Mounir Hemam

Meriem Djezzar

Moufida Maimour

Tarek Djouad

Asma Aounallah

An Ontological Approach for Domain Knowledge Modeling and Querying in Wireless Sensor Networks

Keywords: sensor networks, knowledge representation, ontology, OWL, SPARQL

Wireless sensor networks (WSNs) generate large volumes of raw data which increases the difficulty for applications to manage and query sensor data. WSNs are normally application specific with no sharing or reusability of sensor data among applications. In order for applications to be developed independently of particular WSNs, sensor data need to be enriched with semantic information. Ontologies are widely used as a means for solving the information heterogeneity problems because of their capability to provide explicit meaning to the information. This paper presents our work towards the development of a wireless sensor network ontology. Based on the proposed ontology we use the SPARQL query language to enable querying of sensor data. We present the description of the development of the proposed ontology, partial evaluation of the early prototype ontology, a discussion of design and implementation issues, and directions for future research works.

I. INTRODUCTION

Wireless sensor networks (WSNs) have gained a significant research attention in recent years. A Wireless sensor network in its simplest form can be defined as network of autonomous devices known as sensor nodes. In a Wireless Sensor Network, the individual nodes are capable of sensing the environments, processing the information locally, or sending it to one or more collection points through a wireless link [START_REF] Akyildiz | Wireless sensor networks: a survey[END_REF].The development of WSNs is motivated by several systems such as military monitoring system, smart home system, etc.

WSNs are largely used to collect data of various domains. They deploy large number of heterogeneous sensing nodes and generate large volumes of raw data which increases the difficulty for applications to manage and query sensor data. WSNs are normally application specific with no sharing or reusability of sensor data among applications. In order for applications to be developed independently of particular WSNs, sensor data need to be enriched with semantic information.

Semantics can play a role in assisting users to manage and query sensors and data. Indeed as the scale and complexity of sensing networks increases, machine interpretable semantics may allow autonomous or semi autonomous agents to assist in collecting, processing, reasoning about and acting on sensors and data. For their own part, users generally want to operate at levels above the technical details of format and integration, and rather work with domain concepts and restrictions on quality, allowing technology to handle the details.

The problem dealing with heterogeneity, even semantic has been deeply investigated in the field of ontology. Ontologies describe a shared and common understanding of a domain that can be communicated between people and heterogeneous software tools.

One of the most important projects based on ontologies use is the Semantic Web, which consists of adding to the Web a layer of knowledge, allowing in a first time, a information research at the semantic level and not at a syntactic one [START_REF] Berners-Lee | The semantic web[END_REF]. This led to the proposal of various XML-based languages whose objective is to facilitate the contents representation of electronic documents and the ontology exchange on the Web. Among them, is OWL language. The latter, offers the primitives of a frame-based representation, and adding the formal rigor of a Description Logic [START_REF] Baader | Description Logic Handbook: Theory, implementation and applications[END_REF]. This paper focuses on the semantics of the wireless sensor network. Semantics will be attached to the data to make the sensory data be understood more easily. It enables information exchange between sensors in multiple wireless sensor networks. So, our main goal is to propose the development of a wireless sensor network ontology for facilitating sensor data processing, interpretation and sharing between heterogeneous devices in WSNs.

In the development ontology process, the conceptualization step is based on the work results realized in knowledge engineering which seeks to determine how to find the good concepts and the good concepts interrelations. In addition, the step of formalization and checking are based on the research works in AI. These works give importance to the formal representation and reasoning aspect, and use in particular the description logics which have a clear semantics and provide powerful inferences.

The proposed ontology is implemented using Protege APIs [START_REF]Protégé Programming Development Kit (PDK)[END_REF] which the famous ontology editor. In addition, based on the proposed ontology we use the SPARQL query language [START_REF] Pérez | Semantics and complexity of SPARQL[END_REF] to enable querying of sensor data. Equipped with term representations and relationships definitions defined by the ontology, the SPARQL query engine will have information about the meaning of terms. Moreover, these relationships can be used to capture synonyms of a term to retrieve all information available for particular concept.

The remainder of this paper is organized as follows. Section 2 covers background information on sensor networks, ontologies design and the semantic Web technologies. Section 3 highlights some related work. In section 4 and 5 we present our approach for building and querying a sensor network ontology. Section 6 concludes this paper and talks about further work.

II. BACKGROUND

A. Sensor Network

A wireless sensor network is a group of specialized transducers with a communication infrastructure intended to monitor physical phenomena like temperature, sound, light intensity, location, motion of objects and so on. Each sensor network is deployed to serve a specific purpose and uses its own protocols. This heterogeneity in sensor networks makes it impossible to communicate with each other or to reuse and share their data with different applications.

The term WSN is used since most sensors are connected using wireless technologies such as Zigbee, Bluetooth and Wibree. The typical architecture of WSN consists of several nodes communicating with the outside world through a gateway. Sensor nodes are responsible for detecting and monitoring of certain phenomena and sending the raw measurement data to an end user. Further analysis and processing of sensor data is application dependent.

Sensor networks are characterized by two main features [START_REF] Kim | Building a serviceoriented ontology for wireless sensor networks[END_REF]:

 First, they are highly dense so that hundreds or thousands nodes may be deployed in limited geographical areas. These nodes return huge amount of data that must be efficiently searched to answer user queries. Unfortunately, classical information retrieval techniques showed poor performance in searching sensor networks data as they return many false positives/negatives.

 Second, many of the captured data are analogous in nature making the chance of finding a specific term quite good. Most sensors are characterized by similar calibration mechanisms that can be described using different terms. String matching search techniques may not retrieve all relevant data because different words/terms were used that did not match directly the term. For example temperature measurement is specified with the name temp or min temp and max temp or low temp and high temp. Relative Humidity is a name used in some domains and Humidity in other domains. This compromises the performance of the search engine. A big improvement in search engine performance could be achieved if these relationships are captured and utilized, and this is exactly what an ontology can do.

B. Ontology Building

Ontologies have been developed to capture the knowledge of a real world domain. "Ontology is defined as a formal and explicit specification of a shared conceptualization of a domain" [START_REF] Guarino | An Ontology of Meta-Level Categories[END_REF]. An ontology comprises three components: (1) classes or concepts that may have subclasses to represent more specific concepts than in super-classes, (2) properties or relationships that describe various features and properties of the concepts, also named slots or roles, and (3) restrictions on slots (facets) that are superimposed on the defined classes and/or properties to define allowed values (domain and range). Individuals can be defined simply as instances of the classes and properties. The ontology together with a set of instances of classes and slots constitute the knowledge base.

Many advantages of ontology building are explained in [START_REF] Noy | Ontology Development 101: A Guide to Creating Your First Ontology[END_REF], including: (1) sharing common understanding of the structure of information among people or software agents, (2) enabling reuse of domain knowledge, (3) making domain assumptions explicit, (4) separating the domain knowledge from the operational knowledge, and (5) analyzing domain knowledge. On the other hand, there exist several arguments and challenges, among which are the lack of an agreed-upon taxonomy and quantitative evaluation procedures.

Building a well-developed and usable ontology represents a significant challenge. A range of methods and techniques have been reported in the literature regarding ontology construction methodologies [START_REF] Lopez | Overview and Analysis of Methodologies for Building Ontologies[END_REF]. Mike Uschold's methodology [START_REF] Uschold | Ontologies: principles, methods and applications[END_REF], Michael Grüninger and Mark Fox's methodology [START_REF] Grüninger | Methodology for the Design and Evaluation of Ontologies[END_REF], and Methontology [START_REF] López | Building a Chemical Ontology Using METHONTOLOGY and the Ontology Design Environment[END_REF] are the most representative. These methodologies have in common that they start from the identification of the ontology's purpose and the need for domain knowledge acquisition. However, having acquired a significant amount of knowledge, Uschold's methodology and Grüninger and Fox's methodology propose coding in a formal language and Methontology proposes expressing the idea as a set of intermediate representations. These representations bridge the gap between how people think about a domain and the languages in which ontologies are formalized. Thus, Methontology enables experts and ontology makers unfamiliar with implementation environments to build ontologies.

For our purpose, we have chosen the Methontology for the application ontology building. Methontology enables the construction of ontologies at the knowledge level. It includes the identification of the ontology development process and a life cycle based on evolving prototypes.

C. The Semantic Web Technologies

Semantic Web is an extension to the current Web in which the meaningful relationships between resources is represented in machine processable formats [START_REF] Berners-Lee | The semantic web[END_REF]. The main idea in the semantic Web is to provide well defined and machine accessible representation of the resources and their relationships rather than simple links as they are offered by the link structure on the current Web (i.e. href links in HTML). Ontologies are utilised by the semantic Web applications to offer conceptualised representation of domains and to specify meaningful relationships between the resources. The World Wide Web Consortium (W3C) has defined different standards for representing the semantic Web data in machine accessible and processable formats.

Semantic technologies can improve interoperability and integration, as well as facilitate reasoning, classification and other types of assurance and automation. Semantic interoperability supports high-level, context-sensitive information requests over heterogeneous information resources, hiding systems, syntax, and structural heterogeneity [START_REF] Sheth | Changing Focus on Interoperability in Information Systems: From System, Syntax, Structure to Semantics[END_REF]. Integration and fusion of data into higher-level abstractions is aided by a data representation intended for machine reasoning.

