Seyer and Metzner 



400 800 1200 1600 2000 2400 



Noeb 



Fig. 4 - Dependency of normalized drag 

 reduction upon Reynolds and Deborah 

 numbers (51 ) 



real fluids, can only be used if the Deborah number of the fluid elements to 

 which it is applied (Eq. (5)) is small. This restricts analyses based on Eq. (2) 

 to the small-wavenumber end of the spectrum — in view of Fig. 2 and the com- 

 parable Oliver data to wavenumbers below, say, 10^. 



Two such analyses based on Eq. (2), and both correctly either noting or im- 

 plying the above restrictions, have been reported (19,54), that of Singh being 

 more detailed.* Singh finds what appear to be very small terms leading to de- 

 creased energy levels in this small-wavenumber portion of the spectrum. Such 

 a conclusion is in agreement with the intuitive notion expressed by Eqs. (4) and 

 (5) that major elastic effects are to be expected only in the high-wavenumber 

 region. Elata and Poreh (19) argue that at high Reynolds numbers the terms 

 arising from the Weissenberg and viscoelastic ratio terms (Eqs. (3a) and (3b)) 

 may be large and override in importance the linear viscous terms, but in fact 

 this result stems solely from the evident discrepancy between the quadratic 

 predictions of Eq. (2) when the coefficients of these terms are taken as constants 

 and the actual fluid behavior noted in Fig. 1 and by Oliver, Pruitt and Crawford. 

 K, as suggested by these physical property measurements, the elastic stresses 

 vary linearly with deformation rate, then both the viscous and elastic terms in 

 the Elata-Poreh analysis decrease in magnitude in the same way with increas- 

 ing Reynolds numbers and the effects of elasticity would appear to be small 

 ones, in this sense in agreement with the small effects noted by Singh at these 

 wavenumbers. In this case the results also show no dependence on diameter, 



'The connplete inapplicability of Eq. (2) under flow conditions described by a 

 large Deborah number is perhaps most clearly emphasized in the Coleman- 

 Duffin-Mizel paradox and its resolution (14,40). 



28 



