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We here attempt to give a complete but concise treatment of the
theory of steady viscometric flows of simple (non-Newtonian) fluids
and to use that theory to discuss the design and interpretation of
ex periments. We are able to present the theory with less
mathematical machinery than was used in our original papers, partly
because this Tract has more limited aims than those papers, and
partly because we employ a method, found by Noll and published here
for the first time, for dealing with visco metric flows without the
apparatus of rela tive Cauchy-Green tensors and reduced
constitutive equations. To make the theory accessible to students
not familiar with modern mathematics, we have added to our Tract an
appendix explaining some of the mathe matical concepts essential to
continuum physics. Pittsburgh, July 1965 BERNARD D. COLEMAN HERSHEL
MARKOVITZ WALTER NOLL CONTENTS I. Introduction page 1. Limitations
of the Classical Theory of Navier and Stokes. 1 5 2. Incompressible
Simple Fluids. . . . . . . . . . . . 3. Plan and Scope of this
Monograph . . . . . . . . . 7 II. Theory of Incompressible Simple
Fluids 4. Kinematics. . . . . . . . . . . . 10 5. The Dynamical
Equations . . . . . . . . . . . 12 6. The Principle of Material
Objectivity . . . . . . 14 7. The Definition of an Incompressible
Simple Fluid . 17 8. Static Behavior of Simple Fluids . . . . . . .
. 19 III. General Theory of Viscometric Flows 9. The Kinematics of
Simple Shearing Flow 21 10. The Viscometric Functions . . . . . . .
. . . 22 11. The Dynamics of Simple Shearing Flow; Viscosity 26 12.
The Definition of a Viscometric Flow 29 13. Curvilineal Flows. . .
. . . . . 30 1. Kinematical Description . . . .
by Noll, then scantly known, at the Carnegie Institute of
Technology. An invita- tional meeting on visco-elasticity in the
following April at Lancaster, Pennsyl- vania, brought Coleman and
Noll together. In those days a person went to a meet- ing so as to
learn from a few competent lectures without having to be himself
one more "invited speaker" or to listen to many multiples of ten
minutes of trivial trash. Ericksen lectured on "laminar shear
flows" of incompressible, Rivlin-Erick- sen fluids. That class of
flows contains all those for which Rivlin and others had obtained
exact solutions. Ericksen's paper, with Criminale & Filbey as
co-authors, was to appear soon in Volume 1 of the Archive. At the
meeting, Coleman and Noll found that they had similar views on
thermodynamics. The rheologists there, like those we had
encountered elsewhere, told us that classical thermodynamics was a
complete, closed, perfect science, all in Gibbs's paper, and they
laughed at us. We laughed at them, but silently, for we had read
fundamental parts of Gibbs's work, especially that on the
isothermal and isentropic theories of three-dimensional elasticity,
which, surely, the rheologists could not understand. We knew also
the basic inequality for increase of entropy asserted by Duhem
(1901) and in "The Mechanical Foundations" (1952) called "the
Clausius-Duhem inequality" (Eq. (28. 5", from which Eckart (1940)
had drawn consequences by guessing the signs oftwo terms
("Mechanical Foundations", text following Eq. (31. 1".
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