Viscometric functions

In example 2.2 we obtained that for steady shearing flows the viscometric functions for this constitutive equation are defined by... 

Develop expressions for the steady shear viscometric functions for the White-Metzner model. 

Comment how the viscometric functions for the shear flow of a Lodge rubber-liquid develop in Example 2.4 compare with experimental observations. 

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. 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

The connection between the double value of the slip parameter obtained from the viscometric functions and the violation of the Lodge-Meissner rule becomes more evident when the time-strain separability of the model is considered. For this purpose, the Johnson-Segalman model can be rewritten under the form of a single integral equation, cancelling the Cauchy term, which gives the following form in simple shear flows ... 

The uniaxial extensional viscosity rj(s) and the viscometric functions rj(y) and ki(y), predicted by the Doi-Edwards model for monodisperse melts, are shown in Fig. 3-32. The Doi-Edwards model predicts extreme thinning in these functions the high-shear-rate asymptotes scale as 17 oc oc y , and4 i oc The second normal... 

Appendix 1 Relations for Determining Viscometric Functions in Standard Experimental Arrangements... 

Some of the most difficult material properties of fluid and semisolid foods to determine experimentally are viscometric functions and steady shear rheological properties. The flow properties of a liquid and semisolid food system should be measured in the following instances ... 

Correlation Between Steady-Shear and Oscillatory Data. The viscosity function is by far the most widely used and the easiest viscometric function determined experimentally. For dilute polymer solutions dynamic measurements are often preferred over steady-shear normal stress measurements for the determination of fluid elasticity at low deformation rates. The relationship between viscous and elastic properties of polymer liquids is of great interest to polymer rheologists. In recent years, several models have been proposed to predict fluid elasticity from shear viscosity data. 

The viscosity function rj (referred to as the steady shear viscosity), the primary and secondary normal stress coefficients ij/, and respectively, are the three viscometric functions which completely determine the state of stress in any rheologically steady shear flow. They are defined as follows ... 

The capillary viscometer can only provide the viscosity-shear rate relationship for a polymer. It cannot give other viscometric functions. Viscometric functions associated with normal stress behaviour in steady shear... 

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... 

From Eq. 83 we observe that the viscometric functions are insensitive to the direction of shear and that the primary normal stress coefficient is zero. Hence this is not a realistic model for most shear sensitive fluids. Eq. 82, with i = fi = constant and / 2 = 0 is the Newtonian fluid. If we keep ii shear rate dependent and set /t,2 = 0, we then have the GNF. Several special cases of the GNF are discussed below. 

Eq. 92 is the so-called second order fluid . Proceeding in this fashion yields the class of constitutive relations known as order fluids. They are meant to account for memory effects in an incremental way as one goes to higher orders. Being a subset of the RE fluid, they suffer similar shortfalls. A special member of this class of fluids is the Criminale-Ericksen-Filbey Huid ([4], p. 503) that is specifically designed for viscometric flows, with three material functions taken as the three viscometric functions themselves, i.e. p, and 2 that are functions of the shear rate. 

The fact that both the shear stress and shear rate are independent of position in the gap is what makes the cone and plate arrangement so desirable for steady shear characterization. To get the normal stresses we note that, since cot0 0, Eq. 1672 indicates oee = ( ) Furthermore, since the viscometric functions... 

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. 

Of the viscometric functions, the viscosity is by far the easiest to measure and is thus the most often reported. As in the case of Newtonian fluids, the viscosity of a polymer depends on temperature and pressure, but for polymeric fluids it also depends on shear rate, and this dependency is quite sensitive to molecular structure. In particular, the curve of viscosity versus shear rate can be used to infer the molecular weight distribution of a linear polymer as is explained in Chapter 8. And in certain cases it can also tell us something about the level of long-chain branching. This curve is also of central importance in plastics processing, where it is directly related to the energy required to extrude a melt. 

We note that this is like to the Cox-Merz relationship in that it relates a nonlinear viscometric function to linear behavior. 

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