Ling Jiang
email: ling.jiang@univ-amu.fr

Veronique Cova

Love for Luxury, Preference for Counterfeits -A Qualitative Study in Counterfeit Luxury Consumption in China

Keywords: counterfeiting, brands, consumer behavior, China www.ccsenet.org/ijms

The aim of this study is to deepen the understanding of luxury counterfeit consumption by using the theories of luxury. This study is an interpretative qualitative research in which the social and personal meanings of luxury counterfeit consumption are explored. Both luxury and counterfeit provide a successful meaning transference. A good counterfeit can be regarded as the substitute good of luxury, which provides social meanings (conformity/status seeking/face saving), whereas it may also operate on a personal level (brand experience/fashion/adventure). The process of purchasing and consumption is fun and enjoyable due to the illegal nature of counterfeit. The research aspires to shed light on the essence of counterfeit in a brand perspective.

Introduction

How often do you see a dubious Louis Vuitton handbag or a pair of Prada sunglasses on sale in the street markets? The significant growth of luxury consumption over the past years has been accompanied with a dramatically rise of counterfeit. Consumer decisions to purchase fakes instead of originals have developed into a worldwide crisis. The counterfeit market is responsible for at least 200 billion dollars a year in lost jobs, taxes and sales [START_REF] Furnham | The effect of life values and materialism on buying counterfeit products[END_REF]. China has been acknowledged as the famous "made in" for counterfeit products, where all type of fake can be traced [START_REF] Bian | Consumers' attitudes regarding non-deceptive counterfeit brands in the UK and China[END_REF][START_REF] Cheung | Buyers' perceptions of pirated products in China[END_REF][START_REF] Hung | The Business of Product Counterfeiting in China and the Post-WTO Membership Environment[END_REF]. These products lay over a vast range of goods including apparel electronics, foodstuff, pharmaceuticals, cosmetics products, and others. In particular, as the Chinese are avid consumers of luxury goods [START_REF] Li | How face influences consumption: a comparative study of American and Chinese consumers[END_REF][START_REF] Sonmez | Manchester United versus China: A counterfeiting and trademark match[END_REF][START_REF] Wong | Personal Taste and Family Face: Luxury Consumption in Confucian and Western Societies[END_REF], the counterfeit of luxury brands has reached astronomical levels.

In a Confucian society context, Chinese consumers behave more irrationally in luxury purchase because luxury brands are something "must to have" for them to reinforce their social status. However, the wealth distribution is extremely unjust and the country has one of the largest gaps between the rich and the poor in the world according the statistics data from the World Bank. For most of them, the desire for luxury brands is not achieved due to the lack of affordability to exorbitant price tags of original luxury brands. Hence, the counterfeit manufacturer argue that their production are satisfying the demand of those consumers who unable or unwilling to pay for the originals [START_REF] Lai | Brand imitation: Do the Chinese have different views?[END_REF]. The consumer demand for counterfeit has raised substantial research. The attitude-intention behavior model has been vast studied. However, the explanations provided on the motivation of luxury counterfeit and the association between original luxury and counterfeit luxury consumption are still missing.

Given that the market for counterfeit luxury brands relies on consumers' desire for real luxury brands [START_REF] Hoe | Fakin'It : Counterfeiting and Consumer Contradictions[END_REF][START_REF] Penz | Forget the "Real" Thing-Take the Copy! An Explanatory Model for the Volitional Purchase of Counterfeit Products[END_REF], it is critical for researchers to study the reasons why consumers purchase the fake brand from their perception of real brand, and whether their perception of luxury brand will affects their counterfeit consumption. This is where our paper focuses: based on twenty-fives interviews with the luxury counterfeit consumers, we provide to study the counterfeit consumption into a brand perspective. We aim to explore the brand interpersonal and personal effect in the counterfeit luxury brand consumption, to better understand the reason why consumers buy and own fakes. As such, we contribute to define a set of key drivers of interpersonal and personal meanings in counterfeit products. The connection of luxury brand and counterfeit luxury brand will be highlighted that help to reduce the complexity of counterfeit consumption and enrich the literature.

Literature Review

Counterfeit Definition

Generally the brand of an enterprise is its most valuable asset. But the success of a brand may cause counterfeiting [START_REF] Maldonado | Atitude toward counterfeit products: an ethical perspective[END_REF]. The history of counterfeit brands of luxury products goes back to A.D. 27. In that period, the wine Merchant Gaul filled wine bottle with cheap local wine and sold them as expensive Roman wines. Around 19th century, the counterfeits of valuable trademarks became wider and counterfeiting became a crime that requires punishment in some European countries) [START_REF] Wilcox | Why Do Consumers Buy Counterfeit Luxury Brands[END_REF].

As with any topic, there are various interpretations of the basic concept for the area of counterfeiting study. [START_REF] Kay | Fake's progress[END_REF]'s defined counterfeiting as the production of copies that are identically packaged, including trademarks and labeling, so as to seem to a consumer that the item is the genuine article. [START_REF] Bloch | Consumer accomplices's in product counterfeiting: a demand-side investigation[END_REF] defined counterfeiting as the unauthorized copying of trademarked or copyrighted goods. In the work of [START_REF] Cordell | Counterfeit purchase intentions: role of lawfulness attitudes and product traits as determinants[END_REF], the product counterfeiting is defined as any manufacturing of goods whose special characteristics are protected as intellectual property rights. Eisend and Schuchert-Culer (2006) viewed counterfeiting as meaning that an original product with a remarkable brand value worth copying already existed on the market; the product attributes were copied into another product, which was indistinguishable from the original, and was sold at a lower price as if it were the original. For this study, the definition of Eisend and Schuchert-Culer was used because this definition highlights the key component of a luxury counterfeit: that the characteristics of copied product are identical to the genuine and the original brand value is involved.

From the consumer's perspective, the literature has identified two forms of counterfeiting, as deceptive and non-deceptive. Under deceptive counterfeiting, the consumer is a victim of deception, they are not aware of purchasing a fake product. In our study, we specifically focus on non-deceptive counterfeit product, where consumers intentionally purchase fake products [START_REF] Grossman | Foreign Counterfeiting of Status Goods[END_REF]. This form of counterfeiting is more commonplace in luxury brand markets [START_REF] Nia | Do counterfeits devalue the ownership of luxury brands?[END_REF], where consumers are often able to distinguish counterfeits from genuine brands based on differences in price, the distribution channels, and the inferior quality of the product itself.

Consumer Demand for Luxury Brand Counterfeit

Previous research has developed many factors to understand the phenomenon of counterfeits activities. It is commonly agreed that the consumers' demand for counterfeits is one of the major reason of the existence of the counterfeiting phenomenon. So a number of researches have called for investigation in consumer demand behavior of counterfeiting study, in particular, the determinants of CBP purchase intention. We classify here the determinants into five categories:

The first category based on characteristics of the product, such as price, investment risk (e.g. [START_REF] Cordell | Counterfeit purchase intentions: role of lawfulness attitudes and product traits as determinants[END_REF], product type (e.g. [START_REF] Albers-Miller | Consumer misbehavior: why people buy illicit goods[END_REF], product attribute variables, and style/fashionability (e.g. [START_REF] Bloch | Consumer accomplices's in product counterfeiting: a demand-side investigation[END_REF]. Obviously, the price difference is the best advantage of the counterfeit product, those consumers buy counterfeits because they are getting prestige without paying for it [START_REF] Bloch | Consumer accomplices's in product counterfeiting: a demand-side investigation[END_REF]Albers-Millier, 1999;[START_REF] Harvey | Laboratory markets in counterfeit goods: Hong Kong[END_REF]; The second category includes consumers' demographic and psychographic variables, as well as consumer's attitudes toward counterfeiting. For example, previous studies have found that consumers who purchase counterfeit products are usually have a lower social status (e.g. [START_REF] Bloch | Consumer accomplices's in product counterfeiting: a demand-side investigation[END_REF][START_REF] Wee | Non-price determinants of intention to purchase counterfeit goods: an exploratory study[END_REF] and have more favorable attitudes toward counterfeiting (e.g. Penz and Stottinger, 2005); The third category refers to the social and cultural context in which the counterfeit purchase decision is made from cultural norms [START_REF] Lai | Brand imitation: Do the Chinese have different views?[END_REF], there is a significant relationship between the country of origin, consumers' ethnocentrism and consumers' perceived risk when buying a counterfeit [START_REF] Chakraborty | Exploring consumers evaluations of counterfeits the roles of country of origin and ethnocentrism[END_REF]. Likewise, the social functions served by consumers' luxury brand attitudes can influenced the demand for counterfeit. [START_REF] Wilcox | Why Do Consumers Buy Counterfeit Luxury Brands[END_REF]; The fourth category refers to the mood and situational context. For example, consumers are likely to purchase a counterfeit brand when they are abroad or on holiday. The tourist consumers are more likely to purchase the counterfeits products as an authentic experience (e.g. Gentry & al., 2001); The fifth category refers to the consumers' ethical and lawfulness cues. For some, they holding lax attitude about the legal protection of intellectual property (e.g. Tom et al., 1998). The consumers who are more lawful-minded are less willing to buy counterfeit. In contrast, those who have lower ethical standards are less feel accountable for a counterfeit buying behavior (e.g. [START_REF] Cordell | Counterfeit purchase intentions: role of lawfulness attitudes and product traits as determinants[END_REF].

