Carreau—Yasuda equation

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) ... 

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. 

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 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... 

This is often called the Carreau-Yasuda equation. 

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. 

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

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 ... 

Using the Carreau-Yasuda constitutive equation, this yields a nonlinear first-order differential equation for the velocity field... 

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 ... 

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