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".