Luxury Brand and Counterfeit Luxury Brand Relevance

Among the many product categories that could be counterfeited, one of the most counterfeited products is luxury item [START_REF] Grossman | Foreign Counterfeiting of Status Goods[END_REF]. The brand is always associated with the counterfeiting, because a counterfeited luxury brand must copy an existed famous trademarked brand [START_REF] Cordell | Counterfeit purchase intentions: role of lawfulness attitudes and product traits as determinants[END_REF]; meanwhile, a successful brand should have the highest attractiveness level to counterfeiters. In fact, according to a website statistic, the most desirable brands for Chinese consumers are LV, Gucci and Burberry. The trademark Louis Vuitton turn into the primary target of counterfeiters, which can be abundantly found in street markets, small shops, flea markets, and through web selling.

In the context of non-deceptive counterfeiting, several previous studies proposed if consumers don't attracted by brand product, counterfeited brand product will not be an issue [START_REF] Bloch | Consumer accomplices's in product counterfeiting: a demand-side investigation[END_REF][START_REF] Cordell | Counterfeit purchase intentions: role of lawfulness attitudes and product traits as determinants[END_REF]. Few studies argued that the decision to buy a counterfeit product not only represents a product choice decision, but also represents a brand decision [START_REF] Eisend | Explaining Counterfeit Purchases: A Review and Preview[END_REF][START_REF] Gentry | The effects of counterfeiting on consumer search[END_REF]. Since the quality of counterfeit products has been steadily improving over the years, some counterfeits are recognized as the same quality, same design and even better durability than the originals. [START_REF] Turunen | Diffusing the boundaries between luxury and counterfeits[END_REF] indicate that the perceived authenticity of luxury product is the major difference to counterfeit product. The counterfeit is regarded as the pursuit of luxury by imitating its attributes, but it is can be also placed on the brand level. Since it is debated that luxuries are divided into different level depend on the context and the people concerned, a counterfeit brand can be regarded as the lowest level of luxury, and the consumers may profit the "look-alike" brand instead of the original one as the props aiding them to get desired benefits.

From this point of view, it is necessary to link up the luxury consumption value and counterfeit luxury consumption. [START_REF] Vigneron | A Review and a Conceptual Framework of Prestige-Seeking Consumer Behavior[END_REF] argue that luxury brand items contain both interpersonal and personal meanings. This interpersonally oriented motivation is much more externally driven and represents the consumers' desire to impress others. The personal orientation such as hedonism and materialism are internally driven and reflect the self-fulfillment goals. Hence, we propose in this study to understand the counterpart of luxury brand from interpersonal and personal aspect:

-We suggest that consumption of luxury brands' counterfeits may result in the construction of a positive image for the social needs.

-We suggest that the experience of consuming luxury products' counterfeits may be motivated by self-need or self-pleasure.

Methodology

The empirical part of this study aims to identify what kind of meanings contained by counterfeit luxury brand. To carry out this research, we opted for a qualitative research design. A total of twenty-five in-depth interviews were collected. The participants were selected based on experience with fake luxury brands. They came from four Chinese first tier cities (Beijing, Shanghai, Guangzhou and Wuhan), where the exclusive stores selling international luxury brands are found and the counterfeits can be abundantly found in street markets, small shops, flea markets, and through web selling. 60% of participants were female, 40% male; about half of the respondents were younger than 28 years, the other half older. The younger participants were college students or recently graduated professionals. Most of the older participants were married and had children; they were business people or employees. All the participants had considerable disposable incomes and were interested in luxury and counterfeit brand. The participants' data is presented in Table 1. Since the topic is sensitive, we assured the participants that their anonymity would be preserved. In the interviews, we began with several demographic questions to warm up, and then explored respondents' general views towards luxury product and finally talk about counterfeits' consumption. The counterfeit interpreted especially as counterpart of luxury brand. We asked for the beliefs, feelings, expectations and reasons present during the processes of consuming the products, and let the participants describe their recent consumption experience of luxury brand counterfeit product. The product attributes and shopping context described by the respondents can reflect the interpersonal and personal meanings about the product. Each interview lasted between 30 and 70 minutes. All interviews were conducted in Chinese and recorded, transcribed, then translated into English by a professional linguister in Rhewin Translation at the city of Beijing. For the analysis, we categorized the data by the thematic content analysis, and then developed comparisons between the coded reports of the software tool NVivo9 in order to identify the emerging themes.

Results: Emerging Themes in Consumption of Counterfeit Luxury Goods

In the following, we present the results along two dimensions that emerged from the key topics discussed in the focus groups. The first dimension refers to interpersonal aspect of consumption of counterfeit luxury goods related to conformity, status seeking and face consciousness; the second dimension rests on personal aspect related to fashion consciousness, brand experience and adventure. We shall now analyze each theme in detail.

Interpersonal Aspect

Conformity

Social conformity was defined as "the change in their product evaluations, attitude, purchase intentions, or purchase behavior as a result of exposure to the evaluations, intentions, or purchase behaviors of referent others" [START_REF] Lascu | Consumer conformity: Review and applications for marketing theory and practice[END_REF]. According to Lascu and Zinkhan, conformity plays an important role in shaping individuals' decisions toward product choices based on the influence of referent others.

Be part of the Confucianism, the conformist is one of the most important social factors, influencing the individual and social behavior in China. The Chinese culture has traditionally intensified the level of social behavior over the personal, and therefore strengthened the social influence on individual behavior. In the field of consumption, Chinese consumers are very sensitive to attention and the reaction of others.

"…Now you can go down to the Nanjing Road (one famous walking road in Shanghai), all type of ladies, high, thin, fat, or short…they are wearing a LV or at least a Coach. How can I walk between them without a brand bag…"(Female, Xian)

"…Especially to those friends having a lot of money, I won't tell them that the products is a fake. They might mock at me…" (Male, Yuan) Our respondents normally prefer a product that is widely used by their social group than a product they ever sees. They do not want to be too backward to their social group, because the strong social risk of "losing face". This shows that Chinese consumers are more interested in the social necessity of luxury than a necessity itself. This also means that consumers are not simply searching for branded products, but also for further "products" within the luxury brand model. More precisely, our respondents make a brand choice since they want to be similar to their peers or social groups. However, after the brand decision, consumers now faced with another choice, whether to purchase a genuine item or a counterfeit. As a result of financial reason, they could no longer afford the brand, but could offer a counterfeit version in order to auto-classify themselves as one of the luxury brand users and be able to conform to their social groups.

"Without a brand, nobody looks you in the street." (Female, Xian)

Conspicuousness

According to Goldsmith, Flynn, and Eastman [START_REF] Goldsmith | Status consumption and self-image: A replication with Mexican consumers[END_REF], status consumption is defined as the "motivational process by which individuals strive to improve their social standing through the conspicuous consumption of consumer products that confer and symbolize status both for the individual and surrounding significant others". Status-consumption consumers, regardless of their current class ranking, may desire products, which are indicators of a higher social status [START_REF] Goldsmith | Status consumption and self-image: A replication with Mexican consumers[END_REF].

For the respondents in this study, the luxury brand itself was the main reason for purchase and the meaning associated with the original brand that can be transferred to the users. Individuals seek status to obtain a position or rank given by others in the society. However, Veblen (1899) argument that consumers' desire to gain status is enhanced by conspicuous products indicated that consumers may not always spend higher prices for goods in order to achieve status; they may also purchase a larger quantity of conspicuous goods at lower prices in order to gain esteem. This indicates that consumers may obtain status from consumption of non-expensive products as well. The luxury brands sold at an exorbitant price, which is not affordable for some consumers. This contributes to a dissonance whereby consumer resort to a look-alike luxury brands to satisfy their status seeking needs.

"Here people get impressed if you are wearing an expensive brand…although you know it's a counterfeit." (Male, Paul)

Chinese consumers even behave more irrationally in luxury purchase because luxury brands are something "must to have" for them to reinforce their social status. However, with lower-price acceptability, our respondents have to try a low-grade version with the intent of purchasing the genius item if the trial is successful.

"It was a Hugo boss suit and it had the brand label on the sleeve. My girl friend bought it for me, because I was going to a conference and I have to get well dressed. I thought 'I will look smart in it', the brand was there and it distracts people from looking at me." (Male, Zhihui) Thus, this study considers that the consumers of counterfeiting luxury brand are indeed seeking a higher social status. They take a lower cost to satisfy their social-status needs.

Face Saving

Face refers to a sense of favorable social self-image that a person wants others to obtain in a relational and network context. It contributes to people's socially defined aspect of self, which is usually considered as tightly related to Chinese culture [START_REF] Chan | The constrasting effects of culture on consumer tolerance: Interpersonal face and impersonal fate[END_REF][START_REF] Yau | Chinese cultural values: Their dimensions and marketing implications[END_REF]. Maintaining face is regarded as an important goal in social interactions [START_REF] White | Face threat sensitivity in negotiation: Roadblock to agreement and joint again[END_REF].

