude of the rebound [START_REF] Fuster | Liquid compressibility effects during the collapse of a single cavitating bubble[END_REF].

4.2.6. Non-linear collapse of a single bubble near a wall As a last example we show the collapse of a bubble near a wall. It is well known that the presence of any source of asymmetry leads to the appearance of a high speed jet during the collapse of the bubble [START_REF] Wang | Bubble dynamics in a compressible liquid in contact with a rigid boundary[END_REF][START_REF] Obreschkow | Universal scaling law for jets of collapsing bubbles[END_REF][START_REF] Supponen | Scaling laws for jets of single cavitation bubbles[END_REF]. The problem of the collapse near a wall has particular interest in various applications due to the consequences of jetting on the wall and it has motivated various experimental and numerical works (see for example [START_REF] Lauterborn | Experimental investigations of cavitationbubble collapse in the neighbourhood of a solid boundary[END_REF][START_REF] Popinet | Bubble collapse near a solid boundary: a numerical study of the influence of viscosity[END_REF][START_REF] Yang | Dynamic features of a laser-induced cavitation bubble near a solid boundary[END_REF]). In this work we consider the collapse of a single bubble placed at a distance d/R 0 = 2. In an attempt to reproduce the experimental results of Yang et al [START_REF] Yang | Dynamic features of a laser-induced cavitation bubble near a solid boundary[END_REF], we use the non-dimensional parameters included in Table 3 where the pressure ratio is obtained by assuming that the bubble pressure at the instant of maximum radius is approximately equal to the vapor pressure. The maximum resolution used in the region between the bubble and the wall is R 0 /∆x = 40.

Figure 11 depicts the bubble interface contour during different instants of the primary collapse with the pressure field in the middle plane and also the velocity field. The pressure increases during the instants of minimum radius not only in the region surrounding the bubble but also in the wall. We can observe the generation of a high velocity liquid jet during the bubble collapse dragging some gas and impacting on the wall after the rebound (Figure 12). Figure 13 shows the evolution of the averaged bubble radius computed from the the total gas volume showing excellent agreement with the results reported by Yang et al [START_REF] Yang | Dynamic features of a laser-induced cavitation bubble near a solid boundary[END_REF] at an identical stand-off distance. The long term evolution of the bubble shape in a middle plane (Figure 14) shows similar features than those experimentally observed during the secondary collapse, namely the appearance of gas pockets that are left behind the jetting region due to the asymmetries generated by the presence of the wall, and the large deformation of the bubble shape for long times.

Conclusions

In this manuscript we generalize an all-Mach semi-implicit formulation for the simulation of compressible multiphase flows including viscous and surface tension effects that also accounts for arbitrary values of the thermal dilatation coefficient. One of the particularities of the method is that the advection of conservative quantities is made consistent with the geometrical transport of the color function used in Volume-of-Fluid methods, thus avoiding any artificial transport of mass, momentum or energy between species. The method is fully conservative in the absence of viscous and surface tension forces which allows for the simulation of both linear wave propagation problems and problems involving shock waves.

The proposed formulation takes advantage of previous work developed in the context of incompressible solvers devoted to reducing the impact of spurious currents induced by surface tension forces and shows similar performances.

The method is tested in the context of the simulation of bubble oscillations in slightly compressible fluids. For linear oscillations, the method is shown to correctly predict the oscillation of the bubbles above and below the bubble resonance frequency. The nonlinear effects are discussed in the context of the Rayleigh collapse problem where we show grid convergence in a variety of situations involving surface tension and viscous effects. The results are consistent with the description of the physics of collapsing bubbles discussed in the literature. In particular we observe the appearance of non-spherical modes during the bubble collapse, typically attributed to the development of Rayleigh-Taylor instabilities, which are then damped during the bubble expansion .

The problem of the bubble collapse near a wall shows that, consistent with previous numerical and experimental observations, the method captures the generation of a high velocity jet that impacts on the wall leading to a significant bubble fragmentation. The method is shown to be robust even in the presence of small unresolved structures allowing to investigate the long term dynamic response of collapsing bubbles.

[1] R. H. Cole, R. Weller, Underwater explosions, Physics Today 1 (1948)

35.

Figure 1 :

 1 Figure 1: Top: Evolution of the pressure pulse shape as a function of time for CFL=0.5. Bottom: Effective attenuation induced by the numerical scheme as a function of the wavelength.

