Primary normal stress difference

We define a material function rj, commonly called the elongational or extensional viscosity, through the primary normal stress difference % — %iT, thus, for the case of F e), it is given by... 

Fig. E3.2b The viscosity r and first (primary) normal stress difference xu — t22 of LDPE evaluated using the Weissenberg rheogoniometer (cone and plate). LDPE is Tenite 800 of density 0.918 g/cm3, and M = 25, 800. [Reprinted with permission from I. Chen and D. C. Bogue, Trans. Soc. Rheol., 16, 59 (1972).]...

With the help of Eq. E3.2-11 and the relation Patm = nrr(R), we obtain, after integration of Eq. E3.2-12, the simple relation for the primary normal stress difference function... 

Figure E3.2b shows experimental data for the primary normal stress difference for LDPE.

Figure E3.2b presents the primary normal stress difference data for LDPE, and Fig. E3.2c presents the primary and secondary normal stress-difference data for a 2.5% polyacrylamide solution, again using a cone-and-plate rheometer.

Bird et al. (24) pointed out a simple method of estimating the primary normal stress difference from viscosity data. The method is approximate, originating with the Goddard-Miller (G-M) (25) constitutive equation (Eq. 3.3-8), and it predicts that... 

Torsional Flow of a CEF Fluid Two parallel disks rotate relative to each other, as shown in the following figure, (a) Show that the only nonvanishing velocity component is vg = flr(z/H), where ft is the angular velocity, (b) Derive the stress and rate of deformation tensor components and the primary and secondary normal difference functions, (c) Write the full CEF equation and the primary normal stress difference functions. 

We find the maximum pressure rise at the center of the disk to be proportional to the square of flR/H, which is the shear rate at r = R. Moreover, by comparing Eq. 6.5-18 to Eqs. 6.5-10 and 6.5-11, we find that this pressure rise is the sum of the primary and secondary normal stress-difference functions —[(tn — T22) + (J22 — T33)] at r = R, less centrifugal forces. Since lL is probably negative, it opposes pressurization hence, the source of the pressurization in the normal stress extruder is the primary normal stress difference function ffq. 

Their experimental results are shown in Fig. E6.14b, which plots dimensionless halftime versus dimensionless reciprocal force. Clearly, the Scott equations describe the experimental results given earlier as ti /nX = 1. They recommend that the choice of the parameter X be made on the basis of the Power Law parameters m and n and a similar Power Law relationship of the primary normal stress difference function 4 1 (y) = n 1 as... 

The relationship is experimental. LaNieve and Bogue (36) have related the entrance pressure losses of polymer solutions to the viscosity and primary normal stress difference coefficient. Thus, the works of Ballenger and LaNieve, taken together, seem to imply that the entrance angle (thus the size of the entrance vortices) depends on both the viscosity and the first normal stress difference coefficient. White and Kondo (38) have shown experimentally that, for LDPE and PS... 

Computed values of the primary normal stress difference of a low molecular weight polyisobutelene (PIB) melt we compared with experimentally obtained values, using bire fringence techniques, as shown on Fig. 15.8 they indicate good agreement. 

The effect of changing the longest relaxation time of the K-BKZ and the primary normal stress difference is shown in Fig. 15.9. 

Fig. 15.9 Primary normal stress differences in the calender gap calculated with the K-BKZ model for different relaxation times. [Reprinted by permission from D. Mewes, S. Luther, and K. Riest, Simultaneous Calculation of Roll Deformation and Polymer Flow in the Calendering Process, Int. Polym. Process., 17, 339-346 (2002).]...

In Equation 3.116, is rigorously defined as [(an - 022)I(S 2 > 1 is the sum of a constant term and two oscillating terms, accounted by ijr[ and y i is the strain rate amplitude. Equations 6 to 8 suggest that oscillatory shear stress data are related to oscillatory primary normal stress difference data (Ferry, 1980). Youn and Rao (2003) calculated values of (co) for starch dispersions is applicable to oscillatory shear fields. 

From Eqs. (13.23) and (13.11), the primary normal stress difference can be expressed in terms of viscoelastic parameters as... 

Hence the primary normal stress difference Ni is given by (14,39)... 

Figure 13.32 Primary normal stress difference as a function of the shear rate and molecular weight for 3% (mass) poly(oxyethylene) solutions in water and glycerine, (From Ref. 51.)...

Nonetheless there are some similarities between the results in Eqs. (10.6) and (10.7) and experimental curves. For example (a) all stresses relax more quickly as k0 increases, and (b) the ratio xyx — xyxs)+/ (tyx — xyxs) is always less than the corresponding ratio for the primary normal stress differences. 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

Fig 13. Behavior of the primary normal stress difference function for various... 

In shear flow it can be shown using certain symmetry arguments that there are three independent quantities of stress which depend only on the properites of the fluid. These are the primary normal stress difference (N.) defined as... 

In Figure 4 we have plotted values of the primary normal stress difference (N ) versus time. Here we observe that a single overshoot peak is observed in N. The peak stress occurs at strains of the order of 40 to 6o strain units which is similar to the range observed for the appearance of the second peak in the shear stress. Whereas the shear stress rises rapidly at the start up of flow, the normal stresses rise gradually. It should be pointed out that this is exactly the behavior predicted by the corotational Jeffrey s model. ... 

Fig. 9. Primary normal stress difference versus time for two shear rates for 80 mole % PHB/PET at 320°C.

Fig. 11. Primary normal stress difference at the start up of flow as predicted by Ericksen s theory. Values of are in dimensionless form. 

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