"There are many who are willing to spend one-month salary to buy something, not for quality, not for taste, but only for one thing: face." (Female, Xian) Prior research in consumer behavior shows that face pressure represents an important driver of luxury consumption in China [START_REF] Wong | Personal Taste and Family Face: Luxury Consumption in Confucian and Western Societies[END_REF]. Well, price-insensitive consumers or consumers without financial capacity are always attracted by copies of luxury. With de development of online shopping, Chinese consumers seem to have discovered a new creative way for face saving:

"You can even find paper bags of LV or Chanel sold online. I mean the package of such top brand goods." (Female, Susu) People who cannot afford luxury brand products are buying paper bags with high-end brand names or logos to gratify their vanity. Some are even finding luxury labels, instruction books and receipts sold online. All of these are regarded as the most economic way to get "luxury" goods. Now we can find all the accessories of luxuries of all the brands from Hermes to Prada on Internet. In the shopping page, we found one of the buyer gave a 5 stars satisfaction to the seller and leave a message:

"The paper bag made me in relation with the brand." (Female, from one of the buyer in Tmall) Several respondents mentioned the online shopping Tmall during the interviews. With professional search engine, hundreds of thousands of counterfeit merchandisers are providing low prices, faster delivery times and wider range of delivery for consumers demand. One example, by publishing only 15 days, the selling page attained 290 deal records for one Louis Vuitton handbag. This is only a data of one objet offered by one seller. From the page feedback history, the consumers exchanged their opinions. The degree of satisfaction depends on the degree of similarity. They are happy to gain social image at a much lower cost. However, we discovered not anyone who purchased have made evaluation for the product. A great number of consumers choose anonymous buying, and the evaluation is made by system default. This may explained by one of our respondents:

"In our company, she always wears high-end brand, and never shall the shopping experience with us. On one occasion, I accidently see her web favorite was full of Tmall shops. Well, in fact she's also a big fake fan…"(Female, Pheobe) Saving face in public boost consumers without financial capacity to buy and use counterfeits of well-known luxury brands. However, the great social risk of losing face occurred if others find out that they use counterfeits. Therefore, the face consciousness represents as a double-edged sword for consumers of counterfeits.

4.2 Personal Aspect 4.2.1 Fashion Consciousness [START_REF] Nam | The fashion-conscious behaviours of mature female consumers[END_REF] defined fashion consciousness as "a person's degree of involvement with the styles of fashion products" (p. 103). Researchers have further referred to fashion-conscious consumers as those individuals who are characterized by a deeper interest in fashion brands and products as well as in their physical appearance [START_REF] Gutman | Fashion Life Style, Self-Concept, Shopping Orientation, and Store Patronage: An Integrative Analysis[END_REF][START_REF] Summers | The identity of women's clothing fashion opinion leaders[END_REF]. As the appearance and visibility are particularly sold for fashion items, and the majority respondents felt that the visual brand cue is important rather than any intrinsic feature of the item itself. We propose that a fashion component is important than the functional importance for those consumers of counterfeiting luxury brands.

"I think it is okay, I know is a fake but it's the same style." (Male, Liang)

"The appearance is important when I selected a handbag." (Female, Zixi)

"I bought it because I liked the style and the design and it was not because of the brand." (Male, Hu)

The consumer today can enjoy the same lifestyle at a lower price through a counterfeited version of highfashion luxury product. With a dynamic progress of technology, some counterfeits became as real as possible to the genius. Consumers of counterfeit luxury goods do not view low quality negatively, but pay great positive attention to the appearance of the copycat.

"I'm searching for the model." (Female, Xiao e)

The fashion item is always relative with a short product life cycle, they have to go out of fashion after a certain period, and most people are generally unwilling to spend an exorbitant amount of money on them. The literature conceptualized brand experience as sensations, feelings, cognitions, and behavioral responses evoked by brand-related stimuli [START_REF] Brankus | Brand experience: What is it? How is it measured? Dose it affect loyalty[END_REF].This means brand experience include the psychological and internal consumer reactions. The hedonic dimension of luxury brand could provide the subjective emotional benefit such as sensory gratification and inner pleasure. A counterfeited luxury one may fail to provide such experience due to the low quality associated. But our respondents emphasized that they can obtain really good counterfeits.

"There are many authorized factory in China, we all know that these branded thing were made in china. We can easily find a super well made fake." (Female, Candy) "I've heard of someone in the province of Zhejiang. He produces the counterfeit level AAA that the copy was promised as possible as the same of the genuine article. I think he just knows the manager or someone in the authorized brand factory, and they gave him some important raw and processed materials." (Male, Ming) When two products look alike, such as the counterfeit brand and its real version, they are often perceived as being similar [START_REF] Shocker | Product Complements and Substitutes in the Real World: The Relevance of "Other Products[END_REF].Respondents assumed that luxury brand counterfeit offers a superior quality than non-luxury brand. We found a lot of them know well the originals brands through different channel. Since the exorbitant price hinder their purchase decision, they carefully chose counterfeits version to project a desired brand experience, and the usage of a good counterfeit brand can increase their knowledge of the originals.

"I know the original brand. It costs too expensive and I resort to a copy one." (Female, Lisa) "I wish I could have the real one, but I'm also happy when I wear it (counterfeit version)." (Female, Susu) Our interviews show the consumers satisfied if the fake was similar to the real version. That indicates the luxury counterfeit version could also contribute to consumer's hedonic needs. For this type of consumer, a super well made counterfeited one can provide a similar brand experience but has been bought at an incredibly lower price.

Adventure

Hirschman and Holbrook [START_REF] Hirschman | Hedonic Consumption: Emerging Concepts, Methods and Propositions[END_REF] defined hedonic consumption as those facets of consumer behavior that relate to the multisensory, fantasy and emotive aspects of product usage experience. We found the majority respondents smiled and some even laughed during the interview. They are willing to talk about their counterfeit experience, and the ton is fulfilled with enjoyment.

"With my girlfriends, we love to discuss the model, compare the quality and the copy level. If someone finds a new store that selling good version, we'll go to look for it together. That's totally fun!" (Female, Susu) Not only the funny emotion was expressed during the interview, the participants also described a type of nervous and excited feeling. Their experiences turned to like great stories:

"You know the Qipuroad? If you go shopping there, some stranger will pass by you and whispered 'do you want to buy bags?' Then he will lead you to a store where the sliding wall is closed. After we entered, we saw all the stuff, different models of Gucci, Louis Vuitton, and Channel …that's fantastic! But I do feel nervous, especially when he put the sliding wall closed!" (Female, Cherry) Obviously, there are emotions of pleasure and enjoyment related to the counterfeit consumption experience. But we chose to nominate this theme as "adventure" because the fear of being recognized using a fake also exist during the processes. One of our respondents -Cherry's sense of nervous seemed to due to the illegal aspect of counterfeit experience. Generally, the participant are willing to describe their experience of counterfeit brand because they already achieving the positive benefit or emotion from this type of activities. The enjoyment occurs only when the copy is not discovered. On the one hand, they are having fun with the counterfeits; on the other hand, they feel nervous if the copy is exposed by others.

"I have originals and also fakes. One day when I shopping with my friend, he asked me to test the brand salesman that if he can recognize the real and fake. I said the zipper puller is broken and showed him my fake.

You know what? He just didn't discover it!" (Female, Sky) Thus, just because of this illegal and social tension aspect, the counterfeit contribute to produce endless fun for those consumers.

Discussions, Limitations and Future Research

Arellano [START_REF] Arellano | Informal-underground retailers in less-developed countries: An exploratory research from a marketing point of view[END_REF] states that the exchange of counterfeit products doses not aim to deceive consumers, but to satisfy them. This exploratory study analyses the various ways that consumer draw the external and internal meaning from the consumption of counterfeit luxury brand product. The respondents attained not only the social image that counterfeits help them to make, but also a psychological benefit and value through consumption.

From the interpersonal aspect, the luxury brand's meaning and signaling effect is created through the same product design by the fake versions. The counterpart must represent as real as possible to the originals for a successful meaning transference. A necessity of being similar with the social group and a wish to get status pushed consumers seek for luxury brands. The reason why consumers resort to counterfeits due to the unwilling or unable to pay for the real. Hence, the counterfeit is depending on the existence of the luxury itself. In this context, the "loud luxury", which logo that are visible recognizable are more common place in counterfeit, because the logo interpret meaning and manifest the desired status for the users. If counterfeit don't appear the same design as the originals, if the consumers don't use them in the creation of their identities and the fulfillment of social needs, there would be no consequent market for counterfeit.

The importance of interpersonal issues requires the counterfeit represent as similar as the real. The nature of counterfeit activities depends on the need or desire for the real brand. However, the value consumers draw from a counterfeit as personal meanings is much more linked to the essence of illegal product itself. Compared to original luxury brand, counterfeit luxury brand product represent as more reasonable and more customer-oriented. The personal oriented consumers have a low need to display social status, but manifest much more a self-interest orientation. The process of purchasing and consumption is fun and enjoyable, a much cheaper counterfeit represent a smart choice that made additional happiness for consumers.

