Antoine Feuillet
email: antoine.feuillet@universite-paris-saclay.fr

Amr Alem

Christophe Durand

Mickael Terrien

Nicolas Scelles

What can be learnt from failures in the sports media business? A case study of the Mediapro crash in football

Keywords:

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

Introduction

Sport constitutes one of the major products of the media business. Both industries have influenced each other, so that some business models in sports and the media can be considered interdependent, a phenomenon far from new [START_REF] Andreff | Le sport et la télévision: relations économiques [Sport and television: economic relations[END_REF]. Moreover, both markets are similarly facing a lot of uncertainties [START_REF] Wolfe | Sport and organizational studies: Exploring synergy[END_REF]Murschetz et al. 2020). Television is facing the challenge of the digital revolution and the associated changes in media consumption [START_REF] Pedersen | Routledge handbook of sport communication[END_REF][START_REF] Oliver | Strategic transformations in the media[END_REF]. Understanding the transformation of sports media demand is a critical challenge in implementing a successful business model [START_REF] Karg | Channel preferences among sport consumers: Profiling media-dominant consumers[END_REF]. Sports premium rights are crucial products of media business strategy because of their attractiveness to consumers. Several pay-TV channels have massively relied on sports premium rights to build their business models, such as Sky in the UK, Canal+ in France, and Mediaset in Italy. Since then, the level of turnover of media companies [START_REF] Scelles | What drives sports TV rights? A comparative analysis of their evolution in English and French men's football first divisions, 1980-2020[END_REF] and the competition to acquire these rights have massively increased. Such competition has brought particular attention, questioning a potential winner's curse in the football broadcasting rights market [START_REF] Feuillet | A winner's curse in the bidding process for broadcasting rights in football? The cases of the French and UK markets[END_REF]. For Thaler (1988, p. 192), there are two ways to be considered a cursed winner: "(1) the winning bid exceeds the value of the tract, so the firm loses money; or (2) the value of the tract is less than the expert's estimate, so the winning firm is disappointed."

Securing premium TV rights is costly for companies, which raises concerns about the rationale of the business model behind these massive investments. A business model can be defined as an "articulation between different business model components or 'building blocks' to produce a proposition that can generate value for consumers and thus for the organisation" (Demil & Lecocq, 2010, p. 227). However, there is a lack of research focusing on the business model approach to understand failures in the sports media market, despite several important cases.

In this study, two research questions are addressed: (RQ1) How does the auction system used to commercialise media rights shape a potential negative externality called the winner's curse? (RQ2) How can the business model approach complement the economic approach to media failure? The present study aims to contribute to the literature on the winner's curse by focusing on the winner's business model and not only the auction process. To do so, we compare Mediapro's case with previous occurrences of sports media failure to identify potential (dis)similarities and discuss them.

To respond to these questions, the specific context of the French broadcasting rights market for men's football seems of particular interest. The default payment of Mediapro in France offers an emblematic case to understand the risks associated with investments in sports. This is because it led to a shortfall of earnings of more than €2b for French football clubs.

Furthermore, public access to auction results and direct stakeholder hearings is uncommon.

Besides, despite several cases of bankruptcy in Europe, almost no studies have made an extensive analysis of such events.

The rest of the paper is structured as follows: The theoretical framework presents the historical bankruptcy cases in football media, before shortly reviewing the literature on the winner's curse and media in sport, introducing the concept of business model and linking it to the media, and ending with the evolving demand for sports media. Then, the methodology is described with the case study approach and the data collection. The results are provided, relying first on an economic framework of seven variables leading to the winner's curse. Secondly, we demonstrate the inconsistency in the business model of Mediapro. The discussion reminds major findings, compares them with previous literature to highlight our contribution, provides managerial implications and the limitations of the study. The last section concludes.

Theoretical Framework

Cases of business failure in the sports media rights landscape

In the literature, different examples of the winner's curse in sports media have been briefly analysed. ITV Digital and Setanta in the UK, KirchPayTV media in Germany, Sport7 in the Netherlands, Stream ant Telepiù in Italy and Alpha Digital Synthesis in Greece [START_REF] Smith | The next big match: Convergence, competition and sports media rights[END_REF][START_REF] Evens | Regulating the digital sport broadcasting market: New players, old problems[END_REF]. Other examples are added with Arena Television in Germany and Quiero TV in Spain.

UK market

ITV Digital signed a 3-year contract worth a total of £315 million to broadcast the Football League seasons 2001 to 2004 (lower divisions of English football). The previous contract owned by Sky was worth only £30m per season, which shows major inflation [START_REF] Butler | Has competition in the market for subscription sports broadcasting benefited consumers? The case of the English premier league[END_REF]. ITV Digital failed to meet its breakeven point of 1.7m subscribers, with only 1.2m after one year [START_REF] Emery | Fighting for survival? The financial management of football clubs outside the 'top flight'in England[END_REF]. In addition, technical issues were reported (Iosifidids, 2006).

The Football League did not obtain financial guarantees from ITV Digital's shareholders [START_REF] Szymanski | Insolvency in English professional football: Irrational exuberance or negative shocks[END_REF]. ITV Digital collapsed in March 2002, leaving many Football League clubs in financial despair [START_REF] Scelles | What drives sports TV rights? A comparative analysis of their evolution in English and French men's football first divisions, 1980-2020[END_REF].

Setanta entered the English Premier League market in 2007 thanks to new EU rules relating to competition regulation that led to the unbundling of rights packages [START_REF] Butler | Has competition in the market for subscription sports broadcasting benefited consumers? The case of the English premier league[END_REF] and a reduced number of years of exclusivity (Smith et al., 2018). Thus, Setanta was the first competitor for Sky in the UK since the creation of the Premier League. It obtained 42 of the 138 matches, but the premium packages remained in the hands of Sky [START_REF] Dobson | The economics of football[END_REF]. With low subscriber recruitment, high fixed costs, and an abyssal debt of €250m (Independent, 2013), Setanta UK went into administration two years into its contract with the English Premier League. The economic recession may also have played a role in the failure of Setanta UK.

German market

KirchPayTV went into administration in 2002 just after the bankruptcy of its main subsidiary KirchMedia. Kirch, a giant media group, disappeared with massive debts, notably due to unprofitable investments in football (Solberg, 2006b). Indeed, Kirch had invested €2.3b jointly with ISL to get the rights for the men's football World Cups 2002 and 2006, a six-times increase compared to the price of the three previous World Cups [START_REF] Horne | the world cup and television football[END_REF]. Kirch overestimated its capacity to resell the rights to national free-to-air broadcasters [START_REF] Rowe | Fulfilling the'cultural Mission' Popular Genre and Public Remit[END_REF].

Arena Television tried to challenge the pay-TV channel Premiere by offering € 250 million per season to air Bundesliga 1 and 2 from 2006 to 2009 [START_REF] Dietl | Pay-TV versus Free-TV: A model of sports broadcasting rights sales[END_REF]. After one year, Arena Television sublicensed its rights to its competitor Premiere for the next two seasons.

From the end of 2006 to the first quarter of 2007, Arena Television made massive losses (Manager Magazin, 2007). They were unable to attract a sufficient number of subscribers despite attractive prices. Interestingly, Arena Television was not the highest bidder. Indeed, Premiere offered €280m but lost the auction due to qualitative criteria [START_REF] Dietl | Pay-TV versus Free-TV: A model of sports broadcasting rights sales[END_REF]. This may not be a case of a winner's curse according to the auction outcome but could be explained by the structure of the German broadcasting market in which public free-to-air channels are important players.

Spanish market

QuieroTV started to operate in 2000 and aimed to compete with pay-TV channels by investing in new services such as interactive offers between customers [START_REF] Candel | DTV in Spain digital television in Europe[END_REF]. Technical issues, high subscription costs, underestimation of investments, and a lack of subscribers (Mundoplus, 2022) leads QuieroTV to bankruptcy in 2002. The abrupt arrival of QuieroTV in a competitive market with high investments and the lack of support from their shareholders are likely explanations for such failure. QuieroTV's competitors in the digital satellite market, Canal Satellite and ViaDigital, decided to merge their platforms to reduce the investment costs associated with their investments just after QuieroTV's departure [START_REF] Iosifidis | First the UK, now Greece: "NOVA", the first DTV package[END_REF].

Italian market

Telepiù, created in 1990, was the first pay-TV group in Italy. They challenged the public channel RAI in 1994 to broadcast Italian football before StreamTV entered the Italian market.

StreamTV was launched in 1996 and started to compete with Telepiù in 1999 for Italian football, a battle that shaped the broadcasting system in Italy [START_REF] Bellavita | The digital satellite broadcasting system in Italy: Between mix and hybridism[END_REF]. Due to technical issues of interoperability between set-top boxes and a lack of cooperation between rivals, Italian football fans suffered from the competition in place. As Italian clubs sold their rights individually in these periods [START_REF] Poli | The revolution in the televised soccer market[END_REF], Italian consumers were forced to possess two set-top boxes of Telepiù and Stream, if they wanted to watch all Italian football content [START_REF] Gardini | Italy: Slow penetration, high potential?[END_REF]. It resulted in $300m and $200m losses for Telepiù and StreamTV respectively in 2002 (Solberg, 2006a). Financial difficulties led Vivendi (owner of Canal+) to sell Telepiù, resulting in the creation of Sky Italia in 2003, a merger of Telepiù and StreamTV [START_REF] Iosifidis | Digital switchover in Europe[END_REF].