The primary technologies for the semantic Web include the Resource Description Framework (RDF) 1 and the Web Ontology Language (OWL) 2 . OWL is based on description logic [START_REF] Baader | Description Logics as Ontology Languages for the Semantic Web[END_REF] and facilitates construction of ontologies for different domains. An OWL ontology contains descriptions of classes (or concepts in DL terminology), properties (roles in DL terminology) and individuals. There are two types of properties: object properties (properties for which the value is an individual) and datatype properties (properties for which the value is a data literal).

The OWL data can be accessed by software agents for reasoning and inferencing purposes and to enable systems to derive additional knowledge from the represented data. There are common query languages such as SPARQL 3 , namely SPARQL Protocol and RDF Query Language. It is developed as a query language and a protocol for accessing RDF graphs.

There are categories of tools allow for editing ontologies. The most widely used OWL editor is Protege 4 , a free opensource editing framework developed at Stanford University. By virtue of its open plugin structure, it allows for the easy integration of special-purpose ontology editing components. There are also widely used software systems such as Jena [START_REF] Tzonas | JENA RDF Framework[END_REF] to deploy and manage the constructed ontologies.

Creating a semantic sensor data model for sensor data related to the nature of sensors, their observations and locations is important aspect in designing highly scalable and advanced heterogeneous sensor network applications.

III. RELATED WORK

Ontologies provide a formal, usable and extensible model that is suitable for representing information, in our case sensor data, at different levels of abstraction and with rich semantic descriptions that can be used for searching and reasoning [START_REF] Corcho | Five challenges for the Semantic Sensor Web[END_REF]. Moreover in a highly heterogeneous setting, using standards and widely adopted vocabularies facilitates the tasks of publishing, searching and sharing the data.

Many researchers have realized the problem of semantic integration of sensor data and try to address it using semantic web technologies. A theoretical discussion has been done by Lionel et al. [START_REF] Lionel | Semantic Sensor Net: An Extensible Framework[END_REF], who proposed a concept of Semantic Sensor Net (SSN). They have identified the same associated problems for heterogeneous sensor networks and sensor data. A SSN is a heterogeneous sensor network which enables dynamic tagging of semantic information to sensor data so that it can be integrated and reused across various applications.

The ontologies exploitation in the context of wireless sensor networks, requires a good comprehension semantic, a good design and a good definition of concepts that will be represented. Several sensor ontologies have been proposed in the past, some of them focused on sensor descriptions, and others in observations [START_REF] Compton | A survey of the semantic specification of sensors[END_REF]. The work in [START_REF] Avancha | Ontology-Driven Adaptive Sensor Networks[END_REF] presents an attempt to capture the most important features of a sensor node that describes its functionality and its current state. The ontology describes the main components of a sensor node such as processor CPU and memory, power supply, and radio and sensor modules.

A step further in ontology-based sensor nodes is presented in [START_REF] Jurdak | A Framework for Modeling Sensor Networks[END_REF] and [START_REF] Cybenko | Semantic Agent Technologies for Tactical Sensor Networks[END_REF]. The researchers in [START_REF] Cybenko | Semantic Agent Technologies for Tactical Sensor Networks[END_REF] define an ontology that integrates high level features that characterizes sensor networks for customizing routing behavior. The proposed ontology describes the network topology and settings, sensor description, and data flow. Again, there is no mention of sensor data. Subsequent work like [START_REF] Liu | Towards Semantic Services for Sensor-Rich Information Systems[END_REF] is an effort in the direction of facilitating semantic-service oriented sensor information systems. The notion of ontology used in this research is to capture the information about physical entities that sensors sense and their relationships.

Most of these proposals are, however, often specific to a project, or discontinued, which do not cover many important areas of the sensor and observation domain. Moreover many of these ontologies did not follow a solid modeling process or did not reuse existing standards.

The following sections, present our development process of wireless Sensor Network Ontology (WSNO). In this process, RDF language will be used to specify the requirements specification, results of the first step. METHONTOLOGY methodology is the basic support, for the conceptualization of the Sensor Network ontology to be created. Descriptions logic is the type of formalism which will be adopted for the representation of semi-formal ontology, result of the conceptualization step. Based on this formalization, OWL language will be chosen to implement the ontology. The Protégé editor will be used to design the class and property structure of the proposed semantic data model and also to define the constraint rules for the associations and attributes. Through the use of SPARQL query languages, a semantic sensor network database can be queried to retrieve information about the structure and the nature of sensors, their observations and locations.

IV. THE PROPOSED WSNO BUILDING PROCESS

Our goal is to propose a method for building a wireless sensor network ontology. Thus, we had to face (i) the discovery of relevant concepts, (ii) their organization in taxonomy, and (iii) the non-taxonomic relationships between concepts.

The method that we propose is complete, insofar as, starting from brut data it allows to arrive at an operational ontology represented in OWL language. To do this, four main steps are followed in order to explain and to guide the building of the SN ontology.

A. Requirements Specification Step

The proposed process begins with the ontology specification step. The purpose of this step is to establish requirements specification document. This latter, allows to describe the WSNO, to build, through the following four aspects:

1) Domain Knowledge: This aspect consists of delimiting as precisely as possible the domain that the ontology going to cover.

2) Users: This aspect consists of identifying the intended users of the ontology to be created.

3) Operational goal: This aspect precises the purpose of the ontology to be created for the considered domain.

4) Ontology scope: This aspect consists of determining the most important global terms referring entities of domain knowledge to represent. In the context of sensor network we can determine the following global terms: {System, Sensor, Sensing, Data}.

The result of this step is a document of requirements specification represented by the RDF language. The later, associates three types of objects (Cf. TABLE I): A resource (corresponding to the document of requirements specification) defined by properties (corresponding to the four aspects) and the association of a resource to a property by a value of property (corresponding the result of each aspect). For that, we propose the following two phases: 2) Representation by the RDF syntax : RDF also proposes a XML syntax. This phase consists of encoding, in RDF syntax, the various graphs presented in the preceding phase. For example, the serialization in XML/RDF of the graph given in Figure 1 is as follows:

<rdf:Description about=" URI of document" > < Domain-knowledge>WSN </ Domain-knowledge > </rdf:Description>

B. Conceptualization Step

Conceptualization deserves particular attention because it determines the rest of the ontology construction. The objective is to organize and structure knowledge, using semi formal representations (tables and graphs) that are independent of the paradigms of knowledge representation in which the ontology will be formalized.

During this step, we construct a set of semi-formal representations, that we call "conceptual ontology". The latter identifies and define the domain vocabulary, independently of any implementation language. We were inspired of the conceptualization phases proposed by METHONTOLOGY methodology. To do so, we distinguish the following main activities:

1) Building a Glossary of Local Terms:

A term may be a relevant representation of a domain entity called, concept or a binary relationship between two concepts. This activity consists in building a glossary of local terms. This last collects terms of the domain and associates to each term identified a natural language description, see TABLE 2.

TABLE II GLOSSARY OF TERMS

Term Description

System

System is a unit of abstraction for pieces of infrastructure for sensing. May be: Device, Sensor and Wireless Sensor Network that can be mounted on platforms.

Wireless Sensor Network

A WSN is a communication system consisting of a large number of sensors.

Sensor

A sensor is an entity that can follow a Sensing method for observing a property with Measurement Capability Humidity Sensor A device which used to assess the humidity in a specific field.

Wind Sensor

A device used to evaluate wind speed.

Sensing

A process that results in the estimation, or calculation of the value of a phenomenon, describes the principle behind a sensor and how the observations were made.

Deployment

A process that encompasses all phases of the cycle of a deployed system such as installation, maintenance in a WSN.

Transforming A process that can describe how the transformations of the data were made. The hierarchy of concepts shows the organization of the ontology concepts in a hierarchical order which expresses the relations sub-class and super-class.

We use the relation Sub-Class-Of between the classes to define their classification. C1 class is sub-class of C2 class if any instance of C1 class is an instance of C2 class. We follow a development process from top to bottom. We start with a definition of the general concepts of domain and then continue by the specialization of concepts. For example, we can start by creating classes for the general concepts: System, Process, Property, State, Event, Outputs and Situation . For example, figure 2 shows the hierarchy of Process concept.

3) Building a binary relations diagram:

We will build our diagram in two principal steps; initially, we determine the organization of concepts, then we will connect the concepts by relations so necessary.

We represent the binary relations between classes by a diagram in figure 3. In this diagram the classes are represented by rectangles and the relations by arrows (domain towards codomain) labelled by the name of the relation. The dictionary of concepts consists of describing all concepts represented in the hierarchy of concepts, by representing instances of such concepts, attributes of the concepts and, optionally, concept synonyms (see TABLE 3).

TABLE IIII TABLE OF CONCEPTS DICTIONARY

5) Building a table of binary relations tables:

The binary relations are represented in the form of properties which attach a concept to another. For each relation whose source is in the tree of concepts classification, we define: its name, the name of the source concept, the name of the target concept, cardinality and the name of the inverse relation if it exist (see TABLE 4).

TABLE IV TABLE OF BINARY-RELATIONS

6) Building the logical axioms table:

We will define the ontology concepts by using the logical expressions which are always true. In the table below, we define for each axiom, its description in natural language, the name of the concept to which the axiom refers, attributes used in the axiom and the logical expression (see TABLE 5). In this table, we describe the instances, which are identified in the concepts dictionary. For each instance, it is necessary to specify the instance name, the concept name to belong to it, the attributes and their values.