This study contributes to understand the consumption of luxury counterfeits by establish luxury concept as an antecedent. The correlation between counterfeit and luxury product demonstrate the importance of social issues and the obtainment of personal gratification. The limitations based on the sample size, which was not large enough to draw strong conclusion. As our respondents are extremely price conscious, an investigation with the consumers of high socio-economic identity might get different result. It would be also interesting to investigate the social risk of being "caught" wearing copycat product for future research.

Table 1 .

 1 Profiles of participants in the study

	Number	Name	Gender	A ge	C i t y	Occupation
	1	Zixi	Female	24	Shanghai	College student
	2	Juan	Female	28	Shanghai	Professor
	3	Xian	Female	25	Shanghai	Professor
	4	Fang	Male	34	Shanghai	Engineer
	5	Zhihui	Male	35	Shanghai	Manager
	6	Helene	Female	32	Beijing	Accounting
	7	Hu	Male	40	Beijing	Manager
	8	Liang	M a l e	2 6	B e i jing	Salesman
	9	Lisa	Female	20	Beijing	College student
	10	Yi	Female	49	Wuhan	Interpreter
	11	Susu	Female	39	Wuhan	Professor

  Our respondents argument how rational they are, and the cheaper copy is a smart choice, cause the money they save can invest to a new counterfeit version at the next season. Consumers were inclined to purchase a much cheaper counterfeit version of expensive luxury brands to stay up to date and show their tastes.

	"You could never keep the same speed with the fashion. I buy a new jacket today, it will be old-fashioned next
	autumn, and I should buy another new jacket with the a little improve of its cuff for example. But this tiny change
	is changed in our eyes." (Female, Yingni)
	"I have an original Louis Vuitton handbag; it's the classic style that will never go out of fashion. But if it's a
	trendy one, a new collection, I'll choose the fake one." (Female, Pheobe)
	4.2.2 Brand Experience


Alexey Ph Polikarpov

Irina A Graur

Heat and mass transfer in a rarefied gas confined between its two parallel condensed phases

The heat and mass transfer in a rarefied gas between its two parallel condensed phases is considered on the basis of linearized and non-linear S-model kinetic equations. The profiles of the macroscopic parameters in the gap between gas-liquid interfaces are obtained for several Knudsen numbers and for the cases of complete and non-complete evaporation and condensation. The linearized Navier-Stokes equations and energy equation, subjected to the temperature and pressure jump boundary conditions, are solved analytically and the expressions for the macroscopic parameters are obtained. The comparison of three approaches allowed us to establish the limit of the application of the linearized approaches in term of the saturation temperature ratio between the condensed phases.

Introduction

Evaporation and condensation phenomena are interesting not only from the fundamental point of view, but they are also largely present in various industrial applications. During the evaporation and condensation the liquid-vapor interface is in non-equilibrium state and its detailed modeling requires the use of the molecular based approaches, like the kinetic theory of gases [START_REF] Sone | Kinetic Theory and Fluid Mechanics[END_REF] or the molecular dynamics [START_REF] Meland | Molecular dynamics simulation of the inverted temperature gradient phenomenon[END_REF]. The adequate description of the evaporation-condensation was in focus of interest from a long time [START_REF] Hertz | Uber die verdungstung der flussigkeiten, inbesondere des quecksilbers, im luftleeren raume[END_REF], [START_REF] Knudsen | Die maximale verdampfungsgeschwindigkeit des que-cksilbers[END_REF], [START_REF] Schrage | A Theoretical Study of Interphase Mass Transfer[END_REF], [START_REF] Pao | Application of kinetic theory to the problem of evaporation and condensation[END_REF], [START_REF] Pao | Temperature and density jumps in the kinetic theory of gases and vapors[END_REF], [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF], [START_REF] Sone | Kinetic Theory and Fluid Mechanics[END_REF]. However, the complete understanding of these phenomena is just far to be completely achieved. In addition, the implementation of the kinetic models or the molecular dynamics based approaches is not so easy for the practical problems. This is why it is also interesting to use the continuum approaches, like the classical Navier-Stokes equations, subjected with the temperature and pressure jumps boundary conditions [START_REF] Cipolla | Kinetic theory of condensation and evaporation. II[END_REF], [START_REF] Onishi | Kinetic theory of slightly strong evaporation and condensation -hydrodynamic equation and slip boundary condition for finite reynolds numberevaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF] to simulate the evaporation-condensation problems. Recently the alternative macroscopic approach based on the regularized 13 moments equations [START_REF] Struchtrup | Evaporation boundary conditions for the r13 equations of rarefied gas dynamics[END_REF] was also proposed and applied for the simulation of the evaporation and condensation phenomena.

The main objective of this study is the development of the numerical approach to simulate the behavior of the vapor phase between two parallel liquidvapor interfaces. Contrarily to the authors of Ref. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF], where the BGK equation is implemented, the linearized and non-linear S-model [START_REF] Shakhov | Generalization of the Krook kinetic relaxation equation[END_REF] kinetic equations are solved numerically to simulate the evaporation-condensation phenomena appearing between these two plane condensed phases. The main advantage of the S-model is its capacity to provide the correct Prandtl number, equals to 2/3, for the monatomic gases, which can be important, when the problems involving the vapor flows and the heat flux are considered. In addition, the linearized Navier-Stokes equations and energy equation subjected to the temperature and pressure jumps boundary conditions [START_REF] Cipolla | Kinetic theory of condensation and evaporation. II[END_REF], [START_REF] Onishi | Kinetic theory of slightly strong evaporation and condensation -hydrodynamic equation and slip boundary condition for finite reynolds numberevaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF] are also solved analytically and the simple relations are obtained for the profiles of the macroscopic parameters of vapor in the gap between two condensed phases. The influence of evaporation-condensation coefficient on the macroscopic parameters is also analyzed. Comparing the results obtained by three approaches we establish the limit of the applicability of the linearized kinetic equation as well as of the linearized Navier-Stokes equations for the simulation of the evaporation and condensation phenomena.

Problem statement

We consider two parallel plane condensed phases at rest, maintained at temperature T 1 and T 2 , on the bottom (y = 0) and top (y = H) interfaces, respectively, y is the coordinate normal to both condensed phases. Let p 1 and p 2 be the saturation gas pressures at temperatures T 1 and T 2 , (T 1 > T 2 ), respectively. We investigate here the behavior of the monatomic gas motion caused by evaporation and condensation on the condensed phases, first on the basis of the kinetic theory.

If we consider that the ratios p 1 /p 2 and T 1 /T 2 are independent, then the evaporation-condensation problem is completely characterized by three parameters: p 1 /p 2 and T 1 /T 2 ratios and by the rarefaction parameter δ. However, in practice, the saturation pressure and temperature are related by the Clausius-Clapeyron equation [START_REF] Landau | Statistical physics[END_REF] and therefore cannot take the arbitrary values. In this case the evaporation-condensation problem depends on two parameters only: the slop of p (T ) function and the rarefaction parameter δ. In the following we consider both cases: first the situation when two parameters are independent, and then, the cases of Argon and several other monatomic gases, where the saturation pressure and temperature are related through the Clausius-Clapeyron equation.

The distance H between two interfaces is taken as the characteristic dimension of the problem, so the rarefaction parameter δ is defined as following: where is the equivalent mean free path:

δ = H , (1)
= µ 2 v 2 p 2 , (2)
which is defined using the reference parameters with subscript 2, corresponding to the upper plate. The choice of the parameters related to the upper plate as the reference parameters was done without lost of generality. In Eq. ( 2)

µ 2 = µ(T 2 )
is the dynamic viscosity of the vapor phase

µ(T ) = µ 2 T T 2 , ( 3
)
v 2 is the most probable molecular speed,

v(T ) = √ 2RT , (4)
calculated at the temperature

T 2 : v 2 = v(T 2 ), R = k B /m
is the specific gas constant, k B is the Boltzmann constant, m is the molecular mass.

