Poreh and Dtmant 
Except for small values of BCR RV*/,) and large values of 
Aut, the contributions of both the viscous and elastic layers to the 
integral of ut are small (see table 3, Virk (1971) ). In these cases, 
the details of the sublayer are insignificant and Virk's model gives 
the same friction coefficient as the model of Meyer and Elata. Virk 
termed this case - the polymeric regime. (Note that the last term in 
Virk's friction factor relation for this regime, Eq. (12) in Virk (1971), 
is identical to Aut in Eq. (10). 
In the other extreme case where the elastic sublayer be- 
comes large and the contribution of the log region to the integral of 
+ is negligible, the friction coefficient is described by a universal 
law obtained by integration of (9), 
u 
1/£V2 = 19.0 log (Ref/2) - 32.4 (11) 
Equation (11), termed the maximum drag reduction asymptote, de- 
scribes reasonably well the maximum values of drag reduction ob- 
tained in many investigations at small values of R* . 
A very similar, but slightly more complicated 3-layer 
model, has been offered independently by Tomita (1970). 
Virk's analysis of data in the polymeric regime has yielded 
an additional contribution. He has correlated semi-empirically the 
dependence of the slope of the straight lines in Prandtl's coordinates, 
which are described by Eq. (5), to identifiable polymeric parameters. 
Defining a fractional slope increment A in Prandtl's coordinate sys- 
tem, which is proportional to @ in Eq. (2), 
” A ene 2 
A= oo S)/S8, /\f 28° (12) 
Where Sp is the slope with polymers and S, = An = 4.0 is the 
Newtonian slope, Virk showed that 
nel if 3a. = (A S/O Nee (13) 
where A is Avogadro's number 6, 02 x ii" C concentration as a 
weight fraction, M molecular weight, N number of backbone chain 
links and K a characteristic constant of the species-solvent com- 
bination. The parameter A appears as well in an expression which 
Virk derived theoretically for the turbulent strain energy of the ma- 
cromolecules, 
1308 
