Normal-stress coefficients difference, second

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. 

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... 

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 ... 

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). 

Vrc y,y r 11 o - volume fraction of dispersed and matrix phase, respectively - volume fraction of the crosslinked monomer units - volume fraction of phase i at phase inversion - maximum packing volume fraction - percolation threshold - shear strain and rate of shearing, respectively - viscosity - zero-shear viscosity - hrst and second normal stress difference coefficient, respectively... 

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. 

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)... 

The first and second normal stress differences Ni and N2 can be expressed in terms of two coefficients, j/i and j/2, defined as follows ... 

Rg. 9.20 First and second normal stress differences plotted against the shear rate on a logarithmic scale, nsing a = 1 as a typical example. The second coefficient is negative and its ratio is virtually constant over a wide range of shear rates, (a) Pq = 0, (b) = 1. (Reprinted with permission from... 

Figure 9.13 Logarithmic plots of normalized shear viscosity tj/jJq versus dimensionless shear rate y/D., normalized first normal stress difference coefficient f l/ f io versus yjD, and normalized second normal stress difference coefficient 0 versus y/D. for concentrated...

Equation 10.1 is a second-rank tensor with transpose symmetry. The normal components of stress are the diagonal elements and the shear components of stress are the nondiagonal elements. Although Eq. 10.1 has the appearance of a [3 x 3] matrix, it is a physical quantity that, for one set of axes, is specified by nine components, whereas a transformation matrix is an array of coefficients relating two sets of axes. The tensor coefficients determine how the three components of the force vector, /, transmitted across a small surface element, vary as different values are given to the components of a unit vector / perpendicular to the face (representing the face orientation) ... 

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