Carreau-Yasuda

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). 

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. 

The Bird-Carreau-Yasuda Model. A model that fits the whole range of strain rates was developed by Bird and Carreau [7] and Yasuda [72] and contains five parameters ... 

To simulate the behavioiu of the LLDPE melt, we retained the following Carreau-Yasuda relationships for the viscosity and the relaxation time ... 

Often the Carreau-Yasuda "a" parameter (designated CY-a hereafter) can also serve as an indicator of LCB within a series of similarly produced polymers. The CY-a value is considered as a measure of the relaxation time distribution [526,527]. Because the relaxation time is influenced by LCB much more than by changes in the MW distribution, CY-a can be a sensitive LCB indicator. High levels of LCB tend to produce low CY-a values. Usually, the CY-a data reinforce the conclusions drawn from JC-a values. It is an especially reliable indicator if the MW distribution does not vary much within a series of experiments, or when the two indicators of "breadth" move in opposite directions (GPC narrows whereas CY-a declines). 

Lee et al. [2007] studied the rheological behavior of poly(ethylene-co-vinyl acetate) (EVAc 40 wt% VAc) and its CPNC with < 10 wt% C30B the tests were conducted under steady-state and small oscillatory shear flow. The samples were prepared by melt compounding at 110 C for 25 min, which resulted in a high degree of dispersion. The flow behavior was quite regular, well described by the Carreau-Yasuda equation [Carreau, 1968,1972 Yasuda, 1979] ... 

Here the two equation parameters, r and 6, are defined in terms of the Carreau-Yasuda equation (Carreau 1968, 1972 Yasuda 1979, 1981) ... 

The pseudoplastic flow relation for molten polymer follows the Carreau-Yasuda... 

Derezinski solved the problem numerically with the barrel temperature as the boundary condition and a power law viscosity [337], Later the analysis was extended to include a Carreau-Yasuda viscosity model [338],... 

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. 

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) ... 

The empirical Carreau-Yasuda (or, sometimes, Carreau-Yasuda-Elbirli) model contains two additional parameters and is extensively used to correlate melt data ... 

Finally, the shear rate as deflned by Equation 2.36b is clearly the appropriate argument for the viscosity function only for one-dimensional flows like the one used here. We need a quantity that reduces to dvx/dy for the one-dimensional flow but is properly invariant to the way in which we choose to deflne our coordinate system. The appropriate function, which follows directly from the principles of matrix algebra, is one half the second invariant of the rate of deformation, which is usually denoted Ud- Ud is shown in Table 2.6, where it is identical to the dissipation function O divided by r] for the special case of Newtonian fluids. (It is important to keep in mind that the function /) in Table 2.6 is the proper form for the dissipation only for a Newtonian fluid, whereas IId is a universally valid definition that depends only on the velocity field.) For an arbitrary flow field, then, the power-law and Carreau-Yasuda equations would be written, respectively. 

For an analysis of converging flow with a Navier slip condition for Newtonian, power-law, and Carreau-Yasuda fluids, see... 

Figure 12.8. Streamlines for a Carreau-Yasuda fluid with n = 0.3 and w = 4.0. The dashed line is the streamline for a Newtonian fluid ((v = 0), while the dotted line is the streamline for the corresponding power-law fluid. Reprinted from Joshi and Denn, /. Non-Newtonian Fluid Mech., 114,185 (2003).

A second dimensionless group, w = qP/l abY, arises for the Carreau-Yasuda (C-Y) shear-thinning fluid, Equation 2.40b ... 

Figure 14.2. Simulation of the mixing pattern for two identical Carreau-Yasuda fluids with n = 0.1 after passing through a static mixer with six blades with a twist angle of 180°. Reprinted with permission from Galaktionov et al., Int. Polym. Proc., XVIII, 138 (2003).

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) ... 

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...

Figure 5 shows steady shear viscosity data for a carbon black filled high cis-1.4 polybutadiene compound, as obtained using various rheometers. The Carreau-Yasuda equation was used to yield fit parameters given in the lower right inset the shear viscosity function q = f(y) is drawn in the left graph. As can be seen, a... 

An attractive mathematical model for such a dynamic viscosity function is again the Carreau-Yasuda model, i.e. ... 

Fig. 9 Dynamic Viscosity function of gum EPDM2504 at 100 °C authors s experimental data and fitted Carreau-Yasuda model...

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