Second Normal Stress Function

Values of p22 — P33 = N2 appear to be negative and approximately 10-30% of Nj in magnitude (82). The conventional bead-spring models yield N2=0. Indeed, N2 in steady shear flow is identically zero for all free draining models, regardless of the force-distance law in the connectors (102a). Thus, finite extensibility and, by inference at least, internal viscosity do not in themselves provide non-zero N2 values. Bird and Warner (354) have recently analyzed the rigid dumbbell model with intramolecular hydrodynamic interaction, the latter represented by the Oseen approximation. In this case N2 turns out to be non-zero but positive. 

It is interesting to examine the bead-spring models to see what flow-induced configurational changes would be required in order to develop N2 values of the proper magnitude and sign. In the Rouse model, the components of the stress tensor are related directly to averages of the internal coordinates of the beads. For the simplest case of the elastic dumbbell  

The positive value of N2 thus reflects an expansion of the distribution of configurations in the flow direction, relative to the distribution in the direction of the velocity gradient. Negative N2 values imply an effective contraction of the distribution along the direction of the velocity gradient relative to the 3 direction. 

The sign of N2 is correct but the dependence on y 3 at low shear rates is rather contrary to expectations. 

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Of major interest in this review are t](y) and (O) for which a large quantity of data has now been accumulated on well-characterized polymers. Some limited information is also available on the shear rate dependence of The second normal stress function has proved to be rather difficult to measure N2 appears to be negative and somewhat smaller in magnitude than N2 82). 

The Entanglement Concept in Polymer Rheology 8.4. Second Normal Stress Function... 

Second normal stress function, p22 — p33 at steady state in steady simple shear flow. 

These two expressions correspond to a suspension of a in and a suspension of p in a, respectively. In each case, is the interfacial tension of a droplet of fluid i in the medium j y° is the interfacial tension in the absence of flow a is the droplet radius and 2 the second normal stress function of the fluid (Figure 9.10). 

The second normal stress function ( 2) relates the stress tensor T in the direction of flow and the stress tensor T<00> at an angle normal to the plane defined by the direction of flow and the velocity gradient ... 

The symbols Nt and N2 denote the normal stress functions in steady state shear flow. Symmetry arguments show that the viscosity function t](y) and the first and second normal stress coefficients P1(y) and W2(y) are even functions of y. In the... 

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. 

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. 

Since second normal stresses are generally difficult to obtain from the experimental point of view, it may seem attractive to cancel the Cauchy term of the K-BKZ equation setting h2di, I2) = 0 and to find a suitable material function hidi, I2). Wagner [26] wrote such an equation in the form ... 

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. 

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. 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

The second normal-stress difference as a function of time obtained from the simulations on the five-bead Fraenkel chain is nonzero as shown in Fig. 18.2. By averaging over all orientations, the initial value of the second normal-stress difference calculated from... 

Figure 17 shows the first and the zz normal stress differences (top row), the flow alignment angle and the in-plane alignment (bottom row) as functions of the shear rate for Ak = 1.05 and k = 0.4, at the temperature = 0. Comparison should be made with Fig. 14. The main qualitative difference is in the plots for the zz normal stress difference So which is non-zero for k 0 and rather similar to the in-plane alignment. As a consequence, the magnitude of the second normal stress difference becomes smaller than 0.5 lVi. ...

The two normal stress functions, N y) and A(2(y), are referred to as the first and second normal stress differences, respectively. The former is positive... 

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