Turbulence in Viscoelastic Fluids 



to possess a spectrum of relaxation times in the limit of infinitesimal deforma- 

 tions, not just a single value (24). The implications of this fact have been con- 

 sidered lucidly by Slattery (55). The second exception is that in steady laminar 

 shearing flows one of the two normal stress functions is incorrectly predicted 

 to be zero. It is not zero (27,28,30,35), but in fact appears to be only a small 

 fraction of the other stresses; hence it is properly negligible in many problems. 



2. The simple fluid formulations of Coleman and Noll (10-12,15) possess 

 both a conceptual and mathematical elegance not found in empirical equations 

 such as Eq. (1). Unfortunately, this is lost almost immediately as one applies 

 them to problems of interest, as only simple approximations to the behavior of 

 these fluids have been worked out in detail. In the important case of steady 

 laminar shearing flows the Rivlin-Ericksen approximations are applicable and 

 give (10,11,15,39): 



T^-pI + Zld+aijd^ + aJjdj, (2) 



in which the three material property functions Jl, Wj, and co^ are all arbitrary 

 functions of the invariants of d and d^. It is possible to show that in the limit- 

 ing case of very low deformation rates these functions become constants (15,39). 

 The resulting "second-order-fluid" approximation, being conceptually attractive 

 and mathematically simple has been extensively employed (15,19,26,54,59,61). 

 However, if it is used, a good approximation to the behavior of real materials is 

 obtained only at such small deformation rates as to make even the experimental 

 determination of the coefficients very difficult (28,35). For the slow flows of 

 interest to a number of workers (26,59) it thus represents the logical choice; 

 for the rapid flows of interest in turbulent fields reference to experimental re- 

 sults (28,35,51,53)* shows that the second-order approximation is clearly in- 

 valid. However, Eq. (2) with the coefficients m^ and ^^ chosen as variables so 

 as to portray correctly the approximate linearity of the stresses, would appear 

 to be a sound choice, provided the flows are nearly steady. 



In the case of unsteady flows integral expansions representing first- and 

 second-order approximations to the behavior of simple fluids have been devel- 

 oped and applied to several engineering problems (2,39,40,60). The complexity 

 of these suggests that full approximations, not limited to specific asymptotic 

 flow conditions, are not likely to be forthcoming. However, in rapidly acceler- 

 ating velocity fields such as those encountered in turbulent studies, asymptotic 

 approximations may be of value. 



3. The theory of anisotropic fluids advanced and developed by Ericksen (21) 

 possesses a natural conceptual attractiveness not found in either of the earlier 

 approaches, as the mathematical basis appears to be the only one developed 

 which possesses a counterpart in the actual molecular structure of the fluid. 

 Unfortunately its applicability is at present restricted to flows in which the fluid 

 properties are described adequately and completely by Eq. (2); hence use of 

 such anistropic fluid theories in engineering analyses would appear to introduce 

 unnecessary analytic complications (16). 



These have recently been extended by Oliver (44) into the range of concentra- 

 tions of interest in turbulent drag reduction. 



21 