TABLE V LOGICAL AXIOMS TABLE

Concept name

C. Formalization Step

In this phase, we use the DL (Description Logic) formalism to formalize the conceptual model that we obtained it at the conceptualization phase. DL forms a language family of knowledge representation; it represents knowledge relating to a specific area using "descriptions" which can be concepts, relations and instances. The relation of subsumption (⊑) allows organizing concepts and relations in hierarchy; classification and instantiation are the basic operations of the reasoning on description logic, or terminological reasoning. Classification permits to determine the position of a concept and a relation in their respective hierarchies.

The result of this step is a formal ontology made of two parts: terminological part TBOX in which we define concepts and relations; and an assertion part ABOX in which we introduce the instances (see Table 7).

-TBOX construction: We define here concepts and relations relating to our domain, by using the constructors provided by description logic to give structured descriptions at concepts and relations.

-ABOX construction: The assertion language is dedicated to the description of facts, by specifying the instances (with their classes) and the relations between them.

TABLE VII EXAMPLE OF DL EXPRESSIONS

D. Implementation Step

The implementation deals with building a computable model. To implement the formal ontology Protege tool is used. The knowledge representation language for modeling the various ontological knowledge of sensor network is OWL DL. The constructed class hierarchy is shown in Figure 4.

For checking, we need to use the inference services provided by many systems such as RACER [START_REF] Haarslev | Racer System Description[END_REF]. It can work well with large ontologies. The use of the RACER system can make possible to read OWL file and to convert it in the form of a DL knowledge bases. Once the constructed ontology is validated, it is ready to be invoked by users' requests. As a query language, we used SPARQL query language which is provided and plugged-in to Protege. For example, Figure 5 shows a SPARQL query allowing to find the URIs and locations of temperature sensors. The semantic representation of wireless sensor networks data is an exciting vision that enables structured information to be interpreted unambiguously. Precise interpretation is a necessary prerequisite for automatic search, retrieval, and processing of sensor data.

Name concept

In this paper, we have proposed a process for building an wireless sensor networks ontology. The proposed process is complete, insofar as, starting from brut data it allows to arrive at an operational ontology represented in OWL language. For the future work, we need to concentrate on the ontology enrichment with new concepts to be more meaning and thus really reflects the modelled domain.

CommunicatingA

  process that can describe how communications were made. Its inputs are the outputs of Transformation or Sensor Output. ....... ........... 2) Building a Hierarchy of concepts: URI of the document WSN Domain knowledge

Figure 2 .

 2 Figure 2. Hierarchy of Process Concept.

Figure 3 . 4 )

 34 Figure 3. Binary relations diagram

7 )

 7 Building the instances table:

TBOX Sensor ⊑

  Device Detecting_Sensor  Sensor ⊓  Detect.Event ABOX Detecting_Sensor (s1) Detect (s1, e1)

Figure 4 .

 4 Figure 4. Hierarchy of the proposed ontology

Figure 4 .

 4 Figure 4. SPARQL Query Statements

TABLE I

 I

	REQUIREMENTS SPECIFICATION DOCUMENT AS AN RDF DECLARATION
	Subject (resource)	Needs	specification	document
		identified by a URI
	Predicate (property)	Aspect
	Object (value)	Result of an aspect

1) Representation by the RDF graph: This phase consists of representing the result of each aspect by a RDF graph (diagram). For example, Figure

.

1 illustrates the diagram allowing to represent the following assertion: "the domain knowledge of the ontology to be created is: 'WSN' " Figure 1. RDF graph concerning the first aspect

TABLE VI INSTANCES TABLE

 VITABLE

	Instance	Attributes	Values
		serial_number	111
	wind_sensor_1	creation_date	2016-02-12T14:24:16
		identification	W1

http://www.w3.org/RDF/

http://www.w3.org/TR/owl-features/

http://www.w3.org/TR/rdf-sparql-query/

http://protege.stanford.edu/


Emmanuel Clouet

Maylise Nastar

Classical nucleation theory in ordering alloys precipitating with L1 2 structure

By means of low-temperature expansions (LTEs), the nucleation free energy and the precipitate interface free energy are expressed as functions of the solubility limit for alloys which lead to the precipitation of a stoichiometric L12 compound such as Al-Sc or Al-Zr alloys. Classical nucleation theory is then used to obtain a simple expression of the nucleation rate whose validity is demonstrated by a comparison with atomic simulations. LTEs also explain why simple mean-field approximation like the Bragg-Williams approximation fails to predict correct nucleation rates in such an ordering alloy.

Since its initial formulation in 1927 by Volmer, Weber and Farkas and its modification in 1935 by Becker and Döring the classical nucleation theory (CNT) [START_REF] Martin | Solid State Phase Transformation in Metals and Alloys[END_REF][START_REF] Kelton | Solid State Physics[END_REF][START_REF] Kashchiev | Nucleation : basic theory with applications[END_REF] has been a suitable tool to model the nucleation stage in phase transformations. The success of this theory relies on its simplicity and on the few parameters required to predict the nucleation rate. Recently, the use of computer simulations have allowed to assess the applicability of the theory for solid-solid phase transformations [START_REF] Clouet | [END_REF]5,6,7 . Thanks to a precise control of simulation conditions, it is possible to get accurate estimations of CNT parameters and thus to make a direct comparison between theory predictions and quantities observed during simulations. One thus gains a deeper understanding of the validity of the different assumptions used by the CNT.

Previous studies have shown that the capillary approximation, which CNT relies on, gives a precise description of cluster thermodynamics. Within this approximation, the free energy of a nucleus is written as the sum of a volume contribution, the nucleation free energy, and a surface contribution corresponding to the energy cost to create an interface between the nucleus and the solvent. For CNT to agree with atomic simulations, care has to be taken in the way these two energetic contributions are obtained. In particular, we have shown that one has to take into account short range order when calculating the nucleation free energy in an ordering alloy [START_REF] Clouet | [END_REF] . Usual thermodynamic approximations, like the ideal solid solution or the Bragg-Williams approximation, cannot describe short range order and thus can predict values of the cluster size distribution and of the nucleation rate wrong by several orders of magnitude. This is to contrast with more sophisticated mean-field approximations like the cluster variation method (CVM) which provides good predictions of the nucleation rate 4 . However, an easy use of CNT and a clear determination of the missing ingredients in simple mean-field approximations requires an analytical approach which CVM cannot provide. Such an approach has to lead to accurate expressions of the CNT input parameters so as to make the theory predictive without any fitting of its parameters.

In this Letter, we use low-temperature expansions (LTE) [START_REF] Ducastelle | Order and Phase Stability in Alloys[END_REF][START_REF]Phase Transitions and Critical Phenomena[END_REF] to derive an analytical formulation of the nucleation free energy and the interface free energy in a binary system like Al-Sc or Al-Zr, i.e. a supersaturated Al-X solid solution leading to the nucleation of a stoichiometric Al 3 X compound with the L1 2 structure. This structure corresponds to an ordering of the fcc lattice with solute X atoms lying on one of the four cubic sublattices [START_REF] Ducastelle | Order and Phase Stability in Alloys[END_REF] . LTE are well suited to describe short range order in dilute solid solution and nearly stoichiometric ordered compounds [START_REF] Woodward | [END_REF]11,12,13,14 like Al 3 X compound. The use of this method in CNT framework allows to obtain a fully analytical modelling whose only material parameters are the solubility limit and the solute diffusion coefficient.

To do so, we start from the same atomic diffusion model previously developed for Al-Sc-Zr system [START_REF] Clouet | [END_REF]15 . This model relies on a rigid lattice with interactions between first-and second nearest neighbors and uses a thermally activated atom-vacancy exchange mechanism to describe diffusion. Despite its simplicity, it has been shown to lead to predictions in good agreement with experimental data 15,16,17 . Within this atomic model, atoms are constrained to lie on a fcc lattice and the configurations of a binary Al-X alloy is fully described by the solute atom occupation number p n with p n = 1 if the site n is occupied by a solute atom and p n = 0 otherwise. The energy per atom of a given configuration of the Al-X alloys is then given by

E = U Al + 1 N s (U X -U Al ) n p n + 1 N s ω (1) AlX ′ n,m (1 -p n ) p m + 1 N s ω (2) AlX ′′ n,m (1 -p n ) p m (1)
where the first and second sums, respectively, run on all first and second nearest-neighbor pairs of sites, N s is the number of lattice sites, U Al (respectively U X ) is the energy per atom when only Al (respectively X) atoms lie on the fcc lattice and ω

AlX and ω

AlX are the first and second nearest neighbor order energies. Al-Sc and Al-Zr thermodynamics are characterized by the order tendency between first nearest neighbors (ω

(1)
AlX < 0) and the demixing tendency between second nearest neighbors (ω

(2)
AlX > 0). Eq. 1 is a rewriting for binary alloys of the atomic model developed in Refs. [START_REF] Clouet | [END_REF]15 when one neglects vacancy contributions.

The nucleation free energy entering CNT is defined by

∆G nuc (x 0 X ) = 3 4 µ Al (x eq X ) -µ Al (x 0 X ) + 1 4 µ X (x eq X ) -µ X (x 0 X ) , (2)
where µ Al (x X ) and µ X (x X ) are the Al and X component chemical potentials in a solid solution of concentration x X , and x eq X and x 0 X the concentrations of the equilibrium and supersaturated solid solution.

