Normal stresses Primary

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Fig. 28a, b. Shear rate dependence of the primary and secondary normal stress diffemeces (oN1, alN2) a Magda et al. s experimental results [148] for a liquid crsytal solution of PBLG with M, = 23.5 x 10 (N = 0.54) at 25 °C b Larson s theoretical results [154]... 

The material functions, k i and k2, are called the primary and secondary normal stress coefficients, and are also functions of the magnitude of the strain rate tensor and temperature. The first and second normal stress differences do not change in sign when the direction of the strain rate changes. This is reflected in eqns. (2.51) and (2.52). Figure 2.31 [41] presents the first normal stress difference coefficient for the low density polyethylene melt of Fig. 2.30 at a reference temperature of 150°C. 

The cone-plate rheometer. The cone-plate rheometer is often used when measuring the viscosity and the primary and secondary normal stress coefficient functions as a function of shear rate and temperature. The geometry of a cone-plate rheometer is shown in Fig. 2.47. Since the angle Oo is very small, typically < 5°, the shear rate can be considered constant throughout the material confined within the cone and plate. Although it is also possible to determine the secondary stress coefficient function from the normal stress distribution across the plate, it is very difficult to get accurate data. 

Concentrated emulsions can exhibit viscoelasticity, as can gelled foams and some suspensions. Compared with the previous equations presented, additional coefficients (including primary and secondary normal stress coefficients) are needed to characterize the rheology of viscoelastic fluids [376,382]. 

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.

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. 

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... 

We therefore observe that unlike in the Power Law model solution with a single shear stress component, xn, in the case of a CEF model, we obtain, in addition, two nonvanishing normal stress components. Adopting the sign convention for viscometric flow, where the direction of flow z is denoted as 1, the direction into which the velocity changes r, is denoted as 2, and the neutral direction 8 is denoted as direction 3, we get the expressions for the shear stress in terms of the shear rate, the primary, and secondary normal stress differences (see Eqs. 3.1-10 and 3.1-11) ... 

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. 15.8 Calculated and measured differences of the primary normal stresses in the calender gap for two different planes y(x=o) — Hq/2. [Reprinted by permission from D. Mewes, S. Luther, and K. Riest, Simultaneous Calculation of Roll Deformation and Polymer Flow in the Calendering Process, 7nr. Polym. Process., 17, 339-346 (2002).]...

It is well known in polymer rheology that a torsional parallel-plate flow cell develops certain secondary flow and meniscus distortion beyond some stress level [ 14]. For viscoelastic melts, this can happen at an embarrassingly low stress. The critical condition for these instabilities has not been clearly identified in terms of the shear stress, normal stress, and surface tension. It is very plausible that the boundary discontinuity and stress intensification discussed in Sect. 4 is the primary source for the meniscus instability. On the other hand, it is well documented that the first indication of an unstable flow in parallel plates is not a visually observable meniscus distortion or edge fracture, but a measurable decay of stress at a given shear rate [40]. The decay of the average stress can occur in both steady shear and frequency-dependent dynamic shear. 

Additionally, primary and secondary normal stress coefficients and j/2 are defined by the respective relations... 

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 ... 

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... 

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