First normal stress coefficient from viscosity data

In addition to relationships between apparent viscosity and dynamic or complex viscosity, those between first normal stress coefficient versus dynamic viscosity or apparent viscosity are also of interest to predict one from another for food processing or product development applications. Such relationships were derived for the quasilinear co-rotational Goddard-Miller model (Abdel-Khalik et al., 1974 Bird et al., 1974, 1977). It should be noted that a first normal stress coefficient in a flow field, V i(y), and another in an oscillatory field, fri(ct>), can be determined. Further, as discussed below, (y) can be estimated from steady shear and dynamic rheological data. 

Figure 3-48 Double Logarithmic Plot of First Normal Stress Coefficient, versus Shear Rate from Down Ramp Curve of 5% Cross-Linked Waxy Maize Starch Dispersions Exhibited Quasi-Linear Plots. Predictions of Bird et al. (1974, 1977) based on apparent viscosity data for values of the empirical ctor...

Figure 3.15 The frequency-dependent in-phase and out-of-phase components of the dynamic viscosity, rj and rj in small-amplitude oscillatory shear, along with the shear-rate dependence of the first normal stress coefficient hi (y) for a 0.05 wt% solution of polystyrene of molecular weight 2.25 X 10 in a solvent of oligomeric styrene. The lines through the data show the predictions of the Zimm theory for r and 2r)"f(o and the Zimm theory for hi(y) modified to account for finite extensibility, as discussed in Section 3.6.2.2.I. The dashed lines are the contributions of the individual Zimm relaxation modes to 2rj"((o) /

Rg. 9.10 Theoretical curves for the steady shear viscosity and the first normal stress coefficient fitted to the experimental data (symbols) on HEUR in literature [37]. (Reprinted with permission from Ref. [36].)... 

Vfith expressions 4.4.12 ot 4.4.13 for h y) with G(f) obtained from a sum of exponentials with parameters X,- and Gt (given in Table 3.3.1 for melt I), predictions can now be made for various shearing flows in the nonlinear regime. Hgute 4.2.2 shows that the [ edicted steady state shear viscosity t and first normal stress coefficient agree well with the experimental data. 

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