Cathy Cohen

Analysing multimodal resources in pedagogical online exchanges: Methodological issues and challenges

Keywords: multimodal resources, web-mediated pedagogical interaction, units of analysis, webcam, transcription, multimodal corpora

This chapter focuses on the contribution to webconferencing based pedagogical synchronous interactions of meaning-making multimodal resources (spoken language as well as gesture, gaze, body posture and movement). The first part of the chapter explores different methodological approaches to the analysis of multimodal semiotic resources in online pedagogical interactions. Having presented an overview of what research into synchronous web-mediated online interaction can bring to the field of CALL, we discuss the importance of determining the relevant units of analysis which will impact on the granularity of transcription and orient the ensuing analyses. With reference to three of our own studies, we then explore different methods for studying multimodal online exchanges depending on the research questions and units of analysis under investigation. To illustrate the various ethical, epistemological and methodological issues at play in the qualitative examination of multimodal corpora, the second part of the chapter presents a case study that identifies the different steps involved when studying online pedagogical exchanges, from the initial data collection phase to the transcription of extracts of the corpus for publication.

Introduction

As a result of globalization and easy Internet access, opportunities for exposure to foreign languages have greatly increased over the past two decades [START_REF] Kern | Technology as Pharmakon: The Promise and Perils of the Internet for Foreign Language Education[END_REF]. Not only can language learners access all types of documents (e.g., films, audio and video documents, written texts, images) quickly and simply, they can also exchange synchronously or asynchronously with speakers of the target language, opening up seemingly unlimited possibilities for foreign language contact and potential learning. These might be informal social interactions as learners seek out opportunities to use the target language with their peers, but they may also be specifically designed as pedagogical exchanges between a language teacher and learner, or between two learners under the coordination of a language teacher. Indeed, more and more language learning courses take place online, often between language teachers in one country, and language learners in another. Such courses may involve both asynchronous (e.g., email or blogging) and synchronous (e.g., text chat or videoconferencing) tools. As a result, new interaction patterns and norms are constantly developing and these combine a broad range of semiotic modes [START_REF] Sindoni | Spoken and written discourse in online interactions: a multimodal approach[END_REF], which potentially offer new and diverse opportunities for learning.

The current chapter focuses on pedagogical synchronous interactions which use desktop videoconferencing (henceforth DVC), described by Kern as "a quintessential technological support for providing communicative practice with speakers at a distance, since it is the closest approximation to face-to-face conversation " (2014: 344). This powerful instrument to learn languages is an Internet-based system enabling two or more people located in different places to communicate online with simultaneous two-way audio and video transmission [START_REF] Sindoni | Spoken and written discourse in online interactions: a multimodal approach[END_REF]. The video transmission, made possible thanks to a webcam on each participant's computer, gives access to several meaning-making modes, including spoken language, but also other multimodal elements such as gesture, gaze, body posture and movement. With the growing number of online language courses and telecollaboration projects, it is clearly important for CALL practitioners to gain a better understanding of how these multimodal resources contribute to the pedagogical setting and to learning contexts, and also how the different semiotic resources are orchestrated in interactive technology mediated situations [START_REF] Stockwell | Effects of multimodality in computer-mediated communication tasks[END_REF]). This chapter will analyze the contribution of multimodal resources to pedagogical online exchanges. The first part explores, the different methodological approaches to the analysis of multimodal semiotic resources in online pedagogical interactions. We begin by briefly reviewing recent literature in order to take an overview of what research into synchronous web-mediated online interaction can bring to the field of CALL. The issues of determining the relevant units of analysis will be discussed as the latter have a clear impact on the granularity (i.e., the amount of detail provided by researchers) of transcription and orient the ensuing analyses [START_REF] Ellis | Analysing learner language[END_REF]. Then, with reference to three of our own studies, we explore different methods that can be employed to study multimodal pedagogical exchanges depending on the research questions and the units of analysis under investigation. In the three studies, our focus is on the role played by technological mediation in online pedagogical exchanges and in particular, on the affordances provided by the webcam (see Chapter 3 in this volume).

To illustrate the different ethical, epistemological and methodological issues at play in the qualitative examination of multimodal corpora, the second part of the chapter will present a case study that identifies the different steps involved in the study of online pedagogical exchanges, from the initial data collection phase to the transcription of extracts of the corpus for publication. The case study is an extract from Study 2 which is presented in the first part of this chapter.

Methodological approaches to the study of multimodal pedagogical interactions

In this section, we focus on different methodological approaches that can be employed to analyze how multimodal semiotic resources function in online pedagogical interactions. Studies exploring how these interactions are mediated and organized by the webcam are still quite limited and different units of analysis have been the focus of recent research. Determining the relevant units of analysis is important as they have a clear impact on the type of data collected (quantitative or qualitative, see Table 1), on the granularity of transcription and will orient the ensuing analyses [START_REF] Ellis | Analysing learner language[END_REF].

We use the term unit of analysis to refer to the general phenomenon under investigation. Once the unit of analysis has been identified, it has to be operationalized by researchers who must then select the variable(s) which they are going to investigate. These are the features which the researchers believe constitute the unit of analysis (see Table 1 for examples). Several examples taken from the field of pedagogical DVC interactions are provided here to illustrate this. The unit of analysis for [START_REF] Wang | Task Design in Videoconferencing-Supported Distance Language Learning[END_REF] study was design principles for videoconferencing tasks. One of the components she explored was the role played by the webcam image in task completion. Using personal observation and post-session interviews with the small group of learners who participated in the study, she concluded that facial expressions and gestures visible via the webcam were key features that facilitated task completion. [START_REF] Satar | Multimodal language learner interactions via desktop videoconferencing within a framework of social presence: Gaze[END_REF] focused on how social presence was established in online pedagogical DVC interactions. She explored how the trainee teachers interacting with one another used gaze, and how they compensated for the impossibility of direct eye contact. She identified a range of different uses of the webcam and highlighted the importance of eye contact for the establishment of social presence in online multimodal interactions. Guichon and Wigham (in review) explored the potential of the webcam for language teaching, focusing particularly on the unit of analysis of framing, in other words how trainee teachers framed themselves in front of the webcam and, as a result, what information was made visible to their learners within the frame of the video shot. So, they investigated how trainee teachers made use of the affordances of the webcam to produce non-verbal cues that could be beneficial for mutual comprehension (see Study 3 below for more details). Their results emphasized the need for trainee teachers to enhance their critical semiotic awareness, including paying closer attention to framing, thus enabling them to gain a finer perception of the image they projected of themselves. In so doing, it is hypothesized that they should be able to take greater advantage of the potential of the webcam and, as a consequence, increase their online teacher presence.

Different methods can be employed to study pedagogical online exchanges and researchers' choice of method will depend on the research questions they wish to investigate and the objectives of their study. We will take three examples from our own work to illustrate different approaches. In all three, we are interested in the role played by technological mediation in online pedagogical exchanges and our particular focus is on the affordances provided by the webcam. Table 1 provides an overview of these studies which will be discussed in turn below. The first study, reported fully in [START_REF] Guichon | The impact of the webcam on an online L2 interaction[END_REF], adopted a quantitative methodology and had an experimental design. In this study, we explored the impact of the webcam on an online interaction by comparing several dependent variables between an audioconferencing and a videoconferencing condition, using Skype (http://www.skype.com).

In the audioconferencing condition, the webcam was switched off, whereas it was on in the videoconferencing condition. Our objective was to assess the webcam's contribution to the interaction. There were three research questions, each of which explored different units of analysis which we felt might operate differentially in the two experimental conditions. The first was learner perceptions, which were probed using a short post-task Likert scale questionnaire to gauge learners' feelings of: (1) the teacher's psychological and physical presence, (2) understanding of and by the teacher, and (3) the quality, naturalness and enjoyment of the conversation. The second explored the rhythm of the interactions by measuring silences, overlaps, turn duration and number of words. The third focused on word search episodes by measuring their frequency and duration. Before the experiment began, we had clear hypotheses which stated that being able to see one's interlocutor would make a difference to the online pedagogical interaction. In other words, we stated that we expected to find a statistically significant difference between all the dependent measures under investigation in the audioconferencing and videoconferencing conditions. Furthermore, for the dependent measures relating to learner perceptions, we predicted that the videoconferencing condition would be received more favorably than the audioconferencing condition.

The independent variables were strictly controlled before the experiment began. Forty French students who had a similar level of English, the foreign language they were learning at university, took part in the experiment. Twenty of them were put in the videoconferencing condition and 20 in the audioconferencing condition. Indeed, in order to be able to carry out certain statistical tests, it was necessary to have at least 20 participants in each condition. Statistical tests were used to verify that there were no significant differences between the two groups in terms of sex, age, English level, familiarity with online communication tools and attitudes towards speaking English. Had there been differences between the two groups at this stage, we could not have been sure whether our results were due to initial group differences or rather to differences resulting from the testing conditions. In the experiment, each student interacted individually with the same unknown native English-speaking teacher who was always in the same setting. Furthermore, they all did exactly the same task, which consisted in describing four previously unseen photographs. The duration of the interaction for all participants was set at around ten minutes.