Dutch market

Sport7 started to broadcast football matches of the Dutch Eredivisie in 1996 on the initiative of important shareholders such as Philips and ING associated with the Dutch football federation (KNVB). Dutch football was traditionally broadcast on the Dutch Public Broadcasting System (PBS) but the KNVB decided to discontinue the bidding process to create a venture owning Sport7 [START_REF] Sminia | The failure of the Sport7 TV-channel: Controversies in a business network[END_REF]. Sport7 paid a total of fl1.04b for seven years compared to fl900m from competitors. The legal opposition of PBS and the Feyenoord club against Sport7 put pressure on the venture. Moreover, the new pay-TV channel was dependent on the cable network. The business model of the cable operators was to charge pay-TV to give access to their network, while the venture wanted to have free access to the cable system and let consumers pay for additional content [START_REF] Sminia | The failure of the Sport7 TV-channel: Controversies in a business network[END_REF]. These issues put Sport7 in financial trouble, with fl100m losses in just four months, leading to the closure of the channel in December 1996 (Solberg, 2006b).

Greek market

Alpha Digital Synthesis started to compete at the end of 2001 with Nova, which was already running an unprofitable business since 1999 [START_REF] Iosifidis | Digital switchover in Europe[END_REF]. Alpha Digital Synthesis offered €72m to broadcast 10 of the 16 Greek championship clubs, while Nova had paid €44m to broadcast all clubs in the previous period. Alpha Digital Synthesis did not reach its subscription objectives to be financially viable and quickly closed its channel in 2002. According to Iosidifis ( 2006), the low level of subscriptions, the problem of interoperability between Nova and Alpha Digital decoders, and the level of capital required for pay-TV channel led to its failure.

The winner's curse and media in sport

European sports leagues use auction mechanisms to allocate sports TV rights (Solberg, 2006a;[START_REF] Andreff | The winner's curse in sports economics[END_REF]. Therefore, the winner's curse theory is relevant for analysing this market. Seven indices of the winner's curse have been identified in the sports broadcasting market [START_REF] Andreff | The winner's curse in sports economics[END_REF][START_REF] Feuillet | A winner's curse in the bidding process for broadcasting rights in football? The cases of the French and UK markets[END_REF]: a very swift increase in rights fees; financial loss; post-bid bankruptcy; too many unknown and uncertain details about the date, the place, and the participants; outbidding newcomers increasing aggressive competition on the demand side; television rights re-packaging; disappointment. Suffering from the winner's curse due to investment in sports may impact other media such as investments in movies and TV shows. If the winner's curse plays an important role in the failure of a sports media company, the business model may also have an impact as the cases presented above indicates.

Business model and media

The components of the business model can be approached by the RCOV model composed of the interconnection of resources and competencies (RC), organisation structure (O), and the value proposition (V) [START_REF] Demil | Standard and Poor's[END_REF]. These elements determine the structure of an organisation's revenues and costs. The difference between revenues and costs constitutes the margin that determines the (un)sustainability of the business model.

The use of the business model literature helps to understand the importance of the business ecosystem [START_REF] Moore | Dynamic capabilities in media management research: A literature review[END_REF]. [START_REF] Kostovska | Going beyond the hype: Conceptualising "media ecosystem" for media management research[END_REF] identify three levels of the media ecosystem: the first level is the "ecosystem orchestrators" (direct suppliers and distributors), the second is composed of the immediate environment of the firm and the third consists of actors at distance from the firm. In this regard, the relationship with stakeholders is decisive in building a sustainable business [START_REF] Demil | Business model thinking", business ecosystems and platforms: The new perspective on the environment of the organization[END_REF] and the lack of cooperation between stakeholders can provoke business failure [START_REF] Ozcan | The market that never was: Turf wars and failed alliances in mobile payments[END_REF]. The winner's curse theory and the business model approach can be complementary, with a microanalysis about the consequences of the auction outcome and a mesoanalysis through the business model framework [START_REF] Warnier | Le business model: l'oublié de la stratégie [The business model: the forgotten strategy[END_REF]. In particular, this complementary can be used to study the media which constitutes a rich business research field as the industry has known fast-moving transformation in their business model [START_REF] Fagerjord | Mapping the core actors and flows in streaming video services: What Netflix can tell us about these new media networks[END_REF][START_REF] Mcphillips | Media convergence and the evolving media business model: An overview and strategic opportunities[END_REF]. These transformations raise many issues regarding public policy adjustments to innovation [START_REF] Biggam | The challenges for public policy of adjusting to a multi-platform environment[END_REF] or the fit between consumer needs and existing value propositions.

Evolving demand for sports media

Since the rise of the digital platform in the 2000s [START_REF] Maes | Le business des droits TV du foot[END_REF][START_REF] Scelles | What drives sports TV rights? A comparative analysis of their evolution in English and French men's football first divisions, 1980-2020[END_REF], the consumption of media has shifted towards the empowerment of consumers [START_REF] Mcphillips | Media convergence and the evolving media business model: An overview and strategic opportunities[END_REF]. This has led to more volatility in consumption and less willingness to pay for expansive and rigid offers. As fanatic sports fans are mostly middle-aged men [START_REF] Oh | Analysis of the sports broadcasting market in the television broadcasting industry[END_REF], the switch of platforms is less likely to occur for this segment of consumers due to more inertia and consumer habits [START_REF] Evens | Value networks and changing business models for the digital television industry[END_REF]. Compared to other industries, there is a high level of uncertainty about the success of a media product, notably due to the high level of supply from which the consumer has to choose [START_REF] Picard | Unique characteristics and business dynamics of media products[END_REF]. The large diffusion of content through legal or illegal platforms has also disrupted the media right-of-way on content. The heterogeneity of sports consumers [START_REF] Karg | Channel preferences among sport consumers: Profiling media-dominant consumers[END_REF] between early and late adopters [START_REF] Rogers | Diffusion of innovations[END_REF], for example, has increased the complexity of implementing innovative formats coupled with the fragmentation of devices to consume media [START_REF] Mcphillips | Media convergence and the evolving media business model: An overview and strategic opportunities[END_REF].

Material and Methods

The study focuses on the case of Mediapro in France from 2018 to February 2021; and the closure of the channel 'Telefoot' launched by Mediapro.

Methodology

Overview A three-step analysis is applied, starting with (1) a description of the specific context that leads Mediapro to enter the French market for football media rights, (2) an in-depth analysis of the auction results and consequences by evaluating the indices of the winner's curse, and (3) a breakdown of the Mediapro business model based on the RCOV model to understand its inconsistency.

Case study approach. The purpose of this case study [START_REF] Eisenhardt | Building theories from case study research[END_REF][START_REF] Yin | ). Sage. Annex 1: Framework of the case study validity[END_REF] is to analyse Mediapro's failures in the French broadcasting rights market from its arrival in May 2018 to the closing of its channel in February 2021. Our level of analysis is the firm with a focus on its strategic behavior such as decision-making. The case study approach can be helpful to connect theoretical concepts and tell, in the words of Dyer and Walkins (1991, p. 618), "good stories that have theoretical import". The framework of the case study (Annex 1) and the research procedure (Annex 2) are provided in the annex.

Contextualisation. Ambrosini and Bowman (2010, p. 37) state that in a firm resources study, "their context and how they were created or renewed in practice requires fine-grained investigations". [START_REF] Gibbert | What passes as a rigorous case study?[END_REF]Gibbert et al. ( , p. 1467) ) note that 'case studies seek to study phenomena in their contexts'. The context specific to the case of Mediapro is provided in the results section.

The first part of the results is dedicated to describing the entry of Mediapro into the sports media landscape from 2006 in Spain until they attempted to reach the Italian market in 2018.

Data collection. Secondary data was collected to prove how Mediapro suffered from the curse and why its business model led to its premature departure from France. Various data were used to triangulate available information: official reports, credit rating agencies, national/economic press, media press, and sports press. Moreover, Assemblée Nationale (French National Assembly) hearings (2021a, 2021b, 2021c, 2021d, 2021e) help to understand the point of view of the main stakeholders involved. This documentation was also used to contextualise the case study, supported by research articles. A full list of the documentation used is provided in Annex 3.

Results

This section provides an in-depth case study of Mediapro's arrival and failure in the French football TV rights landscape. The results are presented as follows: contextualisation, economic diagnosis, and strategic diagnosis.

Context of the case study

Spanish market. Mediapro's main entry into the sports broadcasting rights market dates back to 2006 in Spain. At that time, Sogecable (Digital+) was in a monopoly as a satellite pay platform [START_REF] De Moragas | Football and media in Europe: A new sport paradigm for the global era[END_REF]. Mediapro convinced some LaLiga clubs to sign with it (individual clubs' sales of broadcasting rights back then), including FC Barcelona and Real Madrid [START_REF] De Moragas | Football and media in Europe: A new sport paradigm for the global era[END_REF]. Mediapro relied on an aggressive strategy to compete against Sogecable, with more and more clubs signing with Mediapro. This led to jurisdictional conflicts between both entities. Eventually, Mediapro broadcasted its premium free-to-air rights on its channel in 2007-2008. Mediapro created its pay-TV platform (GolTV) with the authorization of the Spanish government but the Mediapro's TV rights subsidiary was reported bankrupt in 2010, notably due to the court decision that sentenced Mediapro to pay important damages mostly to Sogecable [START_REF] Maes | La ruine du foot français [The ruin of French soccer[END_REF]. Italian market. Mediapro acquired the Serie A TV rights in 2018. SkyItalia immediately disputed the result, arguing that Mediapro was not an 'independent intermediary' [START_REF] Sportpromedia | From MediaPro to DAZN: Serie A's remarkable rights battle that saw a billion-dollar deal disappear[END_REF], as Mediapro planned to directly resell its rights and broadcast the league on its channel [START_REF] Espn | Commission closes its probe of Audiovisual Sport after Sogecable/Vía Digital merger[END_REF]. Consequently, the court of Milan rejected Mediapro's deal in 2018, for breach of antitrust rules. The court argued that Mediapro was in charge of 'editorial responsibility' and abused a dominant position that could harm consumers (ANSA, 2018). Furthermore, Mediapro failed to provide the one billion euros in bank guarantees requested by Serie A, leading to the official decision of the clubs to cancel the deal (Capital, 2018). Mediapro took a chance on the next period of Serie A rights (2022-2024) by offering a reported €1.3b per season (Forbes, 2019), before being once again rejected in favour of DAZN and SkyItalia, an offer which reached €927.5m.

