Poreh and Dimant 
line (3a) with the Newtonian profile, which is exactly the procedure 
used by Virk. It follows from Eqs. (24) and (25) that 
4 re ney i. (26) 
a / ee ee tO eee 
where ais related to the polymer properties by Eq. (13). 
We have used Eqs. (26) and (12) to calculate the variation 
of f-!2 versus Refil2 for solutions of the polymers AP-30 and Guar 
Gum, (Estimated values of the critical shear and molecular properties 
are given by Whitistt et al (1968) and Virk (1971), table 5). The calcul- 
ated curves for the three solutions, and curves for constant values of 
At are compared with the measurements of Whitistt et al (1968) in 
Figs. 9 - 12. At small and moderate values of V*/V{, the agreement 
between the data and the theoretical calculations (solid lines) seems 
to be satisfactory even for small values of Ref 2 . The agreement is 
not surprising as it merely reflects the adequacy of Virk's correlat- 
ions and the slight improvement due to the use of the continuous 
equation (25) rather than equations (3a) and (3b). The phenomenon of 
maximum drag reduction, however, appears now in a different light. 
One sees that when VipW becomes large, the data deviates from 
Eq. (26) and seem to be correlated with curves of constant At he 
measurements in the concentrated polyox solutions and the smaller 
pipe-diameters seem to be bound by the curve At = 350, which is 
close to Virk's maximum drag reduction asymptote in the range 
Ref V2 < 1000. However, the deviation from the lines which are calcul- 
ated using Virk's polymeric regime correlations, and the approach to 
the maximum value of A* , donot occur only near the maximum drag 
reduction asymptote. It appears that for each solution, there exists 
a maximum value of At (or Aut) approximately independent of the pipe 
diameter. Only when Rt is small the curves coincide in a limited 
region with Virk's maximum drag reduction asymptote (11). This 
evidence is not manifested in Virk's model which predicts drag reduc- 
tion values of the order of 90% for very large shear rates. It is also 
interesting to note that the measurements of drag reduction with alum- 
inium distearate in an organic solvent shown in Fig. 12 (McMillan et 
al, 1971) exceed the maximum drag reduction curve and appear to 
reach values of At = 600. 
In the absence of a theoretical model for drag reduction 
mechanism there is no way at present to determine whether the asymp- 
totic value of At is determined by properties of the particular poly- 
mers used, experimental limitations, a dependence of drag reduction 
on the existence of a minimum level of turbulence necessary to deform 
the macromolecules in solution, degradation or other causes. 
1314 