In order to compare the different dependent variables between the two experimental conditions and assess the contribution of the webcam, it was necessary to carry out a quantitative study. In other words, we had to be able to measure the different variables in the two experimental conditions to see how they compared. So, for example, number of silences and word search episodes were counted and turn durations were measured (see Annotation below for more details as to how this was achieved). All the data were then imported into SPSS (http://www-01.ibm.com/software/fr/analytics/spss/), a computer-based statistical package for analysis, allowing statistical comparisons to be made between participants in the two conditions. Our results showed that, contrary to our predictions, there were fewer differences than we had anticipated between the videoconferencing and audioconferencing conditions on the dependent measures, with few comparisons reaching statistical significance.

The main difference was the greater number of student silences in the audioconferencing condition.

This first study was clearly time consuming in terms of data collection and analysis. It also involved many people: 40 students, a teacher, an assistant who helped organize the data collection sessions, four research assistants to transcribe and annotate the data (see Annotation below) and two researchers who analyzed the data and wrote up the research for publication. Although the differences between the results obtained from the two experimental conditions were far less clear-cut than we had expected, the results were nevertheless thought provoking. Indeed, we considered that although from a quantitative point of view the presence of the webcam did not seem to have a great impact on the pedagogical interactions with regard to the units of analysis which were investigated, the webcam image could nevertheless be facilitative, could modify the quality of the mediated interaction and that the reality was in fact considerably more complex than our findings seemed to show. Hence these results also highlighted the limitations of using quantitative data to grasp the more subtle interactional aspects in a multimodal learner corpus. Furthermore, our results provide a good example of the iterative process of research, with the first more generic experiment being a necessary step to reveal the need to explore particular parts of our corpus using a much finer grained analysis. This led us to conduct our second study.

Study 2: Qualitative approach on experimental data

In this study [START_REF] Cohen | Researching nonverbal dimensions in synchronous videoconferenced-based interactions[END_REF], we carried out a qualitative and descriptive analysis on small sections of the videoconferencing data taken from the first experimental study. In other words, we used part of the same corpus used in Study 1, but this time, to conduct a microanalysis. The analysis focused on short sections of just three of the 20 videoconferencing interactions, in order to examine how the learners and the teacher used the webcam strategically at different times during their exchanges.

Since we were particularly interested in training language teachers to utilize the affordances of the webcam during pedagogical online interactions and to develop their critical semiotic awareness, we considered that only a fine-grained analysis of non-verbal behavior in the videoconferencing condition would enable us to identify when and how the interaction was facilitated by the appropriate use of the webcam by participants.

The methodology employed in Study 2 was quite different from the first. This time, we worked within the Conversation analysis (CA) paradigm, as articulated in work initially conducted by gesture specialists (e.g., McNeill 1992) and more recently pursued by researchers working on gesture in the field of Second Language Acquisition such as McCafferty and Stam (2008) and [START_REF] Tellier | Stratégies verbales et gestuelles dans l'explication lexicale d'un verbe d'action[END_REF]. We adapted the methodology of these authors who focus on face-to-face pedagogical interactions in order to investigate pedagogical computer-mediated interactions. We also integrated an approach from the broader domain of multimodal discourse analysis, as applied by [START_REF] Norris | Analyzing multimodal interaction -A methodological framework[END_REF] and [START_REF] Baldry | Multimodal Transcription and Text Analysis[END_REF] whose work is not conducted in the pedagogical field. Finally, our approach was influenced by recent work carried out by [START_REF] Sindoni | Spoken and written discourse in online interactions: a multimodal approach[END_REF] who has explored nonpedagogical online interactions using a multimodal approach. In other words, the methodological approach we adopted was influenced by work conducted in several domains of scientific research. By combining and adapting elements from these different areas, we created a method suitable for analyses in our own field of investigation, i.e., the study of multimodal resources in pedagogical online exchanges.

In this second study, we explored the contribution to meaning making of several nonverbal semiotic resources other than speech and investigated how they helped the teacher to manage the online pedagogical interaction and how they were orchestrated. Each of these semiotic modes will now be presented briefly, with specific reference as to how they function in an online DVC interaction.

We considered proxemics, that is to say the physical distance individuals take up in relation to one another and to objects in their environment. Proxemics functions quite differently when interacting online using DVC, since interactants are not in the same location. Sindoni observes that "distance is not established by those who interact, but between one participant and one machine. This distance foregrounds the representation of distance among users." (2013: 56). Added to this, whatever position the user chooses, because he has constant access to his own image in the smaller frame on his computer screen, he is able to monitor and manipulate the image he wishes to project to his interlocutor [START_REF] Sindoni | Spoken and written discourse in online interactions: a multimodal approach[END_REF]. This affordance provided by web-mediated communication also gives the user greater control over the construction and negotiation of social space.

We examined different types of gesture, defined as the use of the hands and other body parts for communicative purposes [START_REF] Mode | Glossary of multimodal terms[END_REF]. We focused in particular on those gestures which were visible in the webcam: iconic gestures representing an action or an object; metaphoric gestures illustrating an abstract concept or idea; and deictic gestures used to point towards concrete or abstract spaces. Our objective here was to assess what type of information was communicated by these gestures and to what extent they appeared to facilitate (or not) the online exchange. For instance, were they transmitting some information to the interlocutor to complement or accompany what was said in the verbal channel (co-verbal gestures)? Or were they self-regulatory gestures, produced unintentionally to help speakers to think, thereby allowing them to maintain a sense of coherence for themselves [START_REF] Mccafferty | Material foundations for second language acquisition: Gesture, metaphor and internalization[END_REF]? To what extent were they visible in the webcam?

Head movements, which may convey meaning between interlocutors (e.g., nodding in agreement; shaking one's head from side to side to convey disagreement; holding one's head quite still while fixing one's gaze on someone to indicate concentration and focus), were also considered.

Finally, we were interested in eye contact, gaze and facial expressions. Compared to face-toface conversation, gaze management is very different in online video interactions. With the current state of technology used in DVC systems, it is impossible for speakers to make direct eye contact with one another (see De Chanay 2011). When speakers direct their eyes to their interlocutor's image on their computer screen, their eyes are slightly lowered, so not aimed directly at their interlocutor's eyes. They can choose to look directly at the webcam which gives the interlocutor the impression that he is being looked at straight in the eyes, but in so doing, paradoxically the speaker can no longer focus on the interlocutor's image on the screen (De Chanay 2011). So, not only are there fewer visible gestures to facilitate communication and intercomprehension in videoconferencing interactions, but there is also the impossibility of mutual gaze. [START_REF] Cosnier | Le face à face en ligne, approche éthologique[END_REF] hypothesize that speakers compensate for this through facial expressions which become more important and seem to be more numerous and perhaps over-exaggerated in videoconferencing interactions compared to face-to-face conversations, precisely to compensate for the lack of visible hand and arm gestures.

The different non-verbal semiotic modes have been discussed separately here, but of course during any chosen communicative event, they are operating simultaneously and, as Sindoni (2013: 69) argues "Ensembles of semiotic resources […] produce effects that differ from those produced by a single semiotic resource and from the mere sum of semiotic resources". A transcript and microanalysis taken from this study corpus is provided below (see Transcript and analysis presentation) as an illustration of our approach. Since the study was exploratory, our hypotheses emerged progressively as the data were explored. Three angles of analysis became apparent: (1) self-regulatory versus co-verbal gestures; (2) gestures which contribute something to the construction of the message versus gestures which potentially cause interference and are distracting and (3) redundant gestures which duplicate what is said in the verbal channel versus to complementary gestures which add some new information.

This qualitative study provided us with rich and complex data, enabling us to gain insights into the multimodal orchestration of the different semiotic resources in an online pedagogical interaction. However, we were using data collected for a study carried out in experimental conditions -the interaction duration was fixed; it was the first time that both the teacher and the learners had met and taken part in an online pedagogical interaction. So, the findings may have been attributable, to some degree at least, either to the novelty of the learning situation and/or the task learners were asked to carry out. In other words, the conditions of this second study, and indeed the first, lacked ecological validity. Thus in our third study, we tried to address this methodological shortcoming.

Study 3: Quantitative and qualitative approach on ecological data

As shown in Table 1, the corpus for the third study was collected in ecological conditions. The context was a telecollaborative project1 in which 12 trainee teachers of French as a foreign language met for online sessions in French with undergraduate Business students at an Irish university. Each trainee teacher met with the same learner (or pair of learners) once a week for approximately 40 minutes over a six-week period. Over this period, the trainee teachers proposed a range of different interactional tasks to their learners. So, unlike Study 2, which was conducted in experimental conditions, i.e., it was set up with the sole purpose of conducting an experiment to test our different hypotheses, Study 3 used data collected from an online course that was set up between two universities with learner training in mind: helping Irish learners to develop their interactional skills in French, and helping students training to be French teachers to develop their online teaching skills. Thus this teaching and learning situation was not set up initially for research purposes but the data collected from the online sessions were used subsequently to conduct research.