Figure 2 :

 2 Figure2: Influence of the CFL number on the effective attenuation. The method is shown to converge with the time-step (CFL) at first order. The cut-off distance below which the numerical dissipation saturates also depends linearly on the CFL.

Figure 3 :Figure 4 :

 34 Figure 3: Structure of the shockwave at t=1 for the Bell-Colella-Glaz (BCG) advection scheme and a simple 1st-order upwind method.

Figure 5 :Figure 6 :

 56 Figure 5: Evolution of the velocity fluctuations of a 2D bubble as a function of the Laplace number using (a) the nondimensional viscous time for the all-Mach solver and (b) the incompressible solver of [18].

Figure 7 :

 7 Figure 7: Top: Temporal evolution of the bubble volume for ω/ω res = 2 as a function of the grid resolution. For comparison, the solution obtained with a Rayleigh-Plesset (RP) equation is included. Bottom: Solution obtained for ω/ω res = 0.1 (right). As reference we include the solution of the RP model.

10 - 2

 102 10 400 4.3 × 10 -2

p l, 0 pσ.Figure 8 :

 08 Figure 8: Temporal evolution of the bubble radius for the bubble collapse in a viscous fluid. SP: legend?

Figure 9 : 10 - 3 ∞

 9103 Figure 9: Temporal evolution of the radial disturbance a for the bubble collapse in a viscous fluid. p l,0 /p b,0 r Re We Ma 20 10 -3 ∞ 1900 2.5 × 10 -2

3 ∞

 3 3800 2 × 10 -2

Figure 10 :

 10 Figure 10: Bubble collapse problem in the case of a strong collapse: p l,0 /p b,0 = 20 and We = 1900. Top: 3D VOF surface colored as a function of the distance to the bubble center at t/t R = 0.75, 1.00, 1.25, 1.50. The color scale range is [0 : 1]. Bottom: Temporal evolution of the bubble radius (red) and standard deviation of the quantity a/R 0 = σ(|x I -x c |/R 0 ), which is a measure of the magnitude of the interface disturbance with respect to the averaged radius.

Figure 11 :

 11 Figure 11: Bubble collapse near a wall (bottom boundary). From top to bottom: t/t R = 0, 0.96, 1.20, 1.40. Top: Bubble interface and pressure colormap (scale p/p b,0 = [1 : 200]) and isocontours on the middle plane. Bottom: Middle plane of the velocity color map (scale |u|t R /R 0 = [0 : 3]) and bubble interface.

Figure 12 :Figure 13 :

 1213 Figure 12: Evolution of the bubble shape and velocity fields for t/t R = 0.96, 1.30, 1.65, 2.20. Same representation and ranges than those of Figure 11.

Figure 14 :

 14 Figure 14: Evolution of the gas fraction in a middle plane for different instants during the collapse and later rebounds. Consistent with the experimental observations of Yang et al [41] gas pockets are left behind the jet during the secondary collapse and afterwards.

  5 for each direction x do Evaluate ∂Y i ∂x (Eq. 23) Obtain geometrical VOF fluxes F C Obtain Y i,adv (Eq. 22) Obtain fluxes for the conservative variables F C i Y i (Eq. 17) Update C and C i Y i (Eqs. 15-16) end for Computeρu = q 1 + q 2 if µ > 0 then Evaluate µ(C) Obtain provisional velocity u * = ρu c 1 ρ 1 +c 2 ρ 2Solve Eq. 30 for p t+∆t Update u f (Eq. 27) Update ρu (Eq. 35) Update c 1 ρ 1 E 1 (Eq. 36) using p 1 (Eqs. 37) Update c 2 ρ 2 E 2 (Eq. 36) using p 2 (Eqs. 38) end while

	Solve viscous terms (Eq. 25)
	end if
	Obtain face velocity u k+1/2 = u k +u k+1 2
	if σ > 0 then
	Obtain curvature
	Update velocity fields (Eq. 26)
	end if
	Obtain p * from provisional values using the EOS
	Update p * applying the source term appearing in Eq. 28
	Evaluate 1 eff ρc 2	from provisional values

Table 1 :

 1 Non-dimensional parameters for the collapse of a bubble in a viscous liquid used in Section 4.2.4.

Table 2 :

 2 Non-dimensional paramters for the collapse of a bubble in an inviscid liquid used in Section 4.2.5.

Table 3 :

 3 Non-dimensional paramters for the collapse of a bubble close to a wall used in Section 4.2.6.