French market. At the auction organised by the French Professional League (LFP) for the 2020-2024 seasons, LFP's objective was to reach the one-billion threshold. LFP attributed the Ligue 1 packages to Mediapro, BeIN, and Free for €1.162b. Competition between broadcasters was perceived to be at its peak notably because of the fragmentation of the offer.

Winner's curse applied to Mediapro

Bidding details. The auction was particularly sophisticated so uncertainty and pressure were at their maximum. Peculiarities include the sequential process, with lots being attributed one after the other with 15 minutes to make an offer. Moreover, bidders were not able to communicate and not aware of the winner for each lot. Also, for the first time in France, the winners of each lot were able to sublicence their rights. The first lot awarded was lot 1 (Table 1), which is the most critical. Not knowing who had won this lot, Canal+ decided to 'scuttle' the tender so that it would not reach the overall reserve price of €800m for lots 1, 2, and 3 by offering bids of €10m for lots 2 and 3 (Assemblée Nationale, 2021e). The micro-level of analysis confirms the winner's curse endured by Mediapro on its determinants (swift increase in rights fees, new entrant in the French market, television rights re-packaging, and uncertain details linked to the value of the product) and consequences (financial losses, post-bid bankruptcy, and disappointment).

Mediapro's business model inconsistency in France

The Mediapro business model is detailed below using each component of the RCOV model.

Structures of revenues and costs are also detailed to demonstrate their level of disequilibrium.

Resources and competencies. Mediapro's core business came from its marketing agency activities (e.g., LaLiga international TV rights) and production of content (e.g., TV shows, movies). Mediapro externalised its market research to a French consulting firm. costs and uncertain revenues (Solberg, 2006b). In this regard, the support of Mediapro's main shareholder was uncertain.

Organisational structure. On the supply side, Mediapro was keen to resale at least a substantial part of its rights to competitors. Nobody was interested in acquiring them at the prices asked by Mediapro. This is because major telecommunications players were disappointed by their previous overinvestments in football broadcast rights (Orange, RMC Sport). As a result, the Structure of revenues. Mediapro aimed to attract 1.5m subscribers in the first year with an objective of 5m in 2023, and a break-even point between 3-3.5m (Assemblée Nationale, 2021d).

Telefoot had only 480,000 subscribers three months after its debut. Mediapro wanted 'minimum guarantees' from distributors above €100m (Capital, 2020) which were refused (Assemblée Nationale, 2021d).

Structure of costs.

Ligue 1 rights for Mediapro were at €798m per year, with the following to be added: €34m per year for Ligue 2 rights; €175m paid to RMC Sport for co-broadcasting European Cups for 2020; €5m paid per year to TF1 Group to use 'Telefoot' brand (Capital, 2020); Production, marketing costs, and salaries were estimated between €50m and €150m.

Thus, Mediapro's total costs for the sole 2020-2021 season were projected to be at least €1.1062b (LesEchos, 2020).

To sum up, Mediapro tried to enter a market that was not its core activity by investing a total amount that was way above its annual turnover. The relationships with external stakeholders were a massive issue, which undermined the ability to quickly reach a sufficient number of potential consumers. The value proposition was not sufficiently idiosyncratic to have a competitive advantage. The comparison between revenues and costs highlights the reasons leading to the prompt meltdown of Mediapro.

Discussion

Major findings

This paper aimed to understand the reasons leading to the failure of a business model in the media. The recent case of Mediapro in France was developed, adding to historical cases of bankruptcy in the industry. The findings confirm that Mediapro suffered from all seven indices of the winner's curse identified in previous research. The details of the auction confirmed that the winner's curse is more likely to occur for the second mover [START_REF] Feuillet | A winner's curse in the bidding process for broadcasting rights in football? The cases of the French and UK markets[END_REF]. Also, the study of the Mediapro business model shed light on its major inconsistencies leading rapidly to financial difficulties. Although Mediapro's core business is not to run a channel, its journalistic quality was not questioned. Furthermore, the overall package offered with the main national football rights and the Champions League associated with Netflix was certainly attractive.

Nonetheless, it lacked idiosyncrasy and the price that could have triggered a change in consumer habits. Mediapro's strategy was certainly to at least a large part of its rights, consistent with its historical core business as a marketing agency. This was ultimately not a relevant strategy in this specific market with a dominant actor Canal+ and market trends

showing the difficulties of such activities (MP&Silva liquidation in 2018). This may echo what happened in the music industry where the inflexibility of mental models [START_REF] Wikström | The adaptive behavior of music firms: A music industry feedback model[END_REF] led some actors to be unable to adapt their business models to the transformation of their environment shifting towards subscription-based streaming [START_REF] Small | Reshaping the music distribution model: An Itunes Opportunity[END_REF]. Mediapro may have lacked dynamic capabilities [START_REF] Rosenbloom | Leadership, capabilities, and technological change: The transformation of NCR in the electronic era[END_REF][START_REF] Teece | Explicating dynamic capabilities: The nature and microfoundations of (sustainable) enterprise performance[END_REF][START_REF] Maijanen | Managing exploration and exploitation in a media organisation-a capability-based approach to ambidexterity[END_REF][START_REF] Tripsas | Capabilities, cognition, and inertia: Evidence from digital imaging. The SMS Blackwell Handbook of Organizational Capabilities[END_REF] while also having to consider the fast-moving transformation in the media industry. The results confirm that the winner's curse is not only the consequence of irrational behaviour [START_REF] Nelson | Evolutionary theorizing in economics[END_REF] during the auction but also the outcome of an inconsistent business model leading to a rapid downfall.

The findings extend the literature on the winner's curse, which is more focused on the economic consequences of the auction than the management of the business model. Ultimately, Mediapro may have applied the greater fool theory [START_REF] Maes | Le business des droits TV du foot[END_REF] in its failed attempt to resell at higher prices the Ligue 1 rights. Overall, this case shows the importance of being a platform in the media business strategy [START_REF] Evens | Value networks and changing business models for the digital television industry[END_REF]. It might explain why the relationship between Canal+ and Mediapro was decisive in building a sustainable business model. This can illustrate the 'media ecosystem' in which Mediapro failed to satisfy stakeholders at the first level of the media ecosystem proposed by [START_REF] Kostovska | Going beyond the hype: Conceptualising "media ecosystem" for media management research[END_REF]. The concept of coopetition may be relevant to further study the importance of networks in a crowded market (Daidj & Jung, 2015;[START_REF] Feuillet | Valuation and negotiation of sport broadcast rights[END_REF] with fierce competition to create successful businesses [START_REF] Küng | The impact of the Internet on media organisation strategies and structures[END_REF].

Comparison with previous cases and contribution

The comparison between historical cases of bankruptcy in football media and the Mediapro case strikes several similarities between media companies that have failed to build a sustainable business model. This helps to suggest several additional endogenous and exogenous indices of the winner's curse that could enrich further analysis.

Regarding endogenous indices, there have been several cases of companies using a high level of borrowing combined with high debts such as Mediapro when investing in football. The limited turnover of certain companies compared to the millions invested to acquire football rights is also a potential index to look at. The ability to sustain important losses at the start is Regarding exogenous indices, negative demand shocks caused by economic crises may be a factor in the winner's curse. Another exogenous factor may be a new regulation (e.g., the no-single-buyer rule), which can play a major role in creating competition artificially. This has been questioned, particularly for its potential negative impact on consumer welfare [START_REF] Butler | Has competition in the market for subscription sports broadcasting benefited consumers? The case of the English premier league[END_REF]. The decision to individually or jointly sell club media rights is also important

Managerial implications

Risk management protocols such as an economic threshold might be helpful for rights owners before organising a call for tender. Qualitative criteria in France counted for only 10%

of the evaluation of the offers, of which only 1% was for guarantees (Assemblée Nationale, 2021e). These criteria should be more decisive, especially the visibility of the product and the level of guarantees. The CEO and president of the LFP were rewarded with incentive payments on the contract signed with Mediapro. This illustrates the principal-agent problem [START_REF] Jensen | Theory of the firm: Managerial behavior, agency costs, and capital structure[END_REF] where the agent (CEO and president) wanted to maximise their revenue at the potential risks of the principal (LFP).

The tender system is one of the leading causes of the winner's curse. Thus, direct negotiation with media partners and the extension of the contract period (currently four years maximum) are simple alternatives to avoid it. This would require changing the law at the European level, which makes it necessary to organise a call for tender (Assemblée Nationale, 2021e

). An over-the-counter negotiation system could help media companies and rights owners to share the financial risk associated with massive investments. Rights owners should have more interest in the business model of their media partner(s).

