Nicolas Tchertchian
email: nicolas.tchertchian@supmeca.fr

Dominique Millet

Akram El

A El Korchi

El Korchi

Design for Remanufacturing: What performances can be expected?

Keywords: Eco-Design, Remanufacturing, LCA, Economic Assessment, DfX, RSC

come

Design for Disassembly and Recycling

Dismantling and recycling are increasingly important in the majority of the industrialized countries because of the increase of amount of products put in discharge. The dismantling of the worn products is regarded as necessary for a recycling economically viable with current technologies of reprocessing. The purpose of DfD is to integrate the constraints of product dismantling in order to facilitate its treatment at the end of the life: extraction of components or modules being able to be re-used, of dangerous substances in order to be able to treat them specifically, of the components to incinerate… dismantling makes it possible to be able to achieve the goal of "recycling" and supports also the taking into account of other scenarios of valorization, in particular the re-use of components or subsets. The transformation of the products at the end of the lifetime for the re-use of the components and the valorization of the secondary materials and energy saving, constitute a means of limitation of the environmental issues but also an economic and social development strategy. To reach this challenge, several research showed the need for breaking up the products upstream to every process of valorization. The study of [START_REF] Haoues | Contribution à l'intégration des contraintes de désassemblage et de recyclage dès les premières phases de conception de produites[END_REF] focused on the contribution to the integration of dismantling and recycling constraints as of the first phases of products development process. A model made up of minimal data from which the disassemblability as soon as possible in the process of design can be defined: this sequence of dismantling is used thereafter as reference for the phase of detailed design. So that a component is disassembled of a product at the end of life, [START_REF] Haoues | Contribution à l'intégration des contraintes de désassemblage et de recyclage dès les premières phases de conception de produites[END_REF] considers that three conditions must be checked:

 Valorisability: This first condition depends primarily on materials constitutive of the component,  Accessibility: It is the condition which depends essentially on the structure of the product and on the dimensions around valorisables components,  Separability: This condition depends primarily on the types of joints between components. Indicators of dismantling assessment for valorization were also elaborate: Material Indicator, Structural Indicator and Connection Indicator corresponding respectively to the condition of valorisability, accessibility and separability. In addition to the engineering problems of dismantling presented above, there exist critical problems on the "back system": tools, the time of dismantling and then the costs (labor, industrial equipment, energies etc). [START_REF] Das | An approach for estimating the end-of-life product disassembly effort and cost[END_REF] indexes the costs relating to the treatments and dismantling process: (1) collection; (2) sorting; (3) product handling; (4) disassembly training for the workers; (5) disassembly; (6) treatment of the parts and materials.

It is noted that "Design for Disassembly" and "Design for Recycling" have common parts. In entry, the use of the recycled materials reduces the quantity of necessary materials. In Output, the recycling of non-desired production and scrap reduce the quantity of waste resulting from the production process or of energy consumption. [START_REF] Coppens | Méthode de conception en vue d'optimiser la valorisation des véhicules hors d'usage[END_REF] proposed the integration of a tool specialized in the design process, to take into account the recycling constraints for vehicles end of the life (VHU). If recycling appears tempting, it has should be considered additional cost and impact indeed the consumption of energy cannot be decreased effectively because additional energy is used in the processes of the recycling of materials. Economically, recycling decreases the benefit of the companies because of the additional costs due to the process of recycling. Consequently, simple recycling could not be a good strategy for some product end-of-life [START_REF] Umeda | Study on life-cycle design for the post mass production paradigm[END_REF].

Design for Modularity

"Design for Modularity" is an approach aiming at subdividing a system in smaller parts which can be created in an independent way and then used in various systems. In addition to the reduction of the costs, the modularity offers flexibility in the design (the addition of new technical solutions). The literature on the topic shows that strong competition on the market pushed the companies to design a larger variety of products to satisfy the customers' needs. However, the frequent variations of product specifications lead to an assembly and a dismantling of the components and modules increasingly complicated. In the study of [START_REF] Tseng | Modular design to support green life-cycle engineering[END_REF], were added to the graph of the connections some attributes allowing the designers to characterize the connections between components during the design process. With a genetic algorithm, it is then possible to gather the components, to consider their modifications taking into consideration need for the modular concept [START_REF] Tseng | Modular design to support green life-cycle engineering[END_REF]. [START_REF] Kimura | Product modularization for parts Reuse in inverse manufacturing[END_REF] proposes a modular design method which takes account of the successive generations of products based on common functionalities and similar life cycles. [START_REF] Umeda | Product modularity for life cycle design[END_REF] consider that the modular concept is important to improve the design of life cycle, for example, maintainability, upgradability, reutilisability and recyclability. They propose a modular design methodology which rises from the structure of the modules based at the same time in the characteristics of the life cycle and geometrical information. Considering the whole life cycle of a product, the components which undergo the same life cycle should be gathered in a module. These modules can then undergo maintenance and processes of recycling without disassembling. That led to an easy management of the components throughout the life cycle and to the reduction of the environmental impact and the costs of logistics and recovery. But the modular concept implies also constraints. A module must be made up of elements belonging to the same group. The violation of this constraint can radically destroy the advantages of the modularity, such as the desassemblability, the reutilisability, and the recyclability. Moreover, one module should not be geometrically surrounded by components belonging to another group; if not, the element cannot be extracted without dismounting the module; this violation of this constraint reduces also the advantages of modularity [START_REF] Umeda | Product modularity for life cycle design[END_REF].

Design for Remanufacturing

Tchertchian N, Millet D & El Korchi A

The origin of Remanufacturing goes back to the World War II during which many products used were remanufactured in order to maintain the company in operating condition. During last decades, the concept and the activity of remanufacturing were largely spread in the industrial and academic fields. There exist various synonyms of the remanufacturing. Thus, in the sector of the remanufacturing of the automobile components, the term "rebuilding" or standard exchange is preferred. In the same way, the manufacturers of tires are called "retreaders", while the remanufacturer of toner cartridges regards himself as "rechargers" [START_REF] Lund | The Remanufacturing Industry: Hidden Giant[END_REF]. If the "rebuilding" of the product is not extensive, i.e., if only some parts must be replaced, the terms of Reconditioning/"refurbishing" is more suitable: the product is then only remanufactured with its original specification [START_REF] Ijomah | Remanufacturing: Evidence of Environmentally Conscious Business Practice in the UK[END_REF]. In the literature, we find a robust definition of the remanufacturing:

 "Remanufacturing is the process of restoring discarded products to useful life." [START_REF] Lund | The Remanufacturing Industry: Hidden Giant[END_REF]  "Remanufacturing is "The process of returning a used product to at least Original Equipment Manufacturer performance specification and giving the resultant product a warranty that is at least equal to that of a newly manufactured equivalent." [START_REF] Ijomah | Remanufacturing[END_REF] The industrial sector which has the most experience in the field of the remanufacturing is automotive industry (for example the clutch of car, the starter…) but more and more, the remanufacturing is propagated with other sectors, like those of the electrical appliance, toner cartridges, photocopiers, apparatuses, of the machine tools, cell phones etc. In the state of the art, we observe that the remanufacturing is currently mainly centered on the following aspects:

 The evaluation of the remanufacturability and the redesign of the product, concerning architectures, connections, joints, modules, materials etc., to facilitate the processes of remanufacturing (Van der Laan and Salomon, 1997) (Zwolinski2006) [START_REF] Gray | Remanufacturing and Product Design[END_REF] [START_REF] Ijomah | Development of design for remanufacturing guidelines to support sustainable manufacturing[END_REF],  Improvement of the processes of remanufacturing (dismantling, inspection, cleaning, reconditioning, re-assembly, etc.) in terms of technologies of treatment (dismantling, inspection, cleaning, reconditioning, re-assembly etc.), of equipment etc. [START_REF] Lund | The Remanufacturing Industry: Hidden Giant[END_REF] [START_REF] Steinhilper | Remanufacturing -The Ultimate Form of Recycling[END_REF],  Coordination between the remanufacturing and the business (functional sale, rent service, etc.). [START_REF] Sundin | Product and Process Design for Successful Remanufacturing[END_REF] [START_REF] Mont | Clarifying the Concept of Product-Service System[END_REF] [START_REF] Mont | A new business model for baby prams based on leasing and product remanufacturing[END_REF],  Coordination between the remanufacturing and reverse logistics [START_REF] Bras | Product, process, and organizational design for remanufacture -an overview of research[END_REF] [START_REF] Mont | A new business model for baby prams based on leasing and product remanufacturing[END_REF] [START_REF] Lindahl | An Interactive Design Methodology for Service Engineering of Functional Sales Concepts -A potential Design for Environment Methodology[END_REF],  Benefit analysis of the remanufacturing from economic and environmental point of view. [START_REF] Kerr | Remanufacturing and eco-efficiency: A case study of photocopier remanufacturing at Fuji Xerox Australia[END_REF] [START_REF] Ijomah | Remanufacturing: Evidence of Environmentally Conscious Business Practice in the UK[END_REF] [START_REF] Sundin | Product and Process Design for Successful Remanufacturing[END_REF]) (Smith and Keoleian, 2004) [START_REF] Kerr | Eco-efficiency gains from remanufacturing. A case study of photocopier remanufacturing at Fuji Xerox Australia[END_REF]. Many research in the field of the remanufacturing is focused on one or more abovementioned aspects. In all the connections, one of the most significant work is that of Robert Lund at the end of the years 1970 -beginning of the year 1980 which launched the debate around the benefit of the remanufacturing and the many aspects which must be considered including the selection of the products, the marketing strategy, the technology of reconditioning, the financial aspects, the organizational factors, and the legal considerations [START_REF] Lund | Guidelines for an original equipment manufacturer starting a remanufacturing operation[END_REF]. Another researcher considered in this field, [START_REF] Steinhilper | Remanufacturing -The Ultimate Form of Recycling[END_REF] illustrated the appropriatenesses and the challenges remanufacturing:

Design for Remanufacturing: What performances can be expected?



A strategy of business based on the remanufacturing can support the world of manufacture by opening new business appropriatenesses in the service market of sales by creating new solutions with total costs lower for their customers, 

The manager of waste will appreciate if the remanufacturing can generate a profitable business in closed loop to replace an expensive process of elimination, 

The engagement of the public authorities for a strategy privileging the remanufacturing must allow the development of new markets and job creation, 

The economists regard the market of the remanufacturing as having a strong potential, the companies engaged today in this way being regarded besides as "hidden champions" in industry, 

For the consumers, the remanufacturing is not only more ecological, but also more economic because it is then possible to reach products with a technology updated with an affordable price.