The research carried out in this study (Guichon and Wigham, in review) focused on very specific elements taken from the sizeable corpus that was collected. As in the previous two studies, we were interested in how participants used the affordances of the webcam, but this time the particular focus was on framing, i.e., how the trainee teachers framed themselves in front of the webcam and, as a result, what information was made visible to their learners within the frame of the video shot. For the qualitative part of the study, the same method of analysis was used as in Study 2. Two questions were explored here. Firstly, in order to study teachers' framing choices, screenshot images were taken of the 12 trainees each week over six weeks, at around minute 17 of their online interaction. A quantitative approach was adopted to provide an indication of the frequency of the trainees' different framing choices along a continuum, from extreme close-up shot, to close-up, to head and shoulder shot, to head and torso shot. In parallel, a qualitative approach was used to conduct a fine-grained analysis on the same data and, in particular, how the trainees positioned their gestures in relation to the webcam over the six-week course.

The findings revealed that, head and shoulder shots, followed by close-up shots of themselves were those most favored by the trainee teachers. Furthermore, qualitative analysis of the data showed that certain trainee teachers adjusted the position of some of their gestures, in particular highly communicative iconic and deictic gestures, so that they were framed and therefore more likely to be visible to learners and, therefore, potentially helpful for learner comprehension. Furthermore, quantitative analyses revealed that these gestures were held for longer in front of the webcam. So such teaching gestures, which clearly had a communicative purpose, appeared to be produced by these trainee teachers quite intentionally, and consequently were aimed at the webcam and remained visible to the language learners for some time.

The second question investigated in this study explored the communicative functions of gestures that were visible or invisible in the frame. For technical and practical reasons explained fully in the study, data were collected for just three participants for just one session each. The teacher trainees were filmed using DVC with their learners with two distinct recordings. One captured the on-screen activity, so what was visible and audible through the webcam, and an external camera was used to film what lay outside the webcam's view (the hors champ). When the two sets of recordings were compared, it became clear that the trainee teachers continued to perform many potentially co-verbal gestures which were either invisible or only partially visible in the webcam recordings which only captured a close-up of the head and upper torso area. In contrast, extra-communicative gestures, such as touching their hair or scratching their ear, become much more visible because of the magnifying effect provided by the very restricted view offered through the webcam. Such gestures, which may have gone unnoticed in a face-to-face interaction because of the presence of other broader contextual elements, were more difficult to miss when communicating using DVC. Indeed, if numerous, they could become rather distracting and interfere with communication.

So, the findings of this study highlighted the need to train teachers "to become critically aware of the semiotic effect each type of framing could have on the pedagogical interaction so that they made informed choices to monitor the image they transmitted to their distant learners according to an array of professional preoccupations" (Guichon and Wigham, in review). This ecological study provided valuable information which could be reinvested in future teacher training courses.

Synthesis

We have explored three different studies, each of which explores the role of HCI in online pedagogical exchanges, with a particular focus on the affordances provided by the webcam.

From the first generic study which was experimental and quantitative, through to the third study which had a specific focus, was ecological and combined both quantitative and qualitative analyses, we have shown that the method adopted will depend on the research questions under investigation. Both quantitative and qualitative analyses are valid means to explore the data collected, as long as the method is sound and the objective clearly stated.

The qualitative microanalysis of a much broader range of units of analysis investigated within the field of webconferencing-supported teaching is certainly to be encouraged in order to further enhance our knowledge of HCI in a pedagogical setting. By putting certain elements of the interaction into the spotlight, we may progressively untangle the complexity of these online pedagogical exchanges.

In the first part of this chapter, we have explored different methods for studying multimodal resources in pedagogical online exchanges. However, in order to be able to conduct the type of analyses presented above, researchers have to ensure that their data are collected and stored in such a way that they can be later transcribed and annotated. Whether the study is quantitative and experimental or qualitative and ecological, numerous transformations are required to progress from the initial data collection stage to the creation of a corpus that can be presented in academic publications or at conferences, and also perhaps be shared among researchers.

In the next section of this chapter, we examine these different stages and investigate the opportunities and challenges concerning the study of data relating to synchronous mediated language learning and teaching.

Reflections on a multimodal approach to synchronous pedagogical online interactions

From the traces of mediated activity to a corpus that can be studied from different perspectives Any mediated learning activity produces traces: digital traces, currently much used in the field of Learning Analytics, can be computer logs that provide quantitative information (frequency of access, time spent on a task, number of times a given functionality is used, etc.). The aim of these digital traces is to understand and optimize learning and learning environments [START_REF] Siemens | Learning Analytics and Educational Data Mining: Towards Communication and Collaboration[END_REF]. Digital traces can also be comprised of "rich histories of interaction" (Bétrancourt, Guichon and Prié 2011: 479) that provide multimodal data and time stamps that can be gathered from digital environments in order to gain an insight into certain teaching and learning phenomena. This second form of traces has been studied by researchers in the field of computer-mediated communication (CMC) for the last 20 years (see for instance [START_REF] Kern | Restructuring classroom interaction with networked computers: Effects on quantity and quality of language production[END_REF][START_REF] Kost | Use of communication strategies in a synchronous CMC environment[END_REF]Pelletieri 2000). Thus, traces collected in forums, blogs, emails, audiographic platforms and DVC have been built into corpora to study the specificities of mediated language learning usually by using conversation and/or interaction analytic tools.

The present section focuses on mediated learning interactions to illustrate how technology helps fashion methodological and scientific research agendas in the field of mediated interactions. Several operations are at play when researchers deal with a data-driven study of multimodal learning and teaching, when they strive to create a corpus that can offer different types of analyses as was illustrated in the first part of this chapter.

If we take the example of a corpus composed of recordings of online learning interactions mediated by a DVC facility, four main operations can be identified: corpus building, annotation, data transcript and presentation. Each of these operations will be explained and illustrated by a case study using data that were initially collected for a larger research project [START_REF] Guichon | The impact of the webcam on an online L2 interaction[END_REF], discussed in Study 1 above). However, before we do this, it is important to underline the ethical aspects that researchers must respect when dealing with data which include participants' images.

Ethical considerations

Ethical issues are relevant to all research involving humans. In the case of the type of studies we have described above, which may involve the publication of participants' images, certain issues should be considered very carefully.

Before recording begins, researchers must obtain written informed consent from participants: first, that they agree to be recorded; second, that they agree to be recorded for research purposes; and third, that they agree that recordings (or screenshots) may be displayed publicly or published (ten Have 1999). If participants consent to all three, they must understand fully what is at stake. For example, will they be recognizable from the recordings (visual, auditory)? Will their faces be blurred/pixelated to avoid recognition? Where will the recordings be shown and where will they be published? Will they be available freely online to anyone (for an (un)limited period of time)? Will participants have access to the recordings before they are used, in order to confirm or cancel their informed consent? (see [START_REF] Yakura | Informed Consent" and Other Ethical Conundrums in Videotaping Interactions[END_REF] for an excellent discussion of the issues at stake here).

The above questions present real challenges for researchers. First and foremost, if recordings or screenshots are to be used publicly, anonymity cannot be ensured at every stage [START_REF] Yakura | Informed Consent" and Other Ethical Conundrums in Videotaping Interactions[END_REF]). Secondly, depending on what participants have consented to, researchers may be more restricted in what they can present and/or publish. If, for instance, researchers wish to provide a finely grained analysis of the different non-verbal semiotic modes employed by participants, but they are only authorized to publish faces which have been blurred, displaying eye contact, gaze and facial expressions becomes impossible, thus "rendering the data unusable for certain lines of linguistic inquiry" (Adolphs and Varter 2013: 149).

How can researchers circumvent this problem in order to preserve and communicate to others some of the richness of their data? To compensate to some extent for the loss of visual information, researchers could provide very detailed written descriptions [START_REF] Lamy | Online communication in language learning and teaching[END_REF]. In a recent study by [START_REF] Sindoni | Spoken and written discourse in online interactions: a multimodal approach[END_REF], because of reservations expressed by certain participants about the publication of screenshots, she opted to use drawings instead. However, she recognises the drawbacks of this, stating that "they are time-consuming and require specific expertise, so that they can be used selectively, only for very brief and fine-grained analyses. Furthermore, drawings incorporate the researcher and artist's bias that represent participants in their interactions." (Sindoni 2013: 71).

Multimodal data collection

Several applications, for instance Camtasia (http://camtasia-for-mac.en.softonic.com) or Screen video recorder (http://www.dvdvideosoft.com/fr/products/dvd/Free-Screen-Video-Recorder.htm#.VHBk8Eve5g0) can be used to capture on-screen activity in an online interaction and this can be converted into a video (for more details, see Chapter 7 in this volume). The advantage of such applications is that they can be installed beforehand on each participant's computer and once switched on, they capture everything that is visible on the screen and audible around the screen, thus providing researchers with access to all the actions and utterances produced by the participants during the online interaction. Hence, whether the study is experimental or ecological (see above), traces of the mediated activity can be collected with little or no interference on the ecology of the learning situation. This is quite different from classroom-based research that requires more intrusive devices (i.e., video cameras) to collect traces of the observable teaching and learning activities.

