Table Of ContentLecture Notes
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Edited yb .H Araki, Kyoto, .J Ehlers, MSnchen, .K Hepp, ZSrich
.R .L ,effaJ Cambridge, ,AM .R ,nhahneppiK ,nehcn~LM .D ,elleuR ettevY-rus-seruB
H.A. ,rell~LmnedieW Heidelberg, .J Wess, Karlsruhe dna .J Zittartz, nISK
Managing Editor: .W BeiglbSck
183
.J Casas-V&zquez .D uoJ (Eds.)
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lacimanydomrehT dna
lacitsitatS Approaches
Proceedings of eht Meeting Held ta the
Bellaterra School of scimanydomrehT
Autonomous ytisrevinU of Barcelona
Sant Feliu de Guxols, Catalonia, Spain
24-28 September 1990
Springer-Verlag
Berlin Heidelberg New York London Paris
oykoT Hong Kong Barcelona Budapest
Editors
Jo$6 Casas-V~zquez
David Jou
Departament de Ffsica, Universitat AutSnoma de Barcelona
E-08193 Bellaterra, Catalonia, Spain
ISBN 3-540-53996-4 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-38?-53996-4 Springer-Verlag NewYork Berlin Heidelberg
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Preface
The analysis of rheological systems provides stimulating challenges to the
development of statistical mechanics and nonequilibrium thermodynamics. A
sound understanding of the observed behaviour of such systems in terms of
microscopic models and the use of nonequilibrium thermodynamics to restrict the
possible forms of constitutive equations are goals which have been a source of
inspiration in the respective fields. nI nonequilibrium thermodynamics, rheological
systems do not admit the simple local-equilibrium hypothesis: a number of internal
variables must be considered, so the form of the entropy cannot be assumed a
priori; but it can be studied by prescribing a functional dependence on a set of
these internal variables. nI statistical mechanics, the geometrical complexity of the
macromolecules requires new techniques for the analysis of transport phenomena.
Furthermore, even those fluids composed of the simplest monatomic particles
exhibit several kinds of rheological behaviour when they are subjected to high
frequencies or to intense thermodynamic forces, so that rheological analysis
becomes an important ingredient ni the study of fast and nonlinear phenomena.
These were the motivations for devoting the fifth session of the Bellaterra
School of Thermodynamics (BEST) to the relation between rheological modelling,
statistical mechanics and nonequilibrium thermodynamics. The aim of BeST is to
gather together leading scientists in these areas, graduate students, and other
researchers interested ni the current developments ni these fields. The school is
addressed to fundamental questions, both theoretical and experimental, rather
than to direct applications, To join the pedagogical features of a school with the
stimulus of fresh information on recent advances is the goal pursued at each
meeting of the Bellaterra School.
The lecturers invited to the School are well known for their contributions to
different aspects of rheology, statistical mechanics or nonequilibrium
thermodynamics. Bird provides a clear introduction to the relation between kinetic
models, macroscopic rheological equations and rheometric measurements.
Joseph .presents an account of the mathematical and physical problems associated
with hyperbolicity arising in the context of viscoelasticity. Hess studies the
microscopic consequences of shear flows on the structure of liquids and compares
the results obtained with kinetic theory and nonequilibrium molecular dynamics
simulations. Rubi considers the role that time dependent hydrodynamic interactions
play in the dynamics of polymer solutions. Grmela studies the interrelations
between several levels of description of rheological systems, from the macroscopic
thermodynamics to the statistical descriptions based on distribution functions, and
the conditions of compatibility between such different descriptions. Carreau and
Grmela discuss conformation tensor rheological models, in which distribution
functions of molecular state variables are used as internal variables ni the theory.
VI
Quemada analyses the complex phenomenology concerning the viscous
behaviour of concentrated suspensions, with emphasis on biological fluids. Wolf
describes the changes that a shear flow produces on the phase diagram of
polymer solutions. Jongschaap contributes a broad account of the problems of
rheological modelling using a thermodynamic approach which may be applied to
many types of models in a unified and systematic way. Finally, de Gennes deals
with new approaches to the problem of adhesion, ni particular various forms of the
constitutive law relating the applied stress and the rate of opening of the junction
width ni a thin ribbon.
The reader will also find the text of some of the seminars delivered at the
School, which complement the insights into rheology that may eb gained from the
perspectives of statistical mechanics and nonequilibrium thermodynamics. Thus,
the book includes such topics as hard sphere suspensions, an EIT approach to
rheology, and the formalism of fractional derivatives.
We gratefully acknowledge the financial support of the DGICyT of the
Spanish Ministry of Education and Science, of the DGU and the CIRIT of the
Government of Catalonia, of the Autonomous University of Barcelona, of the
savings bank La Caixa, and, last but not least, of the City council of Sant Feliu de
Guixols and .~lruM Park Hotel, ni the Costa Brava, where this meeting was held.