"Design for remanufacturing" relates to not only the product but more largely the rendered service and can bring to reconsider the business model company. [START_REF] Gray | Remanufacturing and Product Design[END_REF] identified the constraints to characterize the most useful methods "Design for "the X"" ("X" represents for example dismantling, cleaning, the inspection, etc.). In other work [START_REF] Kerr | Eco-efficiency gains from remanufacturing. A case study of photocopier remanufacturing at Fuji Xerox Australia[END_REF], the concept of eco-effectiveness is used to evaluate the economic and environmental benefit remanufacturing.

Tchertchian N, Millet D & El Korchi A

The democratization of the remanufacturing in industry runs up against obstacles because it has also constraints for the industrialist.

The sensitive products to the change of tendencies and which evolve very quickly are little adapted to the remanufacturing because the users prefer the new products, it does not matter the quality and the cost of the alternative remanufactured.

Practices such as the patents, the intellectual property laws and anti-competitive can also block the remanufacturing. For example, some manufacturers of printers designed their inkjet cartridges so that they are auto destructive when they are empty what prevents to them remanufacturing [START_REF] Ijomah | Development of design for remanufacturing guidelines to support sustainable manufacturing[END_REF].

The processes of remanufacturing are difficult to manage because of uncertainties, like those on times and the operations of treatment or those on the choices of the quantity and the quality of the components to be recovered. Moreover, flows which feed the process of recovery are also unertain in quantity, quality, and terms of date [START_REF] Van Nunen | Enabled Closed-Loop Supply Chains[END_REF].

In a survey of the American car industry, [START_REF] Hammond | Issues in the Automotive Parts Remanufacturing Industry -A Discussion of Results from Surveys Performed among Remanufacturers[END_REF] identified that the costs and the availability of the part (including the modules) are principal concerns for the remanufacturers. However in the last decade, the variety of the products increased significantly because of a faster development of products, of an exacerbated and mondial competition bringing to an increased segmentation of the product ranges, a personalization of product of mass production. That led to an increase in the variety and a reduction in volumes of the parts causing a lower availability of the parts of replacement.

In other words, the problem of the availability and the replacement cost of part are strongly related to the problem of the diversity of the products. The remanufacturor must deal with small batches of products including a range of alternatives and generations, which complicates the operations of remanufacturing, the management of flows, and the evolution of the tools for dismantling and processes of assembly…

The problematic to assess the performance of the Design for Remanufacturing

The review of literature made here watch in particular that the methods Design for Remanufacturing have advantages in their capacity to take into account the needs for the designer (the DfX methods are structured around a mode of reasoning very close to that implemented by the designer) and to a lesser extent on economic dimension (the economic evaluations do not take into account the concept of multiples life cycle). They have however weaknesses on customer dimension (the DfX approaches in general regard the functionalities of the product as being fixed; a fortiori, they do not take into account the update of the functions which could take place on the second life cycle of the product and on the following cycles) like on environmental dimension (in the methods DfX, the environmental evaluations are generally centered on the reduction of waste; they seldom integrate a variety of environmental impacts categories what is a requirement for a rigorous work in eco-design). The continuation of the article aims at filling a lack of information on the performance of the remanufacturing (environmental and economic). Indeed, the study of the performances requires the taking into account of the number of life cycles of the product, of the number of remanufactured modules, the localization of the remanufacturing operations, the range considered, the number of center of regrouping…

Modeling of the life cycle parameters

To model the various scenarios, we defined the parameters of the life cycle. Actually, there exist several alternatives of life cycles integrating the parameters of the "remanufacturing". For example the number of modules remanufactured, lifespan of the modules and the machine, the number and the distribution of grouping centers, the re-use of the modules in the same range or various generations of products, etc. To simplify the modeling of the product life cycle, we focused ourselves on the three principal parameters: the number of remanufactured modules (X), the number of cycles (N), duration of cycles (Z), and the place of the remanufacturing center (Y).

Parameter 1: Number of remanufacturable modules

The definition of the remanufacturables modules is based on multiple criteria: Technical feasibility -Environmental respect -Economic profitability. However before doing any evaluation, the modules, potentially remanufacturables must be given to build the various scenarios in order to choose the strategy which will answer the aims of the study.

We chose, for the case study in the next paragraph, seven modules to combine the various scenarios of remanufacturing. By considering the level of disassembling, we built the

Tchertchian N, Millet D & El Korchi A

scenarios from 1 to 7. For the electronic modules, scenarios 8 and 9 are individually defined. Scenarios are defined according to the following relations: Scenario 0: corresponds to WEEE scenario (no module is remanufactured) Scenario i:

Model for the environmental assessment

The modeling of a life cycle of a product whose certain modules are re-used once is carried out with SIMAPRO® thanks to functions REUSE and DISASSEMBLY. The modeling of several cycles of use using a LCA software is different starting from the second cycle: indeed starting from the second cycles the modeled system depends on the z S * 1 ; 0 Design for Remanufacturing: What performances can be expected? parameters imputed by the user according to the number of life cycles, of the end-of-life of the modules (re-use or recycling) or of the logistic scenario [START_REF] Tchertchian | The Influence of the Multiple Life Cycles on the Environmental Impact of a Product[END_REF]. The systems modeled starting from the 2nd cycle are systems which in "theory" are manufactured without the modules that one wish remanufacture. Of course it is about a fantastic notion because in reality the re-used modules are assembled at the same time as the other components, but the impact of the manufacture of this module "is avoided" by re-using it. The coefficients α k : Recycling rate of material k and β m : Reuse rate of module m give an account of the materials transfers in a cycle and between two cycles. For example the material share which can be recycled in a cycle is reinjected in the following cycle as the transfer of material flow shows it in the diagram below Figure 4). That relates to the modules which are not remanufacturables but also reusable modules whose the performance and quality standards are not respected. In order to have an analysis detailed on the whole of the life cycle, we initially broke up the multiple cycles into phases: Production, Distribution, Use and end-of-life. The method of calculating used is the Method Eco Indicator 99 for all simulations because it makes it possible to obtain a single score which simplifies interpretations (method CML added with 4 complementary indicators, whose results will not be presented, was used in order to bring an "objective" support additional and thus to consolidate the results obtained with the method indicating Eco 99). Then we will enumerate all the elements for each scenario which one defined.

Tchertchian N, Millet D & El Korchi A

For example the Scenario

(1 cycle of Z years, end-of-life undergoes DEEE) is the scenario of reference used to compare the various scenarios of remanufacturing: ), (

z X Fr p U E 2 ; * 3 ; , ) ( z X F m m F E  m 2 ; * 3 ; , ) ( z X Ch m P E m 2 ; * 3 ; / ), ( z X Fr Ch p D E 2 ; * 1 ; ), ( z X Fr p U E z Fr p F E * 1 ; 0
), ( Design for Remanufacturing: What performances can be expected?

Model and assumptions for the economic assessment

The objective of this study is to evaluate the cost of end-of-life of machine. This cost covers all the reverse logistic process since the recovery from the customer until the final treatment (remanufacturing) of the machine. This study is based on the ABC method (Activity Based Costing) and it follows the following step. Initially, the identification of the principal activities which compose the reverse logistic process machines at the end of the contract; these activities were classified in four principal stages. Then, the resources mobilized by these activities are identified. Then, each resource to the activity, which consumes it, is assigned. Lastly, the activities developed with the cost price of the product end-of-life are assigned. The use of the method ABC would make it possible to calculate the real costs of the "remanufacturing" by distinguishing it from the costs of recovery of the other machines at the end of contract or the representatives of the failures. It would also make it possible to identify the consumer activities of resources and which affect the profitability of the "remanufacturing". In order to study the profitability of the remanufacturing from the life cycle point of view we chose modules as the remanufacturing level like limit of the phase of end-of-life. Beside the method ABC which allows a good indirect burden-sharing, we also carried out another calculation based the consideration of the wages charged which cover the majority of the indirect loads. An analysis of sensitivity which makes it possible to show the difference between the two methods of calculating was carried out at the end. Ten scenarios of calculation of the cost of end-of-life of the product were analyzed. The constitution of the scenarios was based on the variation of the number of modules with remanufacturer. In scenario 0, machine is sent directly from production/ remanufacturing center to the recycling center of without passing by any treatment. In the scenarios from 1 to 9 the machines are disassembled in order to recover the remanufacturables modules, the non remanufacturables modules are sent to the recycling center. As we underlined previously the comparison of the various scenarios should not be restricted with the end-of-life phase but it must be led from the complete life cycle point of view. For this reason we constituted three different scenarios of life cycle by varying the configuration of supply chain and reverse supply chain. It is considered that the supply chain deals with the life cycle of the new machines (1st cycle) while the reverse supply chain treats the life cycles of the remanufactured machines (2nd and 3rd cycles). The supply chain and the reverse supply chain are made up of the five following phases: Provisioning -assembly or re-assemblydistribution -use -end-of-life.

Each of the three supply chain scenarios comprises the ten sub-scenarios of end-of-life treated previously. The costs of the sub-scenarios of remanufacturing as their profits compared to sub-scenario WEEE were modeled on two and three life cycles. Below we define the modeling of the costs and the profits for the various scenarios by According to these criteria we made a preselection of seven modules out of 12 on the whole. We then will analyze the interests of the remanufacturing module by module. The product analyzed is a franking machine. The product is composed by 11 modules which 7 are considered as reusable

Definition of life cycle scenarios

The modeling of life cycle scenarios in the preceding paragraph makes it possible to model a total of 72 scenarios while varying the number of modules remanufactured the number and the duration of cycles and the localization of the centers of remanufacturing.

The two examples below make it possible to give an account of logistic flows and material flow for a remanufacturing center located in France/China and for 2 or 3 cycles: -The machines containing the modules remanufactured and new machines follow the same distribution system; the only difference which exists between the two circuits it is transport between China and France in order to place the new machines to France.

-The remanufactured machines and the new machines have the same customers and the same leasing agreements. -The machines containing the modules remanufactured and new machines follow the same distribution systemo d-The remanufactured machines and the new machines have the same customers and the same leasing agreements.