LTE are more easy to handle in semi-grand-canonical ensemble where all quantities are written as functions of the effective potential µ = (µ Al -µ X ) /2. Definition of the nucleation free energy then becomes

∆G nuc (µ) = A(µ eq ) -A(µ) + 1 2 (µ eq -µ) , (3)
where µ eq is the effective potential corresponding to equilibrium between the Al solid solution and the Al 3 X L1 2 compound. We have defined in Eq. 3 the solid solution semi-grand-canonical free energy

A = (µ Al + µ X ) /2 = F (x) + (1 -2x) µ, F (x)
being the usual canonical free energy.

A LTE consists in developing the partition function of the system around a reference state, keeping in the series only the excited states of lowest energies. Use of the linked cluster theorem [START_REF] Ducastelle | Order and Phase Stability in Alloys[END_REF][START_REF]Phase Transitions and Critical Phenomena[END_REF] allows then to express the corresponding semi-grand canonical free energy as (4) where the energy of the ground state is A 0 (µ) = U Al + µ for the Al solid solution and A 0 (µ) = 3/4 U Al + 1/4 U X + 3ω

A(µ) = A 0 (µ) -kT i,n g i,n exp (-∆E i,n (µ)/kT ),
(1) AlX + µ/2 for the Al 3 X L1 2 compound. In the sum appearing in Eq. 4, the excited states have been gathered according to their energy state i and the number n of lattice sites involved. LTE parameters corresponding to the excited states with the lowest energies are given in Tab. I. All excitation energies only involve a set of isolated atoms or in second nearest neighbor position since flipping two atoms at nearest neighbor position produces an excited state with a much higher energy.

The solute concentration in a given phase is obtained by considering the derivative of the corresponding semigrand-canonical free energy. For the solid solution, one gets

x X (µ) = 1 2 1 - ∂A(µ) ∂µ = i,n ng i,n exp (-∆E i,n (µ)/kT ). ( 5
)
The solid solution and the L1 2 compound are in equilibrium when both phases have the same semi-grand canonical free energy. Considering third order LTE

AlX -2δµ 1 6ω

(2)

AlX + 2δµ 1/4 2 2 10ω

(2)

AlX -4δµ 3 10ω

(2)

AlX + 4δµ 3/4 3 2 12ω

(2)

AlX -4δµ -19/2 12ω (2)
AlX + 4δµ -7/8 4 3 14ω

(2)

AlX -6δµ 15 5 4 16ω

(2)

AlX -8δµ 3 5 3 16ω

(2)

AlX -6δµ -96 6 4 18ω

(2)

AlX -8δµ 83 6 3 18ω

(2)

AlX -6δµ -774 7 5 20ω

(2)

AlX -10δµ 48 7 4 + 20ω (2)
AlX -8δµ -1569/2

(i = 3), this happens for the effective potential

µ eq = (U X -U Al )/2 + 6ω (1) AlX + 3 2 kT exp (-6ω (2)
AlX /kT ) + 3 exp (-10ω

AlX /kT )

- 13 2 exp (-12ω (2)
AlX /kT ) , (6) corresponding to the solubility

x eq X = exp -6ω (2)
AlX /kT + 6 exp -10ω

AlX /kT

-16 exp -12ω (2)
AlX /kT . (7) As these expressions have to be consistent with the expansion of A, terms with larger exponential arguments than -12ω

AlX are discarded. For equilibrium phases, one does not need to go further in the expansion than the third order. Indeed, thermodynamic properties are already well converged as the solid solution and the L1 2 compound in equilibrium only slightly deviate from their respective ground states. On the other hand, Fig. 1 show that an expansion beyond the third order of the semi-grand canonical free energy A(µ) of the supersaturated solid solution and of the corresponding concentration x X (µ) improves the convergence of the nucleation free energy. When only the first excited state is included in the expansion, LTE leads to the same value of the nucleation free energy as the ideal solid solution model. As more excited state are included in the expansion, the value deduced from LTE converges to the one calculated with CVM in the tetrahedronoctahedron approximation [START_REF] Clouet | [END_REF] . This is to contrast to the Bragg-Williams approximation which leads to a worse prediction of the nucleation free energy than the ideal solid solution model. So as to understand why Bragg-Williams approximation does so bad, it is worth going back to the canonical ensemble. When considering only the third order LTE, thermodynamic quantities can be expressed as functions of the solid solution nominal concentration x 0 X . In particular, the nucleation free energy is given by

∆G nuc LTE (x 0 X ) = kT q (x 0 X ) -q (x eq X ) + 3kT exp 2ω (2) AlX /kT q (x 0 X ) 2 -q (x eq X ) 2 - 1 4 kT ln q (x 0 X ) -ln [q (x eq X )] , (8)
where we have defined the function

q (x) = 2x 1 + 1 + 4x 6 exp 2ω (2)
AlX /kT -19

.

This expression developed to first order in the concentrations x 0 X and x eq X leads to

∆G nuc LTE (x 0 X ) ∼ 3 4 kT ln 1 -x eq X 1 -x 0 X + 1 4 kT ln x eq X x 0 X + 1 4 kT 1 + 6e 2ω (2) AlX /kT x 0 X -x eq X . (10)
Doing the same development for the nucleation free energy calculated within the Bragg-Williams approximation [START_REF] Clouet | [END_REF] , we obtain

∆G nuc BW (x 0 X ) ∼ 3 4 kT ln 1 -x eq X 1 -x 0 X + 1 4 kT ln x eq X x 0 X + 6ω (1)
AlX + 3ω

(2) AlX

x 0 X -x eq X . (11)
Comparing Eq. 10 with Eq. 11, we see that these two thermodynamic approximations deviate from the ideal solid solution model by a distinct linear term. In the LTE (Eq. 10), the nucleation free energy is only depending on the second nearest neighbor interaction and the coefficient in front of the concentration difference is positive. On the other hand, the Bragg-Williams approximation (Eq. 11) incorporates both first and second nearest neighbor interactions into a global parameter ω AlX = 6ω

AlX + 3ω

AlX . This leads to a linear correction with a coefficient which can be negative due to the oscillating nature of the interactions. In particular, this is the case for both binary Al-Zr and Al-Sc alloys 4 . Bragg-Williams approximation thus leads to a wrong correction of the ideal model because it does not consider properly short range order. In the case of a L1 2 ordered compound precipitating from a solid solution lying on a fcc lattice, one cannot use such an approximation to calculate the nucleation free energy. On the other hand, Eq. 8 is a good approximation and can be used to calculate the nucleation free energy even when the second nearest neighbor interaction ω

(2) AlX is not known. Indeed, this parameter can be deduced from the solubility limit x eq X by inverting Eq. 7, leading to the relation

ω (2) AlX = - 1 6
kT ln (x eq X ) + kT x eq X 2/3 -8 3

x eq X .

This relation combined with Eq. 8 provides a powerful way for calculating the nucleation free energy from the solid solubility. LTE can be used too to calculate the plane interface free energy σ 100 corresponding to a [100] direction. Due to the inhomogeneity perpendicular to the interface, the main contribution arises from broken bonds and excited states, whose energies are lower than in bulk phases, only bring a small correction. One thus does not need to go further than the second order in the expansion. At 0 K, the isotropic interface free energy σ is obtained by multiplying σ 100 with the geometric factor (6/π) 1/3 corresponding to a perfect [100] facetting of the precipitates. For low temperatures, this is a good approximation to assume that the same linear relation holds between both quantities [START_REF] Clouet | [END_REF] . The isotropic interface free energy given by LTE is then

a 2 σ = (6/π) 1/3 ω (2)
AlX -2kT exp (-4ω

AlX /kT )

-kT exp (-6ω (2)
AlX /kT ) , (13) where a is the fcc lattice parameter.

LTE thus allow to calculate all CNT input parameters from the knowledge of the solubility limit. The nucleation rate is then obtained from the equation where D X is the X impurity diffusion coefficient in Al. We thus obtain a fully analytical expression of the nucleation rate. Using the same experimental data, i.e. solubility limits and diffusion coefficients, as the ones used to fit the atomic diffusion model of kinetic Monte Carlo simulations, we can compare CNT predictions with nucleation rate observed in simulations [START_REF] Clouet | [END_REF] . A good agreement is obtained both for Al-Zr and Al-Sc binary alloys (Fig. 2).

J st (x 0 X ) = -16N s ∆G nuc (x 0 X ) √ kT a 2 σ D X a 2 x 0 X exp - π 3 (a 2 σ) 3 kT [∆G nuc (x 0 X )] 2 , ( 14
)
The combination of LTE with CNT thus allows to build a quantitative modeling of nucleation relying on a very limited number of material parameters. Such a model can be directly applied to aluminum alloys where a L1 2 compound precipitates from a supersaturated solid solution as this is the case with Zr, Sc or other rare earth elements like Er, Tm, Yb and Lu 18 . Li too precipitates in aluminum with a L1 2 structure, but this system requires another statistical approximation than LTE. Indeed, this approach based on LTE, requires that the precipitating phase only slightly deviates from its perfect stoichiometry and that the solute solubility remains low. Provided these conditions are fulfilled, it could be applied to alloys other than aluminum alloys. More generally, LTE demonstrate that the oscillating nature of the interactions in an alloy with an ordering tendency has to be taken into account by the CNT and requires a better statistical description than the Bragg-Williams approximation which treats all interactions on the same footing.

FIG. 1 :

 1 FIG. 1: Variation with the nominal concentration x 0Zr of the nucleation free energy ∆G nuc at T = 723 K obtained with different thermodynamic approximations: CVM, ideal solid solution, Bragg-Williams and low-temperature expansions (LTE) to different orders.

