Carreau viscosity model

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

A characteristic relaxation time for the Carreau viscosity model or relaxation... 

Out of the various classes of non-Newtonian flows discussed as above, shear-viscosity-dominated fluid flows are possibly the most common ones for typical biomicrofluidic applications. The Carreau viscosity model is one of the standard constitutive models used for many such applications. The apparent viscosity, as per this model, is given by [3]... 

Fig. 1 Apparent viscosity as a function of the rate of shear strain for the Carreau viscosity model...

It is well known that the rheological properties of partially hydrolyzed polyacrylamide depend on the stresses associated with a given flow field. In a simple shear flow, the apparent viscosity is constant at low shear rates (Newtonian behavior). At a critical shear rate, the apparent viscosity decreases as the shear rate is increased, i.e., a shear thinning behavior [48]. The viscosity shear-rate data of water soluble-polymers are commonly fitted using the Carreau viscosity model [49]. According to this model, the apparent viscosity, p, is a function of the shear rate, Y, as follows ... 

Finally, one other approach is to use the White-Metzner model (see Table 3.1), as at least the viscosity function can be fit to the viscosity data (parameters for the Carreau viscosity model are found using nonlinear regression and are rio = 23,000 Pa-s, n = 0.587, and A = 19.7 s). Values of N cannot... 

A stress-dependent viscosity model, which has the same general characteristics as the Carreau model, is the Meter model (Meter, 1964) ... 

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). 

Table 2.6 presents constants for Carreau-WLF (amorphous) and Carreau-Arrhenius models (semi-crystalline) for various common thermoplastics. In addition to the temperature shift, Menges, Wortberg and Michaeli [50] measured a pressure dependence of the viscosity and proposed the following model, which includes both temperature and pressure viscosity shifts ... 

Many mathematical expressions of varying complexity and form have been proposed in the literature to model shear-thinning characteristics some of these are straightforward attempts at cmve fitting, giving empirical relationships for the shear stress (or apparent viscosity)-shear rate curves for example, while others have some theoretical basis in statistical mechanics - as an extension of the application of the kinetic theory to the liquid state or the theory of rate processes, etc. Only a selection of the more widely used viscosity models is given here more complete descriptions of such models are available in many books [Bird et al., 1987 Carreau et al., 1997] and in a review paper [Bird, 1976],... 

Based on the molecular network considerations, Carreau [1972] put forward the following viscosity model which incorporates both limiting viscosities fio and Mco ... 

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],... 

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...

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...

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... 

For the more current simulations, we implemented an improved viscosity model of the Carreau-Yasuda type ... 

We investigate the influence of different destabilizing parameters on the primary breakup of non-Newtonian jets to analyze the primary breakup process. During our research our numerical simulations have constantly been improved. The more accurate Carreau-Yasuda viscosity model was implemented in FS3D and validated against experimental data. The earlier simulations used aqueous PVP solutions as model fluids due to their status as model fluid inside SPP 1423 and the good... 

C.1 Carreau-Yasuda Model Parameters from Regression Analysis. Use either Solve in Excel or the IMSL subroutine RNLIN (Example 2.1) to find the Carreau-Yasuda model parameters [Eq. 2.8 with (n — l)/2 replaced with (n - )/a] for LLDPE at 170 °C (viscosity data are given in Appendix A.3) and compare the results with those given in Table 2.1. 

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