Environmental assessment

WEEE regulation for the valorization of the electronic devices of categories 2 (small household appliances) recommends a valorization of the machine to a total value of 75%, when one speaks about valorization it is necessary to take into account recycling, the recovery of parts but also the energy recovery (which consists in recovering the heat released by the combustion of waste). With the method Eco-indicator 99, the total environmental impacts (single score) for the Extraction + Manufacture phases are obtained by incorporating the impacts from the various indicators. Obviously, the five electronic modules (the interface, the food, the modem, the mother chart and the module of print) are more impacting. For these five modules, the major part of the impact concentrates on the category "Fossil fuels" (the greenish yellow part on Figure 8) Design for Remanufacturing: What performances can be expected?

The summary of the results (Figure 10) shows that more the modules are remanufactured, less the generated environmental impact is important. In general the scenario of 3 4 years cycles, remanufacturing of 7 modules in France is optimal. In this scenario, the seven modules generate 98.0% of impact of the complete machine. Two re-uses out of three cycles for these seven modules bring a great profit for the environment. In more its utilization period over 12 years cause a drop in its annual impact. At the same time, the scenario of 2 cycles of 5 years with remanufacturing of 7 modules in France has a performance very similar to the precedent. That is explained owing to the fact that the more one module lasts a long time in a cycle, the more it will be powerful for the environment.

Erreur ! Référence non valide pour un signet. summarizes the results of the 72 scenarios (9 scenarios of remanufacturing*4 Multiples Life cycle*2 Location for remanufacturing center). The remanufacturing is more powerful when it is carried out locally (France). When the center of remanufacturing is localized in France logistic flows are less impacting. The scenarios of remanufacturing are then compared with corresponding scenario WEEE. For 3 life cycles of 3 years and a remanufacturing center localized in France allows a profit of 44.4% compared to equivalent scenario WEEE.

Economical assessment

The cost of each scenario is composed of the costs of the four following stages: stage of collection, sorting in Le Lude, treatment and recycling. The costs of the first two stages are constant because they are independent of the number of the modules with remanufacturer. The variation of the costs for the stage of treatment is mainly related to time spent to extract and treat each module. Design for Remanufacturing: What performances can be expected?

It is noticed that the cost of scenario 0 (WEEE) is lowest of all the scenarios because the machine does not undergo any treatment Lude. On the other hand, the scenario of remanufacturing 7 has the cost highest but it allows in against part recovery of a big number of modules strong added value and which can lead a second life cycle. One can conclude from that the comparison of the various scenarios should not be restricted with the phase of end-of-life but it must be led from the point of view of life cycle. The economic benefit compared to scenario WEEE lies between 25 and 35% (Figure 12).

Synthesis of economic & environmental simulation results

Among the 72 scenarios, we retained the scenarios corresponding to 2 5 years cycles with remanufacturing in France (they allow the best environmental performances according to the Figure 10). The 9 scenarios are treated on a hierarchical basis according to the environmental and economic profits compared to the scenario of reference WEEE. The scenarios of remanufacturing 6 and 7 are the powerful ones on the two criteria. In these two scenarios the number of modules re-used is largest. The other remarkable point

Tchertchian N, Millet D & El Korchi A

relates to the scenarios 8 and 9, which integrate electronic modules, they are very good environmentally but their low costs of manufacture do not allow strong economic gains.

Conclusion

The state of the art on the DfX approaches supplies to the eco-designer solutions to improve the product. For example by adopting a modular architecture, by designing the product so as to return the components modules most easily extractable and by gathering the modules having identical end-of-life it is possible to optimize recycling and to re-use certain components modules. However the literature gives an outline of the operations necessary to the remanufacturing and logistic flows but it do not make it possible to emphasize the potentialities of the remanufacturing because of difficulty in apprehending the multiple products with life cycle. This difficulty can be explained by not taken into account of the methods of eco-design and the management of the RSC simultaneously.

The study suggested in this article is pressed on a product whose diagram of distribution in France is the hiring. This product is recovered at the end of the contract and given in state. The exploratory field offered by this company enabled us to carry out evaluations economic and environmental complete system: Product and logistic structure. This experimentation enabled us to develop an economic model and an environmental model for products with multiple life cycles. From this model of simulations were realized. The simulations treated in this article give an account of the influence of the parameters: many remanufactured modules, many life cycles and duration of each cycle and geographical location of the center of remanufacturing. While taking into account simultaneously the characteristics of the product and logistic flows we managed to consider the economic gains and environmental for a mechatronic product on several life cycles. This case study carried out on franking machines associated with the state of the art of the DfX approaches highlights various dimensions of the problem of remanufacturing. All these elements reveal the need for placing at the disposal of the team of design, a method of assistance to the design of new concepts of systems remanufacturables.

Figure 1 :

 1 Figure 1 : The generic process of remanufacturing according to Steinhilper (1998)

  of cycles and duration of cyclesConcerning the parameter of the number of cycles and duration of cycles, we considered 4 possibilities: 2 cycles of 4 years or 5 years (Figure2); 3 cycles of 3 years or 4 years (Figure4). The stages of each cycle and their composition are defined in the following figures:

Figure 2 :Figure 3 :

 23 Figure 2 : Scenario with 2 Life cycles

Figure 4 :

 4 Figure 4: Model for the environmental assessment

  0 (WEEE), Y=1 is defined by the following model:Tchertchian N, Millet D & El Korchi AAnd the total costs of the scenario are defined by the following function: scenario X of remanufacturing with Y=1 is defined by the following model :And the total costs of the scenario are defined, for two 4 years cycles, by the following function: of scenario X compared to scenario 0 is given by the formula: of the remanufacturables modules For this study, the modules (regrouping of components fulfilling a function) were evaluated according to the criteria above in order to select the modules most suited to a remanufacturing: Technical criteria  Environmental criteria  Economic criteria  Failure rateDesign for Remanufacturing: What performances can be expected? Lifetime and reliability  Easy to disassembly  Easy to repair  Modules cost  Modules with high value added  Extraction + Manufacture impact  End-of-life cost (recycling, incineration, discharge, etc.)  End-of-life impact (recycling, incineration, discharge, etc.)

Figure 5 :

 5 Figure 5: Modules definition

Figure 6 :

 6 Figure 6: Remanufacturing M1+M2+M3+M4+M5+M6+M7 in France, 2cycles of 4 years

Figure

  Figure 7: Remanufacturing M1, assembled in China, 3 cycles of 3 years Contract span =3 years

Figure 8 :

 8 Figure 8: Life cycle assessment for WEEE scenario

FigureFigure 9

 9 Figure 9: Environmental impact one complete product (Extraction + Manufacture)

Figure

  Figure 11: Detailed results for the 72 scenarios

Figure 11

 11 Figure11summarizes the results of the 72 scenarios (9 scenarios of remanufacturing*4 Multiples Life cycle*2 Location for remanufacturing center). The remanufacturing is more powerful when it is carried out locally (France). When the center of remanufacturing is localized in France logistic flows are less impacting. The scenarios of remanufacturing are then compared with corresponding scenario WEEE. For 3 life cycles of 3 years and a remanufacturing center localized in France allows a profit of 44.4% compared to equivalent scenario WEEE.

Figure

  Figure 12: Calculation of the overall costs (2 cycles of 5 years)

Figure 13 :

 13 Figure 13: Economic and environmental gain compared to WEEE (2cycles of 5 years)

Figure 10: Total environmental impact for 36 scenarios (Remanufacturing in France)

	M1	M2	M3	M4	M5	M6	M7	M8	M9	M10	M11
	Module		Weight (g) Environmental impact (Pt)
	M1-User interface		526	28,3
	M2-ALD		208	0,2
	M3-Service station	430	7,2
	M4Power supply		562	62,7
	M5-Modem		80,4	5,8
	M6-Main board		96,8	22,1
	M7-Main motor		344	0,2
	M8-Top module		1596	0,7
	M9-Lower module		2108	1,1
	M10-Lower housing	284	0,2
	M11-Cover		1117	0,7

Aknowledgement

We wish to thank Neopost industry to have opened to us carry from their production center and remanufacturing to France and to have given us access to the necessary informations to the realization of this study. We also thank members of MACPMR project as well as foundation CETIM and the ADEME for their financial support.


K L Nguyen

F Treyssède

C Hazard

A.-S Bonnet-Bendhia

Finite element computation of leaky modes in straight and helical elastic waveguides

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, waveguides are often embedded in a large solid domain, considered as unbounded. The waves can attenuate strongly along the guide axis due to the energy leakage into the surrounding medium, which reduces the propagating distance. Searching modes with low attenuation becomes necessary. The goal of this work is to propose a numerical approach to compute modes in embedded elastic waveguides (straight and helical structures), combining the so-called semi-analytical finite element method (SAFE) and a perfectly matched layer (PML) method. The application of this work is the non destructive evaluation of multi-wire strands, which constitute cables in civil structures.

Introduction

For embedded structures, the energy leakage into the surrounding medium yields wave attenuation along the guide axis, which limits the application of guided wave technique for non destructive evaluation (NDE). An accurate determination of low leakage modes becomes necessary to maximize the inspecting range. Analytical tools have been developed to model simple geometries (plates and cylinders) [START_REF] Lowe | Plate waves for the NDT of diffusion bonded titanium[END_REF][START_REF] Pavlakovic | Leaky guided ultrasonic waves in NDT[END_REF].

For complex geometries, modeling often relies on numerical approaches. A classical method is the so-called semi-analytical finite element (SAFE) method restricting the computational domain on the discretized waveguide cross-section. Yet, two difficulties are encountered for the numerical simulation of embedded waveguides : the unbounded cross-section and the presence of leaky modes whose amplitudes grow in transverse directions. To circumvent these difficulties, the SAFE method K. L. Nguyen, F. Treyssède IFSTTAR; Route de Bouaye, 44344 Bouguenais, France, e-mail: fabien.treyssede@ifsttar.fr C. Hazard, A.-S. Bonnet-BenDhia ENSTA Paris Tech; 828, Boulevard des Maréchaux, 91762 Palaiseau, France 1 has been combined with other techniques such as absorbing layer [START_REF] Castaings | Finite element model for waves guided along solid systems of arbitrary section coupled to infinite solid media[END_REF] and boundary element methods [START_REF] Mazzotti | A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary crosssection[END_REF]. An alternative approach proposed in this paper is the perfectly matched layer (PML) method. A SAFE-PML approach has already been applied to model embedded solid plate waveguides in two dimensions [START_REF] Treyssède | On the use of a SAFE-PML technique for modeling two dimensional open elastic waveguides[END_REF].

