Turbulence in Viscoelastic Fluids 



physical property parameters are taken as constants, the one additional variable 

 appearing in Eq. (1) (o) requires one additional dimensionless group, and the 

 two additional variables of Eq. (2) (w, and w^) require two. Physically these 

 are necessary to consider the additional forces arising in deformation of visco- 

 elastic materials. They have been considered in some detail in laminar bound- 

 ary layer analyses (39,61), where it is shown that they may be expressed as 



^1 V 

 Nu,„ = — - 



'Ws 



fl D 



elastic forces 

 viscous forces 



(3a) 



and 



Nvr = — 



= ratio of several elastic forces (3b) 



if Eq. (2) is employed." Correspondingly if Eq. (1) were used, the Weissenberg 

 number N^j,^ may be expressed as 



N - ^ (3c) 



and represents the only additional dimensionless group. 



If the physical property terms co^ and co^, or 6, are chosen as variables, 

 then correspondingly additional dimensionless groups as required to consider 

 the parameters which govern the dependency of these variables on the deforma- 

 tion rates must be introduced. 



In the case of laminar boundary layer in two-dimensional flows it may be 

 shown that the viscoelastic ratio term of Eq. (3b) does not appear and that while 

 the Weissenberg number (Eq. (3a) or (3c)) does, its influence on the flow field 

 will frequently be too small to be measured experimentally unless the ratio of 

 the physical property coefficients co^/ji or is greater than found experimen- 

 tally for most real fluids (61). This conclusion arises because of the high and 

 predominating level of the inertial effects, a factor which effectively introduces 

 consideration of the velocity field as well as the fluid properties. Conversely in 

 low Reynolds number flows the elastic effects are likely to exert greater influ- 

 ences and must be considered. This is brought out clearly in the case of slow 

 flows about rotating spheres as studied by Giesekus (26) and Walters and Savins 

 (59), for example. 



Considering now those more sudden deformations in which the effects of 

 fluid elasticity may intuitively be expected to be more dramatic than in rapid 



These same dimensionless groups were also employed by Elata and Poreh(19). 



23 



