Notebook - LHE

1 C O L E P O L Y T EC H NI Q UEF D R A L E D E L A US A N NEChristopheAnceyLaboratoire hydraulique environnementale(LHE) cole Polytechnique F d rale de Lausanne cublensCH-1015 LausanneNotebookIntroduction to Fluid Rheologyversion of 4th July ,EPFL, ENAC/ICARE/LHE,Ecublens, CH-1015 Lausanne, to Fluid Rheology / travail est soumis aux droits d auteurs. Tous les droits sont r serv s ; toute copie, partielleou compl te, doit faire l objet d une autorisation de l gestion typographique a t r alis e l aide du : S bastienWiederseineret MartinRentschlerpour la relecture Le physicien ne peut demander l analyste de lui r v ler une v rit nouvelle ; toutau plus celui-ci pourrait-il l aider la pressentir. Il y a longtemps que personne nesonge plus devancer l exp rience, ou construire le monde de toutes pi ces surquelques hypoth ses h tives.

2 De toutes ces constructions o l on se complaisaitencore na vement il y a un si cle, il ne reste aujourd hui plus que des les lois sont donc tir es de l exp rience, mais pour les noncer, il fautune langue sp ciale ; le langage ordinaire est trop pauvre, elle est d ailleurs tropvague, pour exprimer des rapports si d licats, si riches et si pr cis. Voil donc unepremi re raison pour laquelle le physicien ne peut se passer des math matiques ;elles lui fournissent la seule langue qu il puisse parler. HenriPoincar , inLa Valeur de la Science4 TABLE OF CONTENTS5 Table of contents1 How does a rheometer operate? .. A long history .. Anatomy of a modern rheometer .. Typical performance of modern lab rheometers .. Principles of viscometry.

3 Fundamentals of rheometry .. Flow down an inclined channel .. Standard geometries .. Inverse problems in rheometry .. A typical example: the Couette problem .. Earlier attempts at solving the Couette problem .. The wavelet-vaguelette decomposition .. Practical example .. Rheometers and rheometrical procedures .. How to determine the flow curve? .. Stress/strain step .. Typical rheological behaviors .. Outlining a flow curve .. Shear-thinning/thickening .. Yield stress .. Viscoelasticity .. Normal stress effects .. Thixotropy .. Problems encountered in rheometry .. Problems with rheometers .. Limitations of the viscometric treatment .. Technical issues related to the derivation of the flow curve.

4 Problems related to sample preparation .. 556 TABLE OF Non-standard techniques: what can be done without a rheometer? .. Viscosity: free fall of a bead .. Yield stress: Slump test .. 562 Rheology and Continuum Why is continuum mechanics useful? An historical perspective .. Paradoxical experimental results? .. How to remove the paradox? .. Fundamentals of Continuum Mechanics .. Kinematics .. Stress tensor .. Admissibility of a constitutive equations .. Specific properties of material .. Representation theorems .. Balance equations .. Conservation of energy .. Jump conditions .. Phenomenological constitutive equations .. Newtonian behavior .. Viscoplastic behavior.

5 Viscoelasticity .. 813 Fundamentals of rheophysics .. Movement of a single sphere and consequences on the flow regime .. From a single sphere to a bulk: averaging .. Averaged balance equations .. Passing from volume averages to ensemble averages .. Dilute suspensions .. Dilute suspension in a Stokes regime: Stokesian theory .. Computations of the constitutive equation .. Concentrated suspensions .. Constitutive equations for concentrated suspensions .. Regime diagram and computation of individual contributions .. 107 References111 TABLE OF CONTENTS7 ForewordObjective of the courseThe objective of this course held in the framework of the doctoral schoolMechanics of Solidsand Fluidset EPFL is to provide the student with the modern tools needed to investigate therheological behavior of complex fluids.

6 Emphasis will be given to particle suspensions. The coursewill start with an introduction of experimental procedures. Phenomenological description of howmatter flows will then be presented. The last part of the course will be devoted to the rheophysicalapproach to modelling the rheological behavior of particle Notebook will focus on materials encountered by geophysicists (mud, snow, magma, etc.)and in industrial or civil-engineering applications (concrete, slurries, etc.): in most cases we willconsider only homogeneous and suspensions of particles within an interstitial fluid without lossof generality. Other complex fluids such as polymeric liquids are rarely encountered in geophysicsand therefore they will not be addressed of the notebookThe mere description of what the term rheology embraces in terms of scientific areas is noteasy.

7 Roughly speaking, rheology distinguishes different areas and offshoots such as the following:rheometry, formulation of constitutive equation, computational rheometry, microstructural ana-lysis and interpretation of bulk rheological behavior, etc. Here we will focus on the followingpoints1:Rheometry. The term rheometry is usually used to refer to a group of experimentaltechniques for investigating the rheological behavior of materials. It is of great importance indetermining the constitutive equation of a fluid or in assessing the relevance of any proposedconstitutive law. Most of the textbooks on rheology deal with rheometry. The books by Colemanet al.(1966), Walters (1975), and by Birdet al.(1987) provide a complete introduction to theviscometric theory used in rheometry for inferring the constitutive equation.

8 The book by Coussot& Ancey (1999b) gives practical information concerning rheometrical measurements with naturalfluids. Though primarily devoted to food processing engineering, Steffe s book presents a detaileddescription of rheological measurements; a free sample is available on the web (Steffe, 1996).In Chapter 1, we will review the different techniques that are suitable to studying variousfluids. Emphasis is given both to describing the methods and the major experimental problemsencountered with materials made up of particles and Other aspects of rheology, such as complex flow modelling and computational rheology, are notaddressed in this introductory OF CONTENTSC ontinuum mechanics. The formulation of constitutive equations is probably the early goalof rheology.

9 At the beginning of the 20thcentury, the non-Newtonian character of many fluidsof practical interest motivated Professor Bingham to coin the termrheologyand to define it asthe study of the deformation and flow of matter. The development of a convenient mathematicalframework occupied the attention of rheologists for a long time after the Second World War. Atthat time, theoreticians such as Coleman, Markovitz, Noll, Oldroyd, Reiner, Toupin, Truesdell,etc. sought to express rheological behavior through equations relating suitable variables andparameters representing the deformation and stress states. This gave rise to a large number ofstudies on the foundations of continuum mechanics (Birdet al., 1987). Nowadays the work of thesepioneers is pursued through the examination of new problems such as the treatment of multiphasesystems or the development of nonlocal field theories.

10 For examples of current developmentsand applications to geophysics, the reader may consult papers by Hutter and coworkers onthe thermodynamically consistent continuum treatment of soil-water systems (Wang & Hutter,1999; Hutteret al., 1999), the book by Vardoulakis & Sulem (1995) on soil mechanics, andthe review by Bedford & Dumheller (1983) on suspensions. A cursory glance at the literatureon theoretical rheology may give the reader the impression that all this literature is merely anoverly sophisticated mathematical description of the matter with little practical interest. In fact,excessive refinements in the tensorial expression of constitutive equations lead to prohibitivedetail and thus substantially limit their utility or predictive capabilities.