The goal of this work is to extend this technique to model three-dimensional elastic waveguides buried in a solid matrix. First of all, the modeling of embedded straight waveguides is considered. The next step consists in applying the PML method in transverse twisting coordinates [START_REF] Treyssède | Investigation of elastic modes propagating in multi-wire helical waveguides[END_REF] to compute modes in embedded helical structures.

Straight waveguides

A PML applied along the Cartesian transverse directions (X,Y ) consists in extending the elastodynamic formulation into complex coordinates ( X(X), Ỹ (Y )) to absorb waves in the surrounding medium: X

= X 0 γ X (ξ )dξ , Ỹ = Y 0 γ Y (ξ )dξ .
A change of variable is then performed to transform the formulation back to real coordinates. In addition to the PML technique, the SAFE method is applied, which assumes an e ikZ dependence, where k is the axial wavenumber and Z is the axial direction. The FE dicretization yields the quadratic eigenvalue problem :

{K 1 -ω 2 M + ik(K 2 -K T 2 ) + k 2 K 3 }U = 0 (1)
Numerical results are computed for a steel bar in concrete. The eigenspectrum at a given frequency shows that the PML method not only computes physical modes (leaky) but also non-intrinsic modes (radiation modes) which depend on PML characteristics and resonate mainly in the PML region. In order to eliminate radiation modes, a filtering criterion is applied, defined by the ratio of kinetic energy in the PML over the kinetic energy in the whole domain. As shown in Fig. 1, dispersion curves of leaky modes are well identified after filtering. Numerical results obtained by the SAFE-PML method are in agreement with Refs. [START_REF] Castaings | Finite element model for waves guided along solid systems of arbitrary section coupled to infinite solid media[END_REF][START_REF] Mazzotti | A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary crosssection[END_REF].

Helical waveguides

Helical waveguides must be considered in twisting coordinates (x, y, z) defined as :

x = X cos(τZ) +Y sin(τZ), y = -X sin(τZ) +Y cos(τZ), z = Z (2)
where τ denotes the torsion of the (x, y) plane around the Z-axis. The twisted PML method transforms twisted transverse coordinates (x, y) into complex ones ( x(x), ỹ(y)). Similarly to the previous section, a twisted SAFE-PML approach can then be applied.

Validation

Since a twisted cylinder remains a cylinder, this test case can be considered to validate the twisted SAFE-PML method.

The comparison between the axial wavenumbers computed in both coordinate systems (with twist and without) shows that the wavenumbers of compressional and torsional modes are unchanged while those of flexural modes are translated by ±τm in the twisting system, where m denotes the circumferential order (details in [START_REF] Treyssède | Investigation of elastic modes propagating in multi-wire helical waveguides[END_REF]).

Although the axial wavenumbers are different in twisting and straight coordinates, the physics of cylindrical modes are left unchanged. Figure 1 shows the agreement between the energy velocities in both coordinate systems.

Examples

Two examples of embedded helical waveguides are now considered : a steel helical wire and a steel seven-wire strand buried in concrete.

Figure 2 compares the axial attenuation of compressional L(0, n) modes in an embedded helical wire with those of an embedded cylindrical bar. This comparison concentrates on compressional modes, which have low attenuations and high energy velocities. Compressional modes are more attenuated in a helical wire than in a straight one. This difference becomes more significant for high order modes. This result shows that the curvature of helical wires reduces the wave propagation distance and can make their NDE more difficult. This phenomenon has been observed for other kinds of curved waveguides [START_REF] Beard | Guided wave inspection of embedded cylindrical structures[END_REF].

The dispersion curves of energy velocity for a steel seven-wire strand in concrete is given by Fig. 3. The strand consists of one cylindrical bar surrounded by six helical wires. Comparing with the results of a free strand [START_REF] Treyssède | Investigation of elastic modes propagating in multi-wire helical waveguides[END_REF], the modal behaviour is strongly modified due to the introduction of the surrounding medium. The wave attenuation is shown in Fig. 4. F(1, 1) ± and L(0, 1) modes have the lowest attenuations in the frequency range [0, 1.5] and [1.5, 2] respectively, which may be of interest for NDE. In addition to the attenuation due to the energy leakage, the surrounding medium has the other effect to make helical wires almost in contact, which further modifies the dispersion curves.

Fig. 1

 1 Fig. 1 Energy velocity for an embedded cylindrical bar obtained by SAFE-PML (circles) and twisted SAFE-PML (crosses) methods.

Fig. 2

 2 Fig. 2 Attenuation of modes in an embedded helical wire (black) and in an embedded cylindrical bar (gray).

Fig. 3

 3 Fig. 3 Energy velocity of an embedded sevenwire strand.

Fig. 4

 4 Fig. 4 Attenuation of modes in an embedded seven-wire strand.


Gaël Prado

Alexander Farutin

Chaouqi Misbah

Lionel Bureau

Viscoelastic transient of confined Red Blood Cells

The unique ability of a red blood cell to flow through extremely small microcapillaries depends on the viscoelastic properties of its membrane. Here, we study in vitro the response time upon flow startup exhibited by red blood cells confined into microchannels. We show that the characteristic transient time depends on the imposed flow strength, and that such a dependence gives access to both the effective viscosity and the elastic modulus controlling the temporal response of red cells. A simple theoretical analysis of our experimental data, validated by numerical simulations, further allows us to compute an estimate for the two-dimensional membrane viscosity of red blood cells, η 2D mem ∼ 10 -7 N•s•m -1 . By comparing our results with those from previous studies, we discuss and clarify the origin of the discrepancies found in the literature regarding the determination of η 2D mem , and reconcile seemingly conflicting conclusions from previous works.

I. INTRODUCTION

The study of blood flow properties is a highly active field of research, both experimentally [START_REF] Baskurt | Blood rheology and hemodynamics[END_REF] and theoretically [START_REF] Fedosov | Predicting human blood viscosity in silico[END_REF][START_REF] Fedosov | Multiscale modeling of blood flow: from single cells to blood rheology[END_REF][START_REF] Ghigliotti | Rheology of a dilute two-dimensional suspension of vesicles[END_REF]. Identifying the relevant (bio)physical parameters controlling such properties is of great interest in the context of diagnosis of blood disorders, and also represents an important fundamental challenge in numerical and theoretical modeling of complex fluids. Blood is a suspension of cellular elements (Red and White Blood Cells, Platelets) in a carrier fluid (Plasma). It is known to behave as a non-Newtonian fluid, the complex rheology of which is mainly due to the presence of Red Blood Cells (RBC), its major cellular component [START_REF] Baskurt | Blood rheology and hemodynamics[END_REF]. The flow properties of blood are thus essentially governed by the concentration of RBCs (the so-called hematocrit), their mutual interactions, and their individual mechanical properties. Over the past fifty years, the latter have been the focus of numerous studies which aimed at characterizing or modeling the viscoelastic behavior of individual RBCs, and in particular that of the cell membrane which separates the outer suspending plasma from the inner cytoplasm [START_REF] Chien | Red cell deformability and its relevance to blood flow[END_REF][START_REF] Hochmuth | Erythrocyte membrane elasticity and viscosity[END_REF][START_REF] Kim | Measurement techniques for red blood cell deformability: recent advances[END_REF][START_REF] Betz | ATP-dependent mechanics of red blood cells[END_REF][START_REF] Fedosov | A multiscale red blood cell model with accurate mechanics, rheology, and dynamics[END_REF][START_REF] Li | Cytoskeletal dynamics of human erythrocyte[END_REF][START_REF] Peng | Lipid bilayer and cytoskeletal interactions in a red blood cell[END_REF][START_REF] Dimitrakopoulos | Analysis of the variation in the determination of the shear modulus of the erythrocyte membrane: effects of the constitutive law and membrane modeling[END_REF]. The membrane region comprises a lipid bilayer, an external glycocalyx layer, and an inner twodimensional (2D) cytoskeleton composed of a spectrin network. The inner and outer layers are connected to the bilayer through transmembrane proteins. Mechanical properties of the membrane are typically described in terms of bending and expansion moduli of the bilayer, a 2D elastic shear modulus of the spectrin network (µ s ), and a 2D membrane viscosity (η 2D mem ) [START_REF] Hochmuth | Erythrocyte membrane elasticity and viscosity[END_REF][START_REF] Kim | Measurement techniques for red blood cell deformability: recent advances[END_REF]. Aside from their influence on bulk rheology, the membrane properties of RBCs are also of utmost importance in microconfined flows, i.e. in the microvascular network, where they control the ability of cells to adapt their shape and flow through channels having dimensions on the order of or smaller than the size of unstrained RBCs. Such a capacity of RBCs to flow through very small channels is * lionel.bureau@ujf-grenoble.fr central in their physiological function of oxygen delivery to tissue, and alterations of the viscoelasticity of RBCs are recognized to be associated with various pathologies [START_REF] Mohandas | Red cell membrane: past, present and future[END_REF]. In this context, in vitro studies of microconfined flows are currently viewed as a potential diagnosis tool to discriminate between healthy and pathological cells [START_REF] Kwan | Microfluidic analysis of cellular deformability of normal and oxydatively damaged red blood cells[END_REF][START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF][START_REF] Tomaiuolo | Microfluidics analysis of red blood cell membrane viscoelasticity[END_REF].