Modeling based on the kinetic equation

To model the evaporation and condensation phenomena between two parallel condensed phases maintained at different temperatures first the S-model kinetic equation [START_REF] Shakhov | Generalization of the Krook kinetic relaxation equation[END_REF] is used. The evaporation-condensation problem between two parallel condensed phases is considered here as steady state and one dimensional in physical space, so the S-model kinetic equation is written as

v y ∂f ∂y = υ f S -f , (5)
where f (y , v) is the one particle velocity distribution function, v = (v x , v y , v z ) is the molecular velocity vector, υ is the collision frequency, υ = p /µ. In the frame of S-model the equilibrium distribution function f S in Eq. ( 5) is defined as following

f S (y , v) = f M 1 + 2mVq 15n (y )(k B T (y )) 2 mV 2 2k B T (y ) - 5 2 , (6)
here T (y ) is a gas temperature, n (y ) is a gas number density, u = (0, u , 0) is a bulk velocity vector, V = vu is the peculiar velocity vector, q = (0, q , 0) is a heat flux vector, f M is the Maxwellian distribution function [START_REF] Kogan | Rarefied gas dynamics[END_REF]. The macroscopic parameters are defined as follows:

n (y ) = f (y , v)dv, u (y ) = 1 n f (y , v)v y dv, (7)
T (y ) = m 3k B n f (y , v)V 2 dv, q (y ) = m 2 f (y , v)V 2 (v y -u y )dv . (8)
By integrating Eq. ( 5) multiplied by (1, v y , v 2 i ) we obtain the integrals expressing the number of molecules, y component of momentum and energy transported in y direction per a unit area of a plan, parallel to the condensed phases. The evaporation flow rate, expressed in the number of molecules per time and per surface unit, J n , and the evaporation mass flow rate, expressed in kilogram per time and surface units, J ρ , are defined as followings:

J n = v y f (y , v)dv, J ρ = m v y f (y , v)dv. (9)
The second definition of the evaporation mass flow rate is usually provided from the experiments. The energy flux is defined as

J E = v y v 2 f (y , v)dv. ( 10
)
The constancy of the integrals J n and J E will be further used to estimate the accuracy of the numerical calculations and for the comparison with the results of Ref. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF].

Linearized S-model equation

To linearize S-model kinetic equation [START_REF] Schrage | A Theoretical Study of Interphase Mass Transfer[END_REF] we assume that deviations between the temperatures of two condensed surfaces and the corresponding saturation pressures are small:

X P = p 1 -p 2 p 2 1, X T = T 1 -T 2 T 2 1. (11)
In previous expressions X P and X T can be related to the thermodynamic forces [START_REF] Sharipov | Onsager-casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. II Application of the theory for single gas[END_REF]. As it was mentioned in Section 2, for a given gas the pressure and temperature differences are coupled by the relation

p 1 -p 2 = β(T 1 -T 2 ), ( 12
)
where β is a positive constant corresponding to the slop of the Clausius-Clapeyron curve at T 2 , so X P and X T are not independent quantities. However here we consider two forces separately, to see clearly the impact of each force to the evaporation-condensation process.

For further derivation we introduce the following dimensionless quantities:

y = y H , c = v v 2 , u = u v 2 , n = n n 2 , T = T T 2 , p = p p 2 , q = q p 2 v 2 .
(13) When X P and X T are small enough compared to 1 we can linearize the distribution function as following:

f = f M 0 (1 + h P X P + h T X T ), (14)
where h P and h T are the perturbation functions, related to the pressure and temperature difference, respectively, f M 2 is the absolute Maxwellian distribution function with the reference number density, n 2 and temperature, T 2 :

f M 2 = n 2 m 2πk B T 2 3/2 exp(-c 2 ). (15)
For the real gases, for any value of β we can obtain the solution as a superposition of two functions as h = (βh

P + h T )∆T, (16)
where ∆T = T 1 -T 2 . However, as it was mentioned above, we consider here X P and X T as independent quantities. The S-model kinetic equation [START_REF] Schrage | A Theoretical Study of Interphase Mass Transfer[END_REF] in its linearized form can be separated into two equations with reference to the thermodynamic forces X P and X T :

c y ∂h i ∂y = δ ν i + 2c y u i + c 2 - 5 2 τ i + 4 15 q i c y -h i , i = P, T. (17)
The macroscopic parameters in Eq. ( 17) can be calculated through the following relations:

n = 1 + ν P X P + ν T X T , u = u P X P + u T X T , (18)
T = 1 + τ P X P + τ T X T , q = q P X P + q T X T . (19)
The dimensionless moments of the distribution function are given as:

ν i = 1 π 3/2 exp(-c 2 )h i dc , (20)
u i = 1 π 3/2 exp(-c 2 )h i c y dc , (21)
τ i = 1 π 3/2 exp(-c 2 )h i 2 3 c 2 -1 dc , (22)
q i = 1 π 3/2 exp(-c 2 )h i c y c 2 - 5 2 dc , i = P, T. (23)
The distribution function of the molecules outgoing from the condensed phases is supposed to be decomposed into two parts [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF]: σ e part of molecules evaporates from the surface with saturation parameters while another part, (1-σ c ), reflects diffusively from the surface. We assume here that the evaporation σ e and the condensation σ c coefficients are equal each to other σ = σ e = σ c . The linearized form of evaporation-condensation boundary conditions reads: for the bottom surface, y = 0:

h P = σ + (1 -σ)ν Pw , h T = σ c 2 - 5 2 + (1 -σ) c 2 - 3 2 + ν Tw , (24)
ν Pw = - 2 π cy<0 h P c y dc, ν Tw = - 1 2 - 2 π cy<0 h T c y dc (25)
and upper surface, y = 1:

h P = (1 -σ)ν Pw , h T = (1 -σ)ν Tw , (26)
ν Pw = 2 π cy>0 h P c y dc, ν Tw = 2 π cy>0 h T c y dc. (27)
To reduce the number of variables in a velocity space we use the standard procedure [START_REF] Chu | Kinetic-theoretic description of the formation of a shock wave[END_REF]: we multiplied first Eqs. [START_REF] Broadwell | Shock structure in a simple discrete velocity gas[END_REF] successively by 1 and (c 2 z -1 2 ), and then we integrated them over c z velocity. Doing so we obtain the system of two equations for reduced distribution functions:

c y ∂φ i ∂y = δ ν i + 2c y u i + c 2 x + c 2 y -1 τ i + 4 15 q i c y c 2 x + c 2 y -2 -h i , (28)
c y ∂ψ i ∂y = δ 1 2 τ i + 2 15 q i c y , i = T, P, (29)
where these reduced distribution functions read:

φ i = 1 √ π ∞ -∞ exp(-c 2 z )h i dc z , ψ i = 1 √ π ∞ -∞ exp(-c 2 z ) c 2 z - 1 2 h i dc z , i = T, P . (30)
The system of equations ( 28), (29) with the boundary conditions, Eqs. ( 24) - [START_REF] Chernyak | Non-linear effects in gases in couette problem[END_REF], is solved by the Discrete Velocity Method (DVM) [START_REF] Broadwell | Shock structure in a simple discrete velocity gas[END_REF], see Section 4.

Thermodynamic analysis

The author of Refs. [START_REF] Sharipov | Onsager-casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. I General theory for single gas[END_REF], [START_REF] Sharipov | Onsager-casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. II Application of the theory for single gas[END_REF] have obtained the Onsager-Casimir reciprocity relation in general form for rarefied gas flows. It was shown [START_REF] Sharipov | Onsager-casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. II Application of the theory for single gas[END_REF] that for the case of evaporation-condensation of a gas between two parallel condensed phases the "fluxes" corresponding to the driving "forces" are defined as follows:

J P = n 2 u , J T = 1 k B T 2 q , (31)
where u y and q y do not depend on y . The thermodynamic fluxes are related to the thermodynamic forces in the matrix form:

J P J T = Λ P P Λ P T Λ T P Λ T T × X P X T . ( 32
)
In this case, the Onsager-Casimir relation, Λ P T = Λ T P , yields the coupling between the mass flux caused by temperature drop and the thermal flux caused by the pressure drop [START_REF] Sharipov | Onsager-casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. II Application of the theory for single gas[END_REF].

Non-linear S-model equation

To establish the limit of applicability of linearized S-model kinetic equation to model two condensed surfaces problem we solve numerically the non-linear S-model kinetic equation. By using Eqs. ( 13) the dimensionless form of Eq. ( 5) reads:

c y ∂f ∂y = δn √ T f S -f . (33)
As for the linearized case, see Section 3.1, we assume that only one part σ of the incident molecules evaporates immediately and (1 -σ) part reflects from the surface diffusively. Therefore the dimensionless boundary conditions for the distribution function of the reflected molecules at the liquid-gas interfaces can be written as

y = 0 c y > 0, f = (σn s1 + (1 -σ)n r1 )f M 1 , f M 1 = 1 π 3/2 p 1 p 2 T 2 T 1 5/2 exp - T 2 T 1 c 2 y = 1 c y < 0, f = (σn s2 + (1 -σ)n r2 )f M 2 , f M 2 = 1 π 3/2 exp -c 2 . ( 34
)
Here n si , i = 1, 2 is the number density of the saturation vapor near each plane interface, calculated from the equation of state using the values of the saturation temperature and corresponding saturation pressure. The number density n ri , i = 1, 2, can be calculated from the impermeability condition on the corresponding interface:

n r1 = -2 π T 1 T 2 cy<0 c y f dc, n r2 = 2 √ π cy>0 c y f dc. ( 35
)
To minimize computational efforts, the c z variable is eliminated by introducing the reduced distribution functions as in [START_REF] Graur | Comparison of different kinetic models for the heat transfer problem[END_REF].