FIG. 2 :

 2 FIG. 2: Variation with nominal concentration and temperature of the steady-state nucleation rate J st for Al3Zr (top) and Al3Sc (bottom) precipitations. Symbols correspond to kinetic Monte Carlo simulations and lines to classical nucleation theory

TABLE I :

 I Coefficients entering in the low temperature expansion (Eq. 4). The first seven excited states are considered for the solid solution and the first three excited states for the Al3X L12 compound. The effective potential is written as µ = (UX -U Al )/2 + 6ω

		(1) AlX + δµ.
		Solid solution	L12 compound
	i n	∆Ei,n(µ)	gi,n	∆Ei,n(µ)	gi,n
	1 1	6ω

The authors would like to thank Y. Le Bouar and A. Finel for helpful discussions on LTE, and B. Legrand, F. Soisson and G. Martin for their invaluable help.


Mohamed Farhloul
email: mohamed.farhloul@umoncton.ca

Abdelmalek Zine
email: abdel-malek.zine@ec-lyon.fr

A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law

Keywords: AMS (MOS) subject classification 65N30, 65N15, Mixed Finite Element, Quasi-Newtonian, Power law, Carreau law

The aim of this work is a construction of a dual mixed finite element method for a quasi-Newtonian flow obeying the Carreau or power law. This method is based on the introduction of the stress tensor as a new variable and the reformulation of the governing equations as a twofold saddle point problem. The derived formulation possesses local (i.e. at element level) conservation properties (conservation of the momentum and the mass) as for finite volume methods. Based on such a formulation, a mixed finite element is constructed and analyzed. We prove that the continuous problem and its approximation are well posed, and derive error estimates.

Introduction

We propose a dual-mixed formulation for non-Newtonian fluid flow where the fluid viscosity is assumed to be nonlinear function of the rate of strain tensor. The governing equations arise in modeling flows of, for example, biological fluids, lubricants, paints and polymeric fluids. In [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF][START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF] we have introduced and analyzed a dual-mixed finite element method for quasi-Newtonian fluid flow obeying to the power law. A priori error estimates for the finite element approximation were proved in [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], while a posteriori error estimation was provided in [START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF]. However, in both [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF][START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF] the analysis used the assumption that the equation describing the stress tensor in terms of the rate of strain tensor was invertible to give the rate of strain tensor as function of the stress tensor. The mixed finite element method developed in [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF] possesses local (i.e., at element level) conservation properties (conservation of the momentum and the mass) as in the finite volume methods. Furthermore, it allows the approximations of all the physical variables (stress, velocity and pressure).

The aim of this work is to extend our investigations by avoiding the assumption of expressing the rate of strain tensor as function of the stress tensor. As example of such a situation is a non-Newtonian fluid flow obeying the Carreau law. For this purpose, we introduce an additional variable for the rate of strain tensor and reformulate the governing equations as a twofold saddle point problem. It must be noted that this kind of approach has been introduced and analyzed in [START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: A priori error analysis[END_REF][START_REF] Ervin | A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem[END_REF] for a class of quasi-Newtonian Stokes flows. However, in both [START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: A priori error analysis[END_REF][START_REF] Ervin | A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem[END_REF] the tensor gradient of the velocity was used instead of the rate of strain tensor. The fact to use the rate of strain tensor introduces a major difficulty in the construction of mixed finite element methods (for more details, see [START_REF] Boffi | Mixed Finite Element Methods and Applications[END_REF]). The difficulty lies essentially in the symmetry of this tensor. One way to overcome this difficulty is to relax the symmetry of this tensor by a Lagrange multiplier. We will present our dual-mixed formulation and establish well posedness. The mixed finite element for this formulation will be provided and the associated a priori error estimates are then derived. The error estimates are optimal and are the same as the ones obtained in the particular case of Power law ( [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF]).

The outline of the paper is as follows: In Section 2 the governing equations and the mixed formulation of non Newtonian flows whose viscosity obeys a power law or the Carreau law are presented. Existence and uniqueness results are given in Section 3. In Section 4 we introduce our finite element approximation and establish well posedness. The a priori error estimates are then derived and last section is devoted to the conclusion.

Governing equations and mixed formulation

Governed by the classical Stokes problem, the Newtonian fluid flows are a reasonable approximation of the more realistic non-Newtonian fluids (quasi-Newtonian or Viscoelastic). In the case of quasi-Newtonian fluids, the viscosity is a function of gradient tensor, temperature, time, etc. For a steady and creeping flow of an incompressible quasi-Newtonian fluid, the most used formulation is based on the strain rate tensor.

In that case, for Ω a bounded domain of R 2 with a Lipschitz boundary Γ and a given mass forces f defined in Ω, the combination of the conservation equations leads to the following Nonlinear Stokes problem:

-div 2ν(|d(u)|) d(u) + ∇p = f in Ω, div u = 0 in Ω, (1)
where u and p, the unknowns of the problem, are the velocity and pressure, respectively.

d(u) = 1 2 ∇ u +(∇ u) t
is the strain rate tensor, and |d(u

)| 2 = 2 i,j=1 d(u) 2 ij .
For ν 0 > 0 a reference viscosity and r a fluid characteristic real parameter verifying 1 < r < ∞, the viscosity function ν(•), depending on |d(u)|, is usually given by one of the two following famous models: ν(x) = ν 0 x r-2 , ∀x ∈ R + , for the Power law model, and

ν(x) = ν 0 1 + x 2 (r-2)/2
, ∀x ∈ R + , for the Carreau model.

Finally, system (1) is supplemented by a set of boundary conditions.

Remark 2.1. For r = 2, both models provide the well known classical Stokes problem:

-2ν 0 div(d(u)) + ∇p = f in Ω, div u = 0 in Ω,
corresponding to a Newtonian fluid flow.

The generalized Stokes problem (1) and its approximation by standard finite elements was first studied in Baranger and Najib [START_REF] Baranger | Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau[END_REF]. Extensions and improvements of the error bounds have been obtained in Sandri [START_REF] Sandri | Sur l'approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau[END_REF] and Barrett and Liu [2,[START_REF] Barrett | Quasi-norm error bounds for the finite element approximation of non-Newtonian flow[END_REF].

In these works, only the primal variables velocity and pressure are taken into account. But, for various reasons, we need information on other variables as velocity gradients ∇ u, strain rate tensor d(u), extra-stress tensor σ = 2ν(|d(u)|) d(u) etc. In that case, it is necessary to build appropriate mixed formulations.

On the other hand, in connection with the use of the gradient tensor ∇ u which corresponds to the Ladyzhenskaya model [START_REF] Ladyzhenskaya | New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them[END_REF]:

ν(|∇ u|) = (ν 0 + ν 1 |∇ u|) r-2 , ν 0 ≥ 0, ν 1 > 0, r > 1,
a large amount of work is available in the literature. Among these works, there may be mentioned Manouzi and Farhloul [START_REF] Manouzi | Mixed finite element analysis of a non-linear three fields Stokes model[END_REF], Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a Ladyzhenskaya model[END_REF], Gatica et al. [START_REF] Gatica | On the numerical analysis of nonlinear twofold saddle point problems[END_REF][START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: A priori error analysis[END_REF] and Ervin et al. [START_REF] Ervin | A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem[END_REF].

The major drawback of formulations using the gradient lies in the fact that we can not deal with natural boundary conditions. To overcome this drawback related to the boundary conditions, we have introduced and analyzed a dual-mixed finite element method for quasi-Newtonian fluid flow obeying to the Power law, in Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF][START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF]. A priori error estimates for the finite element approximation were proved in the first paper, while a posteriori error estimation was provided in the second work. In both, the analysis is based on the fact that the equation describing the extra-stress tensor in terms of the rate of strain tensor:

σ = |d(u)| r-2 d(u) (ν 0 = 1/2),
is invertible and give the rate of strain tensor as a function of the stress tensor,

d(u) = |σ| r -2 σ,
r being the conjugate of r.

The aim of this work is to extend our investigations by avoiding the assumption of expressing the rate of strain tensor as function of the stress tensor. We may be then able to deal with both problems associated with Power law and Carreau Model.