This research provides broader implications for media in the era of digital transformation. First, a period of innovation seems more conducive to business failure in sports media. The introduction of satellite television at the start of the 2000s coincided with several cases of bankruptcy linked to investments in football. This could be due to the attempt of new entrants to contest the first-mover advantages of early entrants [START_REF] Feuillet | A winner's curse in the bidding process for broadcasting rights in football? The cases of the French and UK markets[END_REF]. Access to free or cheap illegal streaming sites and devices (IPTV) is of particular concern, as it inevitably harms strongly the media return on investments. Although legal and technical responses are necessary to block or try to limit access to piracy, they are not sufficient. The bottom line is related to the willingness to pay for sports that has been impacted by the additional costs associated with the fragmentation of the offer. In this regard, the media business could be on the verge of its 'Napster moment' which revolutionised the music industry (Dennis et al., 2009;[START_REF] Maes | Le business des droits TV du foot[END_REF]. Subscription fatigue is also a factor when the content transits regularly from one broadcaster to another. In this era of platforms, exclusivity may not be as decisive as before [START_REF] Kenney | The rise of the platform economy[END_REF] compared to the aggregation of content strategy (D'Arma, 2011).

Moreover, the expansive battle for rights exclusivity appears to harm the consumer and limit potential profit maximisation. Thus, as argued by [START_REF] Evens | Value networks and changing business models for the digital television industry[END_REF], if rights holders were willing to sell their rights without exclusivity, consumer access to content would be significantly better, and the competition between media would be focused on innovation about how to broadcast rather than who is willing to pay more in the auction. The price reduction of selling nonexclusive rights for rights holders may be compensated by negotiating incentive payments from media for additional subscribers.

Limitations

It is acknowledged that this research has three main limitations. First, the methodology of the case study may limit its potential generalisation. However, historical cases of business models in the football media landscape were reviewed to identify potential similarities with the current study. Second, secondary data were used for the analysis. The sources were triangulated, but the use of secondary data only has certain limitations. However, it must be noted that primary data such as interviews with key actors would have been difficult to arrange, as it is unlikely that such actors would have accepted to be interviewed. Additionally, such interviews may have been redundant with the declarations of key actors identified in the secondary data.

The period of the case study is also limited, considering the short duration of Mediapro's presence in France. However, the case was contextualised by looking at the activities of Mediapro in markets outside France since 2006.

Conclusion

The [START_REF] Terrien | COVID-19-Crise sanitaire et vulnérabilité financière [COVID-19-Health crisis and financial vulnerability[END_REF]. The tension between the LFP and the historical broadcaster Canal+ may have precipitated the former to look aggressively for another broadcaster despite the associated risks.

Further research is necessary to describe the dynamics of media transformation, notably in sports, where they are still scarce. The accelerated shift in consumption patterns generated by the health crisis might be an interesting avenue for research, even if live sports remain important for TV networks [START_REF] Fudurić | Understanding the drivers of cable TV cord shaving with big data[END_REF]. Longitudinal studies over several decades could also be relevant to understanding the success of some media business models in sustaining their competitive advantages. Consumer access to the product should probably be at the heart of future research. It can raise research questions about the distribution or the price to fit consumer expectations.

Criteria of validity

Details of criteria [START_REF] Gibbert | What passes as a rigorous case study?[END_REF](Gibbert et al., , p. 1467) ) Framework for the case study

Internal validity

Causal relationships between variables and outcomes

(1) Verification of seven variables (indices) proposed in the literature leading to the winner's curse [START_REF] Andreff | The winner's curse in sports economics[END_REF][START_REF] Feuillet | A winner's curse in the bidding process for broadcasting rights in football? The cases of the French and UK markets[END_REF][START_REF] Thaler | Anomalies: The winner's curse[END_REF].

( (1) The rationale for the case study selection is presented in the introduction (p. 3).

(2) Contextualisation is provided the theoretical framework by reviewing similar cases and at the start of the results. A countryspecific analysis is given to ensure clarity in the process leading to our case study.

Reliability

Case study protocol & database Organization name

(1) The protocol and database of data are provided in Annex 1.

(2) The actual name of the organization studied is given as there is no confidentiality required.

Annex 2: Research procedure

From 2008 ,

 2008 Mediapro held an important position in the Spanish media market[START_REF] Alcolea-Díaz | Football broadcasting rights in Spain in the digital age: Between pay television and streaming services[END_REF]. GolTV closed as a pay-TV channel in 2015. BeIN Sports replaced it thanks to a joint venture between Al Jazeera and Mediapro that lasted until 2019. Mediapro operated the BeIN Sports channel through a central pay-TV platform such as Movistar (Telefonica), Vodafone, and Orange[START_REF] Sportpromedia | [END_REF]. For the auction of the 2019-2022 rights (LaLiga, 2018), Mediapro did not acquire any premium rights, obtaining only free-to-air highlights and rights for bars and restaurants. Thus, Telefonica established a dominant position shared withDAZN (LaLiga, 2021). Mediapro remains in charge of the international broadcast rights of LaLiga until 2024(La Liga, 2018).



  a very swift increase in rights fees: a 62% increase in Ligue 1 rights from 2016-2020 (€726.5m/year) to 2020-2024 (€1.162b/year) (Figure1);  outbidding newcomers: Mediapro was a new entrant in the French market and had to start from scratch in terms of the number of subscribers; Their main competitors during the auction were Canal+, the historical broadcaster; and BeIN Sport, a recent entrant. television rights re-packaging: The number of lots (seven) was just above that of the previous auction (six). The content of the lots differed only in the fact that candidates bid first on time slots before selecting their match choices.  too many unknown and uncertain details: the early termination of French football decided by the French government at the beginning of the Covid pandemic; the late creation of the channel 'Telefoot' in the summer of 2020 and late distribution deals with the Internet services provider.  Financial loss: Moody's Rating Agency (2020a, 2020b, 2021) on Mediapro, from B1 (highly speculative) in early 2020 to B3 (highly speculative) in April 2020 t hen Caa1 (substantial risks) in October 2020 and Caa2 in November 2020 (extremely speculative);  post-bid bankruptcy: the Commercial Court of Bobigny pronounced the judicial liquidation of Mediapro Sport France on November 20 th , 2021 (Capital, 2021); the curse on Mediapro was the result of its specific financial fragility and the non-bailout of its major shareholder Orient Hontai Capital;  disappointment: according to Capital (2021), during the hearing at the Commercial Court of Bobigny, Julien Bergeaud, the managing director of Mediapro France, rejected responsibility and blamed 'Covid-19', 'Canal+' for the abuse of their dominant position, and 'government and public authorities for their inaction against illegal streaming'.

Figure 1 :

 1 Figure 1: Domestic TV rights of the Ligue 1.

  Financialresources are critical to support costly investments in an activity characterised by high fixed 0

  Mediapro channel was inaccessible to 20 million of Free, Orange, and Canal+ subscribers at the start of the season (Assemblée Nationale, 2021e). No deal was concluded with Canal+, which deprived Mediapro of approximately four million consumers.On the demand side, Mediapro started from scratch in the context of supply fragmentation and the downgrading of payment consent. This might explain why the main French Ligue 1 event Olympique de Marseille versus Paris Saint-Germain was reportedly more followed through illegal streaming (500,000 viewers without IPTV) than on Telefoot (420,000 viewers) (L'Equipe, 2022).Value proposition. Mediapro proposed a 100% football thematic channel. Their offer was first on their own OTT platform channel, 'Telefoot'. It included the content of Ligue 1 and Ligue 2, the Champions League, and the Europa League. The introductory offer was €25.90 / month or €29.90/month without obligation. The price for Telefoot plus Netflix was €29.90/month. The Telefoot channel was also distributed non-exclusively through major Internet providers and available on the App Store of Apple.

  important, meaning the uncertainty surrounding shareholder financial guarantees in such cases might indicate potential failure. The arrival in the crowded pay-TV market might indicate high risks of failure. Relationships with external stakeholders are of significant importance. Hence, the relationships with rights owners (league, federation, football clubs), consumers (price positioning), and suppliers (satellite operator, internet distributor) are decisive. Mediapro did not reach the level of cooperation needed in its ecosystem, similar to Sport7 in the Dutch market.

  to consider. It has played a role in the arrival of new competitors in the Italian and Greek markets. Finally, innovation is of strategic interest. A bid on mass diffusion of innovation to propose a new platform may indicate a winner's curse. It had an impact at the beginning of the 2000s in Italy and Greece when satellite television started to be implemented.

  present study explains Mediapro's failure in the French market for football broadcasting rights by a mix of indices of the winner's curse and an inconsistent business model. As demonstrated in this paper, Mediapro failed to create a sustainable business model. For its part, the LFP failed to recognise a potential financial threat in Mediapro's arrival. Yet, several indices should have worried the LFP back in 2018. The solvability of Mediapro in France was only contingent on the potential bailout from its new Chinese shareholders (Orient Hontai Capital), which never happened. The absence of substantial guarantees and the historical habits of Mediapro to take legal actions should have alerted the LFP. Therefore, 91 Mediapro France (production) and 57 Mediapro Sport France (Telefoot channel) employees lost their job, and the finances of French clubs were impacted even if the state-guaranteed loan and social contribution exemption during Covid-19 massively helped clubs in the short term

Table 1 :

 1 Results of the Ligue 1 auction in 2018

	Lot n°	Details of Lot	Lots	Mediapro	Canal+	BeIN
			awarded to	offers	offers	offers
	1	Matches of Sunday	Mediapro (€330m) €330m	€262m	€100m
	2 *	Matches of Friday	Mediapro (€250m) €160m	€10m	€100m
		and Saturdays
	3 **	Matches of Saturday	BeIN (€332m)	No offer	€10m	€142m
		and Sunday
	4	Matches of Sunday	Mediapro (€190m) €190m	€170m	No offer
		and Multiplex Sunday
	5	'Trophée des Champions" and	Not attributed	No offer	€2m	€5m
		Multiplex matches n° 19, 37,	(reserve price not
		38. Ligue 1 play-off matches	reached) ***
	6	Near live (mobile highlights)	Free (€42m)	No offer	€41m	No offer
	7	Ligue 1 'magazines' of	Not attributed	No offer	€2	€7m
		Tuesdays and Thursdays	(reserve price not
			reached) ***
	Sources: Assemblée Nationale (2021f, p. 54, 59-61), L'Equipe (2019; 2020)