Numerical technique

To solve numerically both linearized and non-linear S-model kinetic equation the same Discrete Velocity Method (DVM) [START_REF] Graur | Comparison of different kinetic models for the heat transfer problem[END_REF] is used. First, the Discrete Velocity Method is applied to split the continuum (-∞, ∞) molecular velocity space c x , c y in the governing equation ( 5) into discrete velocity sets c xm , c y k where m, k = -N c , .., -1, 1, 2, .., N c . These velocities, c xm and c y k , are taken to be the roots of the Hermite polynomial of order N c . Then, the set of 2N c kinetic equations, corresponding to 2N c values of discrete velocities, is discretized in space by Finite Difference Method (FDM). In presented here simulations N c is taken to be equal to 20, so the number of points in the molecular velocity space is equal to 40 in each direction. The grid-independence is checked with finer grid with N c = 25 points: the difference in all macroscopic quantities was less than 1%.

The spacial derivatives are approximated by the first-order accurate upwind type scheme, see Ref. [START_REF] Graur | Comparison of different kinetic models for the heat transfer problem[END_REF] for more details. The number of uniformly distributed numerical grid points in physical space, N y , is equal to 1000. This number of grid points insures the calculation accuracy of 1%. The convergence criterion for iteration process is employed in following form

min i q (l+1) i -q (l) i / q (l) i ≤ ε, ( 36
)
where l is the iteration number, i = 1, N y ; ε = 10 -7 is used here. The described above numerical approach was used to solve the linearized and non-linear S-model kinetic equation for the large range of the rarefaction parameter and for several values of the evaporation coefficients. The obtained results are discussed in Section 6.

Solution of the linearized Navier-Stokes equations with the jump boundary conditions

The numerical solution of the model kinetic equation provide very accurate information about the evaporation-condensation process. However, from the practical point of view it is interesting to have the simple and easy to use estimation of the vapor behaviors. Below the analytical solution of the linearized Navier-Stokes equations with the energy equation subjected with the temperature and pressure jump boundary conditions is derived.

For one dimensional in the physical space case, considered here, the linearized Navier-Stokes equations, completed with the energy equations read:

du y dy = 0, dp dy = 0, d 2 T dy 2 = 0. ( 37
)
To take into account the evaporation-condensation on the upper and down interfaces, equations (37) have to be subjected to the special boundary conditions [START_REF] Cipolla | Kinetic theory of condensation and evaporation. II[END_REF], [START_REF] Lang | Evaporation and condensation for general gas-liquid surface scattering[END_REF], [START_REF] Sone | Kinetic theory of evaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF], [START_REF] Onishi | Kinetic theory of slightly strong evaporation and condensation -hydrodynamic equation and slip boundary condition for finite reynolds numberevaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF]. We will use here the most complete form of the boundary conditions suggested by the authors of Ref. [START_REF] Cipolla | Kinetic theory of condensation and evaporation. II[END_REF] and which relate the temperature and pressure jumps on the liquid-vapor interface, the driving "forces", with the "fluxes", evaporation rate and heat flux through interface. For the down plate this relation between "fluxes" and "forces" reads:

(p 1 -p )/p 1 (T 1 -T )/T 1 = a 11 a 12 a 21 a 22 × n 1 u q /(k B T 1 ) . ( 38
)
The heat flux in system (38) reads:

q = -κ dT dy , ( 39
)
where κ is the thermal conductivity coefficient, which can be calculated as:

κ = γ γ -1 1 P r Rµ, ( 40
)
where γ is the heat capacity ratio, P r is the Prandtl number. For the monatomic gas considered here, γ = 5/3 and P r = 2/3. The matrix in Eq. ( 38) is the inverse Onsager matrix, so it is symmetric and a 12 = a 21 . For further derivation it is convenient to introduce the reduced form:

a ij = a ij /(n 1 v 1 ) (41)
The values of these coefficients were calculated in Refs. [START_REF] Cipolla | Kinetic theory of condensation and evaporation. II[END_REF], [START_REF] Lang | Evaporation and condensation for general gas-liquid surface scattering[END_REF], [START_REF] Sone | Kinetic theory of evaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF], [START_REF] Onishi | Kinetic theory of slightly strong evaporation and condensation -hydrodynamic equation and slip boundary condition for finite reynolds numberevaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF] for the monatomic gases and for the case of the diffuse reflection of the molecules which do not condense on the surface:

a 11 = 2 √ π σ -1.4195228, a 12 = a 21 = 0.446658, a 22 = 1.0422028. ( 42
)
It is worth to note that only a 11 coefficient depends on the evaporation coefficient σ.

In our case of the evaporation-condensation between two parallel condensed surfaces maintained at different temperatures the boundary conditions, Eqs. (38), are write as following:

for the bottom condensed surface y = 0:

p = p 1 1 -a 11 u v 1 -a 12 q v 1 p 1 , T = T 1 1 -a 21 u v 1 -a 22 q v 1 p 1 , (43)
for the upper condensed surface y = 1:

p = p 2 1 + a 11 u v 2 + a 12 q v 2 p 2 , T = T 2 1 + a 21 u v 2 + a 22 q v 2 p 2 , (44)
Taking into account the conditions of the smallness of the pressure and temperature gradients, Eqs. [START_REF] Struchtrup | Evaporation boundary conditions for the r13 equations of rarefied gas dynamics[END_REF], linearized system of the Navier-Stokes equations (37) with the boundary conditions (43), (44) was solved analytically. The analytical dimensionless expressions read:

T (y) = Ay + B, (45)
A = a 12 a 11 CX P -CX T , B = 1 - a 12 2a 11 CX P -1 + η 2δ M CX T , (46)
p = 1 + 1 2 X P , (47)
u = 1 2a 11 + a 22 a 11 η 2δ CX P - a 12 a 11 η 2δ CX T , (48)
q = - a 12 a 11 η 2δ CX P + η 2δ CX T , (49)
J n = u, (50)
η = γ γ -1 1 P r , M = a 22 - a 2 12 a 11 , C = 1 + η δ M -1 . (51)
From the analysis of analytical solutions, Eqs. ( 45) -( 49), it is clear that the temperature profile is linear while the evaporation velocity, pressure and heat flux in the gap are constant. The form of the analytical solutions is analogous to the form of Eqs. [START_REF] Sharipov | Onsager-casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. I General theory for single gas[END_REF], where the temperature, velocity and heat flux are split into two parts: one part related to the pressure gradient and the second one related to the temperature gradient. In Section 6 both solutions: the analytical expressions, Eqs. ( 45) -(49), and the numerical solutions of the linearized Smodel equation are compared in their separated (according to the corresponding gradients) form.

It is worth to note that the Onsager-Casimir relation leads here to the equality between the mass flux caused by the temperature drop u T and the thermal flux caused by the pressure drop q P [15]:

u T = q P , (52)
see analytical expressions, Eqs. ( 48), (49).

From the boundary conditions on the down plate, Eqs. (43), we can see that the temperature jump on the liquid-vapor interface depends on both evaporation velocity and heat flux. According to Eqs. (43) the temperature of the gas over an evaporating surface is lower than that of the condensed phase [START_REF] Sone | Kinetic theory of evaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF]. Because of this temperature drop the "negative temperature gradient" phenomenon is observed when the rate of evaporation (or X P /X T ) is sufficiently large [START_REF] Pao | Application of kinetic theory to the problem of evaporation and condensation[END_REF], [START_REF] Sone | Kinetic theory of evaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF]. However, when X P /X T is very small, the mass flow rate is directed from the colder to the hotter condensed phase. This situation was observed in several recent measurements, see [START_REF] Ward | Interfacial conditions during evaporation or condensation of water[END_REF], [START_REF] Badam | Experimental and theoretical invesigations on interfacial temerature jumps during evaporation[END_REF], [START_REF] Kazemi | Experimental and numerical study of the evaporation of water at low pressures[END_REF]. However in all these papers the positive temperature jump the vapor temperature is higher as that of the liquid phase, was observed while the linearized approach predicts the negative one.

Results and discussion

We present the results in following manner. First, the results obtained numerically from the linearized S-model kinetic equation and the analytical solution of the linearized Navier-Stokes equations subjected to the jumps boundary conditions are considered in order to show the limits of the applicability of this analytical solution in term of the rarefaction parameter. The cases of complete and non-complete evaporation-condensation are also presented and compared. Then, the linearized and non-linear numerical solutions of the S-model kinetic equation are compared to show the limits of the linearized approach application. Finally, the numerical solutions obtained from the S-model kinetic equation are compared with those giving by the BGK model in linearized and non-linear formulations.

Linearized approaches, complete evaporation

In this Section we will present the results obtained from the numerical solution of the linearized S-model kinetic equation and the analytical solution of the linearized Navier-Stokes equations completed by the pressure and temperature jump boundary conditions. Both solutions are determined only by two parameters: the gas rarefaction δ and the evaporation-condensation coefficient σ. Figure 2 shows the variations of the macroscopic vapor parameters: dimensionless number density and temperature in the gap between two condensed phases, obtained from the numerical solution of the linearized S-model equation. The case of the complete evaporation-condensation is considered, σ = 1, for four values of the rarefaction parameter: δ = 0.1, 1, 10 and 100. The macroscopic parameters are separated in two parts: one part due to the pressure gradient (subscript P ) and another one due to the temperature gradient (subcript T ), so we consider here these two gradients to be independent. From Figure 2 we can see that both dimensionless number density and temperature vary quasi linearly in the gap. The profiles of the number density, related to the pressure gradient, ν P , are almost coincide for the values of the rarefaction parameter δ = 1, 10 and 100, while the profiles of the number density variation due to the temperature gradient, ν T , are different for different values of the rarefaction parameter. The profiles of the dimensionless temperatures, τ P and τ T , show the "mirror" behavior in the gap: τ P increases while τ T decreases from down to upper condensed phases.