Along this line, several works have investigated the steady-state behavior of confined RBCs, from the theoretical [START_REF] Secomb | Flow of axisymetric red blood cells in narrow capillaries[END_REF][START_REF] Secomb | Analysis of red blood cell motion through cylindrical micropores: effects of cell properties[END_REF], numerical [START_REF] Fedosov | Deformation and dynamics of red blood cells in flow through cylindrical microchannels[END_REF][START_REF] Kaoui | Twodimensional vesicle dynamics under shear flow: effect of confinement[END_REF][START_REF] Thiébaud | Prediction of anomalous blood viscosity in confined shear flow[END_REF] and experimental [START_REF] Tomaiuolo | Microfluidics analysis of red blood cell membrane viscoelasticity[END_REF][START_REF] Forsyth | The dynamic behavior of chemically "stiffened" red blood cells in microchannel flows[END_REF][START_REF] Rosenbluth | Analyzing cell mechanics in hematologic diseases with microfluidic biophysical flow cytometry[END_REF][START_REF] Shevkoplyas | Direct measurement of the impact of impaired erythrocyte deformability on microvascular network perfusion in a microfluidic device[END_REF][START_REF] Hou | Deformability based cell margination-A simple microfluidic design for malaria-infected erythrocyte separation[END_REF] points of view, in order to probe e.g. the influence of mechanical properties on cell shape or velocity. On the other hand, a limited number of studies have investigated the behavior of RBCs in transient situations. Early studies have investigated the relaxation of RBC deformation following the removal of an applied mechanical stress: the pioneer work by Hochmuth et al. focused on shape recovery in micropipette experiments [START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF]; Baskurt and Meiselman later performed rheo-optical strain relaxation experiments upon cessation of shear of concentrated RBC suspension [START_REF] Baskurt | Determination of red blood cell shape recovery time constant in a Couette system by the analysis of light reflectance and ektacytometry[END_REF], and Bronkhorst et al. pioneered the use of optical tweezers to conduct shape recovery experiments on single cells [START_REF] Bronkhorst | A new method to study shape recovery of red blood cells using multiple optical trapping[END_REF]. More recently, Guido and Tomaiuolo have performed measurements of RBC response time upon flow start in microchannels [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF], and Braunmüller et al. have investigated RBC shape relaxation using microfluidic tools [START_REF] Braunmüller | Hydrodynamic deformation reveals two coupled modes/time scales of red blood cell relaxation[END_REF].

The above studies have provided consistent results for the timescale controlling shape relaxation (or establishment) of healthy RBCs: τ c ≃ 0.1 -0.2 s. Moreover, chemical or physical treatments known to affect the mechanical properties of RBCs were observed to clearly modify this characteristic time [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF][START_REF] Baskurt | Determination of red blood cell shape recovery time constant in a Couette system by the analysis of light reflectance and ektacytometry[END_REF]. From such measurements, authors concluded that τ c was indeed governed by the viscoelastic properties of RBCs. Following Evans and Hochmuth [START_REF] Evans | Membrane viscoelasticity[END_REF], τ c has been commonly related to the mechanical properties of the cell as:

τ c = η 2D mem /µ s .
Using typical values µ s ∼ 5 -10 µN•m -1 taken from the literature [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF][START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF], a 2D membrane viscosity η 2D mem ∼ 0.5 -1 × 10 -6 N•s•m -1 has been computed from the measured τ c [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF][START_REF] Tomaiuolo | Microfluidics analysis of red blood cell membrane viscoelasticity[END_REF][START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF][START_REF] Baskurt | Determination of red blood cell shape recovery time constant in a Couette system by the analysis of light reflectance and ektacytometry[END_REF]. Puzzlingly, such studies of the transient response of RBCs conclude to a value of the membrane viscosity that contrasts with that coming from other groups of experiments, which rather yield η 2D mem 10 -7 N•s•m -1 [START_REF] Chien | Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane[END_REF][START_REF] Tran-Son-Tay | Determination Of Red Blood Cell Membrane Viscosity From Rheoscopic Observations Of Tank-Treading Motion[END_REF][START_REF] Riquelme | Determination of the complex viscoelastic parameters of human red blood cells by laser diffractometry[END_REF].

The origin of such a discrepancy between the few experimental determinations of the membrane viscosity is a long-standing issue and is still an open question. However, the membrane viscosity is often a required input parameter, either for experimental data analysis [START_REF] Fischer | Tank-treading frequency of the red cell membrane: dependence on the viscosity of the suspending medium[END_REF] or in advanced numerical models of RBCs [START_REF] Fedosov | Predicting human blood viscosity in silico[END_REF][START_REF] Peng | Lipid bilayer and cytoskeletal interactions in a red blood cell[END_REF][START_REF] Secomb | Two-dimensional simulation of red blood cell deformation and lateral migration in microvessels[END_REF]. In the perspective of, e.g., quantitative numerical studies of the flow behaviour of suspensions of RBCs, there is therefore a clear need for a more accurate knowledge of the membrane viscosity.

We address this question in the present paper. We present an analytical model that describes the shape evolution of a RBC in shear flow, which we validate using 3D numerical simulations. We use this framework to analyze the results of startup flow experiments. The latter are performed in the spirit of the study by Tomaiuolo and Guido [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF]: we extend the work of these authors, and probe the effect on τ c of (i) the flow strength, (ii) the viscosity of the suspending fluid (η out ), (iii) a chemical treatment known to affect the intrinsic mechanical properties of RBCs.

We find that τ c depends on the flow strength, and exploit this dependence to extract, in an original way, both the effective viscosity and the elastic modulus which govern the characteristic transient time of RBCs. We then demonstrate that such an effective viscosity is not identical to the membrane viscosity, but can be used to determine the actual η 2D mem . Doing so, we obtain a value of η 2D mem which is in good agreement with the low values reported by Tran-Son-Tay et al. [START_REF] Tran-Son-Tay | Determination Of Red Blood Cell Membrane Viscosity From Rheoscopic Observations Of Tank-Treading Motion[END_REF]. By combining theoretical, numerical, and experimental efforts in this study of viscoelastic transients of RBCs, we are thus able to reconcile the seemingly conflicting results regarding membrane viscosity of RBCs.

II. THEORETICAL FRAMEWORK A. Qualitative discussion

The membrane of a RBC as well as the internal and external fluids are characterized by their respective viscosities η mem , η in and η out , with η mem = η 2D mem R, where R designates the radius of a sphere having the same surface area as that of a RBC. RBC deformation due to external flow is accompanied by dissipation in the three fluids. Since one expects the slowest mechanism to govern the dynamics, this entails that the three regions act as dashpots in parallel, so that the total effective dissipation coefficient can be written as η ef f = αη out +β ′ η in +γ ′ η mem where α, β ′ and γ ′ are dimensionless numbers to be specified below. Shape deformation of a RBC occurs on a time scale (to be determined) of order τ c and deformation is typically on the order of the RBC radius, R, so that the viscous tension created by RBC deformation is of order η ef f R/τ c (this is a force per unit length). This force counterbalances the combined effects of the force due to external flow, of order η out V (V being a typical applied velocity), and of the non-dissipative part due to elasticity of the cell. The membrane is endowed with bending and shear elastic energy due to cytoskeleton. Calling κ the bending modulus (having the dimension of an energy), the typical force per unit length associated with it is of order κ/R 2 . The shear elastic modulus is denoted as µ s and represents a force per unit length (a 2D shear elastic modulus), so that total elastic force per unit length is of order K ef f ≃ κ/R 2 + µ s . The force balance yields

τ -1 c = K ef f + η out V η ef f R , (1)
Eq. 1 provides us with the typical time scale of deformation of a RBC. Let us rewrite (1) as

τ -1 c = a 1 V + a 0 , (2)
where

a 1 -1 = Rα(1 + βλ + γλ ′ ) (3)
a -1 0 = Rη out α(1 + βλ + γλ ′ ) K ef f (4)
with λ = η in /η out , λ ′ = η mem /η out , and where β = β ′ /α and γ = γ ′ /α. The different coefficients above (α, β, γ) can only be determined numerically (see below). However, when the shape of a cell is not very far from a sphere, an analytical calculation is possible [START_REF] Misbah | Vacillating Breathing and Tumbling of Vesicles under Shear Flow[END_REF][START_REF] Danker | Dynamics and rheology of a dilute suspension of vesicles: Higher-order theory[END_REF][START_REF] Vlahovska | Dynamics of a viscous vesicle in linear flows[END_REF][START_REF] Lebedev | Nearly spherical vesicles in an external flow[END_REF][START_REF] Vlahovska | Dynamics of a non-spherical microcapsule with incompressible interface in shear flow[END_REF][START_REF] Danker | Vesicles in Poiseuille Flow[END_REF]. This has been done for the case of vesicles in a linear shear flow [START_REF] Lebedev | Nearly spherical vesicles in an external flow[END_REF], taking into account η out , η in and η mem , and yields α = 16/3, β = 23/32, and γ = 1/2. A similar study has been performed for a Poiseuille flow [START_REF] Danker | Vesicles in Poiseuille Flow[END_REF], incorporating only the contributions of the outer and inner fluid viscosities, which provides α ≃ 9 and β = 76/85 for the flow of a vesicle in a channel of internal radius 5 µm, as it is the case in our experiments. We discuss in the following section our choice for these parameters.

B. Numerical simulations

We have performed a systematic numerical study of a model of RBC. The imposed flow is given by

V ∞ (R) = e x V 1 - y 2 + z 2 (D/2) 2 , ( 5
)
where V is the velocity at the center of the flow, the x axis is along the flow direction, while the y and z axes are along the transverse directions. We consider here an unbounded flow. The curvature of the imposed Poiseuille flow alone, without the no-slip boundary conditions at the channel walls, has proven to account for several experimental facts provided that the RBC is not too confined, as shown in [START_REF] Coupier | Shape Diagram of Vesicles in Poiseuille Flow[END_REF]. D/2 in Eq. 5 specifies at which distance from the center the imposed velocity falls to zero. D/2 was set equal to 5 µm as in the experiments described below.

Numerical simulations for 3D model of RBC are based on the boundary integral method, as originally described for vesicles in [START_REF] Biben | Threedimensional vesicles under shear flow: Numerical study of dynamics and phase diagram[END_REF]. In a recent work [START_REF] Farutin | 3D numerical simulations of vesicle and inextensible capsule dynamics[END_REF], we have extended this study to include membrane shear elasticity mimicking the spectrin network of RBCs (membrane viscosity is not accounted for at present). We have also made several new numerical improvements that allowed us to study very deflated shapes, as required for simulation of real RBCs. We only provide here the main results concerning numerical determination of shapes and relaxation time scales, while details of the numerical techniques can be found in [START_REF] Farutin | 3D numerical simulations of vesicle and inextensible capsule dynamics[END_REF]. The RBC model takes into account both bending and shear elasticity. The Helfrich model is adopted for the bending energy, while the shear elasticity of the cytoskeleton is described using a FENEM-like (Finite Extensibility Non-linear Elasticity) strain hardening model, which has revealed to capture some realistic features of RBCs (for details see [START_REF] Farutin | 3D numerical simulations of vesicle and inextensible capsule dynamics[END_REF]).