J. Casas-V&zquez
.D Jou
Conl~nts
Lectures
Viscoelastic Behavior of Polymeric Liquids
R.B. Bird
Experimental Evidence for Non-Newtonian Behavior of Polymeric Liquids
Rheometry and Material Functions
Continuum Mechanical Ideas and Empirical Constitutive Equations
Kinetic Theory Ideas and Dumbbell Models
Kinetic Theory Ideas and Chain Models
Problems Associated with the Elasticity of Liquids 22
D.D. Joseph
Physical Phenomena Associated with Hyperbolicity and Change of Type
Conceptual Ideas
Mathematical Theory
Rheology and Shear Induced Structure of Fluids 15
S. Hess
Introduction
Pressure Tensor, Viscosity Coefficients
The Structure of Streaming Fluids
Concluding Remarks
On the Dynamics of Polymers in Solution 47
J.M. Rubl, J. Bonet Avalos and D. Bedeaux
Introduction
Equation of Motion of a Polymer
Polymer Dynamics
Conclusions
Mesoscopic Dynamics and Thermodynamics: Applications
to Polymeric Fluids 99
M. Grmela
Introduction
Hierarchy of Descriptions
Thermodynamics - Geometry of the State Space
Dynamics - Physical Foundation of Thermodynamics
Rheological Modelling
Thermodynamics and Dynamics of Driven Systems
Concluding Remarks
LV
Conformation Tensor Rheological Models 621
P.J. Carreau and M. Grmela
Introduction
Governing Equations
Predictions and Comparison with Experiments
Concluding Remarks
Biofluids as Structured Media: Rheology and Flow
Properties of Blood 851
D. Quemada
Introduction
Main Characteristics of Blood Flows
Blood as a Concentrated Dispersion
Blood as a Shear-Thinning Fluid
Blood as a Thixotropic and Viscoelastic Fluid
Modelling of Blood Microcirculation
Concluding Remarks
Phase Separation of F16wing Polymer Solutions 491
B.A. Wolf
Introduction
Procedures and Observations
Calculation of Phase Diagrams
Discussion
Towards a Unified Formulation of Microrheological
Models 512
R.J.J. Jongschaap
Introduction
Theory
Applications
Discussion
Adhesion and Rheology (Abstract)
P.G. de Gennes
Sem|nAeS
Rheology of Hard Sphere Suspensions 052
B. .U Felderhof
Vii
Extended Irreversible Thermodynamics Versus
Rheology 752
.G Lebon, D. Jou and J. Casas-Vdzquez
Objectivity and the Extended Thermodynamic
Description of Rheology 872
P.C. Dauby
Convection in Viscoelastic Fluids 292
C.P~rez-Garcia, J. Martinez-Mardones and J. Milldn
Fractional Relaxation Equations for Viscoelasticity
and Related Phenomena 903
T.F. Nonnenmacher
Relaxation Functions of Rheological Constitutive
Equations with Fractional Derivatives: Thermodynamical
Constraints 123
.C Friedrich
A Simple One Dimensional Model Showing Glasslike
Dynamical Behavior 133
J.J. Brey and M.J. Ruiz-Montero
Statistical Conformation of a Polymer in a Nematic Medium
Under a Shear Flow Using the Rouse Model 443
Y. Thiriet, R. Hocquart, F. Lequeux and JoF. Palierne
On the Modelling of Stationary Heat Transfer by the
Use of Dissipative Networks 653
.G Brunk
Thermomechanics of Porous Media Filled with
a Fluid 763
.W Derski
VISCOELASTIC BEHAVIOR OF
POLYMERIC LIQUIDS
R. Byron Bird
Chemical Engineering Department and Rheology Research Center
University of Wisconsin-Madison, Madison, WI 53706-1691, USA
A survey is given of the three main approaches to the study of the non-linear
constitutive equations for polymeric liquids: rheometric measurements; continuum
mechanics results for special flows and useful empiricisms; and kinetic theories, which
make use of various mechanical models to represent the polymer molecules. For
further discussions see 1,2.
.1 Experimental evidence for non-Newtonian behavior of polymeric liquids 1,
Chapter 2
For incompressible fluids the laws of conservation of mass and momentum lead to
the equations of continuity and motion 3, Chapter 3:
Eq. of Continuity: (V-v) = 0, (1.1)
Eq. of Motion: p ~tt = -V.rq + pg. (1.2)
Here p is the fluid density, v(r,t) the velocity field, and g the gravitational acceleration.
The stress tensor g may be written as the sum of two terms,
= p5 + ,c" (1.3)
in which p is a pressure (not uniquely determined for incompressible fluids) and x is the
"extra stress tensor", which is zero at equilibrium; 5~ is the unit tensor.
