Poreh and Dtmant 
A : tr 
shear approximation, namely 7 =7,, or 7 = 1. Denoting the veloci- 
ty obtained in this manner by uy, , one can write that 
ao 
a é (17) 
a 
\ | + + 
ty Jeep A snp Ale ty =f - exp(-y /a*)]? 
Integration of Eq. (17) gives for large values of oH the log law 
ussaoiet iin yt+ B (Eq.2) where the value of B is a function of 
At . Very close to the wall, where exp(-y7/At) ~ 1, the solution 
of -Eqy(l7)ors wie S y* (Eq. 1). A comparison with measurements in 
Newtonian fluids (van Driest 1958) shows that the velocity profile 
obtained from Eq.(17) is in good agreement with measurements in 
the sublayer, buffer zone and the log region in zero pressure gra- 
dient boundary layers and pipe flows. A deviation of the data from 
the log law is observed in the outer region of the flows. 
We have already seen that the effect of drag reducing addi- 
tives is to change the value of B in the log law. It is therefore na- 
tural to examine the possibility of describing the velocity distribu- 
tion in such flows by the integral of Eq.(17) with values of At 
larger than 26. We have also seen that the contribution of the velo- 
cities in the viscous sublayer and the buffer zone to the calculation 
of the friction or drag coefficient in the polymeric regime is small. 
Thus the proposed model would be useful only if it can describe the 
velocity distribution near and in the maximum drag reduction regime. 
Now, the maximum drag reduction regime corresponds to large va- 
lues of At and small values of R*, andone sees from Eq.(17) that its 
asymptotic solution for small values of R‘/At is given by ut = ae 
Since we do not expect the velocity at any point in the pipe to exceed 
the velocity given by the parabolic distribution in a laminar flow, 
ut = yta - yt/ar’), (18) 
one has to disqualify this solution. The reason for the failure of this so- 
lution is of course theassumption fT = T\,, which is valid only close to 
the wall. We shall show later that although the error introduced by 
this assumption in Newtonian flows is small, it is large for small 
values of Ban . In view of this difficulty, we shall modify van 
Driest's solution by taking into account the variation of the shear 
stress in the pipe as well as the different character of the flow near 
the center of the pipe. 
The proposed model for drag reducing flows in pipes assu- 
mes that the velocity distribution is composed of two parts 
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