Indices of the winner's curse. Mediapro can be considered a spectacular example of a winner's curse. Mediapro obtained 80% of the French Ligue 1 matches for a sum of €788m, the other 20% being attributed to BeIN Sports for €330m. The case of Mediapro fits the indices of the winner's curse detailed below:

  2) Exhibit of the business model of Mediapro based on RCOV model[START_REF] Demil | Standard and Poor's[END_REF]

	Construct	Data triangulation	(1) Secondary data were collected (government reports, official
	validity	Review of transcripts	reports, economic reports, actors' hearings, credit rating
		and draft by peers	agencies) to triangulate information with notably different types
			of press articles (national/economic press; media press; sport
			press) (see Annex 2).
			(2) The contextualisation and the review of European precedent
			cases relies on peers' research papers,
	External	The rationale for case
	validity	study selection
		Details of the case
		study context (industry
		context, business
		cycle, financial data)


Cyprien Soulaine
email: csoulain@stanford.edu

Michel Quintard

Hervé Allain

Bertrand Baudouy

Rob Van Weelderen

A PISO-like algorithm to simulate superfluid helium flow with the two-fluid model

Keywords: cryogenic, two-fluid model, superfluid helium, simulation, PISO algorithm

This paper presents a segregated algorithm to solve numerically the superfluid helium (He II) equations using the two-fluid model. In order to validate the resulting code and illustrate its potential, different simulations have been performed. First, the flow through a capillary filled with He II with a heated area on one side is simulated and results are compared to analytical solutions in both Landau and Gorter-Mellink flow regimes. Then, transient heat transfer of a forced flow of He II is investigated. Finally, some two-dimensional simulations in a porous medium model are carried out.

Introduction

The physical phenomena of superfluid liquid helium can be macroscopically represented in a general framework best illustrated by Landau's two fluids model [START_REF] Landau | Theory of the superfluidity of Helium II[END_REF], [START_REF] Khalatnikov | An introduction to the theory of superfluidity[END_REF]). In this model, He II is seen as though it were a mixture of two different fluids. One of these is a normal viscous fluid, the other is a superfluid and moves with zero viscosity along a solid surface. Besides this particularity, it must be noted that the superfluid flow does not carry entropy. Moreover, the momentum equations include a thermo-mechanical force that occurs when a temperature gradient exists. When the superfluid velocity reaches a certain critical value, some turbulence phenomena arise and the above equations are no longer valid. In such a case, one introduces the so-called Gorter-Mellink mutual friction term, which has been proposed to estimate the interaction between the two components [START_REF] Gorter | On the irreversible processes in liquid Helium-II[END_REF]; [START_REF] Gorter | Discussion of the properties of Helium-II on the basis of the 2 fluid model[END_REF]).

Only few multi-dimensional codes that solve the complete two-fluid model are reported in the literature.

Actually, as pointed out by [START_REF] Kitamura | A numerical model on transient, two-dimensional flow and heat transfer in He II[END_REF], authors have met numerical difficulties, probably due to the fact that the thermo-mechanical and the Gorter-Mellink mutual friction terms are several orders of magnitude larger than the other terms in the superfluid momentum equation. The heat wave propagation intrinsic to superfluidity, the so-called second sound, may also lead to additional instabilities. To overcome these numerical difficulties, authors have either focused on one-dimensional solutions [START_REF] Rao | Transient characteristics of He II forced flow heated at the center of a pipe line[END_REF], [START_REF] Zhang | Study of the transient thermal wave heat transfer in a channel immersed in a bath of superfluid helium[END_REF]) or proposed simplified multi-dimensional models. For instance, [START_REF] Ramadan | Natural convection in large He II baths[END_REF] have modified the momentum equations as the thermo-mechanical and the mutual friction term are larger than the others in balancing each other (hence they can be dropped from the equations) to create a heat flux with the «familiar» |∇T | 1 /3 dependence. Assuming also that these two terms are approximately equal, [START_REF] Kitamura | A numerical model on transient, two-dimensional flow and heat transfer in He II[END_REF], and later [START_REF] Suekane | Heat transfer and flown of He II in narrow channels[END_REF] and [START_REF] Pietrowicz | Numerical study of the thermal behavior of an nb3sn high field magnet in He II[END_REF], proposed a simplified model made of a conventional continuity equation, a modified Navier-Stokes equation for the total velocity and a heat equation for temperature. [START_REF] Tatsumoto | Numerical analysis for two-dimensional transient heat transfer from a flat plate at one-side of a duct containing pressurized He II[END_REF] proposed a numerical segregated solution to solve superfluid equations. In their algorithm, the total fluid velocity and the pressure fields are first solved, then the superfluid momentum is computed from the resulting pressure field, after what the normal velocity is deduced point-wise. Finally, the temperature equation is solved and the whole sequel iterated for a new time step.

Actually, the discussion regarding the choice of using segregated or coupled algorithms is still an open debate. On one hand, the superfluid equations are strongly coupled, mainly from the thermo-mechanical and the mutual friction terms, and coupled algorithms offer suitable solutions [START_REF] Bottura | Finite element simulation of steady state and transient forced convection in superfluid helium[END_REF], [START_REF] Zhang | Study of the transient thermal wave heat transfer in a channel immersed in a bath of superfluid helium[END_REF]). On the other hand, the segregated approach are clearly less memory consuming.

However, in that case, how can one chose the optimal sequence? Why the normal velocity should be deduced from the superfluid velocity as suggested by [START_REF] Tatsumoto | Numerical analysis for two-dimensional transient heat transfer from a flat plate at one-side of a duct containing pressurized He II[END_REF] and not the opposite? In this paper, we propose an alternative segregated approach to solve superfluid equations. Our algorithm is an extension to the two-fluid model of the Pressure Implicit Operator Splitting (PISO) algorithm by [START_REF] Issa | Solution of the implicitly discretised fluid flow equations by operator-splitting[END_REF]. In our method, a pressure equation is directly derived from the total mass balance and both momentum equations. We called this algorithm Super-PISO. It is at the core of HellFOAM, the superfluid code we have developed using the OpenFOAM® technology.

The paper is organized as follows. In Section 2, we present the superfluid equations and the main assumptions. In Section 3, we introduce the discretization of the superfluid equations and the Super-PISO algorithm. Then in Section 4, we solve numerically the superfluid equations in different configurations

to validate and illustrate the potential of HellFOAM code.

Mathematical model and assumptions

In this section, we introduce the mathematical model on which our numerical study relies. In what follows we denote by subscripts n and s all quantities related to the normal and superfluid flow respectively.

Hence, ρ n and ρ s are the normal and superfluid densities and their sum is the actual density of He II, The mass conservation for the whole fluid reads

ρ = ρ n + ρ s . (1
∂ρ ∂t + ∇. (ρ n v n + ρ s v s ) = 0. (2)
Introducing Γ, the rate at which the superfluid particles become normal during the heating process, we can write the continuity equations for the normal and superfluid components as

∂ρ n ∂t + ∇. (ρ n v n ) = -Γ, (3)
and

∂ρ s ∂t + ∇. (ρ s v s ) = Γ. (4)
in which Γ is an unknown variable that has to be solved and varies in time and space. One notices that Γ = 0 kg/m 3 /s when no species conversion occurs. In fact, it is not granted that the rate of superfluid particles becoming normal (r sn ) is equal to the rate of normal particles that move to the superfluid state (r ns ). Ideally, these two phenomena should be expressed separately and Γ should be replaced by Γ = r sn -r ns in the continuity equations Eqs (3)-(4). One example of derivation of such a non-equilibrium theory can be found in [START_REF] Woods | The Thermodynamics of Fluid Systems[END_REF] where separate expressions are given for r sn and r ns . However, in the absence of such additional equation linking r sn and r ns the problem has an extra unknown variable compared to the number of equations available and the problem turns ill-posed. In what follows, we will consider a symmetry of normal/superfluid mass transfer (i.e., r sn = -r ns ).

The momentum balance equation for normal component reads,

∂ρ n v n ∂t + ∇.(ρ n v n v n ) = - ρ n ρ ∇p -ρ s s∇T + ∇. (µ n ∇v n ) -Aρ n ρ s |v n -v s | 2 (v n -v s ) + [r ns v s -r sn v n ] , (5)
and for superfluid component reads,

∂ρ s v s ∂t + ∇.(ρ s v s v s ) = - ρ s ρ ∇p + ρ s s∇T + Aρ n ρ s |v n -v s | 2 (v n -v s ) -[r ns v s -r sn v n ] . (6)
In these equations, ρ s s∇T represents the thermo-mechanical force which occurs when a temperature gradient exists. It is responsible for creating the counterflow in which normal fluid moves down the temperature gradient from the heat source to the bath, while the superfluid flows toward the heat source.