In order to obtain a dual-mixed formulation of (1), we first formulate problem (1) as:

   -div σ -p I = f in Ω, div u = 0 in Ω, u = 0 on Γ, (2)
and then, we introduce two new variables t = d(u), the strain rate tensor, (3)

A(t) = 2ν (|t|) t = σ, the extra stress tensor. (4)
Suppose f ∈ [L r (Ω)] 2 , 1 < r < ∞. Let r being the conjugate number of r, i.e. 1 r + 1 r = 1. Let ω = ω(u) = 1 2 (∇ u -(∇ u) t ) be the vorticity tensor. Then, for all (τ , q) ∈ [L r (Ω)] 2×2 × L r 0 (Ω), such that div τ -q I ∈ [L r (Ω)] 2 , and for all u ∈ W 1,r (Ω) such that div u = 0, it is easy to see that:

(t, τ ) = (d(u), τ ), = -(div(τ -q I), u) -(ω, τ ),
where, from now on, ( • , • ) denotes the duality pairing between L r (Ω) and L r (Ω). The space L r 0 (Ω) is defined by

L r 0 (Ω) = q ∈ L r (Ω); Ω q dx = 0 .
In order to derive the mixed formulation of (2), we define the following spaces:

T = [L r (Ω)] 2×2 , Σ = (τ , q) ∈ [L r (Ω)] 2×2 × L r 0 (Ω); div(τ -q I) ∈ [L r (Ω)] 2 , M = (v, η) ∈ [L r (Ω)] 2 × [L r (Ω)] 2×2 ; η + η t = 0 ,
equipped with the following norms:

s T = s 0,r,Ω = Ω |s| r 1 r , (τ , q) Σ = τ r 0,r ,Ω + q r 0,r ,Ω + div(τ -q I) r 0,r ,Ω 1 r , (v, η) M = v r 0,r,Ω + η r 0,r,Ω 1 r .
The dual-mixed formulation of problem (2) reads as follows:

Find t ∈ T , (σ, p) ∈ Σ and (u, ω) ∈ M such that      (A(t), s) -(σ, s) = 0 ∀ s ∈ T , (t, τ ) + (div(τ -q I), u) + (τ , ω) = 0 ∀τ ∼ = (τ , q) ∈ Σ, (div(σ -p I), v) + (σ, η) + (f , v) = 0 ∀v ∼ = (v, η) ∈ M . (5)
Remark 2.2. From the last equation of (5),

(div(σ -p I), v) + (σ, η) + (f , v) = 0, ∀(v, η) ∈ M , one gets (σ, η) = 0, ∀η ∈ [L r (Ω)] 2×2 such that η + η t = 0.
This corresponds to the symmetry relaxation of the extra-stress tensor σ by a Lagrange multiplier.

Remark 2.3. As stated above, the use of the rate of strain tensor enable to handle different type of boundary conditions, such as mixed boundary conditions. More precisely, assuming that we consider the following boundary conditions:

u = 0 on Γ D and (2ν(|d(u)|) d(u) -p I) n = 0 on Γ N ,
where Γ = Γ D ∪ Γ N , Γ D = ∅ and n is the unit outward normal vector field along the boundary of Ω.

Then, the only change to be made is to replace the space Σ by the following one:

Σ = (τ , q) ∈ [L r (Ω)] 2×2 × L r 0 (Ω); div(τ -q I) ∈ [L r (Ω)] 2 ; (τ -q I) n = 0 on Γ N .
However, we choose to make the analysis in the case of Dirichlet boundary conditions only for the sake of clearness.

To formally rewrite (5) as a twofold saddle-point problem, see Gatica et al. [START_REF] Gatica | On the numerical analysis of nonlinear twofold saddle point problems[END_REF][START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: A priori error analysis[END_REF], we define the following operators:

A : T -→ T , B : T -→ Σ and C : Σ -→ M ,
where for a Banach space X, X denotes the dual space with associated norm

• X . [A(t), s] = (A(t), s) , ∀ s, t ∈ T , (6)
[B(s), τ ∼ ] = -(s, τ ) , ∀ s ∈ T , ∀τ ∼ = (τ , q) ∈ Σ, (7)
[C(τ ∼ ), v ∼ ] = -(div(τ -q I), v) -(τ , η), ∀τ ∼ ∈ Σ, ∀v ∼ ∈ M . (8)
Remark 2.4. Recall that the operator A is defined by ∀ t ∈ T , A(t) = 2ν(|t|) t, ν being given by either Power or Carreau law.

The problem (5) can be then written in the following twofold saddle-point form:

Find t ∈ T , σ ∼ ∈ Σ and u ∼ ∈ M such that        [A(t), s] + [s, B (σ ∼ )] = 0 ∀ s ∈ T , [B(t), τ ∼ ] + [τ ∼ , C (u ∼ )] = 0 ∀τ ∼ ∈ Σ, [C(σ ∼ ), v ∼ ] = [F , v ∼ ] ∀v ∼ ∈ M , (9)
where, [F , v ∼ ] = (f , v) , ∀v ∼ ∈ M , and B and C denote the dual operators of B and C, respectively.

Solvability of the continuous problem

To prove existence, uniqueness and stability conditions of

t, σ ∼ , u ∼ = (t, (σ, p), (u, ω)) ∈ T × Σ × M
solution of ( 9), we shall recall some technical results given bellow. These results concern the properties of the operators A, B and C. Mainly:

• A is bounded, continuous and strictly monotone,

• B verifies the inf-sup condition on the Kernel of C,

• C verifies the inf-sup condition. Here we give several technical lemmas that establish the appropriate conditions on the operators A. These properties are given for both Power and Carreau laws. In the following, to distinguish the two models, we set δ = 0 for Power law and δ = 1 for Carreau law. For details on the following three Lemmas, see Sandri [START_REF] Sandri | Sur l'approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau[END_REF].

Lemma 3.1. Let s, t ∈ T = [L r (Ω)]
2×2 , we have the following results: For

1 < r < 2, [A(s) -A(t), s -t] ≥ C s -t 2 0,r,Ω δ + s 2-r 0,r,Ω + t 2-r 0,r,Ω , A(s) -A(t) 0,r ,Ω ≤ C Ω |A(s) -A(t)| |s -t| dx 1/r . Lemma 3.2. Let s, t ∈ T = [L r (Ω)] 2×2 , we have the following results: For r ≥ 2, [A(s) -A(t), s -t] ≥ C s -t r 0,r,Ω + Ω δ + |s| r-2 + |t| r-2 |s -t| 2 dx , A(s) -A(t) 0,r ,Ω ≤ C Ω δ + |s| r-2 + |t| r-2 |s -t| 2 dx 1/2 δ + s (r-2)/2 0,r,Ω + t (r-2)/2 0,r,Ω . Lemma 3.3. Let s, t ∈ T = [L r (Ω)]
2×2 , we have the following inequality: For r > 1,

[A(s) -A(t), s -t] ≥ C Ω |A(s) -A(t)| |s -t| dx.
Remark 3.1. The above Lemmas 3.1, 3.2 and 3.3, imply that the operator A is bounded, continuous and strictly monotone on reflexive Banach spaces.

We also state two known results that will be utilized. B: (X/V ) -→ M the quotient operator associated to B. Then, the three following properties are equivalent:

(i) there exists β > 0, such that

inf q∈M sup v∈X [B(v), q] q M v X ≥ β , (ii) B is an isomorphism from M onto V o and B q X ≥ β q M ∀q ∈ M , (iii)
.

B is an isomorphism from (X/V ) onto M and .

B

.

v M ≥ β . v (X/V ) ∀ . v∈ (X/V ).
The proof of this Lemma is given in Sandri [START_REF] Sandri | Sur l'approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau[END_REF] (see also Remark 4.2, p. 61, in Girault and Raviart [START_REF] Girault | Finite Element for Navier-Stokes Equations, Theory and Algorithms[END_REF]). Lemma 3.5. ( [START_REF] Renardy | An introduction to Partial Differential Equations[END_REF], Theorem 9.45, p. 361, Browder-Minty). Let X be a real reflexive Banach space and let K : X -→ X be bounded, continuous, coercive and monotone. Then for any g ∈ X there exists a solution u of the equation K(u) = g; i.e. K(X) = X . Now, we establish the appropriate inf-sup conditions for the operators B and C. Define Z 1 , the null space for C:

Z 1 = τ ∼ ∈ Σ; [C(τ ∼ ), v ∼ ] = 0, ∀v ∼ ∈ M .
Lemma 3.6. There exists a positive constant

β 1 such that inf τ ∼ ∈Z1 sup s∈T [B(s), τ ∼ ] s T τ ∼ Σ ≥ β 1 . ( 10
)
Proof. Let τ ∼ = (τ , q) ∈ Z 1 . Then, τ ∈ [L r (Ω)] 2×2 , q ∈ L r 0 (Ω) and div(τ -q I) = 0. Now, let s * = -|τ | r -2 τ . We have s * ∈ T and s * T = τ r -1 0,r ,Ω . Thus, [B(s * ), τ ∼ ] s * T = -(s * , τ ) s * T = τ r 0,r ,Ω τ r -1 0,r ,Ω = τ 0,r ,Ω .
On the other hand, since div(τ -q I) = 0, there exists a constant C > 0 such that (see Lemma 4 in [START_REF] Manouzi | Mixed finite element analysis of a non-linear three fields Stokes model[END_REF]) q 0,r ,Ω ≤ C τ 0,r ,Ω . Therefore, for all τ ∼ ∈ Z 1 , we have sup s∈T

[B(s), τ ∼ ] s T ≥ [B(s * ), τ ∼ ] s * T = τ 0,r ,Ω ≥ C τ ∼ Σ ,
which completes the proof.

Lemma 3.7. There exists a positive constant

β 2 such that inf v ∼ ∈M sup τ ∼ ∈Σ [C(τ ∼ ), v ∼ ] τ ∼ Σ v ∼ M ≥ β 2 . ( 11
)
Proof. The proof of this result is the same as the one of Proposition 2.2 in [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF].

The main result of this section can be established now.

Theorem 3.1. Problem (9) admits a unique solution (t, σ ∼ , u ∼ ) ∈ T × Σ × M satisfying the following stability condition, t T + σ ∼ Σ + u ∼ M ≤ C(f ),
where C(f ) is a positive constant depending on f .

