Flows wtth Drag Reduction (Veloctty and Frtctton) 
Virk's correlations describe a large volume of the data in the 
polymeric regime and in the maximum drag reduction regime. It 
should be noted, however, that the correlations proposed for the two 
regimes are not related. The equations proposed for the polymeric 
regime are unaffected by the details of the elastic sublayer, whereas 
Eqs.(9) and (11) proposed for the maximum drag reduction regime, 
are independent of the polymer properties. Thus, it appears to the 
authors that the correlations do not prove the existence of an elastic 
sublayer, which is described by a universal law and is fundamentally 
different in character from the corresponding layer in a Newtonian 
fluid. We shall show that this transitional zone in dilute polymer so- 
lutions is similar to the conventional buffer zone in a Newtonian fluid 
by deriving the entire velocity profile in the wall region for both cases 
using van Driest's mixing length model. 
f= xy [1-exp(-y'/aA’)] (14) 
letting AY be a function of the polymer-solvent properties and the shear. 
The model which gives a continuous velocity distribution can be easily 
applied to other boundary layer flows and to problems of heat transfer 
and diffusion. 
A MODEL FOR CALCULATING THE MEAN VELOCITY DISTRIBUTION 
In analogy to the damping of harmonic oscillations near a wall, 
van Driest (1958) proposed that the turbulent mixing length near a 
wall be described by Eq. (14) where At = 26 isa dimensionless uni- 
versal constant for smooth boundaries and k = 2. eye: = 0.4. There 
is some doubt whether Ay, and By are truly Reynolds number inde- 
pendent. Coles (1954) for instance, suggests that A, slightly inc- 
reases at low Reynolds numbers. Accordingly, the shear stress ina 
turbulent pipe flow, given by + = p (v+£2|du/dy | ) du/dy, can be 
described by the equation 
+ + +), + 4 on LD PPLE 4 fe 
mY } +key" |du'/ay’ | [i - exp(-y /A’) | leu /day (15) 
+ 
where7T = T io and Tp yi " . Equation (15) may also be written as 
a - 
du J 2% (16) 
+ 
d NA 
y 1 + 1 + 4k? yf - exp(-y'/A’)]*r* 
In order to find the mean velocity profile, van Driest used the constant 
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