While the traces of the mediated learning activity constitute the main material of the study, complementary data have to be collected via consent forms, researchers' field notes, pre-and post-interviews or questionnaires with the participants to gather crucial information about:

• Ethical dimensions (as discussed above);

• Socio-demographics and learner profiles: age of the participants, gender, relations to one another (in case of an interaction), familiarity with the given program or application, level in target language and motivations, experience in learning or teaching online; • Pedagogical dimensions: nature of the interaction, tasks, themes, documents used, instructions, place within the curriculum; • Temporal dimensions: length of each interaction, frequency of interactions (e.g., once a week), duration of module (e.g., a semester); • Methodological dimensions: how participants were recruited for the study, how their level was assessed, how they are divided (in case of an experimental study that compares two or several groups), what they were told of the aim(s) of the study, precisely how the data collection was organized, how ethical considerations were taken into account (see above); • Technological dimensions: type of software and hardware used (e.g., desktop or laptop, devices used for recording, etc.).

The conjunction of field notes, questionnaires, interviews, consent forms with the main data thus help create "a dynamic constellation of resources, where meanings are produced through inter-relationships between and within the data sets, permitting the researcher literally to "zoom in" on fine-grained detail and pan out to gain a broader, socially and culturally, situated perspective" (Flewitt et al. 2009: 44).

The data that serve as the illustration for this chapter come from Study 2, discussed above. The reader will need to know a number of elements about the two participants who took part in a larger study (Study 1 discussed above, see [START_REF] Guichon | The impact of the webcam on an online L2 interaction[END_REF]. The learner was 20 at the time of the study. His level had been assessed as B2 (according to the Common European Framework of Reference for Languages) and he described himself as a keen language learner. He used Skype for social purposes but had never used it for language learning. It was the first time he had interacted with the 28-year-old female native teacher and this interaction was not part of his usual class. The teacher had several years of experience teaching non-specialist university students in a classroom setting and was a regular user of Skype, mainly for personal communication. However, this was the first time that she had taken part in an online pedagogical interaction. Neither of the two participants was informed of the study's purpose or hypotheses before the experiment. The task consisted in getting the student to describe four previously unseen photographs. These photographs were chosen because each one contained what were considered to be problematic lexical items likely to trigger word search episodes, chosen as the unit of analysis for this research. The interaction via Skype lasted for about 10 minutes and participants were asked not to use the keyboard.

All the secondary data (field notes, questionnaires and interviews) had to be digitalized and grouped together with the data comprising the traces of the mediated interaction "to reconstitute for researchers, in as many ways as desired, information about the original experience" (Lamy and Hampel 2007: 184) and to enrich subsequent analyses.

Annotation

There are several computer software tools that researchers can use to code audio and video data. Among these, ELAN ((http://tla.mpi.nl/tools/tla-tools/elan/) is a linguistic annotation tool devised by researchers at the Max Planck Institute [START_REF] Sloetjes | Annotation by category -ELAN and ISO DCR[END_REF].

Figure 1 shows a sample of the data that were annotated with ELAN with which the researchers can:

1. Access the video stream of one or up to four participants; 2. Play the film of the interaction at will with the usual functionalities to navigate it; 3. View a time line aligned with media time; 4. Transcribe, on the horizontal axis, the utterances of the participants (one layer per participant); 5. Add a new layer for each element they wish to investigate (indicating for instance the onset and the end of a gesture and its description); 6. View annotations of one layer in a tabular form to facilitate reading. With ELAN, there can be as many layers (called tiers) as is deemed useful for a given study (i.e., words, descriptions, events, translations, etc.). As the case study presented here focuses on the verbal and co-verbal behavior of the learner who has to describe four photographs to a distant teacher via Skype, the direction of the eyes, the gestures that were produced (e.g., points to his ear), the silences between and within turns were annotated because they all were crucial elements when a learner is speaking in his L2 and is confronted with word search episodes. Researchers working on multimodal data can thus align different features of the interaction, accurately transcribe data across modes and then obtain a variety of views of the annotations that can be connected and synchronized. The data from the three studies described in the first part of this article were all transcribed using ELAN. Hence, although the first study was quantitative and the second qualitative, the same annotation tool was used for both although the tiers which were the focus of each study were different.

Annotation corresponds to a necessary transformation of the data in view of further analysis. It is a time-consuming and demanding task that requires devising a coding scheme so that all annotations are consistent across different annotators. As noted by Adolphs and Varter (2013:155), coding schemes have to be carefully explained and recorded so that "they can be shared across different research communities and with different community cultures and different representational and analytical needs". It is methodologically sound to get two different researchers to annotate a sample of the same data in order to ensure that the coding scheme is sound. This can be verified by calculating the inter-rater reliability to determine for instance whether two researchers interpret and code gestures consistently and reach a satisfactory level of agreement. If they fail do to so, the annotation scheme needs to be refined and retested in the same way until satisfactory inter-rater reliability is achieved [START_REF] Allwood | Multimodality in own communication management[END_REF]). Yet, as noted by Calbris (2011: 102), "achieving the ideal of scientific objectivity when coding a corpus is a delusion, because coding depends on perception, an essentially preinterpretative and therefore subjective activity". Furthermore, priorities and research questions have to be carefully defined beforehand so that the granularity of the annotations does not evolve. Researchers such as [START_REF] Flewitt | What are multimodal transcription and data?[END_REF] underline that annotation already corresponds to a first level of analysis since it entails selecting certain features of the mediated interaction and leaving others out according to both a research rationale and agenda.

Once the data have been annotated, they can then be organized into a coherent and structured corpus (see Chapter 10 in this volume for a full account of corpus building and sharing). They may also be put on a server allowing them be shared with other researchers. In order to do this, close attention has to be paid to the formats of the data so that they are compatible with different computer operating systems. Providing researchers with clear information as to how to access the data, specifying all the contextual information (see above) and ethical dimensions (e.g., what can be used for analysis and what cannot be used for conferences or publications because participants have withdrawn their permission) are important steps to make the corpus usable, searchable and sharable. The field of CMC would greatly profit from having more researchers working on the same corpora: not only would it reduce the costs associated with corpus building, transcription and annotation, but it would provide researchers with the opportunity of examining the same data using different tools, methods and research questions and would therefore produce results that can present more significance and reliability to the community at large.

Transcript and analysis presentation

Once the data have been organized into a coherent corpus, analyses can be made starting by the making of the transcript. [START_REF] Bezemer | How to transcribe multimodal interaction?[END_REF] allocates two functions to the making of a multimodal transcription. The first function of transcription is epistemological and consists of a detailed analysis of a sample of an interaction in order to "gain a wealth of insights into the situated construction of social reality, including insights in the collaborative achievements of people, their formation of identities and power relations, and the socially and culturally shaped categories through which they see the world" (Bezemer 2014: 155). The second function is rhetorical in that the transcript is designed to provide a visual transformation of the trace of the interaction that can be shared with readers in a scientific publication. Transcripts chosen and prepared for an article are not illustrations of a given approach or theory but are both the starting-point of the analysis and the empirical evidence that supports an interpretation and can be shown as such to readers. The researcher must therefore find an appropriate time-scale (e.g., a few turns, an episode, a task, a series of tasks, a whole interaction) to study a phenomenon (for instance negotiation of meaning in a mediated pedagogical interaction) and then define the boundaries of the focal episode. Making the transcript may also involve refining the initial research questions and determining what precise features will be attended to.