Proof. Using the Inf-Sup condition [START_REF] Gatica | On the numerical analysis of nonlinear twofold saddle point problems[END_REF] and applying Lemma 3.4(iii) to the operator C, we get the existence of

. σ ∼ = ( . σ, . p ) ∈ Σ / Z 1 such that [C( . σ ∼ ), v ∼ ] = [F , v ∼ ] ∀v ∼ ∈ M (12)
and

. σ ∼ Σ / Z1 ≤ C f 0,r ,Ω . (13)
Let

σ ∼ 0 = (σ 0 , p 0 ) ∈ . σ ∼ such that σ ∼ 0 Σ = . σ ∼ Σ / Z1 ≤ C f 0,r ,Ω . (14)
Setting σ ∼ = σ ∼ * + σ ∼ 0 , i.e., (σ, p) = (σ * + σ 0 , p * + p 0 ), problem (9) may be written in the following way: Find t ∈ T , and σ

∼ * ∈ Z 1 such that    [A(t), s] + [s, B (σ ∼ * )] = -[s, B (σ ∼ 0 )] ∀ s ∈ T , [B(t), τ ∼ ] = 0 ∀τ ∼ ∈ Z 1 . (15)
Now, from the properties of the operator A (see Remark 3.1) and Lemma 3.5, there exists

a t ∈ Z 2 such that [A(t), s] = -[s, B (σ ∼ 0 )] ∀ s ∈ Z 2 , (16)
where

Z 2 = Ker(B) = s ∈ T ; [B(s), τ ∼ ] = 0 , ∀τ ∼ ∈ Σ .
Uniqueness of t is implied by the fact that the operator A is strictly monotone (see Remark 3.1). Furthermore, one can prove that

t 0,r,Ω ≤ C f 0,r ,Ω + f r /r 0,r ,Ω . (17)
Equation ( 16) implies [A(t) + B (σ

∼ 0 ), s] = 0 ∀ s ∈ Z 2 , i.e, A(t) + B (σ ∼ 0 ) ∈ Z o 2 .
Thus, by the Inf-Sup condition [START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF] and Lemma 3.4(ii), there exists a unique σ

∼ * ∈ Z 1 such that [B (σ ∼ * ), s] = -[A(t) + B (σ ∼ 0 ), s] ∀ s ∈ T (18) and σ ∼ * Σ ≤ C A(t) + B (σ ∼ 0 ) 0,r ,Ω . (19)
Therefore, from ( 12) and (18), we get

[A(t), s] + [s, B (σ ∼ )] = 0 ∀ s ∈ T , [C(σ ∼ ), v ∼ ] = [F , v ∼ ] ∀v ∼ ∈ M ,
and, owing to ( 14), ( 17) and ( 19), we have

t T + σ ∼ Σ ≤ C(f ),
where C(f ) is a positive constant depending on f . On the other hand, since [B(t), τ ∼ ] = 0 ∀τ ∼ ∈ Z 1 , we have B(t) ∈ Z o 1 . Thus, by the Inf-Sup condition (11) and Lemma 3.4(ii), there exists a unique u

∼ ∈ M such that [τ ∼ , C (u ∼ )] = -[B(t), τ ∼ ] ∀τ ∼ ∈ Σ and u ∼ M ≤ C B(t) Σ . Therefore, [B(t), τ ∼ ] + [τ ∼ , C (u ∼ )] = 0 ∀τ ∼ ∈ Σ and u ∼ M ≤ C(f )
, which completes the proof.

Discrete mixed formulation and error analysis

We assume that the boundary Γ of the domain Ω is polygonal. We first give some finite element notations. Let h > 0 and T h a triangulation of Ω into triangles. We assume that the triangulation T h is regular in the sense of Ciarlet [START_REF] Ciarlet | The Finite Element Methods for Elliptic Problems[END_REF]. Let K ∈ T h , be an element of the triangulation, we denote by b K the bubble function defined by:

b K (x) = λ 1 (x)λ 2 (x)λ 3 (x), ∀x ∈ K, λ i , i = 1, • • • , 3
being the barycentric co-ordinates with respect to the element K. Let P k (K) denote the space of polynomials of degree less than or equal to k on K, and

R(K) = P 1 (K) 2 ⊕ R curl b K , where curl b K = ( ∂b K ∂x 2 , - ∂b K ∂x 1
).

To write the discrete mixed formulation, we introduce the following finite dimensional spaces:

T h = s h ∈ T ; s h| K ∈ R(K), ∀K ∈ T h , Σ h = τ h ∼ = (τ h , q h ) ∈ Σ; τ h| K ∈ [R(K)] 2 , q h| K ∈ P 1 (K), ∀K ∈ T h , M h = v h ∼ = (v h , η h ) ∈ M ; v h| K ∈ P 0 (K) 2 , η h = θ h χ, θ h| K ∈ P 1 (K), ∀K ∈ T h ,
where

χ = 0 -1 1 0 .
The discrete mixed formulation of problem ( 9) is given by: Find

t h ∈ T h , σ h ∼ ∈ Σ h and u h ∼ ∈ M h such that          [A(t h ), s h ] + [s h , B (σ h ∼ )] = 0 ∀ s h ∈ T h , [B(t h ), τ h ∼ ] + [τ h ∼ , C (u h ∼ )] = 0 ∀τ h ∼ ∈ Σ h , [C(σ h ∼ ), v h ∼ ] = [F , v h ∼ ] ∀v h ∼ ∈ M h . (20)
In order to analyze the discrete problem (20) and prove error estimates, we have to state some intermediate results.

Lemma 4.1. There exists a positive constant β * 1 independent of h, such that

inf τ h ∼ ∈Z h 1 sup s h ∈T h [B(s h ), τ h ∼ ] s h T τ h ∼ Σ ≥ β * 1 , (21)
where Z h 1 is the discrete Kernel of C, i.e.,

Z h 1 = τ h ∼ ∈ Σ h ; [C(τ h ∼ ), v h ∼ ] = 0, ∀v h ∼ ∈ M h . Proof. Let τ h ∼ = (τ h , q h ) ∈ Z h 1 . Then, τ h ∈ [L r (Ω)] 2×2 , q h ∈ L r 0 (Ω) and div(τ h -q h I) = 0. Now, let s * = -|τ h | r -2 τ h and s * h ∈ T h such that Ω s * h φ h dx = Ω s * φ h dx, ∀φ h ∈ T h .
Proceeding as for the proof of Lemma 4.1 in [6], we have s * h T ≤ C s * T , where C is a positive constant independent of h. Thus, s * h ∈ T h and s

* h T ≤ C τ h r -1 0,r ,Ω . Therefore, [B(s * h ), τ h ∼ ] s * h T = -(s * h , τ h ) s * h T ≥ C τ h r 0,r ,Ω τ h r -1 0,r ,Ω = C τ h 0,r ,Ω .
On the other hand, since div(τ h -q h I) = 0, there exists a constant C > 0 such that (see Lemma 4.3 below) q h 0,r ,Ω ≤ C τ h 0,r ,Ω . Therefore, for all τ h

∼ ∈ Z h 1 , we have sup s h ∈T h [B(s h ), τ h ∼ ] s h T ≥ [B(s * h ), τ h ∼ ] s * h T ≥ C τ h ∼ Σ ,
which completes the proof.

For the proofs of the two following results, see Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF].

Lemma 4.2. There exists a positive constant β * 2 independent of h, such that

inf v h ∼ ∈M h sup τ h ∼ ∈Σ h [C(τ h ∼ ), v h ∼ ] τ h ∼ Σ v h ∼ M ≥ β * 2 . ( 22
)
Lemma 4.3. There exists a positive constant C, independent of h, such that

∀τ h ∼ ∈ Z h 1 , q h 0,r ,Ω ≤ C τ h 0,r ,Ω . (23)
Due to the previous Lemma, the discrete inf-sup conditions (21) and ( 22), we obtain, as for the continuous problem, the existence, uniqueness and stability of the discrete solution. More precisely, Theorem 4.1. Problem (20) admits a unique solution

(t h , σ h ∼ , u h ∼ ) ∈ T h × Σ h × M h satisfying the following stability condition, t h T + σ h ∼ Σ + u h ∼ M ≤ C(f ).
wher C(f ) is a positive constant depending on f and independent of h.

To prove the error estimates, we shall use the following result (see Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF]). Proposition 4.1. Let s be a real such that s > 1, then there exists an operator

Π h : [W 1,s (Ω)] 2×2 × W 1,s (Ω) ∩ L s 0 (Ω) -→ Σ h τ ∼ -→ Π h (τ ∼ ) which satisfies [C(τ ∼ -Π h (τ ∼ ), v h ∼ ] = 0, ∀v h ∼ ∈ M h . (24)
Further, for all τ

∼ = (τ , q) ∈ [W m,s (Ω)] 2×2 × (W m,s (Ω) ∩ L s 0 (Ω)), m = 1, 2, we have (τ , q) -Π h (τ , q) 0,s,Ω ≤ Ch m |τ | m,s,Ω + |q| m,s,Ω (25)
where C is a positive constant independent of h, and (τ , q) 0,s,Ω = τ s 0,s,Ω + q s 0,s,Ω

We are now able to state the error estimates for problem (20). In the following

(t, σ ∼ , u ∼ ) = t, (σ, p), (u, ω) and (t h , σ h ∼ , u h ∼ ) = t h , (σ h , p h ), (u h , ω h )
denote respectively the solution of the continuous problem [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF] and the solution of the discrete problem (20). In the sequel, we use the following notation:

u * h = P 0 h u, the L 2 -projection of u onto   K∈T h P 0 (K)   2 . Theorem 4.2. Let 1 < r < 2 and m = 1, 2. Suppose t ∈ [W m,r (Ω)] 2×2 , σ ∼ = (σ, p) ∈ W m,r (Ω) 2×2 × W m,r (Ω), and u ∼ = (u, ω) ∈ W 1,r (Ω) 2 × [W m,r (Ω)] 2×2 .
Then, there exists a positive constant C

independent of h such that

t -t h 0,r,Ω ≤ C h m(r/2) , (26)
σ -σ h 0,r ,Ω + p -p h 0,r ,Ω ≤ C h m(r-1) , (27)
u * h -u h 0,r,Ω + ω -ω h 0,r,Ω ≤ C h m(r/2) , (28)
u -u h 0,r,Ω ≤ C h r/2 if m = 1, and u -u h 0,r,Ω ≤ C h if m = 2. ( 29
)
Proof. By subtracting the last equation of (20) from the last equation of ( 9), we get for v