The evaporation velocity and the heat flux are constant in the gap, so their average over the gap values for different gas rarefaction are presented in Table 1. It is to note that the gas velocity generated by temperature gradient, u T , is oriented in the direction opposite to the evaporation flow generated by the pressure gradient, u P . In addition, the values of u T velocity is much lower than that of u P velocity, especially in the case of large value of rarefaction parameter, δ = 100. In this regime, the evaporation rate is engendered essentially by the pressure gradient. The evaporation velocities, u P and |u T |, decrease with rarefaction parameter increases. It can be seen from Table 1 that the heat flux generated by the temperature gradient, q T , is positive and directed from the hot bottom plate to the cold upper plate, while the heat flux generated by the pressure gradient, q P , has the opposite sign and is much smaller in absolute value. The values of the heat flux related to the pressure gradient, q P , are not provided in Table 1, because their values can be found from the Onsager-Casimir reciprocity relations q P = u T , see Eq. ( 52). Both heat fluxes, |q P | and q T , decrease with δ increasing, see Table 1.

S-model

By analyzing the results provided in Table 1 one can conclude that the difference between the analytical solution of the linearized Navier-Stokes equations subjected by the jump boundary conditions and the numerical solution of the S-model kinetic equation is very small. The difference between both solutions for δ = 10 is of the order of 0.1%. For larger value of the rarefaction parameter, δ = 1, this difference keeps surprisingly small, around 5%. Therefore, analytical expressions, Eqs. (45) -(49), can be used for estimation of the evaporation rate with a good accuracy up to δ = 10.

Linearized approaches, non-complete evaporation

Two approaches, the numerical solution of the S-model kinetic equation and the analytical solution of the Navier-Stokes equations subjected to the jump boundary conditions, are analyzed now for the case of non-complete evaporation. The cases of σ = 0.2, 0.5, 0.8 and 1 are considered. Figure 3 shows the influence of the non-complete evaporation-condensation on the macroscopic profiles of the number density and temperature, obtained from the numerical solution of the linearized S-model kinetic equation. The results for four values of the evaporation coefficient, σ = 0.2, 0.5, 0.8 and 1, are presented on Fig. 3 for rarefaction parameter δ = 10. It is worth to note that the profiles of the number density and temperature, caused by the temperature gradient, ν T and τ T , are independent from the evaporation coefficient, so they depend only on the temperature gradient between the condensed surfaces.

As it was mentioned above, the evaporation velocity and the heat flux are constant in the gap, so their averaged over the gap values, obtained from the numerical solution of the linearized S-model kinetic equation and the analytical solution of the linearized Navier-Stokes equations, Eqs. (45) -(49), are presented in Table 2 for the case of non-complete evaporation, σ = 0.5, and different values of the rarefaction parameter δ. Comparing the values of the evaporation velocity, provided in Tables 1 and2 one can conclude that the decrease in the evaporation coefficient decreases the absolute value of both evaporation velocities. The heat flux caused by temperature gradient, q T , remains practically the same with σ decreases, while the values of the heat flux caused by pressure gradient, q P = u T , see Table 2, decrease in approximately two times when the evaporation coefficient reduces from 1 to 0.5, see also Table 1. This behavior of the heat flux is in agreement with the temperature behaviors, presented on Fig. 3: temperature τ T , due to the temperature gradient, remains the same when the evaporation coefficient changes. It is worth to note that as for the case of the complete evaporation, the agreement between numerical solution of the linearized S-model kinetic equation and the analytical solution of the Navier-Stokes equation is very good: the difference between both solutions for δ = 10 is of the order of 1%.

S-model

Navier

Comparison of the BGK and S-model numerical solutions

As it was underlined in the Introduction, the S-model is chosen here because it provides the correct Prandtl number, equal to 2/3 for monatomic gases, contrary to the BGK model, for which the Prandtl number is equal to 1. In this Section both linear and non-linear BGK and S-model solutions are compared to see the influence of the applied models on the macroscopic profiles.

First, the numerical solutions of the linearized BGK and S-model equations are shown on Fig. 4 for the case of weak evaporation and condensation, T 1 /T 2 = 1.001, p 1 /p 2 = 1.01 and for the large value of the rarefaction parameter, δ = 250. It can be seen from Fig. 4 that both linearized solutions are in good agreement, except for the heat flux profiles, as expected. However, the heat flux profiles obtained by the linearized S-model equation and linearized Navier-Stokes equations coincide, because both models have the same Prandtl number.

Let us compare now the solutions of the non-linear BGK and S-model equations. Tables 3 and4 provide the numerical values of the dimensionless evaporation rate, Eq. ( 9), J n /(n 2 v 2 ) and energy flux, Eq. ( 10), J E /(p 2 v 2 ). Both quantities are obtained by integration of the kinetic equation, Eq. ( 5), and show the conservation of momentum and energy. Both quantities are independent from the type of the kinetic model and show the reliability of applied numerical methods to obtain the numerical solutions. The temperature and pressure ratios were chosen according to Ref. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF]. As it is clear from Tables 3 and4 very good agreement was found between the presented here results (BGK and S-model) and the BGK results from Ref. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF] for the dimensionless evaporation rate, J n and energy flux J E . The difference between three solutions is of the order of 1%.

The profiles of the macroscopic parameters, obtained from the numerical solution of the BGK and S-model equations are however slightly different, especially, as it was expected, the profiles of heat flux, which was also observed when comparing the linearized versions of both models. Figure 5 illustrates this difference in the macroscopic profiles of the evaporation velocity and heat flux in the gap between condensed phases. The comparison with the measurements are needed to conclude definitively about the advantages of one or other models.

Evaporation-condensation of Argon

In this Section we establish the applicability domain of the linearized S-model kinetic equation and of the analytical expressions (45) -(49). Contrarily to the linearized equation, to solve the non-linear S-model equation we need to fix the pressure and temperature ratios between two condensed surfaces. Usually the authors [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF] consider the variations of the saturation temperature and pressure independently, but here we consider more realistic case.

The S-model kinetic equation describes the behavior of the monatomic gases. Therefore, we fixed three temperature ratios, T 1 /T 2 = 1.001, 1.01 and 1.05 and estimated the corresponding saturation pressures for some monatomic gases, taking into account the Clayperon-Clausius equation [START_REF] Landau | Fluid Mechanics[END_REF]. The fixed temperature ratios of two condensed phases, T 1 /T 2 = 1.001, 1.01 and 1.05, lead to corresponding pressure ratios p 1 /p 2 = 1.01, 1.1, 1.55 for Krypton, starting with temperature 120K, to pressure ratios p 1 /p 2 = 1.01, 1.1, 1.57 for Argon, if we start with the temperature T 2 = 84K and to pressure ratios p 1 /p 2 = 1.004, 1.04, 1.2 for Helium. Some approximative tendency is so deduced: the increase of saturation temperature in 1% causes the increase of saturation pressure in approximately 10%. Therefore, the three sets of saturation parameters are chosen to solve numerically the non-linear S-model kinetic equation: T 1 /T 2 = 1.001, 1.01, 1.05 and p 1 /p 2 = 1.01, 1.1, 1.5 (see also two first lines in Table 5) to analyze the eventual limits of application for the linearized models.

δ T 1 /T 2 = 1.001 1.01 1.05 p 1 /p 2 = 1.01 1.1 1.5 Lin S
Lin NS Non-l S Lin S Lin NS Non-l S Lin S Lin NS Non-l S 1 0.00250 0.00245 0.00248 0.02599 0.02545 0.02469 0.15235 0.14842 0.12092 10 0.00239 0.00238 0.00237 0.02486 0.02480 0.02367 0.14573 0.14473 0.11684 100 0.00236 0.00236 0.00235 0.02458 0.02452 0.02344 0.14410 0.14310 0.11654 Table 5: Dimensionless evaporation rate Jn obtained by: the numerical solution of the linearized S-model kinetic equation (Lin S), the analytical expression Eq.( 50), (Lin NS), and the numerical solution of the non-linear S-model kinetic equation (Non-l S).

Table 5 provides the values of dimensionless evaporation rate, J n , obtained by: numerical solution of the linearized S-model kinetic equation, analytical expression, Eq.( 50), and numerical solution of the non-linear S-model kinetic equation. By comparing the results obtained from these three approaches it is clear that for two first pairs of saturation temperature/pressure values all the approaches gives very similar results. The difference between two linearized approaches does not exceed 0.2%. It is to note that the values of analytical solution of the linearized Navier-Stokes equations are provided in Table 5 also for δ = 1, even if theoretically this approach is not valid for δ < 10. But for given conditions the good agreement was found between all three approaches also for δ = 1. However, the difference between numerical solutions of the linearized and non-linear S-model kinetic equations becomes of the order of 4.6% for the temperature ratio 1.01 and this difference increases up to 20%, when the saturation temperature ratio increases up to 1.05. In the same time the agreement between two linearized approaches remains very good, of the order of 1% even for relatively large temperature ratio of 1.05. From the presented analysis, we can conclude that the linearized approaches work well for the real monatomic gases, i.e. taking into account their respective Clausius-Clayperon relations, only up to very small temperature ratio of the order of 1%, and therefore for the larger temperature ratios the non-linear approaches have to be applied.

