Normal stress coefficient primary

We note that the primary normal stress coefficient P 1 is positive, whereas the secondary normal stress coefficient P2 is negative, but with a lot of scatter in the data. It is difficult to measure (r22 — T33) and its value is in doubt, but the ratio — (tn — X22)/ x22 — T33) appears to be about 0.1. 

Repulsive interparticle forces cause the suspension to manifest non-Newtonian behavior. Detailed calculations reveal that the primary normal stress coefficient [cf. Eq. (8.7)] decreases like y 1. In contrast, the suspension viscosity displays shear-thickening behavior. This feature is again attributed to the enhanced formation of clusters at higher shear rates. 

The slip parameter can be easily determined from various experiments in shear situations by some fit of the steady state shear viscosity and primary normal stress coefficient. Analytic expressions are easily derived in steady state and transient flows in the form ... 

Figure 7.4 Variation of average relative primary normal stress coefficient as a function of the reduced volume fraction of filler. (Reprinted from Ref. 71 with kind permission from John Wiley Sons. Inc., New York, USA)...

Using a filler size distribution, the normal stress difference of the filled polymer system can be altered. The relative primary normal stress coefficient is reduced when a bimodal distribution is used and gives the lowest values at about 10-30% volume fraction of the small particles. 

The normal stresses c,/ cannot be specified on an absolute basis because of the arbitrary hydrostatic pressure in equation 59, but their differences are predicted by continuum mechanics and several molecular theories and can be measured. The primary and secondary normal stress differences are defined as (Th — an and 022 — an respectively. Data are often expressed in terms of the primary normal stress coefficient... 

At low values of 7, the coefficients 1 and 2 approach the zero-shear-rate limits of j,o and 2,0 respectively. All the single-integral equations mentioned above predict for the limiting primary normal stress coefficient ... 

The primary normal stress coefficient during steady-state shear flow is also related to the relaxation of shear stress after cessation of steady-state flow, as shown by combining equations 67 and 74 ... 

The primary normal stress coefficient at low shear rates according to equation 74 of Chapter 3 is given by i,o = 2i . At high shear rates, relations between SJ i(7) and t] y) are provided by some phenomenological theories " from a qualitative molecular standpoint, it may be inferred that the decrease in with increasing y is related to the drop in rj and hence, in the framework of the entanglement model, to a decrease in the steady-state concentration of entanglements. 

The primary normal stress coefficient at low shear rates, gives the same information as 7°, as specified by equation 74 of Chapter 3 ... 

FIG. 17-28. Plot of primary normal stress coefficient against shear rate with logarithmic scales for 20% solution of polyisobutylene in Primol D oil at 25 C. (Data of Huppler, Ashare, and Holmes. ) Reproduced, by permission, from Dynamics of Polymeric Liquids, by R. B. Bird, R. C. Armstrong, and O. Hassager, John Wiley and Sons, New York, 1977. 

Fig. 7.15. PE under steady state shear flow at 150 °C Strain rate dependencies of the viscosity ry, the primary normal stress coefficient and the recoverable shear strain 7e. The dotted line represents Eq. (7.122). Results obtained by Laun [76]...

The relationship tells us that, in the limit of low shear rates, we still have only two parameters which are independent. The primary normal stress coefficient does not represent an independent property, but is deducable from the two parameters controlling the linear response, and r/o-... 

We thus obtain, for the steady state viscosity and the steady state primary normal stress coefficient, the expressions... 

From Eq. 83 we observe that the viscometric functions are insensitive to the direction of shear and that the primary normal stress coefficient is zero. Hence this is not a realistic model for most shear sensitive fluids. Eq. 82, with i = fi = constant and / 2 = 0 is the Newtonian fluid. If we keep ii shear rate dependent and set /t,2 = 0, we then have the GNF. Several special cases of the GNF are discussed below. 

The elasticity of polymer melts is manifested through two material functknis, namely, the primary normal stress coefficient ii and the secondary normal stress coefficient ijia. The secondary normal stress coefficient is not as well duuacter-ized as the primary normal stress coefficient due to its small magnitude. The primary normal stress measurements are themselves difficult and require highly... 

Abdel-Khalik et al. [90] and Bird et al. [91] presented a relation between the steady-state values of the primary normal stress coefficient i >i and the viscosity function q as... 

In the following, Eq. (2.47) is used for obtaining the relationship between primary normal stress coefficient and the shear rate. One of the models described earlier, namely, the Carreau model, is used for the viscoaty function. Thus, from Eqs. (2.43) and (2.47), the primary normal stress coefficient i ii can be readily obtained as... 

Steady state shear viscosity and primary normal stress coefficient for low density polyethylene melt T and from the Kaye-Bernstein, Kearsley, Zapas (K-BKZ) equation wim the double exponential damping function, eq4.4.13 (solid lines) and with the single exponential, eq4.4.12 (dotted Une). Data at different temperatures have been shifted to one master curve by ar T). Replotted from Laun (1978). 

The third curve in the figure, denoted 7e, gives the amount of shear strain recovery subsequent to a removal of the external torque. There is an interesting observation. The linear increase of 7e with 7 at low shear rates is exactly determined by the zero shear rate values of the viscosity and the primary normal stress coefficient, 770 and This is revealed by the coincidence of the limiting curve 70(7 —0) with the dotted line representing the linear function... 

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