Viscometric functions Normal stress differences

This section describes two common experimental methods for evaluating i], Fj, and IG as functions of shear rate. The experiments involved are the steady capillary and the cone-and-plate viscometric flows. As noted in the previous section, in the former, only the steady shear viscosity function can be determined for shear rates greater than unity, while in the latter, all three viscometric functions can be determined, but only at very low shear rates. Capillary shear viscosity measurements are much better developed and understood, and certainly much more widely used for the analysis of polymer processing flows, than normal stress difference measurements. It must be emphasized that the results obtained by both viscometric experiments are independent of any constitutive equation. In fact, one reason to conduct viscometric experiments is to test the validity of any given constitutive equation, and clearly the same constitutive equation parameters have to fit the experimental results obtained with all viscometric flows. 

We shall now examine the effects produced by the stresses generated during the reorientation process by calculating the viscometric functions that relate the shear and normal stress differences. For a planar geometry and using the convention in (Bird R. et al 1971) the first normal stress difference is defined by... 

Rheological measurements are performed so as to obtain a test fluid s material functions. Under viscometric flows we have seen that the shear viscosity and the primary and secondary normal stress differences suffice to rheologically characterize the fluid. If the flow field is extensional and the material is able to attain a state of dynamic equilibrium, then one measures the extensional viscosity otherwise, we measure the extensional viscosity growth or decay functions. In this section, we will examine steady and dynamic shear plus uniaxial extensional tests, since these make up the majority of routine rheological characterization. 

While the cone and plate geometry is the preferred arrangement to obtain the steady viscometric functions, it is limited to low shear rates — usually, to those less than 10 s . At higher shear rates encountered in processing ( 10-10 s ), it is customary to resort to capillary rheometry to measure the shear viscosity. Unfortunately, the normal stress differences cannot be obtained from this test. To get N at high shear rates one can, however, employ a slit device based on the so-called hole pressure effect [21]. 

After a sufficient length of time at constant shear rate, all the stresses become independent of time, and three rheologically meaningful material functions of shear rate can be defined. These viscometric functions are the viscosity and the first and second normal stress differences, which are described in Sections 10.8 and 10.9. 

The second normal stress difference is the most difficult to measure of the three viscometric functions. Several techniques have been proposed for the determination of the second normal stress difference. The most common method involves the use of a cone-plate rheometer, where... 

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... 

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