The RBC was modeled as an inextensible membrane, endowed with a shear modulus µ s = 1.9 µN•m -1 and a bending modulus κ = 2.7 × 10 -19 J. The stress-free state of the membrane was chosen as a biconcave shape with a surface area of 122 µm 2 and a volume of 82 µm 3 (i.e. R ≃ 3 µm). Most simulations were performed using 5120 triangular elements to discretize the membrane (2562 vertices). The viscosity of the inner solution of the cell was chosen as η in = 6 mPa•s [START_REF] Betz | ATP-dependent mechanics of red blood cells[END_REF]. The viscosity of the suspending medium, η out , and the flow rate were varied as in the experiments. The elastic and geometrical parameters of the cell were chosen in order to get a model consistent with optical tweezers experiments [START_REF] Mills | Nonlinear Elastic and Viscoelastic Deformation of the Human Red Blood Cell with Optical Tweezers[END_REF]. Several additional simulations were performed with different values of µ s and κ in order to mimic the chemical treatment of the cell.

The characteristic time of the cell was determined by monitoring the transient behaviour of the cell velocity upon flow startup: in a manner similar to what has been used for experimental data analysis, we define the characteristic time τ c as the time needed for the cell to reach 99% of its steady-state velocity.

III. MATERIALS AND METHODS

A. Materials

Fused silica capillaries of inner diameter 10 µm and outer diameter 150 µm were purchased from BGB Analytik (Germany).

Polydimethylsiloxane (PDMS, Dow Corning Sylgard 184) was obtained from Neyco (France). Bovine Serum Albumin (BSA), Phosphate Buffered Saline (PBS) tablets, dextran of average molecular weight 40 kDa, and diamide were purchased from Sigma Aldrich (France). Solutions of PBS 0.01M and pH 7.4 were prepared using 18.2 MΩ•cm ultrapure water.

The viscosity of PBS and PBS+dextran solutions was measured on a cone/plate rheometer (Anton Paar MCR301), and found to be η out = 1.5 ± 0.15 mPa•s and 5±0.5 mPa•s respectively for pure PBS and PBS + 10%w of dextran 40kDa.

B. Preparation of blood samples

Blood samples from healthy donors were obtained through the Établissement Français du Sang (Grenoble) and stored at 4 • C until use. Red blood cells were extracted from whole blood by successive washes in PBS solution and centrifugation. After each centrifugation, the supernatant was pipetted out and PBS was added to refill the tube. The washing/centrifugation cycle was repeated three times.

RBCs were used as such, or after being exposed to diamide: using a protocol akin to that described in [START_REF] Forsyth | The dynamic behavior of chemically "stiffened" red blood cells in microchannel flows[END_REF], erythrocytes were incubated at room temperature for 30 minutes in a PBS solution containing 5mM of diamide, then washed in PBS before use in the flow cell.

C. Experimental Setup

Experiments were performed using a custom-built flow cell composed of the following elements, as illustrated in Fig. 1: 4 silica capillaries were cut in 3-6 mm long segments and fitted into 4 grooves in the central band of a PDMS spacer. The spacer was then sandwiched between a glass coverslip and a clear polycarbonate plate, and the stack was clamped in an aluminum frame to ensure tight sealing of the cell. The inlet and outlet drilled into the polycarbonate upper plate were connected by silicone tubing to external reservoirs. A solenoid valve was connected between the inlet reservoir and the flow cell, in order to control the start-up of the flow, while the steady-state flow velocity was controlled by adjusting the height difference (∆z) between the liquid free surfaces in the inlet and outlet beakers. The response time upon startup of the whole setup (including valve, tubings and flow cell) was measured to be 10 ms (see Fig. 4a below).

Before the experiments, the flow cell was filled with a PBS solution containing 0.2 %w BSA, in order to coat the inner walls of the capillaries and prevent spurious adhesion of the cells to the walls of the microchannels. A RBC suspension was then added to the inlet reservoir and the concentration adjusted with PBS so that the initial bulk hematocrit was less than 1%.

The flow cell was placed on the motorized stage of an inverted microscope (Olympus IX-71) equipped with a 100x oil-immersion objective. Image sequences of the flowing RBCs were acquired using a high speed camera (Phantom Miro4) at a frame rate of 4000 images per second.

D. Experimental data analysis

The recorded image sequences were analyzed using Im-ageJ software and Matlab toolboxes. Image stacks showing the position of a RBC inside the capillary as a function of time were opened with ImageJ and cropped in order to keep only a rectangular region of interest (ROI), of height 10 µm and length 167 µm, enclosing the lumen of the capillary focused at its midplane (see Fig. 2a-c). The intensity of each image in the stack was averaged over the height of the ROI, and the result of this averaging was plotted as a function of time, in order to obtain a space-time diagram (see Fig. 2d). Such a space-time plot was further processed with a gradient filter and finally thresholded, in order to obtain a binary image showing the position of the front and rear parts of the flowing red blood cell as a function of time. This binary image was then processed with standard Matlab tools in order to extract the position of the front or rear ends as a function of time, x(t), from which the instantaneous velocity V RBC (t) was computed. We have checked that no significant difference was obtained on V RBC (t) when computed from the position of the front or rear end, and that the obtained results were quantitatively in agreement with velocity measurements performed using Particle Imaging Velocimetry (PIV) tools from ImageJ. The method described above has the benefit of being much less time consuming than PIV.

IV. RESULTS

A. Measurements of transient time

An example of cell behavior during flow startup is given in Fig. 3a. It can be seen that a RBC initially at rest in the channel gradually deforms with time, reaching a steady-state shape after a few tens of milliseconds. In the range of pressure drop explored in our work, the steady-state shape of the deformed RBC could be either parachute or slipper-like [START_REF] Skalak | Deformation of Red Blood Cells in Capillaries[END_REF][START_REF] Secomb | A two-dimensional model for capillary flow of an asymmetric cell[END_REF][START_REF] Kaoui | Why do red blood cell have asymmetric shapes even in a symmetric flow?[END_REF], as illustrated in Figs. 3b-d. The time dependence of the velocity of a RBC (V RBC ) is illustrated in Fig. 4a. From such V RBC (t) curves we compute:

(i) the steady-state velocity of the cells (V max ), which depends, as expected for a Poiseuille flow, linearly on the imposed pressure drop ∆P , as shown in the inset of Fig. 4a, and (ii) the transient time τ c as defined in Fig. 4a. In Fig. 4b, we compare the characteristic time τ c determined visually (from shape evolution) and from velocity transients, for all the data collected on RBCs suspended in PBS solution (the set of data corresponds to a total of about 100 cells). We find a good correlation between the times determined from the two methods, with τ c computed from velocity being typically 20% smaller. This difference arises, to a large extent, from the criterion we use to define τ c from velocity transients, which underestimates the time to actually reach steady-state by 10-20%. The results presented in the rest of the paper correspond to τ c determined from velocity transients. Data are presented as average values for τ c and V max , with error bars corresponding to ± one standard deviation, computed on groups of 8 to 12 RBCs.

B. Dependence of τc on flow strength, external viscosity and diamide treatment

We observe that the characteristic time τ c decreases as the flow strength increases. This is illustrated in Fig. 5a, for RBCs suspended in PBS solution, where it is seen that τ c decreases from ∼ 0.1 to 0.03 s as V max increases from 0.5 to 5 mm•s -1 .

Plotting the inverse time 1/τ c as a function of V max , we find that 1/τ c exhibits a linear increase with the steadystate velocity (Fig. 5b).

Upon increasing the viscosity of the suspending medium from that of PBS solution (η out = 1.5 mPa•s) to that of the 10%w dextran solution (η out = 5 mPa•s), we observe that the dependence of 1/τ c on V max remains linear, with a greater slope than for RBCs suspended in PBS and a similar value of the intercept at zero velocity (Fig. 5b). Finally, we show in Fig. 6 the effect of diamide on transient time. It is seen that diamide-treated RBCs still exhibit an inverse transient time 1/τ c which increases linearly with V max , with an overall upward shift of the curve with respect to that obtained for healthy RBCs.

As discussed in detail below, all the above results are in good agreement with the simple heuristic argument given in section II A. Lines are best linear fits of the data for untreated (dashed line, with slope and intercept values as in Fig. 5), and diamidetreated RBCs (full line, a0 = 15.8 ± 1.6 s -1 and a1 = 3.4 ± 0.7 mm -1 ).

C. Numerical results

Fig. 7 shows a snapshot illustrating the temporal evolution of the shape of a RBC obtained from simulations, starting from a biconcave shape at rest. The inverse relaxation time is plotted in Fig. 8 as a function of the steady-state velocity. 1/τ c is seen to vary linearly with the flow rate, and to display a stronger dependence on V max for lower viscosity contrasts λ, i.e. for larger η out (see Fig. 8a). There is no apparent deviation from linearity neither for weak nor for strong flows in the range of explored velocities, in qualitative agreement with experiments. As shown in Fig. 8b, for a given viscosity contrast λ, changing the value of the inner fluid viscosity, η in , merely results in a vertical shift of the τ -1 c (V max ) curve, without affecting the slope. Data corresponding to an inner viscosity η 2 in can be obtained from results computed for η 1 in by simply rescaling τ -1 c and V max by the ratio η 1 in /η 2 in .

The effect of varying the shear or the bending modulus is illustrated in Fig. 8b. Decreasing µ s by a factor of two leaves the slope of the curve essentially unchanged, and results in a twofold decrease of the intercept at zero velocity. Decreasing κ by a factor of two increases the slope by 10% and lowers the offset at zero velocity by about 15%.

V. DISCUSSION

A. Numerical and analytical results

Fig. 7 illustrates the ability of the numerical model to reproduce the shape evolution of a RBC upon startup of a Poiseuille flow: the shape transition from discocyte to parachute obtained numerically is consistent with experimental observations such as that shown in Fig. 3a.