The term

Aρ n ρ s |v n -v s | 2 (v n -v s )
, which is present in the equation only at high velocities, represents the Gorter-Mellink mutual friction term. The last term in the right hand side of Eqs ( 5)-( 6) represents the momentum transfer between the two fluids. It means that the superfluid momentum decreases when particles turn to normal state. Landau and Lifshitz (1969) suggest that there is no momentum transfer while [START_REF] Woods | The Thermodynamics of Fluid Systems[END_REF] who derived superfluid hydrodynamic equations from a thermodynamic point of view keeps it. With the assumption that the rate at which the normal fluid gains mass from the superfluid is equal to the rate at which the superfluid gains mass from normal fluid, then the momentum transfer term reads

[r ns v s -r sn v n ] = Γ 2 (v s + v n ) , (7)
as proposed by [START_REF] Ramadan | Natural convection in large He II baths[END_REF] in their numerical model. This is the formulation we use in our simulations. Actually, several variants of the two-fluid model may be found in the literature, in particular depending on the choice of the reference frame [START_REF] Khalatnikov | An introduction to the theory of superfluidity[END_REF], see discussion in [START_REF] Woods | The Thermodynamics of Fluid Systems[END_REF]). The diffusion energy term ρnρs 2ρ ∇ |v s -v n | 2 appears in the standard Landau's model which is not included in our analysis. In fact, we have included it in another version of our code without major mathematical problem and we found that it does not play a significant role in the test cases presented in this paper. It must be emphasized that other models for superfluid helium flow are under discussion, like for instance those derived from Extended Irrerversible Thermodynamics [START_REF] Mongiovì | A monofluid flow mathematical model of liquid helium ii based on extended non-equilibrium thermodynamics[END_REF]; [START_REF] Mongiovì | Velocity of the fourth sound in liquid helium II via extended thermodynamics[END_REF]) but this is beyond the scope of this paper.

Beside these two momentum conservation equations, we should consider the energy conservation which is only transported by normal fluid. We have an energy conservation equation that writes,

∂ρs ∂t + ∇.(ρsv n ) = ∇. k n T ∇T + Aρ n ρ s |v n -v s | 4 T , (8)
where the last term on the right-hand side expresses the energy dissipation based on the mutual friction between the two components. This term may behave as a source term in case of motion of the superfluid and normal particles [START_REF] Soulaine | Numerical investigation of heat transfer in a forced flow of He II[END_REF]). To obtain a temperature equation, Eq (8) can be recast into the following form, assuming that s and ρ are not pressure-dependent,

ρ ∂s ∂T + s ∂ρ ∂T ∂T ∂t + ∇. (v n T ) = ∇. k n T ∇T + Aρ n ρ s |v n -v s | 4 T -ρs -T ∂ρs ∂T ∇.v n . (9)
We note that, besides the mutual friction term, the equations Eqs (2)-( 9) are strongly coupled by the presence of a temperature gradient in the momentum equations. Additionally, the physical variables s, ρ n , ρ s , k n and A depend on temperature, which also contribute to the coupling of the equations.

This model is valid beyond a critical velocity in the so-called superfluid turbulent regime. To simulate flow at smaller Reynolds number, one has just to set the Gorter-Mellink coefficient A equal to zero and hence obtain Landau's equations.

The superfluid equations must be completed with boundary conditions. They are defined as follows (Landau and Lifshitz (1969)). Firstly, the normal component of the total mass flux must vanish at any surface at rest:

n. (ρ n v n + ρ s v s ) = 0. (10)
Moreover, the tangential component of the normal velocity v n must be zero at a solid surface,

v n -nn.v n = 0. (11)
In the absence of heat transfer between the solid surface and the fluid, we write

n. (ρsT v n -k n ∇T ) = 0. (12)
Neglecting the diffusive term k n ∇T , the component of v n perpendicular to the surface is also zero.

Hence, for this case, we obtain a no-slip boundary condition for the normal velocity (v n = 0) and a slip boundary condition for v s .

At the heater, the energy condition boundary becomes,

n. (ρsT v n -k n ∇T ) = n.q. ( 13
)
At the helium bath entrance, temperature and pressure are fixed values. Actually, the situation regarding the temperature boundary condition at the bath entrance may be more complicated since superfluid equations cause a heat wave phenomenon, the so-called second sound, that has to be carried away from the computational domain. This point will be discussed in paragraph 4.1.2.

Numerical approach to solve the problem

All numerical developments of HellFOAM are performed with OpenFOAM® version 2.3. This code is a C++ library that solves partial differential equations with the method of finite volumes [START_REF] Jasak | Error analysis and estimation for the finite volume method with applications to fluid flows[END_REF]; [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF]). It handles 3D geometry by default. One of its features is to solve equations using segregated approaches. Since the superfluid problem presents a strong coupling between all its unknown variables (v n , v s , p, T and Γ), it is important to develop a suitable algorithm. We introduce in this section the Super-PISO algorithm, a PISO algorithm [START_REF] Issa | Solution of the implicitly discretised fluid flow equations by operator-splitting[END_REF]) adapted to superfluid equations. The question of boundary conditions is treated in a separate paragraph.

Discretization of the superfluid helium equations

The set of equations Eqs ( 2)-( 9) is transformed into a set of algebraic equations after application of any standard discretization procedure. Using the Euler implicit difference scheme and if k and k + 1 denote successive time levels, then the semi-discretized form of the normal momentum equation reads,

V ρ k+1 n v k+1 n P -ρ k n v k n P δt = -a n P v k+1 n P + N P a n N P v k+1 n N P -ρ s s∇T - ρ n ρ ∇p + K ns v k s P . (14)
In this equation, V and δt stand for the cell volume and the time step respectively, subscript P denotes values at the cell center, and a n N P accounts for the influence of neighbor nodes. They are mainly composed by convective and diffusive fluxes across cell faces. They also involve the normal part of the momentum exchange terms characterized by

K ns = Γ 2 + Aρ n ρ s |v n -v s | 2 .
In the spirit of the Rhie and Chow procedure [START_REF] Rhie | A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation[END_REF]), the pressure gradient and the temperature gradient terms are not discretized at this stage.

Introducing the vector S

= Vρ k+1 n δt -ρ s s∇T + K ns v k
s P that contains all the explicit source terms (the source part of the transient term, the thermo-mechanical term ρ s s∇T and the superfluid part of the momentum exchange terms) apart from the pressure gradient, Eq (14) can be rearranged as

Vρ k+1 n δt + a n P v k+1 n P = N P a n N P v k+1 n N P + S - ρ n ρ ∇p. ( 15
)
We can now introduce the diagonal term a n P = Vρ k+1 n δt + a n P of the full matrix resulting from the momentum equation discretization, and the operator H n (X) = N P a n N P X N P + S. Therefore, Eq (15)

turns to,

a n P v k+1 n P = H n (v k+1 n ) - ρ n ρ ∇p. ( 16
)
Following the same process, the semi-discretized superfluid momentum equation can be expressed as

a s P v k+1 s P = H s (v k+1 s ) - ρ s ρ ∇p, (17)
and the energy balance equation as

a T P T k+1 P = H T (T k+1 P ). ( 18
)
Since the diffusive term in Eq (6) and Eq (9) are very small or non-existent, we discretized the convection terms in these equations with a first order upwind scheme. All the laplacians are discretized with a second order scheme.

The Super-PISO algorithm

We present in this sub-section the algorithm we propose to solve the superfluid helium equations. It is based on the Pressure Implicit Operator-Splitting algorithm designed by [START_REF] Issa | Solution of the implicitly discretised fluid flow equations by operator-splitting[END_REF]. The algorithm is extended to the two-fluid model and a pressure equation is derived from the total mass conservation and both momentum equations. We have called this algorithm Super-PISO. The main steps are described below and sketched in Fig. 1.

1. At a time step, we start by updating all physical properties from the temperature field at the latest time step and get ρ k+1 n , ρ k+1 s , ρ k+1 , s k+1 , k k+1 n , µ k+1 n and A k+1 .

2. We continue by predicting Γ k+1 from the normal and superfluid velocities evaluated at the previous time step. From Eq (3) and Eq (4) we have:

Γ k+1 = 1 2 V ρ k+1 n -ρ k n δt + ∇. (ρ n v n ) k -V ρ k+1 s -ρ k s δt -∇. (ρ s v s ) k . ( 19
)
3. Then we evaluate the temperature field solving implicitly Eq (18). It must be noticed that, at this stage, the temperature is transported by v n of the previous time step.

4. We now solve the pressure-velocities (v n , v s , p) coupling by adapting the PISO algorithm [START_REF] Issa | Solution of the implicitly discretised fluid flow equations by operator-splitting[END_REF]). We give here the main steps of the procedure. The first stage consists in the prediction of the normal and superfluid velocities, v * n and v * s , by solving implicitly the momentum equations

a n P v * n P = H n (v * n ) - ρ k+1 n ρ k+1 ∇p k , ( 20
)
a s P v * s P = H s (v * s ) - ρ k+1 s ρ k+1 ∇p k , ( 21
)
where the gradient of the pressure field is evaluated from the values computed at the previous time step. This stage is called the momenta predictor. Actually, we have noticed that due to the small time-step requirement, it is not necessary to solve these equations. We only need to update a i P and H i (v i ) with the latest computed velocities.

5. The predicted velocities v * n and v * s do not satisfy the continuity equation Eq (2) and have to be corrected. This is achieved by looking for (v * * n , v * * s , p * ) that obeys:

v * * n P = 1 a n P H n (v * n ) - ρ k+1 n ρ k+1 ∇p * , (22) v * * s P = 1 a s P H s (v * s ) - ρ k+1 s ρ k+1 ∇p * , (23)
V ρ k+1 -ρ k δt + ∇. ρ k+1 n v * * n + ρ k+1 s v * * s = 0 (24)
Assembling these three equations, the pressure equation can be formulated as

∇. ρ k+1 n ρ k+1 ρ k+1 n a n P + ρ k+1 s ρ k+1 ρ k+1 s a s P ∇p * = ∇. ρ k+1 n H n (v * n ) a n P + ρ k+1 s H s (v * s ) a s P + V ρ k+1 -ρ k δt (25)
and solved implicitly. The corrected velocities v * * n and v * * s are then computed point-wise from Eq (22) and Eq (23). This fifth step may be repeated several times to force the convergence.

