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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 . . . .

Viscometric Flows of Non-Newtonian Fluids: Theory and Experiment

I. Introduction.- 1. Limitations of the Classical Theory of Navier and Stokes.- 2. Incompressible Simple Fluids.- 3. Plan and Scope of this Monograph.- II. Theory of Incompressible Simple Fluids.- 4. Kinematics.- 5. The Dynamical Equations.- 6. The Principle of Material Objectivity.- 7. The Definition of an Incompressible Simple Fluid.- 8. Static Behavior of Simple Fluids.- III. General Theory of Viscometric Flows.- 9. The Kinematics of Simple Shearing Flow.- 10. The Viscometric Functions.- 11. The Dynamics of Simple Shearing Flow; Viscosity.- 12. The Definition of a Viscometric Flow.- 13. Curvilineal Flows.- i. Kinematical Description.- ii. Calculation of the Stress Tensor.- IV. Special Viscometric Flows.- 14. Flow through a Channel.- 15. General Properties of Helical Flows.- 16. Flows between Concentric Cylinders.- 17. Couette Flow.- 18. Flow between Stationary Concentric Cylinders.- 19. Poiseuille Flow.- 20. Normal Stress Effects at Free Boundaries.- i. Climbing in Couette Flow.- ii. Swelling in Poiseuille Flow.- 21. Cone and Plate Flow.- 22. Torsional Flow.- V. Experimental Methods and Results.- 23. General Considerations.- i. The Adherence Condition.- ii. Heat Generation.- iii. The Sample.- 24. Simple Shearing Flow.- 25. Couette Flow.- i. The Velocity Field.- ii. Viscosity.- iii. Normal Stresses.- iv. Climbing.- 26. Other Flows between Coaxial Cylinders.- 27. Poiseuille Flow.- i. The Velocity Field.- ii. Viscosity.- iii. Swelling.- 28. Cone and Plate Flow.- i. The Velocity Field.- ii. Viscosity.- iii. Normal Stresses.- 29. Torsional Flow.- i. Viscosity.- ii. Normal Stresses.- VI. Historical Remarks.- 30. History of the Development of the Theory.- 31. History of Experiments.- A. Appendix on Mathematical Concepts.- A 1. Vectors.- A 2. Bases, Linear Forms.- A 3. Points, Euclidean Space.- A 4. Tensors.- A 5. Multiplication of Tensors.- A 6. Transposition; Symmetric, Skew, and Orthogonal Tensors.- A 7. Traces and Determinants.- A 8. Point, Vector, and Tensor Functions.- A 9. Deformations, Gradients.- A 10. Coordinates.- A 11. Special Coordinate Systems.- References.