Flows wtth Drag Reduction (Veloctty and Frietton) 
FRICTION FACTORS AND RELATION TO POLYMERIC PROPERTIES 
The dependence of the friction factor f£ V2 on Refl2 asa 
function of At , has been obtained numerically and plotted in Fig. 8. 
One sees that at large values of Refl2 the variation of f V2 for constant 
values of At is described by a logarithmic law. The Newtonian case 
A+ = 26 coincides wir the line describing the equation f/@= 4.0 
log Ref 2-0. 4. Integration of the theoretical limit of Eq. (19) for small 
values of R*/At gives the laminar friction law. 
CS SP Y= (23) 
Several data points appearing in Fig. 1 of Virk (1971), near 
Ref 2 = 200(R+= 70), are quite close to Eq. (23). However, the avail- 
able data at larger values of Ref V2 indicate that the values of At 
obtained so far in dilute polymer solutions are bound by pat 1288 50: 
At the polymeric regime, as defined by Virk, an approximate 
relation between A* and the polymeric properties can be found using 
Virk's correlations, At large values of Ref , where the friction 
factor curves for different values of A+ are described by parallel 
lines, Au? is uniquely related to At. From Fig. 8 it was found that 
at this range 
+ 
Au /\f2 ~ 40 log (AT/A” + 4) - 28 (24) 
At small values of Ref 12 the relation between Au* and At depends 
on the values of Refl2 , however, if Aut(A*) is measured along 
straight lines originating at Ref l2 >1000 and having slopes which do 
not exceed the slopes recorded in actual measurements, the deviation 
from Eq. (24) is less than 5%. 
The relation between ae and the shear stress can now be 
obtained from Eq. (3). This equation is composed of two expressions ; 
for V*< VQ, and for V*>V%,. It is suggested that a better descrip- 
tion of the variation of Au+ is obtained by the single equation 
Sas fey Ay ee Le (v/v*_)*] (25) 
Equation (25) deviates from Eq. (3a) at V"> aiVes by less than 3% 
and is practically zero for V*< Mi go /2s The values of al according 
to Eq. (25) should be determined by the intersection of the straight 
13h3 
