Hydrodynamic Aspects of Macromolecular Solutions 



for it turns out that the characteristic angular dispersion time for a molecular 

 mode is about equal to its relaxation time. This fact is of great importance in 

 determining the strain for small shears. For supercritical shears, the elon- 

 gated molecule is convected rapidly through the high shear region, and spends 

 most of its lifetime within an angle to the horizontal which is of the order of 

 l/(Tj 9u/By), as we earlier indicated in Fig. 2, top right side. Under super- 

 critical conditions we find that the strain energy stored there is at first propor- 

 tional to (Tjau/By)^ and for higher shears to (r^ Bu/By)^/ ^ The change in 

 stored energy is particularly rapid at the onset of supercritical conditions, 

 where we estimate it increases by an order of magnitude as (Tj3u/3y) increases 

 from 2 to 4. 



The kind of considerations and conclusions sketchily outlined here can lead 

 to an improved understanding and prediction of rheological effects. For exam- 

 ple, we are led to conclude that the unfolding and shearing over of molecules 

 under supercritical shear conditions is almost certainly responsible for the so- 

 called pseudoplastic behavior of polymer solutions in steady shearing. Beyond 

 that, it suggests that the flow of a macromolecular solution be considered in 

 terms of two separate but coupled flows of discrete fluids: the solvent and the 

 imbedded molecules. A theory involving dual dynamic equations requiring si- 

 multaneous solution would result. 



TURBULENT FLOWS 



Strain energy storage and release will effect turbulent flows. The magni- 

 tudes of elastic shear stresses associated with energy storage due to unsteady 

 straining in a turbulent flow are easily estimated: 



elastic stresses ^ . /*"s^ 

 2 strain 



Reynolds stresses 



normal (second-order) stresses strain 

 elastic stresses 2 



In ratio to the inertial or Reynolds stresses, we see that the elastic 

 stresses depend upon the ratio of c^, the speed of propagation of elastic shear 

 waves, to the turbulent velocity v^ , where I is the characteristic eddy size. 



The effective strain in a turbulent flow may be estimated as the product of 

 the strain rate and the strain lifetime, where strain rate ~ v^/l and strain life- 

 time = -t/v^, so that strain = 0(1). 



In accordance with this rough order of magnitude estimate, the normalized 

 elastic stresses (both shear and normal) are of the order of (c^/v^)^. 



The strain and therefore strain energy stored in elongated molecules sub- 

 jected to a given imposed flow deformation is greater than for relazed mole- 

 cules. As a result, under certain conditions the bulk stiffness associated with 

 the strain of molecules subjected to supercritical shears on a small scale can 



11 