∼ = (v, η) = (v h , η h ) [C(σ ∼ -σ h ∼ ), v h ∼ ) = 0
and then, owing to (24),

div [(σ * h -p * h I) -(σ h -p h I)] , v h + (σ * h -σ h , η h ) = 0, (30)
where (σ * h , p * h ) = Π h (σ, p). Let (u * h , ω * h ) = (P 0 h u, P 1 h ω) denote the L 2 -projection of (u, ω) onto the space M h and t * h = P * h t the L 2 -projection of t onto the space T h . By subtracting the second equation of (20) from the second equation of (9), we get for τ

∼ = (τ , q) = (τ h , q h ) (t -t h , τ h ) + (div(τ h -q h I), u -u h ) + (τ h , ω -ω h ) = 0,
and then

(t -t h , τ h ) + (div(τ h -q h I), u * h -u h ) + (τ h , ω * h -ω h ) = (τ h , ω * h -ω). (31)
This last relation and (30) yield

(t -t h , σ * h -σ h ) = (σ * h -σ h , ω * h -ω) and then (t * h -t h , σ * h -σ h ) = (σ * h -σ h , ω * h -ω). (32)
Now, by subtracting the first equation of (20) from the first equation of ( 9), we get for s = s h

(A(t) -A(t h ), s h ) = (s h , σ -σ h ). (33)
Thus, using this last equation and (32), we have

(A(t) -A(t h ), t -t h ) = (A(t) -A(t h ), t -t * h ) + (A(t) -A(t h ), t * h -t h ) = (A(t) -A(t h ), t -t * h ) + (t * h -t h , σ -σ h ) = (A(t) -A(t h ), t -t * h ) + (t * h -t h , σ -σ * h ) + (t * h -t h , σ * h -σ h ) = (A(t) -A(t h ), t -t * h ) + (t * h -t h , σ -σ * h ) + (σ * h -σ h , ω * h -ω) = (A(t) -A(t h ), t -t * h ) + (t * h -t, σ -σ * h ) + (t -t h , σ -σ * h ) +(σ * h -σ h , ω * h -ω), therefore (A(t) -A(t h ), t -t h ) = (A(t) -A(t h ), t -t * h ) + (t * h -t, σ -σ * h ) + (t -t h , σ -σ * h ) +(σ * h -σ h , ω * h -ω). (34)
On the other hand, owing to (33), we have

(A(t) -A(t h ), s h ) = (s h , σ -σ * h ) + (s h , σ * h -σ h ) ∀ s h ∈ T h and then (s h , σ * h -σ h ) = (A(t) -A(t h ), s h ) + (s h , σ * h -σ) ∀ s h ∈ T h .
Thus, from (30) and the discrete Inf-Sup condition (21), we have

β * 1 σ * h -σ h 0,r ≤ A(t) -A(t h ) 0,r + σ -σ * h 0,r . (35)
Using this last estimate and (34), we get

(A(t) -A(t h ), t -t h ) ≤ A(t) -A(t h ) 0,r t -t * h 0,r + t -t * h 0,r σ -σ * h 0,r + t -t h 0,r σ -σ * h 0,r + C ( A(t) -A(t h ) 0,r + σ -σ * h 0,r ) ω -ω * h 0,r thus (A(t) -A(t h ), t -t h ) ≤ A(t) -A(t h ) 0,r t -t * h 0,r +C A(t) -A(t h ) 0,r ω -ω * h 0,r + t -t h 0,r σ -σ * h 0,r + t -t * h 0,r σ -σ * h 0,r + C σ -σ * h 0,r ω -ω * h 0,r . (36)
Now, owing to Lemma 3.1, Lemma 3.3 and (36), we have

t -t h 2 0,r δ + t 2-r 0,r + t h 2-r 0,r + Ω |A(t) -A(t h )| |t -t h | dx ≤ C Ω |A(t) -A(t h )| |t -t h | dx 1/r t -t * h 0,r + C Ω |A(t) -A(t h )| |t -t h | dx 1/r ω -ω * h 0,r + C t -t h 0,r σ -σ * h 0,r + C t -t * h 0,r σ -σ * h 0,r + C σ -σ * h 0,r ω -ω * h 0,r .
Then, using the Young inequality,

a b ≤ 1 r εa r + 1 r b ε r , for ε > 0, a ≥ 0 and b ≥ 0, we have t -t h 2 0,r δ + t 2-r 0,r + t h 2-r 0,r + Ω |A(t) -A(t h )| |t -t h | dx ≤ ε r r Ω |A(t) -A(t h )| |t -t h | dx + C rε r ( t -t * h r 0,r + ω -ω * h r 0,r ) + C( t -t * h 0,r + ω -ω * h 0,r ) σ -σ * h 0,r + C t -t h 0,r σ -σ * h 0,r .
Choosing 0 < ε < 1 (for example, ε = 1/2), it follows from the previous inequality the following one:

t -t h 2 0,r δ + t 2-r 0,r + t h 2-r 0,r + Ω |A(t) -A(t h )| |t -t h | dx ≤ C t -t * h r 0,r + ω -ω * h r 0,r + t -t h 0,r σ -σ * h 0,r + ( t -t * h 0,r + ω -ω * h 0,r ) σ -σ * h 0,r . (37)
Then, using the stability conditions on t and t h (see Theorem 3.1 and Theorem 4.1), we get, for all ε > 0, On the other hand, due to (30) and (23), we have

t -t h 2 0,r ≤ C t -t * h r 0,r + ω -ω * h r 0,r + t -t h 0,r σ -σ * h 0,r + ( t -t * h 0,r + ω -ω * h 0,r ) σ -σ * h 0,r ≤ C t -t * h r 0,r + ω -ω * h r 0,r + ( t -t * h 0,r + ω -ω * h 0,r ) σ -σ *
p * h -p h 0,r ≤ C σ * h -σ h 0,r
and then we get the following estimation for the pressure p -p h 0,r ≤ Ch m(r-1) .

It remains to prove the error estimates (28) and (29). To this end, using (31) and the discrete Inf-Sup condition (22), we have

β * 2 (u * h -u h , ω * h -ω h ) M ≤ t -t h 0,r + ω -ω * h 0,r ≤ Ch m(r/2)
and then u * h -u h 0,r + ωω h 0,r ≤ Ch m(r/2) . Finally, (29) is a consequence of this last estimate and the interpolation results. independent of h such that tt h 0,r,Ω ≤ C h m(r -1) , σσ h 0,r ,Ω + p -p h 0,r ,Ω ≤ C h m(r /2) , u * h -u h 0,r,Ω + ωω h 0,r,Ω ≤ C h m(r -1) , uu h 0,r,Ω ≤ C h r -1 if m = 1, and uu h 0,r,Ω ≤ C h min{1,2(r -1)} if m = 2.

Proof. The proof of this last Theorem is similar to the one of Theorem 4.2.

Conclusion

We have developed in this paper a mixed approach for the approximation of all the important variables (stress, velocity and pressure) appearing in quasi-Newtonian fluid flow problems. Our approach allows to treat, independently, both the famous laws of Power and Carreau without using the inversion of constitutive laws. The continuous and discrete problems are well posed. The error estimates are optimal and are the same as the ones that we have obtained in the particular case of Power law (see Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF]).

Lemma 3 . 4 .

 34 Let (X, • X ) and (M, • M ) be two reflexive Banach spaces. Let (X , || • || X ) and (M , || • || M ) be their corresponding dual space. Let B : X -→ M be a linear continuous operator, and B : M -→ X its dual. Let V = Ker(B) be the kernel of B; we denote by V o ⊂ X the polar set of V , i.e., V o = {x ∈ X , < x , v >= 0, ∀v ∈ V }, and .

h 0,r +ε t -t h 2 0+≤

 2 ( tt * h 0,r + ωω * h 0,r ) σσ * h 0,r.Thus, using the interpolation results and (25), we obtaintt h 0,r ≤ Ch m(r/2) .From (37) and this last estimate, we getΩ |A(t) -A(t h )| |t -t h | dx ≤ Ch mrand then, by (35) and Lemma 3.1, we haveσ * h -σ h 0,r ≤ C( A(t) -A(t h ) 0,r + σσ * h 0,r ) ≤ C Ω |A(t) -A(t h )| |t -t h | dx Ch m(r/r )thus σσ h 0,r ≤ Ch m(r/r ) = Ch m(r-1) .

Theorem 4 . 3 .

 43 Let r ≥ 2 and m = 1, 2. Suppose t ∈ [W m,r (Ω)] 2×2 , σ ∼ = (σ, p) ∈ W m,r (Ω) 2×2 × W m,r (Ω), and u ∼ = (u, ω) ∈ W 1,r (Ω) 2 × [W m,r (Ω)] 2×2 .Then, there exists a positive constant C