Shear Carreau-Yasuda model

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

The models used are typically either the Cross model or the Carreau-Yasuda model (UNIT Hl.l), if a complete curve is generated. A complete curve has both plateaus present (zero and infinite shear see Figure HI.1.4). 

In order to parameterize the data into a descriptive model, the combined data sets of viscosity and shear rate relationship for linear and linear-branched PLA are fitted to the Carreau—Yasuda model. The form of the model used is given by (Lehermeier and Dorgan 2001) ... 

Figure 4 illustrates how the Carreau-Yasuda model meets the shear viscosity data of Fig. 2. A non-linear fitting algorithm (i.e. Marquardt-Levenberg) was used to obtain the parameters given in the inset. As can be seen the fit curve provides a shear viscosity function that corresponds reasonably well with experimental data so that the high shear behavior is asymptotic to a power law and the very low shear behavior corresponds to the pseudo-Newtonian viscosity po- The characteristic time X (56.55 s) can be considered as the reverse of a critical shear rate (i.e. = Yc = 0.0177 s ) that corresponds to the intersection between the high shear power...

Fig. 4 Shear viscosity function of an unfilled SBR1500 compound at 100 °C as fitted with the Carreau-Yasuda model see Fig. 1 for symbols meaning...

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. 

We then simulate shape oscillations of shear-thinning droplets. At first the implementation of the Carreau-Yasuda model is validated against experimental data. We then analyze the droplet oscillations and compare them to Newtonian droplets with the same Ohnesorge number. We investigate the viscosity distribution... 

Figure 2.16 Shear-dependent viscosity of poly(styrene) melt at 453 K, estimated from the Carreau-Yasuda model.

If a logarithmic ramp is performed, then the data should not be fit with linear models (unit m.i). These data should be plotted as viscosity versus shear rate on logarithmic axes and the Carreau-Yasuda or Cross models (or subsets) should be used instead. It is unlikely that the zero-shear plateau will be seen in these types of tests. For a complete flow curve, the equilibrium tests described in Basic Protocol 2 should be used. 

Capillary rheometers cannot measure accurately at shear rates low enough to determine po directly. For many polymers, this determination would require measurements in the 0.01-0.001 s range. Correlations are sometimes made using the lowest reproducible shear rate possible instead of PO) or models (such as the Carreau or Yasuda model) are used to extrapolate back to Pq. When comparing viscosity curves that extend to sufficiently low shear rates (where the curve begins to flatten out), it is possible to make the statement that material A has a higher M than material B because its low-shear viscosity is higher. 

From a physical standpoint, at small velocities, the polymeric chains of the alginate have a random orientation, increasing the viscosity, while under a sufficient shear they align with the flow, and the viscosity is reduced. Different laws exist for the viscosity of alginate solutions the Carreau-Yasuda law is often used to describe the viscosity of semi-dilute alginate solutions. Similarly, at small velocities, red blood cells form stacks that considerably increase the viscosity. These stacks are dispersed at sufficiently high velocity. Usually blood viscosity is modeled by Cassons law, and an asymptotic value of 4.0 10 Pa.s for the viscosity is obtained when the cells are dispersed. 

As mentioned above, the Newtonian plateau is (or has been) rarely observed with gum rubbers so that po(T) must be obtained by extrapolating experimental data towards zero shear rate, by making use of an appropriate model for the shear viscosity function. In the author s experience, a most flexible model is the so-called Carreau-Yasuda equation, i.e. (at a given temperature T) ... 

The shear thinning behavior, as generally observed with polymer systems, is a typical nonlinear viscoelastic effect, so that by combining the Carreau-Yasuda and the Arrhenius equations a general model for the shear viscosity function can be written as follows ... 

Modeling the shear viscosity function of filled polymer systems by combining two Carreau-Yasuda equations the curve was calculated with the following model parameters Tioj = 8x10 Pa.s X = 500 s Aj = 1.9 = 0.4 rioj = 3x10 Pa.s - O l s = 3 Wj = 0.33. 

By combining two Carreau-Yasuda equations, an eight parameter model is obtained that would meet all the likely typical features of the shear viscosity behavior of filled polymer systems. 

The advantage of these models is that they predict a Newtonian plateau at low shear rates and thus at low shear stresses. We will see back these models in Chap. 16 where an extra term 7700 is added to the equations to account for the viscosity of polymer solutions at high shear rates. At high shear rates the limiting slopes at high shear rates in log r) vs. log y curves are for the Cross, the Carreau and the Yasuda et al. models —m, (n-1) and (n-1), respectively. 

Other equations have been developed to describe the shear thinning behavior of polymer melts, for instance, the Yasuda-Carreau equation, which is written here as Equation 22.19 [41]. In this equation, as in the power-law model, the effect of temperature on viscosity of the system can be taken into account by means of an Arrhenius-type relationship ... 

A common model of shear-tiiinning behavior is that of Carreau and Yasuda (Yasuda et al, 1981), for which the shear-rate dependent viscosity t] y) is... 

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