The general tendency of the evaporation rate behavior can be derived by analyzing the results from Table 5: the evaporation rate decreases slightly when the rarefaction parameter increases for the fixed saturation temperature/pressure ratios. The evaporation rate increases more than in 10 times when the saturation temperature ratio increases in 10 times, this tendency of the proportionality remains up to temperature ratio equal to 1.05.

The numerical solutions of the linearized and non-linear S-model equations and the analytical solution of the linearized Navier-Stokes equations subjected to the pressure and temperature jumps boundary conditions, Eqs. ( 45) -( 49), are presented on Figs. 6789for two values of the rarefaction parameter δ = 10 and 100, for the case of the complete evaporation, σ = 1, and for three values of the saturation temperature/pressure ratios, presented in Table 5. The macroscopic profiles of the temperature, pressure, evaporation velocity and heat flux in the gap are provided.

It is seen from these Figures that all the temperature and pressure profiles for two temperature ratios T 1 /T 2 = 1.001 and 1.01 practically coincide for two values of the rarefaction parameter, δ = 10 and 100, except, the case, shown on Fig. 8(c), where the temperature profile becomes non-linear, in agreement with the findings of Ref. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF]. In this case, the maximum of the temperature/pressure ratio is smaller than the rarefaction parameter and the linearized solutions do not coincide with the non-linear one, even if the deviations of the macroscopic parameters from their equilibrium state is small enough and normally the linearized theory should work.

With further saturation temperature increasing, up to T 1 /T 2 = 1.05, the temperature profiles, provided by the non-linear S-model equation, start to deviate from the linear ones for both value of δ, while the pressure profiles keep their shape and only absolute values of pressure change. The profiles of the evaporation velocity and of the heat flux differ from those provided by the linearized approaches even for small temperature ratio. For larger temperature ratios, T 1 /T 2 = 1.01 and 1.05, the behaviors of the linearized and nonlinear models differ not only quantitatively, but also qualitatively. Especially it is clearly seen near the liquid boundaries for heat flux distributions, Figs. 7 and 9 and for temperature distributions, Figs. 6 and8. The pressure profiles, obtained from all models and for all considered rarefaction levels, have the same qualitative behavior: fully coincide for T 1 /T 2 = 1.001 and have a difference in 2% for T 1 /T 2 = 1.05.

It is to note that the analytical solution of the linearized Navier-Stokes equations, even subjected with the temperature and pressure jump boundary conditions, cannot provide the correct profiles of the macroscopic parameters inside thin layer near condensed phases, the Knudsen layer, see the temperature, evaporation velocity and heat flux profiles. Even for very small pressure and temperature ratios, 1% and 0.1%, respectively, the profiles given by the analytical solutions are different, see Figs. 6789(a) and (b). However, the analytical results provide acceptable accuracy of the macroscopic parameters outside of the Knudsen layer up to temperature ratio T 1 /T 2 = 1.01 and can be used for rough estimations.

An interesting remark can be done concerning the limits of validity of the linearized approach: the consideration of the evaporation and condensation phenomena reduces considerably this limit. It was found in [START_REF] Chernyak | Non-linear effects in gases in couette problem[END_REF] that the heat flux between two parallel solid plates, obtained from the numerical solution of the non-linear S-model equation, still keeps its linear behavior up to temperature ratio equal to T 1 /T 2 ≈ 3, which is much larger compared to the limit found in present analysis.

It is worth to note that the "inversion of the temperature gradient", mentioned for the first time in Refs. [START_REF] Pao | Application of kinetic theory to the problem of evaporation and condensation[END_REF], [START_REF] Pao | Temperature and density jumps in the kinetic theory of gases and vapors[END_REF], is observed in the gap for all considered here cases. With the temperature gradient inversion the temperature near the hotter condensed surface becomes lower than that near the colder one. This phenomenon is obtained in numerical simulations, when the ratio between the pressure and temperature differences, X P /X T becomes larger than 4.772337 [START_REF] Sone | Kinetic theory of evaporation and condensation -hydrodynamic equation and slip boundary condition[END_REF]. Here this criterion is satisfied for all calculated cases, so the vapor temperature is larger near the colder condensed surface, see Figs. 6 and8.

Conclusion

The linearized and non-linear S-model equations are solved for different values of the rarefaction parameters and several saturation pressures and temperatures ratios. The case of the non-complete evaporation-condensation is also considered. The comparison between the BGK and S-model numerical solutions shows the difference only in the heat flux profiles while the other macroscopic parameters have not beed impacted by the model changing. The analytical solution of the linearized Navier-Stokes equations completed by the pressure and temperature jumps boundary conditions is obtained. The numerical solutions of the linearized and non-linear kinetic equations are compared together and with the analytical solutions of the continuum equations. It was found that the analytical solutions of the Navier-Stokes equations provide the excellent agreement with the numerical solutions of the linearized BGK and S-model kinetic equations up to rarefaction parameter equal to 10, in the case of complete and non-complete evaporation-condensation. The case of the real relations between saturation temperature/pressure for some monatomic gases are considered and it was found that the linearized approaches give satisfactory results only for the temperature deviation between condensed phases of the order of 1%, but for the larger saturation temperature ratio the non-linear approach has to be applied.

Acknowledgements

The research work of one of the authors (A. Polikarpov) was funded by RFBR according to the research project no. 18-31-00194. A.Polikarpov also thanks the Ministry of Education and Science of the Russian Federation Agreement no. 02.A03.21.0006 for travel grant. Irina Graur thanks Prof. Aldo Frezzotti for the helpful discussions.

Figure 1 :

 1 Figure 1: Sketch of 1D geometry

δT

  H /T C = 1.01 1.2 P H /P C = 1.02 2 BGK S-mod BGK [8] BGK S-mod BGK [8] 0.090 0.00421 0.00420 0.00421 0.22936 0.22925 0.22891 0.903 0.00427 0.00422 0.00429 0.22151 0.22103 0.22109 9.027 0.00454 0.00450 0.00457 0.22025 0.22006 0.22030

Figure 2 :Figure 3 :

 23 Figure 2: Numerical solution of the linearized S-model equation: the variations of macroscopic quantities, dimensionless number density and temperature, both separated in two parts: due to the pressure gradient (with superscript P , (a) and (c)) and due to the temperature gradient (with superscript T , (b) and (c)) in the gap between two condensed phases for different values of the rarefaction parameter δ and for the case of the complete evaporation, σ = 1.

Figure 4 :Figure 5 :

 45 Figure 4: Temperature (a) and pressure (b), obtained from the numerical solution of linearized BGK and S-model equations, and non-linear S-model equation, T 1 /T 2 = 1.001, p 1 /p 2 = 1.01, δ = 250.

T 1 /T 2 = 1 .05, p 1 /p 2 = 1 5 Figure 6 :

 12112156 Figure 6: Comparison of three results for δ = 10 and complete evaporation, σ = 1: the numerical solution of the linearized S-model kinetic equation, the analytical solution of the linearized Navier-Stokes equations with the temperature and pressure jumps boundary conditions and the numerical solution of the non-linear S-model kinetic equation. The temperature and pressure profiles in the gap are compared for three sets of saturation temperature and pressure ratios: (a), (b) T 1 /T 2 = 1.001, p 1 /p 2 = 1.01, (c), (d) T 1 /T 2 = 1.01, p 1 /p 2 = 1.1; (e), (f) T 1 /T 2 = 1.05, p 1 /p 2 = 1.5.

Table 1 :

 1 Macroscopic parameters obtained from the numerical solution of the linearized Smodel kinetic equation and the analytical solution of the Navier-Stokes equations with the jump boundary conditions for different values of the rarefaction parameter δ and for complete evaporation, σ = 1.

	Navier-Stokes

Table 2 :

 2 Macroscopic parameters obtained from the numerical solution of the linearized Smodel kinetic equation and the analytical solution of the Navier-Stokes equations with the jump boundary conditions for different values of the rarefaction parameter δ and for noncomplete evaporation, σ = 0.5.

	-Stokes

Table 3 :

 3 Dimensionless evaporation rate Jn

		T H /T C = 1.01		1.2
	δ	P H /P C = 1.02		2
	BGK	S-mod BGK [8]	BGK	S-mod BGK [8]
	0.090 0.01395 0.01397 0.01391 0.66267 0.66293 0.66141
	0.903 0.01308 0.01320 0.01296 0.63507 0.63633 0.63287
	9.027 0.01206 0.01217 0.01201 0.62106 0.62161 0.62105

Table 4 :

 4 Dimensionless energy flux J E