There is a strikingly good qualitative agreement between the simulations and the model presented in section II: as inferred from a heuristic argument (Eq. 1), a linear dependence of 1/τ c on V max is predicted numerically, the slope of this linear dependence is controlled by the effective viscosity, hence by the ratio λ, while the elastic moduli affect only the value of τ -1 c at V max = 0. The magnitude of the effects produced by varying either µ s or κ indicates that the elastic behavior is essentially governed by µ s (Fig. 8b). Such an agreement between numerics and theory prompts us to further exploit numerical simulations. Fig. 8a shows that the slope of 1/τ c vs V max , called a 1 in Eq. 2, depends on the viscosity contrast λ = η in /η out . In Fig. 9, we plot the inverse slope, a -1 1 , obtained from numerical simulations, as a function of λ. It is seen that a -1 1 increases linearly with λ. This is again fully consistent with theoretical predictions (see Eq. 3). A linear fit of a -1 1 (λ) therefore allows us to determine the parameters α and β, which set the relative contributions of η out and η in to the effective viscosity (see Eq. 3): we get αR = 17.2 µm and αRβ = 16.8 µm, which yields, with R = 3µm, α = 5.7 and β = 0.98. These values are to be compared with those predicted by theories, namely α ≃ 5.3 and β ≃ 0.72 for the case of a linear shear flow [START_REF] Lebedev | Nearly spherical vesicles in an external flow[END_REF], and α ≃ 9 and β ≃ 0.9 for vesicles in a Poiseuille flow [START_REF] Danker | Vesicles in Poiseuille Flow[END_REF]. The fair agreement between the parameters determined numerically and their theoretical values is all the more satisfactory that the analytical treatments proposed in [START_REF] Lebedev | Nearly spherical vesicles in an external flow[END_REF] and [START_REF] Danker | Vesicles in Poiseuille Flow[END_REF] are based on spherical harmonics expansions that are strictly valid only for small deformations around a reference spherical shape, whereas numerical simulations handle the case of an initially biconcave, highly deflated, cell undergoing large deformations under flow. This provides strong support to the fact that analytical models such as those developed in [START_REF] Lebedev | Nearly spherical vesicles in an external flow[END_REF] and [START_REF] Danker | Vesicles in Poiseuille Flow[END_REF] are robust and able to capture the main contributions to dissipation during vesicle flow.

On this basis, we will now analyze our experimental data within the proposed theoretical framework. Since numerical simulations account for more realistic shapes and deformations of the cells than theoretical models, we will use in the following the values of the parameters α and β determined from the numerical results. As far as the parameter γ is concerned, it cannot be determined from simulations since membrane viscosity is not included in the numerical model. Still, the reasonable agreement between the numerical and theoretical values obtained for α and β suggests that analytical parameters are of the correct order of magnitude, and we will therefore use the theoretical value γ = 1/2 [START_REF] Lebedev | Nearly spherical vesicles in an external flow[END_REF]. A more accurate estimate of γ, i.e. of the exact weight of η mem in η ef f , would call for more advanced simulations accounting for membrane viscosity.

B. Interpretation of experimental results for healthy RBC

We first compare our results with those obtained by Tomaiuolo and Guido [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF]: most of the results presented in their study have been obtained at a single steady-state velocity of 1 mm•s -1 , at which τ c = 0.09 s is found for healthy RBCs in PBS. At the same velocity, we measure a characteristic time τ c = 0.08±0.01 s, which is in excellent quantitative agreement.

Besides, as predicted from a simple heuristic argument (see Eqs. 1 and 2), we find experimentally that 1/τ c is linear with V max (Fig. 5). Linear fits of the data obtained in PBS and PBS+dextran yield the values of the intercept at origin (a 0 ) and slope (a 1 ) reported in Table I. We observe roughly a threefold increase in the slope when η out is multiplied by 3. Recalling that a 1 = η out /(η ef f R) (Eq. 1), this suggests that η ef f is only marginally affected by the external viscosity for η out in the range 1.5 -5 mPa•s, hence that the contribution of η out to η ef f is weak. This is also consistent with the fact that the change in η out does not modify significantly the value of a 0 = K ef f /(η ef f R). In our experiments, the main effect of changing η out is therefore to affect the magnitude of the hydrodynamic forces (η out V ) applied to the RBC.

From a 0 and a 1 , we compute the effective viscosity and elastic modulus which are given in Table I. We thus get a value of η ef f ≃ 120 mPa•s which is in good agreement with the one reported in a recent study by Betz et al., namely η ef f ≃ 100 mPa•s [START_REF] Betz | ATP-dependent mechanics of red blood cells[END_REF]. The value of K ef f ≃ 3 µN•m -1 is consistent with that of 2.5±0.4 µN•m -1 re-ported for µ s by Henon et al. [START_REF] Hénon | A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers[END_REF]. This suggests that, in the experimental conditions used in the present study, K ef f is governed by the shear elastic modulus. This is in agreement with our numerical results, which show that the value of τ -1 c at V max = 0 is essentially controlled by µ s (Fig. 8b).

Furthermore, from the values of a 1 and using Eq. 3, we make the following estimates for the membrane viscosity: η mem ≃ 20 -28 mPa•s (see Table I). From such an estimate, we conclude that, for an outer fluid viscosity in the physiological range (η out ∼ 1 mPa•s), η in and η mem are the two major contributions to the effective viscosity: α(βη in + γη mem ) ≃ 100 mPa•s ≃ η ef f . Finally, if we compute a 2D membrane viscosity as η 2D mem = η mem R, we get η 2D mem ≃ 0.6 -0.85 × 10 -7 N•s•m -1 . To summarize, our results regarding the order of magnitude of τ c are consistent with those from previous works on the transient response of RBCs [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF][START_REF] Tomaiuolo | Microfluidics analysis of red blood cell membrane viscoelasticity[END_REF][START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF], but on the other hand we estimate a membrane viscosity η 2D mem which is about one order of magnitude lower than the values reported in these studies (i.e. η 2D mem ∼ 10 -6 N•s•m -1 [START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF]). However, we note that the latter are based on an analysis proposed initially by Evans and Hochmuth [START_REF] Evans | Membrane viscoelasticity[END_REF], who developed a model assuming that the only dissipation relevant to RBC shape recovery is associated to the membrane viscosity. They thereby neglected dissipation in the inner and outer fluids, and estimated η 2D mem ≃ τ c µ s . By contrast, here we make no such assumption. We find that (i) the effective viscosity is a combination of η mem , η in and η out , and (ii) each viscosity entering η ef f has a weight which is larger than 1 (α ≃ 6). The latter point, which is an outpout of our theoretical and numerical analysis, is consistent with the ad hoc assumption made by Betz et al. in their modeling of membrane fluctuations [START_REF] Betz | ATP-dependent mechanics of red blood cells[END_REF]. Taking into account all the viscous contributions and their respective weights in our data analysis, we get values for η 2D mem in very good agreement with those reported by Tran-Son-Tay et al. [START_REF] Tran-Son-Tay | Determination Of Red Blood Cell Membrane Viscosity From Rheoscopic Observations Of Tank-Treading Motion[END_REF]. These authors deduced η 2D mem from the tank-treading frequencies of RBCs under shear, using a model incorporating all dissipation sources, and obtained values which lie between 0.5 and 1.2×10 -7 N•s•m -1 . Our work, fully consistent with the one of Tran-Son-Tay et al. [START_REF] Tran-Son-Tay | Determination Of Red Blood Cell Membrane Viscosity From Rheoscopic Observations Of Tank-Treading Motion[END_REF], thus strongly suggests that the assumption made by Evans and Hochmuth leads to an overestimate of η 2D mem and is at the origin of the apparent discrepancy between the reported values of RBC membrane viscosity.

ηout a0 a1 η ef f K ef f ηmem (mPa•s) (s -1 ) (mm -1 ) (mPa•s) (µN•m -1 ) (mPa•s) 1.5 8.3 ± 0.

C. Comparison between experiments and numerical simulations

Although numerical simulations show a linear dependence of 1/τ c on V max , in qualitative agreement with experiments, we observe that both the slope (a num 1 ) and the extrapolated value of 1/τ c at zero velocity (a num 0 ) predicted numerically are larger than the experimental values (a exp 1 and a exp 0 ). This is shown in the summary table below (Table II).

Following the data analysis proposed in the previous section, the difference in slopes can straightforwardly be attributed to the contribution of membrane viscosity to the effective viscosity. Indeed, we have used the numerical results regarding a num 1 (λ) to determine the parameters α and β (Fig. 9), and then fitted the experimental data using Eqs. 2 and 3 in order to evaluate η mem , so that the difference between the inverse slopes merely reads 1/a exp 1 -1/a num 1 = αγRη mem /η out . Using the latter expression to compute η mem yields exactly the same values reported before for the membrane viscosity. Now, a much more stringent test is to check whether the difference in the offsets (a 0 ) can be attributed to the contribution of membrane viscosity, because in contrast to a num 1 , a num 0 was not used to determine parameters of the model. From Eq. 4, we expect:

1 a exp 0 - 1 a num 0 = αγη mem R K ef f (6)
In order to check the above equality, we estimate K ef f from Eq. ( 6) and compare it to the effective modulus determined in the previous section. Using the values of a 0 reported in Table II and of η mem determined above, we get: K ef f = 2 -2.5 µN•m -1 respectively for λ = 1.2 (η out = 5 mPa•s) and λ = 4 (η out = 1.5 mPa•s). This is in quantitative agreement with the value of K ef f ≃ 3 ± 1 µN•m -1 determined from experimental data, and therefore supports the fact that discrepancies between simulations and experiments are due to the missing contribution of membrane viscosity in the numerical model.