For liquids made up of small molecules, the mechanical behavior is well described
by the incompressible Newtonian constitutive equation 4 (this is just the linear term in
the ordered expansion in eq. (3.14)):
C" = -~(VV + (Vv) t) = ,)1(t~L,I- (1.4)
in which (Vv)t indicates the transpose of (Vv), and ~/(1) is the rate-of-strain tensor. The
Newtonian viscosity t~ may be determined from any viscometer for which eqs. (1-4) can
be solved (for steady-state or unsteady-state flow).
Polymer solutions and polymer melts cannot be described by eq. (1.4) and are
therefore non-Newtonian. One of the challenges in polymer fluid dynamics is that of
obtaining a constitutive equation for x, containing a small number of physically
interpretable parameters, that is capable of describing a wide range of fluid responses.
Many experiments indicate that polymeric liquids are non-Newtonian :1
(A) In a cone-and plate viscometer the torque needed to turn the cone is
proportional to the angular velocity of the cone for Newtonian fluids, but not so for
polymeric liquids. This means that for polymers the shear stress is not proportional to
the velocity gradient. Consequently, the viscosity (defined as the proportionality factor
between shear stress and velocity gradient) is not a constant. In fact, the viscosity may
decrease by factors of ten to a thousand as the velocity gradient increases. Such a
dramatic decrease in viscosity cannot be ignored in fluid dynamics calculations. Even in
extremely dilute polymer solutions this shear thinning behavior is observed.
(B) A cylindrical container with inside radius R is filled with a Newtonian fluid.
When a rotating disk of radius R is placed in contact with the fluid surface, it imparts a
tangential motion to the liquid (the "primary flow"); at steady-state there is, in addition
to the primary flow, a "secondary flow" outward along the rotating disk, caused by the
centrifugal force, and then downward along the cylinder wall, inward along the bottom,
and upward near the cylinder axis. For polymeric fluids, however, there is a reversal
of the secondary flow in the disk-and-cylinder system: the fluid moves inward along the
rotating disk, down along the axis and then upward at the cylinder wall!
(C) If a Newtonian fluid flows down an inclined semicylindrical trough, the
liquid surface is flat (except near the edges where meniscus effects may be seen).
Polymeric liquids, however, display a slightly convex surface in tilted-trough flow.
(D) When the pumping of a Newtonian fluid through a tube is suddenly stopped,
the fluid comes to rest almost at once. But when one stops supplying a driving force for
the flow of a polymeric liquid in a tube, the polymeric liquid begins to retreat and then
gradually comes to rest. This recoil is symptomatic of a fluid which can "remember" its
past kinematic experience; the fluid does not retreat all the way to its initial
configuration, since as time proceeds it "forgets" its kinematic history. Polymeric fluids
are often referred to as "fluids with fading memory."
(E) When a Newtonian liquid is being siphoned, the siphon ceases to function
when the end of the tube is removed from the liquid. For a polymeric liquid, however,
the tube may be withdrawn from the liquid and the siphoning continues; there may be a
gap of several centimeters between the liquid surface and the end of the tube. This
effect is called the tubeless siphon.
Many other experiments showing that polymeric liquids are qualitatively different
from Newtonian fluids are described in 1,12, where the original literature references
are given. The behavior in expt. A is obviously not explainable by eq. (1.4). Expts. B
and C can be explained if in steady shear flow the normal stresses ,xxX( ~yy,'~zz) are all
different from each other; such "normal-stress phenomena" cannot be explained by eq.
(1.4). To describe expt. D it is necessary that the stress at the present time t depend on
the kinematic tensors at all past times t'. Finding a replacement for eq. (1.4) is an
enormous challenge, requiring the collaboration of the experimental rheologist, the
continuum mechanicist, and the kinetic theorist.
2. Rheometry and material functions 5
Much of our present knowledge about the flow behavior of polymeric liquids has
been obtained from measurements of stress components in carefully controlled flows.
The two most studied flows are swolfraehs for which
Vx = T(t)y; Vy = 0; V z = 0, (2.1)
where Xt) is the "shear rate," and swolflanoitagnole for which
xV =- L 2 ~(t)x; yV =- L 2 ~(t)y; zV = g(t)z
(2.2)
where ~(t) is the "elongation rate." There is a general feeling among polymer fluid
dynamicists that continuum or molecular theories that can describe the above flows will
probably be successful for more complex flows.
2.1 Shear flows
For steady-state shear flow, with constant ,T three material functions ,)/~(lr
,)t~(lW and u/2(~t) are defined by
xyX = (2.3)
'~xx - "Cyy = -~I/l~g 2, (2.4)
zyy- x= = (2.5)
The viscosity ,11 and the first normal stress coefficient 1W are positive, and the second
normal stress coefficient utt2 is negative. The secondary flow in the disk-and-cylinder
experiment is related to W1, and the negative ~IS2 explains the convex surface in the
tilted-trough flow. The ratio w2/w1 varies between about -0.1 and -0.4 for most fluids.
