Normal stresses Secondary

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. 

The authors of [203-205] proposed a theory according to which the normal stresses of the matrix and filler may differ only under one condition i.e. the filler content by volume is above some critical value — when its concentration is sufficient to generate the so-called secondary network. In accordance with Privalko and Lipatov s classification [102], this concentration corresponds to the lower boundary of the high-filled class of composites. 

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]... 

G0 initial module of high elasticity p2 is coefficient of normal stresses taking into account secondary flows. 

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. 

Figure 9.41 presents the predicted secondary flow patterns that result from the vicoelastic flow effects. The Giesekus model with one relaxation time was used for the solution presented in the figure. For the simulation, a relaxation time, A, of 0.06 seconds was used along with a viscosity, r], of 8,000 Pa-s and a constant a of 0.80. Similar results were achieved using the Phan-Thien Tanner-1 model. As expected, when the White-Metzner model was used, a flow without secondary patterns was predicted. This is due to the fact that the White-Metzner model has a second normal stress difference, N2 of zero. 

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]. 

Furthermore, when the cone-and-plate rheometer is outfitted with pressure taps at various radial positions, the experimentally obtained pressure distribution is found to be increasing with decreasing radial distance. This, as we will see later, enables us to compute the secondary normal stress difference, namely, x22 — T33, where direction 3 is the third neutral spatial direction. 

Upon rearrangement and integration, and taking into account that the negative of the secondary normal stress difference, — ngg, is a constant (since jg is constant), and that ng at 0 — n/2 (the plate) is a function of the radius, we have... 

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. 

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. 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

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... 

According to this equation, the thrust profile can, in principle, be measured on the upper cone or lower plate. Note that the normal stress differences are assumed to be independent of position. By plotting He0(r) against —ln r/R), a straight line is obtained from whose slope a combination of the primary ( (jxi) - CTee) and secondary (dee - normal stress differences is obtained. Though the pressure profile is difficult to measure, the primary stress difference can be readily determined from the force F exerted on the cone or plate. The value of F is given by... 

P0 secondary normal stress function at zero shear rate... 

Similar definitions can be made for p and p1 related to the oscillating secondary normal stress difference. The quantities 0 and p1 are real, whereas tf, 8, and (t are complex. It is customary to write these complex quantities thus12 ... 

Equations (10.7), (10.8), and (10.10) show that (xyy — t2z)+ = —(2/7) (xxx — xyy)+ for small Xk0. This result and the result in Eq. (6.3) suggest that the secondary normal stress difference might be easier to observe in an unsteady-state experiment than in a steady-state experiment. Note that, from Eqs. (10.6) and (10.10), the relation... 

Fig. 11. Plot of Stewart and Sorensen (72) results for ft = 0 (no hydrodynamic interaction) and ft = 3/8 (maximum ft value using Stokes law). When ft = 0, )8 is zero for all values of khK. In the limit as k- oo, the limiting slopes of the curves are viscosity, —1/3 primary normal stress function, —4/3, secondary normal...

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