Second normal stress difference

The (CEF) model (see Chapter 1) provides a simple means for obtaining useful results for steady-state viscometric flow of polymeric fluids (Tanner, 1985). In this approach the extra stress in the equation of motion is replaced by explicit relationships in terms of rate of strain components. For example, assuming a zero second normal stress difference for veiy slow flow regimes such relationships arc written as (Mitsoulis et at., 1985)... 

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Quotient of the second normal stress difference Nt) and the square of the shear rate (y) in the limit of zero shear rate... 

A similar arrangement consists of two parallel plates (20). In this arrangement the cone is replaced by a second plate. A combined use of cone-and-plate and parallel plate apparatuses yields, in principle, the possibility for the additional determination of the "second normal stress difference p22 — pi2. As no use will be made of this type of result in the following, it should suffice to state that with the parallel plate apparatus the shear rate is not uniform, but increasing linearly with the distance from the centre. 

In Chapter 1 the validity of a stress-optical law has been presumed. Furthermore, it has been shown for several polymer systems that this law is, at least approximately, valid and that the second normal stress difference (p22 — p33) must be very small compared with the first normal stress difference (pn — 22). In the present chapter some theoretical considerations of a more general character will be reviewed in order to indicate reasons for this special behaviour of flowing polymer systems. Some additional experimental results will be given. 

Kuo Y, Tanner RI (1974) Use of open-channel flows to measure the second normal stress differences Rheol Acta 13 931... 

Measurable normal stress differences, N = txx — ryy and N2 = tvv — tzz are referred to as the first and second normal stress differences. The first and second normal stress differences are material dependent and are defined by... 

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 second normal stress difference is difficult to measure and is often approximated by... 

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. 

In a complex, polymeric liquid, normal stresses as well as the shear stress can be present, and these contributions will influence the shape of the structure factor. The simplest rheological constitutive model that can account for normal stresses is the second-order fluid model [64], where the first and second normal stress differences are quadratic functions of the shear rate. Calculations using this model [92,93,94,90,60], indicate that the appearance of normal stresses can rotate the structure factor towards the direction of flow in the case of simple shear flow and can induce a four-fold symmetry in the case of exten-sional flow. 

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. 

In addition, the second normal stress difference, which for conventional polymers is in general negative with an absolute value of not more than 20% of the first normal stress... 

FIG. 16.34 The first and second normal stress differences in the shear flow of a 12.5% nematic solution of PBLG in m-cresol. From Magda et al. (1991). Courtesy The American Chemical Society. 

Cancellation of the Cauchy term may bring some discrepancies, the more evident one being that, whatever h is, it leads to a zero second normal stress difference. A more subtle one concerns the loss of the thermodynamic consistency of the model. Indeed, it is not possible to find any potential function in the form Udi, I2) with h2di, I2) = 0 unless hi only depends on Ii. As mentioned by Larson [27, 28], this can induce violation of the second principle in complex flows such as those encoxmtered in processing conditions. 

Prediction of the second normal stress difference in shear and thermodynamic consistency obviously requires the use of a different strain measure including of the Cauchy strain tensor in the form of the K-BKZ model. With the ratio of second to first normal stress difference as a new parameter, Wagner and Demarmels [32] have shown that this is also necessary for accurate prediction of other flow situations such as equibiaxial extension, for example. 

Second normal stress difference is predicted with the following dependence ... 

The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses part of its original thermod3mamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. 

Constant viscosity and first normal stress difference. Second normal stress difference is zero. 

As will be shown below, the experiment in the sliding plate rheometer does not allow one to determine Nl, since the normal force is in fact related to the second normal stress difference. For this reason, we studied the stress-optical law in shear by assuming that the principal directions of shear and refractive index are close to each other in the x-y plane. It is then straightforward to express the difference of principal stresses in the x-y plane... 

For this model, the second normal stress difference is zero at all shear rates. For the freely jointed chain, to which the FENE or FENE-P spring is an approximation, the polymer contribution to the shear viscosity at high shear rates is proportional to rather than... 

Laun (1994) has measured the first and second normal stress differences and N2 of suspensions at steady state in the shear-thickened state. He found, remarkably, that Ni is negative and N2 is positive, which are opposite the usual signs for these quantities He also found the following relationship between A i, N2, and the shear stress [Pg.308]

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

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