Rheological models steady shear viscosity

Choi, G. R. and Krieger, 1. M. 1986. Rheological studies on sterically stabilized model dispersions of uniform colloidal spheres II. Steady-shear viscosity. J. Colloid Interface Sci. 113 101-113. 

FIGURE 8 Comparison of rheological model of Eqs. (70)-(72) with experiment on rubber-carbon black compound, (a) Steady shear viscosity, (b) Transient, (c) Shear-rest-shear flow behavior. 

Rheological models have been described for steady shear viscosity function, normal stress difference function, complex viscosity function, dynamic modulus function and the extensional viscosity function. The variation of viscosity with temperature and pressure is also discussed. 

The rheological predictions that derive from this simple molecular model are very similar to the upper-convected Maxwell model see example 4.3.3. Recall that t = Tj + tp = r , 2D + Xp. We obtain Steady shear viscosity... 

The model predicts the steady-state shear rheology to the degree of accuracy of the multi-mode PTT model. This is an obvious result of fitting the multi-mode PTT to the bulk viscosity vs. shear rate data. Figure 1 is a graph of the 7 mode PTT fit to the steady shear viscosity vs. shear rate flow curve. 

In addition to relationships between apparent viscosity and dynamic or complex viscosity, those between first normal stress coefficient versus dynamic viscosity or apparent viscosity are also of interest to predict one from another for food processing or product development applications. Such relationships were derived for the quasilinear co-rotational Goddard-Miller model (Abdel-Khalik et al., 1974 Bird et al., 1974, 1977). It should be noted that a first normal stress coefficient in a flow field, V i(y), and another in an oscillatory field, fri(ct>), can be determined. Further, as discussed below, (y) can be estimated from steady shear and dynamic rheological data. 

The measurement of yield stress at low shear rates may be necessary for highly filled resins. Doraiswamy et al. (1991) developed the modified Cox-Merz rule and a viscosity model for concentrated suspensions and other materials that exhibit yield stresses. Barnes and Camali (1990) measured yield stress in a Carboxymethylcellulose (CMC) solution and a clay suspension via the use of a vane rheometer, which is treated as a cylindrical bob to monitor steady-shear stress as a function of shear rate. The effects of yield stresses on the rheology of filled polymer systems have been discussed in detail by Metzner (1985) and Malkin and Kulichikin (1991). The appearance of yield stresses in filled thermosets has not been studied extensively. A summary of yield-stress measurements is included in Table 4.6. 

In Section 3.4 it was explained that polymers having very well defined structures can be prepared by means of anionic polymerization, and this technique has been widely used to prepare samples for rheological study. This has been a particularly fruitful approach to the study of the elfects of various types of long-chain branching structure on rheological behavior. Linear viscoelastic properties are very sensitive to branching. In this section we review what is known about the zero-shear viscosity, steady-state compliance, and storange and loss moduli of model branched polymers. 

Figure 1. 7 mode PTT model fit to the bulk steady-state viscosity vs. shear rate curve of a short glass fiber filled PP. Rheological tests were performed at 200 C. 

Rheological measurements Two instruments were used to investigate the rheology of the suspensions. The first was a Haake Rotovisko model RV2(MSE Scientific Instruments, Crawley, Sussex, England) fitted with an MK50 measuring head. This instrument was used to obtain steady state shear stress-shear rate curves. From these curves information can be obtained on the viscosity as a function of shear rate. The yield value may be obtained by extrapolation of the linear portion of the shear stress-shear rate curve to zero shear rate. The procedure has been described before (3). 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

During the experiments, the solid concentration was increased to 20% by volume. Except for suspensions with plastic particles, the suspensions showed a Newtonian behavior up to volume contents of 15 %. Suspensions with glass beads and s = 0.2 as well as all examined suspensions with plastic particles showed a shear thinning behavior. Considering the non-Newtonian behavior of these suspensions in the calculation of the time steady flow based on Eqs. (5.9-5.21), the viscosity of the suspension had to be described by a model depending on the deformation speed y. A Carreau-Yasuda model according to Eq. (5.52) fitted well to measurements carried out with a Couette system. The parameters Hq, a, n, and X were determined by the rheological measurements. 

It is seen that the material functions obtained from the covariant convected derivative of a are different from those obtained from the contravariant convected derivative of a. Experimental results reported to date indicate that the magnitude of N2 is much smaller than that of (say -A 2/ i 0.2-0.3). Therefore, the rheology community uses only the contravariant convected derivative of a when using Eq. (3.4), which is referred to as the upper convected Maxwell model. However, the limitations of the upper convected Maxwell model lie in that, as shown in Eq. (3.6), (1) it predicts shear-rate independent viscosity (i.e., Newtonian viscosity, t]q), (2) is proportional to over the entire range of shear rate, and (3) N2 = 0. There is experimental evidence (Baek et al. 1993 Christiansen and Miller 1971 Ginn and Metzner 1969 Olabisi and Williams 1972) that suggests Nj is negative. Also, as will be shown later in this chapter, and also in Chapter 5, in steady-state shear flow for many polymeric liquids, (1) l (k) follows Newtonian behavior at low y and then decreases as y increases above a certain critical value, and (2) increases with at low y and then increases with y (l < n < 2) as y increases further above a certain critical value. 

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