Turbulence in Viscoelastic Fluids 



N', 



Fig. 3 - Typical drag coefficient vs Reynolds 

 number curves for viscoelastic fluids. The 

 laminar curve and purely viscous turbulent 

 curves are shown for reference purposes (51). 



numbers (Eqs. (3a) and (5)) will, in view of Fig. 1, be most pronounced in the 

 smallest tubes. The result of a correlation of the data depicted in Fig. 3, apply- 

 ing these concepts, is given in Fig. 4. The drag reduction parameter F defined 

 as 



fpv-f. 



is normalized with respect to the differences between the purely viscous and the 

 laminar curves. Full details are available elsewhere (51); similar studies have 

 also been performed by Astarita (3) and by Rodriguez, Zakin, and Patterson (43). 



Somewhat more detailed analyses of the underlying mechanisms leading to 

 the observed reductions in the drag coefficient may be developed by means of 

 several distinct approaches. Perhaps most directly, an extension of the Reyn- 

 olds equations for the turbulent motion may be carried out using not the usual 

 Newtonian equation but rather an expression such as Eq. (1) to describe the 

 fluid properties. In fact the convected derivative term of Eq. (1) is inattractive 

 mathematically; hence the much simpler second-order expression (Eq. (2) with 

 all coefficients taken as constants) commends itself mathematically. As noted 

 earlier the only available experimental measurements on dilute solutions, those 

 of Oliver (44) and of Pruitt and Crawford (46), support the general results shown 

 in Fig. 1, viz., that the elastic stress terms increase linearly, not quadratically, 

 with deformation rate under the high deformation rates of interest. Thus, use 

 of the second-order approximation will apparently in general tend to overpredict 

 the magnitude of any real effects. Secondly, it must be noted that Eq. (2), rep- 

 resenting as it does a steady flow asymptotic approximation to the behavior of 



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