Seyer and Metzner 



2. Viscoelastic parameters, namely the Deborah and Weissenberg numbers, 

 which are of importance for ordering purposes and scale-up, 



3. Macroscopic (drag coefficient) results, reviewed in light of the above, 

 and tentative analyses of the turbulent spectrum. 



4. Experimental techniques for the more detailed study of viscoelastic 

 fluids. 



A large number of experimental reports in which drag coefficients have 

 been reported but in which no attempt to either estimate or measure the physi- 

 cal properties of the polymeric solutions used are available. These are of value 

 in distinguishing the systems and conditions to be used in maximizing the prag- 

 matic effects desired. As they do not, however, usually lead to any insight into 

 the mechanisms responsible for the observed results they will generally not be 

 considered in the present report. The authors wish to emphasize that this 

 omission is not intended to reflect a value judgment so much as a need for keep- 

 ing the present paper within prescribed bounds, and a recognition of the fact that 

 those directly involved in extensive test programs are more capable of survey- 

 ing those activities. 



CONTINUUM PROPERTIES OF POLYMERIC SOLUTIONS 



Five distinct mathematical approaches of some generality have been sug- 

 gested for portraying the properties of viscoelastic fluids. All incorporate the 

 intuitive notion that these systems possess a gradually fading memory for their 

 previous deformation history; in order of historical development they are as 

 follows. 



1. The "convected Maxwellian" equations developed primarily by Oldroyd 

 (42,43,62), though empirical in nature, appear to incorporate all of the real ef- 

 fects which are exhibited by materials of interest. Thus they appear to be gen- 

 erally applicable to all flow fields of interest, including the unsteady flows of 

 importance in turbulence studies. While the complexity of these equations may 

 be extended so that all real effects may be portrayed precisely, what appears to 

 be more important is that even the simplest forms of the equations portray 

 those unsteady effects studied to date as correctly as the experiments reveal 

 them (20,27,32,58,60). For purposes of later application to turbulent processes 

 the two-parameter equation* 



r' ^ e ^ = fj.d (1) 



will be used. The relaxation time ff and the viscosity m are taken as arbitrary 

 variables dependent on the invariants of d . 



Equation (1), with but two exceptions, precisely describes all effects which 

 have been observed quantitatively. One exception is that real materials appear 



*A list of symbols follows the text. 



20 