For our study on videoconference-based language teaching, it seemed crucial to understand how the distant teacher helped the learner during word search episodes and used the semiotic resources (such as gestures, facial expressions and speech) at her disposal. It was equally important to examine how the learner used different resources to signal a lack of lexical knowledge and how meaning was negotiated with the native teacher. The interplay of gestures, head and body movements, gaze and facial expressions produced by both participants while the learner was trying to describe a photograph became features that were selected as especially important for the transcript (see Figure 2). Although conventions used for Conversation analysis can be adjusted to multimodal transcription, new questions arise concerning the representation of co-verbal resources (gesture, gaze) with text, drawings or video stills and the alignment of these different representations so that the reader can capture how verbal and nonverbal resources interact (see Figure 2). Ochs (1979 as cited in Flewitt et al. 2009: 45) underlined the theoretical importance of transcription, arguing that "the mode of data presentation not only reflects subjectively established research aims, but also inevitably directs research findings". For instance in Figure 2, the choice of presenting, when relevant, the images of the two interlocutors side by side (e.g., images 5 and 6) was made because we felt that the detail of their facial expressions, smiles and micro-gestures within the same turn was necessary to understand minutely the adjustments that occurred during such an interaction. Such a transcription allows a vertical linear representation of turns and makes it possible to unpack the different modes at play "via a zigzagged reading" (Sindoni 2013: 82). Working iteratively on the transcript and on the accompanying text (see Table 2) helps refine both because they oblige researchers to give saliency to certain features in the transcript (simultaneousness of different phenomena, interaction between different semiotic modes, etc.), while the text that they write has to deploy textual resources to recount them. Neither the transcript nor the text can stand alone; rather they function as two faces of the proposed analysis. In turn 1, certain marks of hesitation, long pauses and self-admonishments ("I don't know") signal a communication breakdown while the learner is trying to find a way to describe the unknown lexical item. By touching his own ear repeatedly and miming a hole with his fingers, the learner is not only making his search visible to the teacher but is negotiating the meaning with her and looking for signs of her understanding. Her smile in image 6 suggests that she seems to understand what he is trying to describe, although he pursues his description in an attempt to be even more precise. As is visible in image 7, the student has what Goodwin and Goodwin (1986) would describe as a "thinking face", indicating to the teacher that he is still searching for the exact term, before he looks directly at the screen in image 8 -suggesting he wants confirmation from the teacher that she understands precisely what he is trying to describe. This search triggers a smile from the teacher and the mirror gesture (image 9) of that of the learner, which indicates that the teacher ratifies the description to a certain extent and that the interaction can continue while she is giving him her full attention by looking directly at the screen. Once the association of the verbal and nonverbal messages seem to have reached their objective, the learner verbally adds an element ("a hole") and gives redundant information by prodding his index finger at his ear again making sure that the teacher has understood the lexical item (she nods in image 11) even if the precise word has not been found. Table 2

: Textual analysis of the episode

There is no stabilized way of making multimodal transcripts although more and more researchers (see for instance [START_REF] Bezemer | How to transcribe multimodal interaction?[END_REF][START_REF] Flewitt | What are multimodal transcription and data?[END_REF][START_REF] Norris | Analyzing multimodal interaction -A methodological framework[END_REF][START_REF] Sindoni | Spoken and written discourse in online interactions: a multimodal approach[END_REF] have devised astute ways of approaching this. Reading these authors, several considerations arise in relation to the units of analysis that can be selected, ethical dimensions that have to be attended to, the readability and the presentation of multimodal transcription.

First, turns of speech that constitute the conventional unit of analysis in Conversation analysis may not be as pertinent for multimodal analysis because, as noted by Flewitt et al. (2009: 45), "as soon as multiple modes are included, the notion of speech turns becomes problematic as other modes contribute meanings to exchanges during the silences between spoken turns". New units of analysis have thus to be devised to capture the specificity of multimodal interactions. For example, what is a speech turn when an individual uses written chat and speech simultaneously? Second, multimodal transcription makes participants identifiable, which makes it even more crucial to be vigilant about ethical considerations (as discussed above). Finally, researchers must establish a careful balance between the representation of all the features that are be considered in a truly multimodal interaction and what a reader -even a seasoned one -is able to capture when confronted with a thick rendering of multimodality. As noted by Flewitt et al., "the perceptual difficulties for the audience of 'reading" genuinely multimodal transcription might outweigh the advantage of its descriptive 'purity '" (2009: 47). Eventually, there will be new ways of presenting multimodal data along with more traditional paper-based publication that will truly render the multimodal nature of such data. How to transform multimodal data in order to make them accessible with various degrees of complexity or presentational choices constitutes one direction for future research.

Drawing conclusions

Once transcriptions are completed, researchers can proceed to analyses such as the one proposed in Table 2. If their approach is quantitative, all the annotations can be exported to "applications that are able to perform statistics on the results" [START_REF] Wittenburg | ELAN: a Professional Framework for Multimodality Research[END_REF].

Quantitative studies can thus give insight into a certain number of phenomena that can be relevant to understanding online learning and teaching. For instance, the number of pauses, the frequency of overlaps and the length of turns can shed light on the rhythm of a given interaction. The number of gestures and facial expressions produced by the participants could also give indications as to the communication potential of videoconferencing. The main outcome of quantitative studies concerns the identification of interactional patterns.

Although some examples of quantitative studies can be found, studies usually rely on qualitative approaches to data and focus on short episodes. As Bezemer says: Making a transcript is an invaluable analytical exercise: by forcing yourself to attend to the details of a strip of interaction you gain a wealth of insights into the situated construction of social reality, including insights in the collaborative achievements of people, their formation of identities and power relations, and the socially and culturally shaped categories through which they see the world. (2014: X) Yet, Adolphs and Varter point out that the community of researchers interested in multimodal analysis might profit from adopting a mixed approach and combining, when possible and pertinent, the conversation analysis of small samples of data with a corpus linguistics-based methodological approach. Thus, with the inclusion of large-scale data sets such an approach could extend "the potential for research into behavioral, gestural and linguistic features" (2013: 145).

Conclusion

In this chapter, we have shown the importance of taking into account the array of technologies that accompany the fabrication, analysis and transformation of interactional data. With ever refined software and transcription techniques, interactional linguistics has come to integrate into its agenda the intrinsically multimodal nature of interactions (Détienne and Traverso 2009). This is even more apparent when the interactions under study are themselves mediated by technologies, as is the case with videoconferencing-based exchanges. Technologies thus facilitate the gathering of interactional data and allow researchers to search them, replay them at will, annotate them with different degrees of granularity, visualize them from different perspectives, and structure them according to different scientific agendas [START_REF] Erickson | Persistent conversation: an introduction[END_REF]. Not only do these technologies change the way researchers approach data, they also require them to develop new technical and methodological skills. As we have seen with the various steps involved in the collection, transcription and analysis of multimodal data, the different techniques at play mostly concern the representation of data. Each transformation of the data results in a new object that can be subject to yet another transformation, until the refinement is complete enough to yield a satisfactory comprehension of the phenomena under study. This points to the essential work of representations that "serve as resources for communicating and meaning-making" to the scientific community and beyond (Ivarsson, Linderoth and Saljö 2009: 201) and are "achieved by combining symbolic tools and physical resources" (ibid: 202).

The kinds of studies we have conducted not only help us to uncover the interplay of the different multimodal semiotic resources in online teaching environments but, ultimately serve to improve the design of teacher training programs (e.g., how to use the affordances of the webcam in online interactions, how to pay attention to learner needs thanks to visual cues) so as to enhance learner computer interaction in online webcam-mediated exchanges.

Figure 1 :

 1 Figure 1: Example of a sample of data annotated with ELAN

Figure 2 :

 2 Figure 2: Multimodal transcript of a word search episode

Table 1 :

 1

	Duration	Type of	Task	Design	Number	Unit of	Features/Variables
		data				of	analysis	studied
						participa
						nts
	Study 1 1	Quantitative Describe	4	Experimental	40	Learner	Feeling of psychological
	interaction		pictures				perceptions	and physical presence;
	lasting						of	online	understanding of and by
	around 10						interaction	teacher; quality,
	minutes								naturalness and
	per								enjoyment of interaction
	student						Rhythm of	Silences; overlaps; turn
							interaction	duration; number of
									words
							Word		Frequency; duration
							search
							episodes
	Study 2 1	Qualitative	Describe	4	Experimental	3	Word		Multimodal
	interaction		pictures				search	orchestration of speech
	lasting						episodes	and non-verbal features
	around 10								(e.g., gaze, nods,
	minutes								gestures, facial
	per								expressions)
	student
	Study 3 1 weekly	Quantitative	Range	of	Ecological	12	Framing	Teachers' semiotic self-
	interaction	and	different tasks			choices	awareness
	lasting	qualitative	and	open-		3	Visibility of
	around 40		ended				gestures in
	minutes		conversation			and out of
	over a 6-						the webcam
	week
	period

Overview of studies on affordances of the webcam

Study 1: Quantitative approach on experimental data

ISMAEL projet: http://nicolas.guichon.pagesperso-orange.fr/projets.html

Software

Camtasia: http://camtasia-for-mac.en.softonic.com ELAN: http://tla.mpi.nl/tools/tla-tools/elan/ Screen Video Recorder: http://www.dvdvideosoft.com/fr/products/dvd/Free-Screen-Video-Recorder.htm#.VHBk8Eve5g0 Skype: http://www.skype.com SPSS: http://www-01.ibm.com/software/fr/analytics/spss/


Keywords: short fiber suspensions, transport equation, Fokker-Planck equation, quadratic closure relation

N umerical modelling of non-Newtonian flows involves usually the coupling between equations of motion, which define an elliptical problem, and the fluid constitutive equation, which introduces an advection problem related to the fluid history. In short fiber suspensions (SFS) models, the extra-stress tensor depends on the fiber orientation whose evolution can be modeled from a transport problem. In all cases the flow kinematics and the fiber orientation are coupled: the kinematics of the flow governs the fiber orientation, and the presence and orientation of the fibers modify the flow kinematics. For example, in a contraction flow of a dilute suspension, large recirculating areas appear (Lipscomb et aI., 1988).

