Poreh and Dtmant 
1966); a viscous sublayer where ae Sy" (1) 
and a log region where 
4 + + 
yu =A lopy + °B, + Au (2) 
n n 
* 
where A, = D4 (5. ae see eas Bs of a) ew y" = yv /» and V~ 
is the shear velocity. The term Au’ , which describes the upward shift 
of the log profile in the conventional law of the wall representation, 
was empirically related to the shear velocity and polymer characterist- 
ics by the equations 
4° * * # * 
Au -*2e log (V/V ea ip ic (3a) 
+ * * 
= (0) : 
Au Pie ae ve (3b) 
where we is the shear velocity at the onset of drag reduction anda 
is a concentration dependent parameter. 
Virk and Merril (1969) correlated measurements of the onset 
of drag reduction in ''thin'' solvents by the semi-empirical relation 
(Ref?) = ee 2. (R/R,) (4) 
where Rg is the polymer radius of gyration in dilute solutions, Qa 
non-dimensional constant characteristic to the polymer species- 
solvent combination, R is the radius of the pipe, Re the Reynolds 
number based on the mean velocity and diameter, and f Fanning's 
friction coefficient. 
Integration of a” over the area of the pipe yields an expres- 
sion for the friction coefficient f. At high Reynolds numbers and small 
to moderate values of Au+, the contribution of the sublayer to the 
integral of ut is negligible, yielding for V hiya the equation 
IE excise ineviletlede Subistecakae Enea \AN2 (5) 
n n CE 
Where a, = 4.0 and b, ~ 0.4. Plotted on Prandtl coordinates, 
rV2 versus Ref 2, Eq. (5) gives straight lines which intersect the 
Newtonian line (@= 0) at (efo). , where V™ = Vo. This result 
has been supported by numerous independent pressure-loss measure- 
ments at large Reynolds numbers for small values of Aut. The data 
deviates from Eq. (5) at large values of Aut, where Aut seems to 
reach a maximum value (Seyer and Metzner 1969, Whittist et al 1968) 
1306 
