Equilibrium viscometric flows

For equilibrium viscometric flows of incompressible fluids (the latter is a reasonable assumption for polymer melts and solutions in most engineering circumstances), the situation is brighter. With the coordinate directions assigned as described for viscometric flows in the last chapter, the shear stresses T12 and T21 are equal (or there would be a rotational flow component), and ti3 = t3i — T32 = T23 = 0, Because ofthe arbitrary nature of the static pressure definition, two independent diflerences of the deviatoric normal stresses are commonly defined  

The first and second normal-stress coefficients px and pz sjre defined in terms of the square of y because the normal stresses are even functions of shear rate  

The determination of (17.5) is discussed in Chapter 16. Techniques for measuring functions (17.6) and (17.7) have been comprehensively reviewed.  

In general, JVi is positive and, like r, increases with y. The two are roughly comparable in magnitude for many polymer melts and solutions. This means that there can be significant tension in the flow direction, Nz is considerably smaller (and therefore much more difficult to measure) and is often negative. 

In nonequilibrium deformations, the response of the material depends not only on its current rate of strain, but on its complete strain history. Nonequilibrium deformations are treated in the next chapter. 

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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. 

In general, each of these parameters depends on the shear rate. This dependence is associated with the fact that the macromolecule rotates around the vorticity axis in the shear flow, and if flexible, at sufficiently high shear rate becomes distorted and oriented in the direction of flow [Hsieh and Larson, 2004 Texeira et al., 2005]. However, at sufficiently low shear rates, there is a regime where any distortion and orientation of the macromolecular structure by the flow field is erased by Brownian motion on a time scale much faster than the flow rate. Here we focus on this Newtonian regime, where t]iy), I (y), and [Pg.19]

Instead of carrying out equilibrium averaging, one can introduce a consistently averaged hydro-dynamic interaction , in which the averaging is not performed with the equilibrium distribution function, but rather with a distribution function that is consistent with the local flow field.This leads to shear-rate dependence for all the viscometric functions, and also to a nonzero (but positive) value of the second normal stress coefficient. 

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