Tulin 



be several orders of magnitude greater than if the molecules are relaxed, and 

 as a result the potency of very small concentrations may become exaggerated. 

 Of course, this does not mean to say that molecular unfolding or shear stiffen- 

 ing is necessary in order that macromolecules in solution affect turbulent flows. 

 It does mean, however, that molecular unfolding may very well be responsible 

 for the onset of those effects which begin to occur in the most dilute polymer 

 solutions studied. 



Some detailed considerations lead us to expect that this shear -stiffening 

 effect will occur in a turbulent flow when the local shears associated with the 

 dissipation scale are supercritical and when at the same time the local turbu- 

 lence Reynolds number for the energy containing scale is less than about 100. 

 The latter condition arises through the necessity that the characteristic time 

 for local reorientation of the shear pattern on the smallest (dissipation) scale 

 not be too much shorter than the duration of large scale strains due to shears 

 on the largest (energy containing) scale. These conditions begin to occur in the 

 turbulent shear flow closest to the wall itself when the rate of strain there be- 

 comes supercritical (t Bu/By > 2). The thickness of the viscous sublayer is, of 

 course, determined in this region. The effects of macromolecules may thus be 

 enhanced by unfolding in a region most sensitive from the point of view of wall 

 friction. 



In Table 1 are shown the theoretical dependencies (constants of proportion- 

 ality are omitted) of bulk fluid stiffness upon molecular concentration c, mo- 

 lecular relaxation time Tj , and molecular weight m for the two cases involving 

 the straining of relaxed and of unfolded molecules and for strain durations t, 

 both greater and less than t ^•, the case of relaxed molecules (first column) 

 corresponds to that treated earlier by Rouse (8), etc. 



Table 1 

 Dependencies of Bulk Fluid Stiffness 



*Cannot occur in turbulent flow. 



The case in the lower right corner (hypercritical) will tend to occur at the 

 highest flow speeds and lowest concentrations; the conditions corresponding to 



12 