If one uses SFS flows in material forming processes, the final fiber orientation state depends on the process (induced anisotropy). In this case we need to compute the coupled model (kinematics-fiber orientation) in order to predict the final mechanical properties of the conformed pieces (which depend strongly on the fiber orientation). Of major industrial interest is process optimization to align the fibers in structural pieces in the main stress directions.

The mechanical model governing the SFS flow is given by the following equations [START_REF] Batchelor | Slender-body Theory for Particles of Arbitrary Crosssection in Stokes Flow[END_REF][START_REF] Batchelor | The Stress Generated in Non-Dilute Suspensions in Elongated Particles by Pure Straining Motion[END_REF][START_REF] Hand | A Theory of Anisotropic Fluids[END_REF]Hinch andLeal, 1975, 1976;[START_REF] Meslin | Viscosite en Cisaillement d'un Composite Fibres Courtes a l'Etat Fondu[END_REF].

Divg = Q
where Q is the stress tensor.

Secondly, the incompressibility condition:

Di~=Q where y represents the velocity field.

(1)

(2)

This paper focuses on an accurate evaluation of short fibers suspensions models coupling the flow kinematics with the fiber orientation evolution. In coupled models the flow kinematics is usually solved using the finite element method, where the fiber orientation is introduced in the constitutive equation through its value in some points (nodes or integration points). In this paper we will compare in a simple steady shear flow, the exact solutions of the extrastresses associated with the fibers' presence with the numerical simulations obtained using both the method of characteristics and the discontinuous Galerkin's method to solve the equation governing the generalized gradient evolution, in order to avoid the introduction of any closure relation. The error introduced if a quadratic closure relation is considered in the constitutive equation will be also quantified.

Ce travail porte sur I'analyse du couplage cinematique -orientation des fibres dans les rnodeles numeriques des suspensions diluees de fibres courtes. Dans les modeles couples, la cinematique est resolue "avec une technique d'elements finis, ou I'orientation des fibres est conslderee dans un certain nombre de points (Ies nceuds oules points utilises pour l'inteqration nurnerique), Dans ce travail, nous comparons les extra-contraintes dans un ecoulement de type cisaillement simple, obtenues nurneriquernent avec la methode des caracteristiques ainsi qu'avec une methode d'elements finisdiscontinus, avec la solution exacte connue pour ce type d'ecoulement Dans tous les cas, I'orientation a ete calculee sans I'introduction de relations de fermeture. Ce fait nous permet egalement de quantifier la validite d'une fermeture quadratique pour Ie tenseur d'orientations.

•Author to whom correspondence may be addressed. E-mail address:

Thirdly, the constitutive equation for a dilute suspension of high aspect-ratio particles is given by:

(3) From a physical point of view, we can consider that the eigenvalues of the second-order orientation tensor (g) represent the probability of finding the fiber in the direction of the corresponding eigenvectors.

And lastly, if we consider spheroidal fibers, we can describe the orientation evolution with the Jeffery equation:

where g is the vorticity tensor, k is a constant that depends on the fiber aspect ratio r (fiber length to fiber diameter ratio):

where p denotes the pressure, 1 the unit tensor, " the viscosity (which depends on the chosen model as explained in [START_REF] Meslin | Viscosite en Cisaillement d'un Composite Fibres Courtes a l'Etat Fondu[END_REF], g the strain rate tensor, N p a scalar parameter depending on both the fiber concentration and the fiber aspect ratio, ":" the double dyadic product, i.e. [~: glij = 0ijkPkl and ~the fourth-order orientation tensor defined by:

= de = gJ + k(Q-(Q: (p ® p))p) d t - ----- (10)
(4)

k=,l-l/,l+l

A solution of this equation is given by:

(11

)
where P is the unit vector aligned in the fiber axis direction, ® denotes the dyadic product (l.e, (.e ® .e)ij = P'pj)' and \jf(.e) is the orientation distribution function satisfying the normality condition:

(12)

(5)

where .eo is the initial orientation Po= .e (t =0) and the transport ~is the solution of the following transport equation:

with

~(t = 0) =J.
The evolution equation of the second-order orientation tensor results:

If \jf(p) = o(p -13), with 00the Dirac's distribution, all the orientation probabilitY is concentrated in the direction defined byp, and the corresponding orientation tensor will be tJ. =P® P® P-® p.

We can also define the second order orien1ation tensor as:

dE =k = (0+ kD)E dt = = =k (13)
It is easyto verify that if \jf(p) = o(pp), the fourth-order orientation tensor can be written as the tensorial product of the second order orientation tensor, i.e.: and then, the second order orientation tensor related to a planar isotropic orientation state is:

For general expressions of \jf(p) the previous relation is not exact, and in that case Equation (7) becomes a closure approximation known as quadratic closure relation. However, other closure relations are usually applied (Advani and Tucker, 1990).

In a planar case, the isotropic orientation state is defined by the uniform distribution function:

' P(p) =_1 - 21t (6) (7) (8) (14)
A similar equation can be derived for the evolution of the fourth-order orientation tensor, which in this case involves the sixth-order orientation tensor.

The solution of Equation ( 14) with a quadratic closure relation (~= ~® ~) can be written as: (15) where ~o is the initial second order orientation tensor Qo = Q(t = 0).

Remark 1: Slender body theories [START_REF] Batchelor | The Stress Generated in Non-Dilute Suspensions in Elongated Particles by Pure Straining Motion[END_REF]Dmh and Armstrong, 1984) assume that the fibers have an infinite aspect ratio, i.e. k = 1 and the suspension viscosity is equal to the solvent viscosity. In that case, to describe the increase of the shear viscosity it is necessary to introduce diffusion effects to avoid the full alignment of the fibers with the flow [START_REF] Folgar | Orientation Behaviour of Fibers in Concentrated Suspensions[END_REF]. The orientation equation becomes: The introduction of the diffusion term Dr (~-1/2) seems no longer necessary if we consider the spheroidal particles model because in that case the fact that k *' 1 induces a continuous rotation of the fibers which can describe the rise in shearviscosity [START_REF] Meslin | Viscosite en Cisaillement d'un Composite Fibres Courtes a l'Etat Fondu[END_REF].

I 0= -=- = 2 (9) dg = Gradva+ ~Gradv)T-2(0 :D) + Dr(O _1J dt -== - ~= =2 (
In the numerical simulation of dilute fiber suspensions there are three kinds of numerical models: model is the necessity of proceeding with a fully Eulerian description, whereas the fixed point coupled models can proceed with a Eulerian-Lagrangian mixed description, using, as previously described, the method of characteristics (the most accurate technique solving advection equations) in the solution of the orientation equation. which verifies exactly the quadratic closure relation ~= ~@ ~.

from which the expression for the second order orientation tensor Q results: 2. Coupled models using the Evans hypothesis solve the flow kinematics assuming that at each point of the fluid domain the fibers are aligned with the flow (local alignment hypothesis) [START_REF] Lipscomb | The Flow of Fiber Suspensions in Complex Geometries[END_REF][START_REF] Chiba | A Numerical Solution for the Flow of Dilute Fiber Suspensions Through an Axisymmetric Contraction[END_REF]. In this case, the flow model is given by Equations ( 1), ( 2) and ( 3), where the fiber orientation is given by:

1. Uncoupled models solve the kinematics without considering the presence of fibers. In this case, the equations of motion result in the Stokes's problem defined by Equations ( 1), ( 2) and (3) with N p =O. From the velocity field obtained by the solution of the Stokes's problem, we can compute the fiber orientation evolution solving any of the models described previously. [START_REF] Givler | Numerical Prediction of Fiber Orientation in Dilute Suspensions[END_REF] solved the Jeffery equation along the flow streamlines. To take into account different initial orientation distributions, a great number of fibers with different initial orientation are considered, and the averaging of their orientations at each point allows us to compute the different orientation tensors.

The first type of model does not allow prediction of the strong variations in the flow patterns that appear for example in contraction or expansion flows, even for dilute regimes [START_REF] Lipscomb | The Flow of Fiber Suspensions in Complex Geometries[END_REF]. The second type of model, extensively used, allows us to predict the existence of recirculating flow areas near the contraction corners [START_REF] Lipscomb | The Flow of Fiber Suspensions in Complex Geometries[END_REF][START_REF] Chiba | A Numerical Solution for the Flow of Dilute Fiber Suspensions Through an Axisymmetric Contraction[END_REF]. Moreover, as observed in several works (Towsend and Walters, 1993), and theoretically proven in [START_REF] Poitou | Numerical Simulation of the Steady Recirculating Flows of Fibers Suspensions[END_REF] and [START_REF] Chinesta | On the Steady Solution of Linear Advection Equations in Steady Recirculating Flows[END_REF], for fibers with infinite aspect ratio the local alignement of the fibers with the flow is the only solution in general recirculating flows. However, the local alignment cannot be applied either to divergent flows for which this solution is not stable, or to fibers with finite aspect ratio whose orientation changes continuously along their trajectory. As expected, the best modeling, widely used, consists of coupling flow kinematics and fiber orientation, as described for the third type of models. Different numerical strategies are used to solve the orientation equation: the method of characteristics, the SUPG or the discontinuous finite element methods or the discontinuous finite volumes method, using the second-or the fourth-order orientation tensors, as well asthe generalized gradient ~asvariables. One of the main difficulties in using Eulerian discretisation techniques to compute the evolution of the tensorial orientation fields is the low accuracy of standard finite element interpolations of these orientation tensors, as discussed in [START_REF] Chinesta | Numerical Prediction of the Fiber Orientation in Steady Flows[END_REF] and [START_REF] Chinesta | A Semi-Lagrangian Strategy to Predict the Fiber Orientation in Steady Flows of Reinforced Thermoplastics[END_REF].

