Stress difference coefficient

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

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. 

Experimental Determination of the Viscosity and Normal Stress Difference Coefficients, 94... 

EXPERIMENTAL DETERMINATION OF THE VISCOSITY AND NORMAL STRESS DIFFERENCE COEFFICIENTS... 

In Section 6.1 we noted that the term V x in the equation of motion is an important source of pressurization. We have further pointed out that this source may be related either to viscosity or to the normal stress difference coefficient. We discuss the latter case in this section. 

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... 

Instrument Shear rate viscosity Normal stress differences/coefficients ... 

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). 

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 ... 

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. 

Here, and 2 the first and second stress difference coefficient functions, and the derivative of the strain rate is the Jaumann derivative, which is related to a frame of reference that translates and rotates with the local velocity of the fluid (this relationship can be numerically evaluated from the deformation and vorticity tensors). 

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

The first normal stress difference coefficient can be similarly obtained from (9.101) as... 

Figure 8.15 First normal stress difference coefficient growth function of a low density polyethylene filled with two different fillers. (Reprinted from Ref. 91 with kind permission from Gulf Publishing Co.. Houston, Texas. USA.)...

Figure 15 The viscosity (dotted lines) and first normal-stress difference coefficient (1st NSE)C, solid lines) as afunction of shear rate for solutions of polyvinyl acetate in ben2yl cohof [7].

Figure 19 The viscosity and first normal-stress difference coefficient (1st NSDC) as a function of shear rate for polystyrene melt at 232 C, with and without rubber filler particles, [11],...

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...

FIGURE 3.4 Viscosity (steady and complex) versus shear rate or angular frequency (primary normal stress difference coefficient versus shear rate for polyphenylenesulfide at 330 °C. Values of Fi (A), rj ( ), and t) (x) were obtained by means of a cone and plate device. All other values of jj were obtained by means of a capillary rheometer. 

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