The resulting values are assimilated to (v k+1 n , v k+1 s , p k+1 ). This iteration process is called the Super-PISO loop. As all the solution algorithms that belong to the PISO family, Super-PISO is not unconditionally stable. This is typical of the predictor-corrector feature of the method since as the solution to the equations is achieved approximately, the residual errors in these approximations may alter the stability characteristics of the overall scheme. To handle the instabilities, the time-step size required some restrictions. A rigorous analysis of the stability is actually a tough task and beyond the scope of this paper, especially with this strongly coupled physics that involves a lot of non-linearities. From a practical point of view, we have noticed that our simulations need very small time steps. Moreover we also noticed that a fixed ∆t was much more stable than an adjustable time-step. The time step management will deserve further investigations.

How do we treat the boundary conditions ?

We discuss here the treatment of boundary conditions at walls, at the heated section and at the vicinity of the bath.

Walls

In absence of heat transfer at the solid walls, boundary conditions are defined as follows in OpenFOAM®:

• v n : fixedValue with value equal to uniform(0 0 0) (no-slip boundary conditions)

• v s : slip

• p : zeroGradient • T : zeroGradient 3.3.2.

Heated section

Regarding the boundary conditions at the heated area, we overcome the difficulty of imposing Eq (10)

to Eq (13) by considering the heat source n.q = q as a volume heat source in the energy equation that is distributed in some cells. Hence, the actual energy equation we have coded in OpenFOAM® is

ρ ∂s ∂T + s ∂ρ ∂T ∂T ∂t + ∇. (v n T ) = (• • • ) + qS heater V α α ( 26
)
where α is a phase indicator equal to 1 for the heated cells and 0 elsewhere. Hence, we can define a heated area within the computational domain. V α is the volume of the heated area and S heater is the surface of the heated boundary. This "trick" offers some advantages. First, Eq (13) involves a non-linear term, ρsT v n , which needs to be solved in the framework of a sequential algorithm. With the proposed method, this non-linear boundary condition is replaced by a volume source term and conditions of zero mass and heat fluxes at the boundary. First, it is no longer necessary to deal with boundary conditions for velocity fields at the heater since they are automatically set up at the α jump. Second, this method allows the heat flux to be relaxed over several cells, which improves the stability of the calculation. In particular, the creation of normal fluid component is now spread over several cells instead on a singularity at the heated surface. Moreover, this "trick" of a heat source distributed in a volume also allows more flexibility to set up He II simulations. For example, to simulate a bath of helium with a heater in the middle of the computational domain, it is not necessary to explicitly mesh the heater and specify boundary conditions.

Instead, only a cartesian grid and a mask function that indicates the heated cells are needed.

Helium bath

At the bath entrance, the boundary conditions are defined as follows in the OpenFOAM® framework

• v n : directionMixed to precise that zero tangential velocity and zero gradient in normal direction

• v s : zeroGradient

• p : fixedValue with value equal to uniform 0

• T : fixedValue with value equal to bath temperature

Some simulation results

In this section, we present some simulations we have performed with HellFOAM. (2008). The goal of theses results is to confront HellFOAM code by comparing its results against known data, quantitatively or semi-quantitatively. Finally, we present some 2D simulations of He II in a porous medium model, as an example of a rather complex situation.

Capillary containing helium II in Landau's regime

This test consists in the simulation of a capillary (2h × L = 1 mm × 15 mm) tube filled with He II, heated at the left-hand side and connected to a He II bath at the right-hand side (see Fig. 2 ). For sake of computing economy, we only simulate the flow on half the geometry using a symmetry plane boundary condition. The grid is made of 200 × 20 hexaedral cells. Bath is assumed to be at 2 K and so does the capillary tube initially. At this temperature, we have the following fluids properties: ρ n = 82.12 kg/m 3 , ρ s = 63.53 kg/m 3 , µ n = 1.489 × 10 -6 kg/m/s, k n = 0.00387 W/m/K and s = 957.7 J/kg/K. Initially, the superfluid and normal components are at rest (v n = 0 and v s = 0), the bath temperature at 2 K, and the pressure field is uniformly zero. The left-hand side is heated to q = 1000 W/m 2 . Timestep is set up to 10 -6 s and the simulation is carried out until steady-state is reached. In order to use

HellFOAM code in Landau's regime, A is set to zero.

Steady-state

To avoid or limit the propagation of the heat waves intrinsic to superfluid equations, the so-called second sound (Landau and Lifshitz (1969)), the left-hand side is gradually heated using a heat-up ramp, q 1 -e -t /τ , where τ = 10 -2 s. We plot in Fig. 3 the simulation results we obtain for the normal and superfluid velocities at steady-state. Normal velocity v n has a parabolic profile while the superfluid component flows in a homogeneous manner at counter-current to the normal component. At steady state, this test-case has analytical solutions (Landau and Lifshitz (1969)). Indeed, in the established regime, both velocities are aligned along the x-axis. From the heat flux equation, the normal velocity averaged over a cross-section is estimated by v n = q ρsT ≈ 0.00358 m/s. Moreover, noticing that the sum of the two momentum equation, Eqs ( 5)-( 6), leads to a Stokes equation for the couple (p, v nx ), one can deduce that the normal velocity has a parabolic profile, v nx (y) = v nmax 1 -y h 2 , where

v nmax = 3
2 v n ≈ 0.00537 m/s. On the other hand, since there is no viscosity in the superfluid momentum equation, Eq (6), then the superfluid velocity is constant over a cross-section. It can be predicted using the zero mass transfer relation v s = -ρn ρs v n ≈ -0.0046 m/s. We see in Fig. 3 that simulation results and analytical solutions are in very good agreement. The temperature elevation between the ends of the capillary tube can be evaluated using London's equation ∆T = 1 ρs ∆p and the pressure loss / velocity relation in a 2D capillary v n = h 2 12µn ∆p L . It follows that ∆T ≈ 6.9 × 10 -9 K which is in good agreement with the temperature profile in the traverse direction obtained by simulation (see Fig. 4). It must be noted that for such a small temperature increase, the fluids properties variation ρ n , ρ s , µ n , k n and s is negligible in agreement with our analytical comparison. We notice a temperature increase of the order of magnitude of 7 × 10 -9 K as expected by the theory.

Transient

We now turn our attention to the transient regime of the superfluid He II problem. One important characteristic of the superfluid equations is the existence of a heat wave called second sound: when a capillary is heated at one side, a temperature front travels toward the other side. Landau and Lifshitz (1969) have estimated its propagation velocity at u = T s 2 ρs cρn . Using the approximation that relates the compressibility and the entropy c ≈ 3s at very low temperature, we get u ≈ 22.2 m/s. It means that the heat front crosses the capillary in approximately 7 × 10 -4 s. To point out this phenomenon, we start again the simulation without the heat-ramp. As depicted in Fig. 5, a heat front travels along the capillary and reaches the bath at about 9 × 10 -4 s in accordance with Landau's predictions. However, we can notice that the heat front is then reflected towards the heater (see 10 × 10 -4 s for instance).

While a reflexion at any place where geometry changes is not unphysical, its amplitude here is probably unrealistic if we consider the presence of a bath starting at this boundary. And it is also very probable that the boundary conditions at the bath side, namely a Dirchlet conditon for the temperature, cannot represent the dampening effect of a real bath. The use of a wave-transmissive boundary condition could overcome this problem but it has not been developed for our superfluid case. We also notice in Fig. 5 that the heat wave amplitude is much higher than the steady-state value predicted by the theory. Indeed, the dissipation mechanisms that lead to the steady-state value are quite slow (since kn ρs ≈ 4.9 × 10 -8 m 2 /s).

Capillary containing He II in Gorter-Mellink regime

We now turn our attention to the investigation of He II flow dynamics in the Gorter-Mellink regime. We keep the same geometry and fluid properties as those used in section 4.1. At 2 K, the Gorter-Mellink coefficient is equal to A = 1280 m.s/kg. As for the other fluid properties, even if they are estimated as a function of the temperature in HellFOAM, for a temperature increase below 1 mK, it is reasonable to consider their value constant in the following analytical formula. Initially, both components are at rest at the temperature of the He II bath and the relative pressure field is uniformly zero. We heat the left hand side to 10 4 W/m 2 . Time step is set up to 5 × 10 -6 s and simulation is stopped when steady-state is reached.

In the Gorter-Mellink regime, the source term in the energy equation, Eq (9), behaves as a strong dissipative process. As expected, the temperature field reaches its steady-state quasi-immediately after the heat-up time, and there are no longer heat wave propagations. The Gorter-Mellink dissipation terms have absorbed these waves. The temperature gradient we found (see Fig. 6) is in agreement with the value analytically obtained by the formula ∆T ≈ AρnL ρ 3 s s 4 T 3 q 3 ≈ 0.0009 K [START_REF] Arp | Heat transport through helium II[END_REF]).

Temperature

Figure 6: Plot of the temperature profile along the capillary at steady-state. We notice a temperature increase up to 0.0009 K as expected by the theory.

Regarding the velocity fields, as plotted in Fig. 7, the normal velocity field remains a parabolic profile while the superfluid velocity field is no longer homogeneous over the cross-section and some higher values are noticed in the walls vicinity. Profiles may be explained merely from the two-fluid model. Indeed, adding the two momentum equations, Eqs ( 5)-( 6), and assuming that the velocities are aligned with the x-axis, one gets a Stokes equation for the normal velocity. Consequently, the normal velocity has a parabolic profile as in the Landau's regime. Regarding the superfluid momentum, it is now well-known that the thermomechanical effect and mutual friction terms are the controlling terms. Neglecting the other ones, we can derive the relationship, v sx (y) = v nx (y) -3 s ρnA T L , that relates both velocities. Numerical results and the analytical theory fit very well, as illustrated by the results presented in Fig. 7.

