Normal-stress coefficients difference, first

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]... 

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. 

Here are the components of the stress tensor as defined in rheology Tn—T22 is the first normal stress difference and T21 the shear stress, equal to Nt and rxsh, respectively. Hence, from dynamic mechanical measurements it is possible to determine the zero shear first normal stress coefficient Fq0 and zero shear viscosity y0. 

These equations show that both the shear stress and the first normal stress difference gradually increase from zero to a steady state value for —>oo. This results in values for the viscosity and the first normal stress coefficient equal to... 

In Sect. 15.4 it was shown how the shear thinning behaviour of the viscosity could be described empirically with the aid of many suggestions found in literature. It was not mentioned there that the first normal stress coefficient also shows shear thinning behaviour. In this Sect. 15.5 it became clear that also the extensional viscosity is not a constant, but depending on the strain rate upon increasing the strain rate qe the extensional viscosity depart from the Trouton behaviour and increases (called strain hardening) to a maximum value, followed by a decrease to values below the zero extensional viscosity. It has to be emphasised that results in literature may show different behaviour for the extensional behaviour, but in many cases this is due to the limited extensions used,... 

FIG. 15.46 Viscosity, 77, and first normal stress difference, Nh of Vectra 900 at 310 °C as functions of shear rate, according to Langelaan and Gotsis (1996). The first normal stress coefficient, Yi, is estimated from N, by the present author. ( ) Capillary rheometer ( ) and ( ) cone and plate rheometer ( ) complex viscosity rj (A) non-steady state values of the cone and plate rheometer. Courtesy Society of Rheology. 

It is amazing that in many cases the viscosity and the first normal stress differences are reported together it would be wise to compare the viscosity and the first normal stress coefficient, Th, because these both are material properties. For that reason in Fig. 15.46, the present author also incorporated Ll. It also seems to reach a Region II. 

Fiber-reinforced polymer systems, 38 Fickian diffusion, 665 Fick s law, 663,684 Field flow fractionation, 20 Filled polymers, 38 First normal stress coefficient, 545 difference, 640 First-order transition, 27,152 Flame-retardant additives, 861 Flammability, 847 Flashing, 804 Flash line region, 807 Flexibility of a chain molecule, 246 Flexible polymer molecules, 706 Flexural deformation under constant load, 825 Flexural formulas, 826 Flexural rigidity, 877 Floor temperature, 751 Flory-Huggins... 

However, it is now worth pointing out that the difference in ax and aN imply that the initial model should be replaced in shear by a pair of two independent correlations for shear stress (eq. 50a and 50c) or for first normal stress coefficient (eq. 50b and 50d). But at this point some questions arise concerning the choice of the proper value (bt or un) to be used in any other flow situation. Though it is possible to imagine equation (49) Including some variation of a with flow history or invariants, it could hardly be different in two equations for the same flow kinematics. 

Figure 1-1 First Normal Stress Coefficient Data of Starch Dispersions with Different Concentrations as a Function of Shear Rate (Genovese and Rao, 2003). Abbreviations cwm, cross-linked waxy maize tap, tapioca gran, granule.

According to Fig. 6-15, this Peclet number is low enough to be in the low Peclet-number limit, where the normal stress differences are quadratic in the shear rate, and hence the first normal stress coefficient is a constant. This constant can be obtained from the low-shear-rate portion of Table 6-1. Since p is large (p oo), this table gives... 

The ability to homogenize resins with widely different molecular weight can be exemplified by UHMWPE/HDPE blends. Addition of high MW polymer is expected to increase G , G , T), and the first normal stress coefficient, Pj. For the linear polymers, these parameters at... 

Alternatively, from steady shearing experiments, which yield rio directly in the limit of low shear rate, the steady-state recoverable compliance can be obtained from the first normal stress coefficient, which is the ratio of the first normal stress difference to the square of the shear rate, measured at low shear rate... 