On the other hand, the ellipticity of the equations of motion, when the orientation field is taken at the previous iteration, allows us to use a standard Galerkin formulation in finite elements. In general, this technique is used in most of the related works. Thus, in each element of the finite element mesh, the fiber orientation is considered only at some points (integration points). When the fibers are assumed of spheroidal shape and finite aspect ratio, it is well known that in a simple shear flow they cannot be fully aligned in the flow direction, rotating continuously. The higher the fiber aspect ratio the longer the fibers remain close to the flow direction, and faster is their rotation. We can expect that considering only the orientation state in some points we can predict an orientation state, which does not correspond with the physical reality. Thus, the probability of finding, at the integration points, the fibers aligned in the flow direction increases with the fibers' aspect ratio. In this case Np' ~:Q .. 0 and the numerical model cannot predict the rise in shtiar viscosity experimentally observed. An alternative way to avoid this over-alignment through modeling is to consider the slender body theory, where, in spite of the fact that the fibers are assumed to have an infinite aspect ratio, the diffusion term introduced in Equation ( 16) avoids the full alignment of the fibers with the flow.

This paper evaluates the accuracy of coupled models using the spheroidal particles theory. We will analyse a simple shear flow, for which we can derive theoretically the rise in shear viscosity. These results will be compared with the different numerical simulations, without using any closure relation, in order to derive a conclusion about their accuracy, (17)

(18) v@v Q==--= = 1I~12 v e=II~1
3. Coupled models take into account both the dependence of the kinematics on the fiber orientation and the orientation induced by the flow kinematics. Usually the coupled models are solved by means of a fixed point strategy. In this case, at each iteration the flow kinematics results from Equations( 1), ( 2) and (3) assuming the fiber orientation field at the previous iteration. From the kinematics just computed, the fiber orientation is updated solvlnq the advection equation governing the evolution of the second-or the fourth-order orientation tensors, but in these cases the consideration of a closure relation is required. These advection equations can be integrated by using any accurate numerical technique for hyperbolic equations: the method of characteristics, SUPG or discontinuous finite element techniques, discontinuous finite volumes, ... [START_REF] Rosenberg | Simulation of Non-Recirculating Flows of Dilute Fiber Suspensions[END_REF][START_REF] Ausias | Etude de l'Extrusion de Tubes en Polyrneres Thermoplastiques Charges de Fibres Courtes[END_REF][START_REF] Altan | Anisotropic Channel Flow of Fiber Suspensions[END_REF][START_REF] Souloumiac | Etude Rheoloqique, Modelisation et Simulation Nurnerique de l'Ecoulernent des Thermoplastiques Charges de Fibres Courtes[END_REF][START_REF] Azaiez | Investigation of the Abrupt Contraction Flow of Fiber Suspensions in Polymeric Fluids[END_REF][START_REF] Chiba | Numerical Solution of Fiber Suspensions Flow Through a Complex Channel[END_REF][START_REF] Poitou | Numerical Simulation of the Steady Recirculating Flows of Fibers Suspensions[END_REF]. Coupled models solving simultaneously the flow kinematics and the fiber orientation (fully coupled models) are rare in literature. The main difficulty of using fully coupled models is the different character of the model equations, which requires specific numerical techniques. Another particularity of this kind of To take into account the effects of the fiber orientation on the flow kinematics, an accurate description of the fibers' orientation state in the equations of motion is required. This paper focuses on this area. The aim of this work is not the computation of coupled solutions, because, as commented previously, these calculations do not introduce additional difficulties if both an accurate strategy for solving advection equations and an accurate description of the orientation field into the equations of motion are performed. For this reason, only simple shear flows are considered in order to evaluate the capabilities of different numerical strategies to introduce accurately the fiber orientation distribution into the motion equations. In spite of the simplicity of such flows, the conclusions can be generalized to more general flows, where an effective coupling between the orientation and motion equations must be carried out for simulating industrial applications.

In the same way, other effects such as the fiber-fiber or wall-fiber interactions, which can be important in semi-concentrated or concentrated flows regimes, are not considered in this work, because to introduce accurately these effects more sophisticated mechanical models are required. In a first approximation, the introduction of diffusion effects in the equation governing the evolution of the fiber orientation distribution could describe the fiber-fiber interaction in semi-concentrated flows. In this case, the fiber orientation field is smoother than the one obtained without considering diffusion effects, and consequently, it can be accurately introduced into the equations of motion to compute coupled solutions.

Finally, this work concerns only planar orientation distributions. However, the analysis proposed in this paper can be applied to 3D orientation distributions as well as to general 3D flows.

Orientation Solver

In this section we will consider the velocity field as a datum. As just mentioned, we will use a numerical strategy to compute the fiber orientation field without the introduction of any closure relation. For this purpose, we propose to solve the evolution equation related to the generalized gradient fk:

where fl is the unit outwards vector defined on the domain boundary an. Now, we must integrate the orientation equation along the characteristics(streamlines for steady flows) from the boundary condition fk ~=fk (~E r -.J =J until we return to the starting point ~. A fourth-order Runge-Kutta scheme with a control step has been used in the numerical integration, which leadsto very accurate solutions.

The main limitation of this strategy is the necessity of reconstructing each trajectory associated with each point where the orientation solution must be computed. In general, this procedure is highly time consuming for the CPU. A semi-Lagrangian strategy that combines an integration by characteristics into each element and a Lagrangian interpolation on the mesh skeleton, reducing drastically the CPU time, was proposed by the authors in [START_REF] Chinesta | A Semi-Lagrangian Strategy to Predict the Fiber Orientation in Steady Flows of Reinforced Thermoplastics[END_REF]. Another fully Eulerian strategy applies the discontinuous Galerkin method to solve Equation (19), as we will describe in the next section.

A First-order Discontinuous Finite Element Discretisation

The simplest discontinuous Galerkin formulation lies in taking a value of fk constant into each element of a finite element mesh. We will denote by g(t) the value of the generalized gradient fk in the element n e at time t. The initial condition is given by g(t

= 0) =1, VeE[l, ... ,N e ],
where N e is the number of elements in the mesh, i.e, n = u~~~ene. The integration of an advection equation requires also a boundary condition on the inflow boundary 1_, which in our case results in fk~e r -.J = J.

The first-order time discretization is given by: (22) Now, a discontinuous Galerkin formulation is used to compute the time derivative. The integral conservation form, associated with the advection Equation (19), results in each element n e , taking into account the flow incompressibility (DillY. =0):

A Lagrangian Discretisation: The Method of Characteristics

To solve this linear advection equation (Equation 19) we consider the method of characteristics, which allows us to compute with very good accuracy the solution at each point. If we look for the solution of fk at a point ~, we need to reconstruct the trajectory, integrating upstream the equation of the characteristics:

dE =k = (n+ kD)E dt = = =k dt dx . ,. =-; with ~(t =0) =~p (19) (20) (23) ve e [1, ..., N e ]
where an e is the element boundary and 11 is the unit-outwards vector defined on the element boundary. Asg is not defined on the element boundaries, we take on the inflow boundary an:, the existing value in the upstream element, whereas on the outflow boundary, an~, we take the value existing in the element n e . Moreover, taking into account that §1 is constant into each element, we can finally write:

until it reaches at the point ~the inflow boundary T_ of the fluid domain n: The main advantage of this explicit strategy is a significant reduction in the CPU time.

(

) (31 ) 32
To obtain a from 8 we use the Jefferyequation:

TIs = cr~2 =TI +2N p TlC1J 212 =TI + A'J1 ' Y ~1(!)e9
e a = 11~1(!)e911

where the symmetry relation 01212 = 01212 has been introduced, and A' J1 represents the rise in shear viscosity, which depends on

°1212'

Remark 2: When the fibers have a quasi-infinite aspect ratio, k = 1, the steady solution is given by the local alignment of the fiber with the flow, i.e. 'II fJt" a) =l>(A, 8 -0), from which it results 01212 = 0, and consequently A' J1 =O.

Remark 3: When the fibers have a finite aspect ratio and the orientation distribution at each point of the fluid domain is given by the Dirac's function '¥(A, 8) = l>~a -cpC4)], then 01212 = 0~2' and consequently, the quadratic closure relation is exact.