Fuzier and van Sciver's experiments

In this section, we simulate transient heat transfer for a forced flow of He II at high Reynolds number inspired from the setup of the experiments performed by [START_REF] Fuzier | Experimental measurements and modeling of transient heat transfer in forced flow of he II at high velocities[END_REF]. In their work, authors measured temperature profiles in a forced flow of superfluid helium in a 1 m long, 9.8 mm inside diameter, smooth tube. The liquid is pushed from a bellows pump to reach a velocity up to 22 m/s. Fuzier and Sciver experiments cannot be entirely reproduced by our numerical model mainly because these experiments were performed at relatively high normal fluid Reynolds numbers, hence with potentially turbulence effects. The two-fluid model developed in this paper is able to simulate inertia effects but not a fully developed turbulence, such as the kind of flow that develop in a straight tube, once hydrodynamic instabilities have started. Therefore, the comparison here will be more qualitative, remembering that the simulated flow remains laminar. In addition, only a 2D geometry will be considered. The computational domain is depicted in Fig. 8. It consists in a 1 m long, 4.9 mm thick, 2D half tube. The grid is made of 500 × 10 hexaedral cells. The heater is placed between 0.30 and 0.31 m from the left-hand side. Liquid helium enters into the domain from the left-hand side at 2 m/s and flows out at the right hand side of the tube. Initially, we assume that the superfluid and normal velocities are equal to 2 m/s within the tube and the temperature is 1.7 K. Several probes are placed along the tube. They correspond to the temperature measurements T3 to T8 of [START_REF] Fuzier | Experimental measurements and modeling of transient heat transfer in forced flow of he II at high velocities[END_REF]. Simulations are performed with the full two-fluid model involving Gorter-Mellink mutual friction terms. The time step is set up to ∆t = 5 × 10 -6 s. After a preliminary simulation where we let the flow to be established without warming, we start the heater at q = 9.9 W/cm 2 for 20 ms and we record the temperature evolution until 250 ms for the different probes as depicted in Fig. 4.3. The results are in very good qualitative agreement with the experimental results shown in Fig. 5 of [START_REF] Fuzier | Experimental measurements and modeling of transient heat transfer in forced flow of he II at high velocities[END_REF]: the temperature peak for each probe are comparable both in amplitude and in time. This comparison example is preliminary and semi-quantitative in the sense that some assumptions do not correspond exactly to the actual situation Finally, simulation results of forced flow of He II at 16 m/s are plotted in Fig. 11. Results are clearly comparable to Fuzier's experiments (see [START_REF] Fuzier | Experimental measurements and modeling of transient heat transfer in forced flow of he II at high velocities[END_REF], Fig. 7).

- These results lead to two conclusions : i) they confirm the possibility of simulating laboratory experiments for relatively simple geometries with the use of a complete two-fluid model, ii) they comfort the conclusions of the analytical validation tests for the HellFOAM code.

Superfluid flow through an array of beads

A lot of devices involving He II superfluid can be seen as a porous medium involving two separate characteristic scales: a pore scale and a macro-scale. At the smaller scale, the flows and other transfer phenomena are simulated through the exact geometry of the material. Basically, to a cell of the mesh grid corresponds either the fluid or the solid structure. On the other hand, at the macro-scale, the flow can be described by volume-averaged equations. For coupled, non-linear equations, developing a macro-scale theory may benefit strongly of the quantitative view offered by direct pore-scale numerical results, which point in the right direction for performing necessary approximation. Such an illustration of this kind of approximation can be found in the paper by [START_REF] Allain | Upscaling of superfluid helium flow in porous media[END_REF] in the case of the Landau regime. In this section, we show the capability of HellFOAM to simulate pore-scale flow of superfluid helium through a model porous medium.

The geometry (see Fig. 12) consists in a capillary (same dimensions as paragraph 4.1 and 4.2) that contains 16 solid beads (mm diameter). While of course the geometry is a very sketchy representation of a porous medium, it has all the fundamental features, i.e., it contains several representative unit cells with some geometrical tortuosity. The heater is placed in a small extension of the capillary at the left-hand side of the array of beads. The capillary is initially filled with He II at 2 K. The left-hand side is then warmed up to a heat flux density of 10 4 W/m 2 and the flow is simulated using the Gorter-Mellink model. The computational domain is gridded with 3500 hexaedral cells. The time-step is set up to ∆t = 10 -6 s.

Fig. 13 to Fig. 15 depict the temperature, the normal and the superfluid velocities at steady-state.

We can notice in Fig. 13 that the temperature increase is much higher (30 times) than in the case of a straight capillary tube. This result is directly related to the presence of the solid structure within the tube and the related tortuosity effect. From Fig. 14 and Fig. 15, we clearly observe cyclic flow patterns. This suggests that one may define a Representative Elementary Volume (REV) of the porescale physics and then apply some upscaling technique like a volume-averaging methodology for instance in order to derive macro-scale equations. Moreover, the results indicate that further simulations could be restricted to a single REV with some kind of periodicity condition for velocities and temperature and pressure deviations. This can be used to significantly reduce the CPU time and, more fundamentally, this suggests also that some kind of decomposition technique in terms of macro-scale variables and deviations can be performed in the context of upscaling. This has already been used in the paper by [START_REF] Allain | Upscaling of superfluid helium flow in porous media[END_REF] in the Landau regime, our results indicate that this can be extended to more inertial flows with also significant Gorter-Mellink effects.

The inserts plotted in Fig. 14 and Fig. 15 represent the normal and superfluid velocity vectors, respec-tively, in two adjacent REVs. In Fig. 14 we can note the presence of an eddy in the region in between two beads and the top wall. In comparison, a Navier-Stokes simulation at high Reynolds number in the same geometry will give two recirculations (one in each symmetrical part) of the flow in the area in between two adjacent cylinders. From Fig. 15 we also notice no detachment for the superfluid components.

Several points may explain this phenomenon. First, the superfluid velocity has a slip condition at the solid boundaries. Then, one can assume that the Gorter-Mellink term in the momentum equations plays an important role. This illustrates the potential complexity of such flows in porous media and deserves further studies, in particular with the perspective to apply a full upscaling methodology. The recent developments of the cryogenic visualization technique [START_REF] Guo | Visualization of two-fluid flows of superfluid helium-4[END_REF]) will be helpful to validate these simulation results.

Conclusions and perspectives

We have presented in this paper the first version of HellFOAM, a helium superfluid simulator based on the OpenFOAM® technology. Given the sequential nature of OpenFOAM® algorithms, we had, in order to solve superfluid equations, to develop a new algorithm (called Super-PISO) extending the type of PISO algorithm used in OpenFOAM®.

Using the implemented code, different scenarii were simulated which correspond to situations of current research interest and with significant complexity. Solutions on straight tubes for which analytical (or partially analytical) solutions are known were used to check the validity of the numerical tool. Other simulations showed the interest of performing numerical simulations as a tool to interpret experiments, or in the framework of a multi-scale analysis of superfluid flow through porous media. These various simulations have proven the robustness and efficiency of HellFOAM.

)

  These quantities vary according to the temperature: ρ s vanishes at the λ-point where the fluid becomes fully normal, ρ n is null at absolute zero. Moreover, as a classical fluid, the normal component has a fluid viscosity µ n and a fluid heat conductivity k n . Besides these variables, the model also requires the Gorter-Mellink coefficient A introduced later. All these physical properties are assumed to be a function of temperature only and the pressure dependency is neglected. Their values are provided by polynomial interpolations from the HePak thermodynamic package by Cryodata Inc.

Figure 1 :

 1 Figure 1: Principle of the Super-PISO algorithm

Figure 2 :

 2 Figure 2: Geometry and mesh of the capillary tube.

Figure 3 :

 3 Figure 3: Plot of the velocity profiles over a vertical cross-section at the middle of the tube in the Landau flow regime. Both components are flowing at counter-current. The normal velocity profile has a parabolic profile and the superfluid component is homogeneous. The results fit very well with the analytical solution.

Figure 4 :

 4 Figure4: Plot of the temperature profile along the capillary at steady-state. We notice a temperature increase of the order of magnitude of 7 × 10 -9 K as expected by the theory.

Figure 5 :

 5 Figure 5: Plot of heat front propagation along the capillary tube filled with He II in the Landau regime.

Figure 7 :

 7 Figure 7: Plot of the velocity profiles over a vertical cross-section in the Gorter-Mellink flow regime. The normal velocity profile has still a parabolic shape while the superfluid component is no longer homogeneous. Indeed, the theory predicts that vs x (y) = vn x (y) -3 s ρnA

(

  for example: 2D geometry as opposed to a cylindrical geometry, laminar flow while some turbulence may have arisen in Fuzier and Sciver experiments). It is intended to show that the expected physics is captured by the proposed numerical tool, i.e, correct evolution of physical variables. Further work is in progress towards a more comprehensive comparison. Additional discussions regarding this comparison can be found in another work by the same authors (Soulaine et al. (2014)).

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Geometry and boundary conditions of the domain.

Figure 11 :

 11 Figure 11: Transient temperature profile recorded by several probes placed along the tube in case of a forced convection flowing at 16 m/s.

Figure 12 :

 12 Figure 12: Geometry and mesh of the domain.

Figure 13 :

 13 Figure 13: Plot of the temperature profile at steady state in a capillary tube filled by 16 beads. The presence of solid materials inside the capillary leads to an increase of the ∆T .

Figure 14 :

 14 Figure 14: Plot of the normal velocity magnitude and of the normal velocity vectors in two adjacent REV. We clearly notice cyclic flow patterns.

Figure 15 :

 15 Figure 15: Plot of the superfluid velocity magnitude and of the superfluid velocity vectors in two adjacent REV. We clearly notice cyclic flow patterns.