λ a num 0 (s -1 ) a num 1 (mm -1 ) a exp 0 (s -1 ) a exp 1 (mm -1 ) 4

D. Effect of diamide treatment

In Fig. 6, we see that treating RBCs with diamide mainly results in an upward shift of the curve τ -1 c (V max ), i.e. diamide-treated RBCs exhibit a shorter transient time than healthy ones. From a linear fit to the data, we obtain η ef f = 147 ± 45 mPa•s and K ef f = 7.0 ± 2.8 µN•m -1 . The major effect of diamide is therefore an increase of the effective elastic modulus K ef f , by roughly a factor of 2 with respect to healthy RBCs. Diamide is known to affect the spectrin network of RBCs, by creating crosslinks between proteins through the formation of disulphide bonds [START_REF] Fischer | Selective alteration of erythrocyte deformability by SH-reagents. Evidence for an involvement of spectrin in membrane shear elasticity[END_REF]. Such a crosslinking process has been reported to induce a stiffening of RBCs [START_REF] Fischer | Selective alteration of erythrocyte deformability by SH-reagents. Evidence for an involvement of spectrin in membrane shear elasticity[END_REF][START_REF] Johnson | Decreased survival in vivo of diamideincubated dog erythrocytes. A model of oxidant-induced hemolysis[END_REF][START_REF] Rodrigues | Red Blood Cells deformability index assessment in a hyperbolic microchannel: the diamide and glutaraldehyde effect[END_REF][START_REF] Shin | Disposable Biosensor for Measuring Red Blood Cell Deformability using Laser-Diffraction Technique[END_REF], via an increase in the membrane shear modulus [START_REF] Fischer | Selective alteration of erythrocyte deformability by SH-reagents. Evidence for an involvement of spectrin in membrane shear elasticity[END_REF]. Our results concerning the effect of diamide on RBC transient time are therefore fully consistent with these previous studies. Furthermore, attributing the increase of K ef f to an increase in µ s is consistent with the numerical results presented in section IV C. However, Forsyth et al., from which we have adapted the protocol for diamide treatment in the present work, recently reported that, in microfluidics experiments relying on cell stretching during transient confinement in a channel constriction, no difference in elongation could be detected between healthy and diamide-treated RBCs [START_REF] Forsyth | The dynamic behavior of chemically "stiffened" red blood cells in microchannel flows[END_REF]. We believe that the apparent contradiction between our finding and the conclusions of reference [START_REF] Forsyth | The dynamic behavior of chemically "stiffened" red blood cells in microchannel flows[END_REF] merely arises from the very different time scales probed in the two studies. Indeed, while we characterize the effect of diamide by measuring the transient viscoelastic time of RBCs, of order 50 ms, Forsyth et al. probe the elongation of RBCs during a transient confinement that lasts between 1 and 5 ms only (calculated from the flow speeds and constriction length reported in [START_REF] Forsyth | The dynamic behavior of chemically "stiffened" red blood cells in microchannel flows[END_REF]). The duration of confinement in the work of Forsyth et al. is therefore much shorter than the time required for RBCs to reach their steady shape. This, added to the fact that we have used both a slightly larger diamide concentration and treatment time in our study, is likely to be at the origin of the differences between our work and the one reported in [START_REF] Forsyth | The dynamic behavior of chemically "stiffened" red blood cells in microchannel flows[END_REF].

VI. CONCLUSIONS

We have performed a study of the characteristic viscoelastic time τ c of red blood cells. We have combined theory and numerical simulations in order to establish a framework for the analysis of flow startup experiments on RBCs confined into microchannels. We have obtained experimental values of τ c in quantitative agreement with those from previous studies, measured either from flow startup [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF], relaxation after cessation of shear [START_REF] Baskurt | Determination of red blood cell shape recovery time constant in a Couette system by the analysis of light reflectance and ektacytometry[END_REF], or micropipette experiments [START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF]. Moreover, we have shown that probing the dependence of τ c on flow strength allows us to determine both the effective viscosity (η ef f ) and elastic modulus (K ef f ), and have obtained values for these two quantities which are consistent with those from other works [START_REF] Betz | ATP-dependent mechanics of red blood cells[END_REF][START_REF] Hénon | A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers[END_REF]. Most importantly, we have shown that, in contrast to the assumption made by Evans and Hochmuth [START_REF] Evans | Membrane viscoelasticity[END_REF] and commonly used later [START_REF] Tomaiuolo | Start-up shape dynamics of red blood cells in microcapillary flow[END_REF][START_REF] Tomaiuolo | Microfluidics analysis of red blood cell membrane viscoelasticity[END_REF][START_REF] Hochmuth | Red-cell extensional recovery and the determination of membrane viscosity[END_REF][START_REF] Baskurt | Determination of red blood cell shape recovery time constant in a Couette system by the analysis of light reflectance and ektacytometry[END_REF], η ef f is not equal to the membrane viscosity η mem . We have identified, from theory and simulations, the relative contributions of the membrane, inner (η in ) and outer (η out ) fluid viscosities to the overall effective viscosity, and used this in order to compute the value of the membrane viscosity from our experimental data. Doing so, we obtain η mem in the range 20-28 mPa•s, which translates into a 2D viscosity η 2D mem = 0.6-0.85×10 -7 N•s•m -1 . We conclude that the difference of up to one order of magnitude that can be found in the literature regarding η mem only results from the fact that assuming η ef f ≡ η mem leads to overestimate η mem by about a factor of ten. The present work thus reconciles previous contrasting reports about RBC membrane viscosity, and provides a range of values for η 2D mem which is in excellent quantitative agreement with the result of Tran-Son-Tay et al. [START_REF] Tran-Son-Tay | Determination Of Red Blood Cell Membrane Viscosity From Rheoscopic Observations Of Tank-Treading Motion[END_REF]. Moreover, we have demonstrated that measuring τ c as a function of flow strength provides valuable information not only on dissipation, but also on the elastic response of RBCs, and is sensitive to elasticity alterations such as those caused by diamide. This type of experimental investigations, analysed within the proposed framework, should therefore prove to be useful in discriminating how physical or biological factors may affect red blood cell elasticity or viscosity.

FIG. 1 .

 1 FIG. 1. Sketch of the flow cell used in the start-up experiments: (a) silica capillaries, (b) PDMS spacer with grooves, (c) glass coverslip, (d) polycarbonate plate.

  The characteristic transient time (τ c ) of RBCs upon flow startup was determined in two different ways:(i) we have estimated τ c from visual inspection of the time-dependent shape of RBCs, defining τ c as the time above which no change in RBC shape was visible by eye;(ii) we have computed the transient time from RBC velocity, defining τ c as the time required for a RBC to reach 90% of its steady-state velocity, as illustrated in Fig.4abelow.

FIG. 2 .

 2 FIG. 2. Image of a RBC in a channel at rest (a), during transient (b), and at steady-state (c). (d) Space-time diagram built from the full image sequence: the diagram shows the averaged intensity along the flow direction (horizontally, flow is from left to right), as a function of time (vertically, from top to bottom). Image size is 167 µm × 85 ms. The steady state velocity is 3 mm•s -1 , for a pressure drop of 2 kPa.

FIG. 3 .

 3 FIG. 3. (a) Montage showing the time evolution of the shape of a RBC upon flow startup. Scale bar is 10 µm. Time is stamped below each image. The steady-state velocity reached by the RBC is Vmax = 2.8 mm•s -1 . (b)-(d) illustration of the variety of initial RBC configurations (left image) and final steady-state shapes (right image) observed in our experiments. The three RBCs shown on (b)-(d) reach the same Vmax = 0.88 mm•s -1 .

FIG. 4 .

 4 FIG. 4. (a) Main panel: Time evolution of the velocity of a RBC going from rest at t = 0 to a steady-state velocity Vmax = 2 mm•s -1 . The arrow indicates the typical velocity overshoot observed upon opening of the solenoid valve, which lasts 10 ms. Inset: dependence of Vmax on the imposed ∆P . (b) τc determined from shape evolution as a function of τc computed from velocity transient. The dashed line is a linear (y = ax) fit to the data, with slope a = 1.22.

FIG. 5 .

 5 FIG. 5. Dependence of τc on experimental parameters. (a) τc as a function of Vmax for healthy RBCs suspended in PBS. (b) 1/τc as a function of Vmax for RBC in PBS (•) and in PBS containing 10%w of dextran 40kDa ( ). The dashed line is a linear fit, τ -1 c = a0 + a1Vmax, to the data obtained in PBS, with a0 = 8.3 ± 0.8 s -1 and a1 = 4.1 ± 0.3 mm -1 . The full line is a linear fit to the data obtained in PBS+dextran, with a0 = 8.4 ± 1.4 s -1 and a1 = 14.0 ± 1.9 mm -1 .

FIG. 6 . 1 /

 61 FIG. 6. 1/τc as a function of Vmax for untreated RBCs suspended in PBS (•), and diamide-treated RBCs in PBS ( ).Lines are best linear fits of the data for untreated (dashed line, with slope and intercept values as in Fig.5), and diamidetreated RBCs (full line, a0 = 15.8 ± 1.6 s -1 and a1 = 3.4 ± 0.7 mm -1 ).

FIG. 7 .

 7 FIG. 7. Numerical results illustrating the shape transformation for a RBC in Poiseuille flow. Color by surface tension. V = 0.5 mm•s -1 , λ = 5.

FIG. 8 . 1 c

 81 FIG. 8. (a) Inverse relaxation time (τ c ) as a function of steady-state velocity (Vmax), computed with ηin = 6 mPa•s, µs = 1.9 µN•m -1 and κ = 2.7 × 10 -19 J, for various viscosity contrasts: ( ) λ = 1.2, (•) λ = 3, ( ) λ = 4, ( ) λ = 6.7. Full lines are best linear fits to the data (τ -1 c = a num 0 +a num 1 Vmax). (b) τ -1 c (Vmax) computed for λ = 3 and parameters as in (a) (•), with ηin = 10 mPa•s (•), with κ = 1.35 × 10 -19 J ( ); with µs = 0.95 µN•m -1 ( ).

FIG. 9 .

 9 FIG. 9. Inverse slope, 1/a1, as a function of viscosity contrast λ ( ). The full line is the best linear fit to the data, with a value of the intercept at λ = 0 of 17.2 µm and a slope of 16.8 µm.

8 4 . 1 ±

 41 0.3 122 ± 21 3.0 ± 0.8 28 ± 8 5 8.4 ± 1.4 14.0 ± 1.9 119 ± 28 3.0 ± 1.2 20 ± 11 TABLE I. Values of a0 and a1 obtained from a linear fit of the experimental data for RBCs suspended in PBS (ηout = 1.5 mPa•s) and in PBS+dextran (ηout = 5 mPa•s). The corresponding values of η ef f and K ef f have been computed from Eqs. 1 and 2 with R = 3 µm. The membrane viscosity ηmem has been calculated from η ef f using Eq. 3 with R = 3 µm, α = 5.7, β = 0.98, γ = 0.5 and ηin = 6 mPa•s.

VII. ACKNOWLEDGMENTS

We acknowledge financial support from CNES (Centre National d'Etudes Spatiales), ESA (European Space Agency), and Université Franco-Allemande, Collège doctoral "Liquides Vivants" (AF).