Computing the Shear Viscosity

Now, if we consider the constitutive equation, Equation (3), and the simple shear flow defined by 'iT = (W,O), the shear viscosity TIs' defined by the ratio between the shear stress and the shear rate y, results:

Numerical Examples

If we consider the simple shearflow defined by the velocity field ':iT = (W,O), with Y= 1 and the fibers with an aspect ratio such that k = 0.95, and we focus our analysis on the orientation solution along the streamline defined by y = 1, then the exact expression of ~, solution of Equation (19), results: Now, if we consider 8 +de, Equation (31) gives us the angle a*, from which we can obtain do. = 10.* -0.1. Thus, we can compute for each angle a, '¥~, 8)d8 = 'IIo~' a) do. , which allows us to obtain numerically the integrals defining both orientation tensors, Equations ( 28) and ( 29). With the value of &e computed at different points with the method of characteristics, or in each element using the justdescribed first-order discontinuous Galerkin method, the expression of the orientation tensors can be computed applying the numerical procedure described in this section.

With the orientation distribution on the inflow boundary 'II oC4e r _, e) done, the expression of f! and f! can be obtained from their definition expressions:

= from the upstream element ~and q+ is the flux leaving the element {le. Taking into account the incompressibility, we obtain:

We will be able to evaluate the previous expressions as soon as the fiber orientation distribution at any point ~, '¥~, p), is computed from the initial one, '11 0 (known on the inflow boundary), applying the following balance equation: where a defines the fiber orientation at point ~, which, when the fiber reaches point ~(both ~and ~are on the same trajectory), its orientation is defined by the angle 8. Moreover, all the fibers whose initial orientation at point ~is in the interval [a, a + 00.] will be in the oriented interval [8, 8 +de] when they reach the position

~• (33) ( Mlx] cos 2 ~1-k 2 . ~1-k 2 x
---Sin ....:.-.:.------

1-k 2

Analysis of the Solution Along a Streamline for Fibers Aligned in the Flow Direction on the Inflow Boundary In this case, at each point x, the fibers will be perfectly aligned in the direction given by the Jefferyequation, i.e.:

From this equation we can come to the conclusion about the periodicity in the fibers' orientation. The spatial fiber orientation period X is in our example X = 41t/~= 40.24. We will considerthe inflow boundary located at x = 0, where in a first time, the fibers are aligned with the flow. Later, an isotropic fiber orientation on the inflow boundary will be considered.

(

) (29) 30
whose components are e(x)T =(P1(x), P2(x». So, 01212 at point

x results:

element. For general flows, an accurate expression of ~at a point requires the use of the method of characteristics, which, aswe have previously indicated, is very expensive when considering CPU time. Moreover, we can expect that, by computing the average value of ~in each element, using for example a discontinuous finite element method in the discretisation of Equation ( 19), an average of °1212 will be obtained implicitly. In this situation we can define:

(35)

The average value in the spatial period X can be considered, using the ergodic principle, as the exact steady local solution: where 01212 is the value found in the element n e , which is assumed to be constant into the element. The numerical application gives us the value aftfl = 0.058 very close again to the exact solution.

_ 1 N,
The main advantage of using a discontinuous finite element method to solve the equation governing the evolution of ~, Equation ( 19), is that it does not require the resolution of any linear system, and moreover, it reduces drastically the CPUtime with respect to an integration by characteristics.

where 0l c 1q1denotes the value of 01212 at the center of gravity of the element of.

When the number of elements increases, the numerical solution aftf.2 converges to the exact one, Le. aftf.2 ~a,e~rtf = 0.059 (in our numerical example) when N e ~00.

Analysis of the Solution in the Whole Domain Using an Eulerian Discretisation

In this section we consider the 2D flow model defined by the following kinematics:

i=N -num 1 ",' cdg,i 0 073 0]212 =-£.J °1212 = . Ne i=1 (37) ~x, y) =r7}(x, y) E [0, Ll x [0, 1] (41)
which is far from the exact value. Until now the generalized gradient ~has been obtained exactly from Equation (33) at the center of gravity of each which is very close to the exact value. Now, we can evaluate °12 at the center of gravity of each element 0f¥,i as well as the mean value of 01212 assuming a quadratic closure relation a,ntn qcr:

Analysis of the Solution Along a Streamline for an Isotropic Fiber Orientation Distribution on the Inflow Boundary

In this case, the variation of °1212 is smoother than the previous one, because even if the fibers rotate very quickly, not all the fibers rotate at the same point, due to their different initial alignments. Thus, for the mesh considered in the previous example, we obtain: If we taker =1, then the period X of ~depends on the y coordinate in the following form:

(42)

X(y)=Ñ

ow, if we define a uniform mesh in the fluid domain [0, X(y = 1)] x [0,1], the characteristic element size near the upper wall, y = 1, will be lower than the fiber orientation period X(y), whereas in the area close to the lower boundary, y =0, the characteristic element size is greater than the fiber orientation period.

For this scenario, very close to actual general flow situations, we would like to compare results obtained by an accurate computation of tensor ~at the center of gravity of the mesh elements (by means of the method of characteristics, for example) with the one obtained solving the evolution equation of ~, Equation (19), by means of the discontinuous Galerkin method (which assumes a value of ~constant into each element, as described previously).

Using a mesh of 940 elements, N e = 940, and imposing the local alignment of the fibers with the flow on the inflow boundary, i.e. eti =(1,0), we obtain the following results:

(38)

(39) N, -num _ 1 '" cdg,i -006 0]212 --£.J °1212 -• N e i=1 N, 2 -num,qcr _ _1_"'(fL C d9 ,i) -0 02 0,212 - £.J -12 -. N e i=l
Now, if we assume on the inflow boundary an isotropic fiber orientation, i.e, '¥~E r -> =1/21t we obtain:

N.

-num 1 ~cdg,; 0 060 Oj212 =-""'" a, 212 = .

Nomenclature

From these results we can conclude that both techniques predict accurately the rise in shear viscosity. The solution accuracy is not significantly degraded when a considerable reduction in the number of degrees of freedom (in the order of 10) is performed. Even if the result obtained with the integration by characteristics seems slightly better, this technique requires much more CPU time than the second numerical technique. In order to improve the accuracy of the finite element technique, a local remeshing is needed. As in this technique we use a discontinuous approximation of the unknown field ~, we can proceed to non-structured local remeshings.

Conclusions

In this paper we have shown that in spite of the fast rotation of the fibers with finite aspect ratio in shear flows, the extrastresses are correctly evaluated in considering the fiber orientation only in a reduced number of points. If we use uniform meshes, the characteristic element length will be, in general, greater or smaller than the fiber rotation period. Even in that case, the solution accuracy is excellent, mainly when an isotropic fiber distribution is assumed on the inflow boundary. We have also noticed that very accurate solutions may be obtained by using the discontinuous Galerkin method, and they open new perspectives in the use of fully Eulerian coupled techniques. The highest deviations were found when the extrastress tensor was computed using a quadratic approximation for the fourth-order orientation tensor.

Other Symbols

I8i dyadic product. Thus, the components of u I8iyare UjV j dyadic product, twice contracted

  Laboratoire de Mkanique des Systemes et des Procmes) CNRS-ENSAM-ESEM. 151 Boulevard de l'Hopital. F-75013 Paris. France and LRTMM-CNAM Paris. 292 rue Saint-Martin. F-75141 Paris cedex 03. France 2 LMT Cachan. CNRS -ENS Cachan. -Universite Paris 6. 61 Avenue du President Wilson. F-94235 Cacluui cedex, France Firstly, the momentum balance equations, without inertia and mass terms:

  16) nos The Canadian Journal of Chemical Engineering, Volume 80, December 2002

=k

  Inel= ~qe; E e; _q+E e Vee[l, ... ,N e ] {~E an== I, ~~) '1~) < 0} 1110 (21 ) where N~is the number of upstream neighboring elements which are identified by the superscript ~gei, is the flux coming The Canadian Journal of Chemical Engineering, Volume 80, December 2002 and

  =~+ V'ee[l, ... ,N e l i If we denote by Pe the unit vector with components pI = (coss, sine), we can then write: -Computin9 the Fiber Orientation Tensors and the Rise in Shear ViscosityComputing the Orientation Tensors from the Generalized Gradient

  ~generalized gradient ~1 is the usual gradient of deformations) defines the fiber axis direction g stress tensor, with components a ij , (Pa) 'P fiber orientation distribution n fluid domain whose volume is denoted by Inl (m 3 ) ~vorticity tensor, with components n ij , (S-I)

  X) including 20 elements, N e = 20, and we compute the values of 01212 only at the center of gravity of each element, the mean value a 1 ntf.2 results:

	fLnum,d-fem -1212	'" n.. e e i=l = N £.J -1212	(40)
	(36)
	that, in our case, results in ale~r~t = 0.059.
	Now, if we consider a uniform one-dimensional mesh on the
	interval [0,

The Canadian Journal of Chemical Engineering, Volume 80, December 2002

Acknowledgements

We want to thank Professor Claude Wolff, director of the research team at the Conservatoire National des Arts et Metiers (CNAM) of Paris, where a part of this work has been done.