Seyer and Metzner 



4. Recent papers by Bernstein, Kearsley, and Zapas (8) and by Bogue (9) 

 develop promising constitutive relationships which, though yet largely unapplied 

 to problems of interest in the context of the present paper, may well serve to 

 extend the asymptotic validity of simple fluid analyses into ranges of greater 

 engineering interest, as well as having unique regions of applicability directly. 



5. Although molecular theories have been used in several instances, their 

 ability to portray effects at finite deformation rates remains untested. Fluid 

 relaxation times are portrayed as strong functions of concentration, which ap- 

 pears unrealistic (44), except possibly at very low concentrations. In the low 

 concentration range they may work, but as no data are yet available in that 

 range, it does not appear possible to draw useful conclusions at this time. 



In summary Eq. (2) represents a valid approximation to the behavior of 

 simple fluids which also portrays correctly all effects noted experimentally un- 

 der the asymptotic steady flow conditions under which it applies. As such it 

 represents the logical choice for analyses of steady flows. The same is not 

 true, however, of the "second-order" approximation to Eq. (2). 



In the case of unsteady flows Eq. (1) appears to be of better than first- 

 approximation utility and will be employed herein. This is the area of primary 

 interest in studies of turbulence, of course, but definitive comments concerning 

 the choice of applicable equations do not yet appear possible. As Eq. (1) also 

 portrays the major effects of interest in steady flows it appears to be of. rather 

 general utility. 



VISCOELASTIC PARAMETERS 



As a further guide in formulating an understanding to problems of interest, 

 and to aid in choosing which constitutive approximation is most useful, it is of 

 interest to formalize our intuitive ideas of what a viscoelastic material is. 

 Generally a viscoelastic material may be viewed as a material possessing 

 properties that are partly those of a purely viscous material and partly those of 

 an elastic solid. Whether either or both of these characteristics are empha- 

 sized depends not only on the material but also on the flow process: if the de- 

 formation is an extremely sudden one, it is intuitively reasonable that the time 

 scale of the process may be such that the viscous modes of the material re- 

 sponse do not have time to be operative. Thus, for ordering purposes dimen- 

 sionless groups which consider the deformational processes in question (i.e., 

 the velocity field) as well as the fluid properties, will evidently be necessary. 



Considering the fluid properties first, application of the pi theorem to any 

 flow process in which Eqs. (1) or (2) are used in place of the Newtonian expres- 

 sion* to describe the fluid properties requires the inclusion of additional dimen- 

 sionless groups in view of the additional variables involved. Thus, if all the 



*If Newtonian fluids are considered, then Eq. (1) with 0=0 and /j. = constant 

 will describe the fluid properties. Alternately, Eq. (2) with /j. = constant and 

 Wj = oJj = may be used. 



22 



