Weissenberg friction

The asymptotic nature of the friction factor is clearly brought out in Fig. 10.24, which shows the measured fully established friction factors taken in three tubes of differing diameters as a function of the Weissenberg number based on the Powell-Eyring relaxation time for fixed values of the Reynolds number for aqueous solutions of polyacrylamide. The critical Weissenberg number for friction (Ws), is seen to be on the order of 5 to 10. When the Weissenberg number exceeds 10, it is clear that the fully developed friction factor is a function only of the Reynolds number. 

FIGURE 10.24 Fully established friction factors for aqueous polyacrylamide solutions in turbulent pipe flow as a function of the Weissenberg and Reynolds numbers. 

It is recommended that either Eq. 10.71 or Eq. 10.72 be used to predict the fully developed friction factor (that is, for xld greater than 100) of viscoelastic aqueous polymer solutions in turbulent pipe flow for Reynolds numbers greater than 6000 and for Weissenberg numbers above critical value. The critical Weissenberg number for aqueous polyacrylamide solutions based on the Powell-Eyring relaxation time is on the order of 5 to 10 [50]. In the absence of experimental data for other polymers, this value should be used for other viscoelastic fluids with the appropriate caution. 

The Fanning friction factor/and the dimensionless heat transfer coefficient jH for the polyacrylamide 1000-wppm solution measured at an xId of 430 and at the Reynolds number equal to 20,000 [37] are presented in Fig. 10.33 as a function of hours of circulation. The dimensionless jH factor is seen to remain relatively constant at its minimum asymptotic value until some 3 hours have passed. On the other hand, the friction factor does not depart from its asymptotic value until some 30 hours of circulation have occurred. Estimates of the critical Weissenberg number based on the Powell-Eyring model yield values that are in good agreement with those given for the once-through system ... 

Above the corresponding critical Weissenberg number, the friction factor and the heat transfer remain at their asymptotic values. 

The fully established friction factor for turbulent flow of a viscoelastic fluid in a rectangular channel is dependent on the aspect ratio, the Reynolds number, and the Weissenberg number. As in the case of the circular tube, at small values of Ws, the friction factor decreases from the newtonian value. It continues to decrease with increasing values of Ws, ultimately reaching a lower asymptotic limit. This limiting friction factor may be calculated from the following equation ... 

However, the first DNS based on a microscopically originated constitutive equation for the polymer dynamics (the finite extensibility nonlinear elastic with the Peterlin approximation (FENE-P) model [46]) was conducted by Sureshkumar et al. [47]. In this work, for a fixed friction Reynolds number and other rheological parameters, drag reduction was observed as the Weissenberg number increased beyond a critical onset value. Moreover, accompanying drag reduction, characteristic changes were observed in the velocity and vorticity mean and rms values, the Reynolds stress, and... 

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