Seyer and Metzner 



8, SEC 



Fig. 2 - Rheological properties of 

 viscoelastic fluids; relaxation 

 time measurements on aqueous 

 solutions. The data are from 

 Seyer and Metzner (51). 



applicable at these concentration levels also. Finally it is clear that while the 

 second-order approximation to simple fluid behavior (Eq. (2) with all parame- 

 ters taken as constants) may be a valid approximation at the lowest shear rate 

 levels studied, at which the shearing stress appears to be linearly dependent on 

 shear rate and the normal stress term quadratically, such an approximation is 

 clearly not very useful at shear rates beyond 1 to 10 sec"^. The shear rate 

 levels of turbulent fields are generally in the range 10^ to 10^ sec"^, and herein 

 the normal stresses depend nearly linearly upon shear rate.* 



Figure 3 depicts typical drag coefficient vs Reynolds number relationships; 

 these are for the polymeric solutions having the time constants depicted in Fig. 

 2. Also shown in Fig. 3 are the turbulent drag coefficients curves for purely 

 viscous fluids (0.4 < n' < 1.0) having flow behavior indices in the range of 

 interest, t 



In contrast to the turbulent behavior shown by purely viscous fluids the ex- 

 perimental drag coefficients shown are all very low and show a pronounced di- 

 ameter effect. This latter is expected, since at a given Reynolds number the 

 fluid velocities are greatest in the smallest tubes. This means that the shear 

 rates are greatest here; hence any effects of either the Weissenberg or Deborah 



*Less directly but possibly more sensitively, Pruitt and Crawford (46) have 

 shown that in laminar ducted flow fields entry region effects are large in dilute 

 solutions and nnay be used to infer fluid properties. Analysis of their results 

 (5) yields elastic effects, for several dilute (250 ppm) polymeric solutions, 

 which also appear to be nearly linearly dependent on the shear rate of the fluid. 



'Obviously purely viscous fluids with shear-dependent viscosities (n' ^ 1,0) do 

 not behave in the manner indicated by drag reducing materials. This has been 

 clear for nearly a decade (17), but the contrary illusion still reappears with 

 disturbing frequency in the fluid mechanics literature. 



26 



