The correlation must be considered only provisional, however, because it does not simplify to the limiting case of extremely dilute solutions. Plausible extensions based on models of steady-flow non-Newtonian viscosity behavior, and other possible correlation schemes utilizing viscoelastic fluid properties, are briefly discussed.Three kinds of mechanical behavior of polymers and polymer solutions have been studied experimentally: the shear-dependent (non-Newtonian ) viscosity 7); the normal stress (Weissenberg) effect, described by the normal stress coefficient i; and the transient response to small amplitude displacements which can be described by the complex viscosity t*. Recently it has been proposed (24, 25, 26) that the transient response of the normal stresses should also receive experimental attention.One of the primary aims of rheology is to formulate constitutive equations capable of describing all of the above phenomena and their interrelations. To this end some very general and elegant equations have been proposed. Despite their all-inclusiveness they may not necessarily offer the most judicious description of systems encountered in engineering applications. For many applications an adequate description of a viscoelastic fluid is provided by the generalized Newtonian model ( 5 , 18)in which q is a function of the second invariant of the viscous momentum flux tensor T or the rate-of-deformationThe Ton-Newtonian viscosity 7 appearing in Equation (1) can usually be curve fitted with a function containing a small number of constant parameters. One then states that the mechanical behavior of the fluid is characterized by specifying values of these parameters. The use of Equation (1) with a curve-fit function for 71 has been studied rather extensively at this and other laboratories for several fluids and flow systems; the following facts seem to emerge regarding the above method:1. The method seems to be adequate for describing steady state laminar flow in a variety of systems; that is when the model parameters are determined in one geome- try, the same parameters are able to describe the flow of the same fluid in a different geometry (17, 10, 19, 20, 23).2. The method seems to be adequate for describing unsteady state laminar flow provided that the dominant time constant of the system is larger than the dominant time constant for the fluid ( 2 ) .3. The method provides a small number of parameters which can be used for the preparation of dimensionless correlations for steady flow in geometrically complex systems (20, 21, 22).4. The parameters in the non-Newtonian viscosity function may be useful for correlating some turbulent flow phenomena (12).5. The method is probably quite adequate for solving steady state heat transfer problems (1 ) .6. The generalized Newtonian model is easily adapted to the use of variational methods for obtaining analytical solutions to complex flow problems (13, 3, 9, 2 7 ) .Hence, although the generalized Newtonian models cannot possibly describe normal stresses or small amplitude...

A correlation of turbulent tube‐flow friction factors for non‐Newtonian polymer solutions, based upon the fluid property τ1/2 defined in Part I, has been found to represent data taken on seven solutions of Natrosol hydroxyethylcellulose in ½‐ and 1‐in, I.D. smooth tubes, with an average accuracy of 10.8%. Also, for these seven solutions, this correlation gives somewhat more accurate predictions of the point of transition into turbulent flow than are made by the Ryan‐Johnson stability theory. The correlation must be considered only provisional, however, because it does not simplify to the limiting case of extremely dilute solutions. Plausible extensions based on models of steady‐flow non‐Newtonian viscosity behavior, and other possible correlation schemes utilizing viscoelastic fluid properties, are briefly discussed.

In recent years there have been quite a few experimental studies on turbulent flow in annuli. In this paper a Prandtl mixing‐length approach is applied to give a friction factor vs. Reynolds number expression for annuli [see Equation (22) and Table 1]; this expression describes tube flow and slit flow as special cases. No new adjustable constants appear in the final result other than those determined earlier for tube flow. The final expression is found to predict friction factors within the accuracy of the existing experimental data. The mixing‐length friction‐factor expression is thus substantially more accurate than the usual hydraulic‐radius procedure and of comparable accuracy to other recent annulus friction‐factor treatments.