Hydrodynamic Aspects of Macromolecular Solutions 



the upper left corner will tend to occur at the lowest flow speeds and largest 

 concentrations. In the hypercritical case it may be shown that for large flexible 

 molecules r^ oc m^^^; the stiffness therefore scales something like conc.xM'^^, 

 a result in good agreement with the results of experiments with disks rotating 

 at high speeds in polyethylene oxide (1). 



RADIATION DAMPING 



What are the precise manifestations of the bulk fluid stiffness? There are 

 no doubt many. For one, I have earlier offered the suggestion (14) that stiffness 

 provides a mode for the extraction of turbulent energy from an eddy through the 

 radiation of elastic shear waves; these, removed from the scene, later decay 

 through the usual action of viscosity. In fact, this radiation may be calculated 

 along the lines of the theory of the generation of sound by turbulence, pioneered 

 by Lighthill (15). We are dealing with an incompressible fluid. The momentum 

 balance is 



where 



,(0) 



Bt 



_d_ (0) ^_ 



(UiU.) 



_ 1 J 



P ~^ 



,( 1) 



\Bx. 



du, 



which is the stress tensor for a normal viscous fluid, including the Reynolds 

 stress (u.Uj), and where T^ M is the additional stress tensor due to the elastic 

 properties of the fluid. At its very simplest, the latter takes the form: 



,( 1) 



de. 



1 



3X: 



Be. 

 I 



BX: 



where e ^ is a suitably defined displacement field. 



Slow changes in displacement much less effectively generate elastic 

 stresses because of the reduction in stiffness with increasing frequency, and we 

 are not therefore concerned with the actual displacements, which become arbi- 

 trarily large with increasing time, but only those e ^ whose time scale is suffi- 

 ciently short. We thus make the short-time approximation 



Be. 

 with the result that 



2(1) 



Bt 



2(1) 



BX: 



Bx: 



(0) 



13 