The ability to homogenize resins with widely different molecular weight can be exemplified by UHMWPE/HDPE blends. Addition of high MW polymer is expected to increase G, G", t, and the first normal stress coefficient, Pj. For the linear polymers, these parameters at low deformation rates, rjo and Pjo, are proportional to M , and Mw , respectively. Thus, the elasticity is more sensitive to the high MW fractions. For this reason, the frequency dependence of the storage modulus ratio G (blend)/G (PE), at 200 °C, for HOPE and its blends with 3 wt% UHMWPE was measured. The blends prepared in TSE had the worst... 

The viscoelastic equivalents to viscosity—the stress divided by the shear rate—are the so-called first and second normal-stress coefficients, F,and These are given by the first and second normal-stress differences divided by the shear rate squared, so... 

Particular examples of the first normal-stress difference Ni or its coefficient are shown in figures 12 - 29, where a comprehensive collection of examples is given for polymer solutions and melts, as well as an emulsion. These are compared with either the equivalent shear stress or the viscosity. All the different possible combinations of shear and normal-stress difference, and viscosity and normal-stress coefficient are displayed to show the way that results are presented in the rheological literature. The figures are set out to illustrate overall behaviour, with especial emphasis on low shear-rate and mid-range (i.e. power-law) behaviour. 

It is evident that the Giesekus model can quantitatively describe the shear thinning behavior of the entangled solutions of rod-shaped micelles. The decrease of the viscous resistance is caused by the alignment of the anisometric aggregates in the streaming solutions. Similar conclusions can be drawn from measurements of the first normal stress difference. This parameter is often represented in terms of the first normal stress coefficient ... 

We noted in Section 10.7.2 that the second-order fluid approximation for flows only marginally removed from the rest state indicates that the first and second normal stress differences are second order in the shear rate, so that the first and second normal stress coefficients Pj q and T z 0 approach non-zero limiting values at vanishing shear rate. The second-order approximation also predicts that the net stretching stress in uniaxial extension is second order in the Hencky strain rate, and this implies that the extensional viscosity approaches its limiting zero-strain-rate value 3t7o with a non-zero slope ... 

This flow is shown in Figure 2(a) where the velocity distribution is given by Vx = yy,Vy = 0,V2 = 0 and y = dv /dy is a constant. For this flow it is possible to measure a shear stress a first normal stress difference x — and a second normal stress difference These three quantities are in general strong functions of the shear rate y — dVx/dyl It is conventional to define three viscometric functions , namely the (non-Newtonian) viscosity rj (equation 1), the first normal stress coefficient Pi (equation 2) and the second normal stress coefficient 2 (equation 3), as follows... 

First normal stress difference Second normal stress difference Third normal stress difference Normal stress coefficient Birefringence Poiseuille flow No-slip condition Capillary flow... 

Because the shear stress is an odd function of the shear rate and the normal stress differences are even functions, it is customary to define the viscosity function and the first and second normal stress coefficients as follows ... 

Figure 2.31 Reduced first normal stress difference coefficient for a low density polyethylene melt at a reference temperature of 150°C.

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... 

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

According to van Oene (34) the Interfacial tension coefficient depends on the difference between the first normal stress differences, N of the blend components ... 

The theory makes it possible to compute the drop aspect ratio, p = a /a, a parameter that can be directly measured in either transient or steady-state flows. Following the derivation by Hinch and Acrivos [1980] the flow-induced changes to the drop aspect ratio were assumed to be proportional to the first normal stress difference coefficient of the matrix fluid. The coalescence was assumed to follow the Silberberg and Kuhn [1954] mechanism. These assumptions substituted into Eq 7.110 gave a simple dependence for the aspect ratio ... 

The terms on the left-hand side of Equation 22.15a and 22.15b are the first and second normal stress differences, respectively. V i and V 2 Ihe first and second normal stress difference coefficients, respectively, and y is the shear rate. 

Laun proposed two equations relating the dynamic storage and loss moduli to the first normal stress difference coefficient for steady shearing ... 

Young s modulus (Pa) power-law consistency coefficient (Pa s") power-law consistency coefficient for first normal stress difference (Pa S ) first normal stress difference (Pa) second normal stress difference (Pa) power-law index (-) pressure (Pa) total normal stress (Pa)... 

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