First normal stress coefficient

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

We can also calculate other viscoelastic properties in the limit of low shear rate (linear viscoelastic limit) near the LST. The above simple spectrum can be integrated to obtain the zero shear viscosity 0, the first normal stress coefficient if/1 at vanishing shear rate, and the equilibrium compliance J... 

Indicating that the convected Jeffreys model gives a constant viscosity and first normal stress coefficient, while the second normal stress coefficient is zero. 

The coefficients used to fit the data are summarized in Table 2.11 [43], The viscosity and first normal stress coefficient data presented in Figs. 2.30 and 2.31 where fitted with the Wagner form of the K-BKZ equation [41],... 

Fig. E3.5 Steady-state shear viscosity rj and first normal stress coefficient i, obtained from dynamic measurements versus shear rate for a low-density polyethylene melt, melt I. [H. M. Laun, Rheol. Ada, 17, 1 (1978).]...

Here are the components of the stress tensor as defined in rheology Tn—T22 is the first normal stress difference and T21 the shear stress, equal to Nt and rxsh, respectively. Hence, from dynamic mechanical measurements it is possible to determine the zero shear first normal stress coefficient Fq0 and zero shear viscosity y0. 

NON-NEWTONIAN SHEAR VISCOSITY AND FIRST NORMAL STRESS COEFFICIENT OF POLYMER MELTS... 

These equations show that both the shear stress and the first normal stress difference gradually increase from zero to a steady state value for —>oo. This results in values for the viscosity and the first normal stress coefficient equal to... 

An important conclusion is that it is clear that Lodge s constitutive equation is not able to describe non-Newtonian behaviour in steady shear, because both the viscosity and the first normal stress coefficient appear to be no functions of the shear rate. 

In Sect. 15.4 it was shown how the shear thinning behaviour of the viscosity could be described empirically with the aid of many suggestions found in literature. It was not mentioned there that the first normal stress coefficient also shows shear thinning behaviour. In this Sect. 15.5 it became clear that also the extensional viscosity is not a constant, but depending on the strain rate upon increasing the strain rate qe the extensional viscosity depart from the Trouton behaviour and increases (called strain hardening) to a maximum value, followed by a decrease to values below the zero extensional viscosity. It has to be emphasised that results in literature may show different behaviour for the extensional behaviour, but in many cases this is due to the limited extensions used,... 

FIG. 15.46 Viscosity, 77, and first normal stress difference, Nh of Vectra 900 at 310 °C as functions of shear rate, according to Langelaan and Gotsis (1996). The first normal stress coefficient, Yi, is estimated from N, by the present author. ( ) Capillary rheometer ( ) and ( ) cone and plate rheometer ( ) complex viscosity rj (A) non-steady state values of the cone and plate rheometer. Courtesy Society of Rheology. 

It is amazing that in many cases the viscosity and the first normal stress differences are reported together it would be wise to compare the viscosity and the first normal stress coefficient, Th, because these both are material properties. For that reason in Fig. 15.46, the present author also incorporated Ll. It also seems to reach a Region II. 

The first normal stress coefficient at low shear rates, vPio, and the elastic shear compliance, /g, are related by Eq. (15.78) which reads... 

FIG. 16.19 First normal stress coefficient T, vs. shear rate q on double logarithmic scales for a 20% solution of PIB in Primol D at 25 °C. Data from Huppler et al. (1967) and reproduced from Bird et al. (1987). 

Fiber-reinforced polymer systems, 38 Fickian diffusion, 665 Fick s law, 663,684 Field flow fractionation, 20 Filled polymers, 38 First normal stress coefficient, 545 difference, 640 First-order transition, 27,152 Flame-retardant additives, 861 Flammability, 847 Flashing, 804 Flash line region, 807 Flexibility of a chain molecule, 246 Flexible polymer molecules, 706 Flexural deformation under constant load, 825 Flexural formulas, 826 Flexural rigidity, 877 Floor temperature, 751 Flory-Huggins... 

Figures 7 and 8 show the predictions of the Wagner model compared to experimental data for transient shear viscosity and first normal stress coefficient of LD. These have been calculated according to ...

However, it is now worth pointing out that the difference in ax and aN imply that the initial model should be replaced in shear by a pair of two independent correlations for shear stress (eq. 50a and 50c) or for first normal stress coefficient (eq. 50b and 50d). But at this point some questions arise concerning the choice of the proper value (bt or un) to be used in any other flow situation. Though it is possible to imagine equation (49) Including some variation of a with flow history or invariants, it could hardly be different in two equations for the same flow kinematics. 

Figure 1-1 First Normal Stress Coefficient Data of Starch Dispersions with Different Concentrations as a Function of Shear Rate (Genovese and Rao, 2003). Abbreviations cwm, cross-linked waxy maize tap, tapioca gran, granule.

Similarly, a power type empirical equation was found for G /o/ ca) and the first normal stress coefficient, (y) ... 

Relationship Between Viscosity and First Normal Stress Coefficient... 

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...

Integral of time-temperature history, t -First normal stress coefficient. Pa s Second normal stress coefficient. Pa s Dynamic frequency, rad s ... 

---