Equations Carreau

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

In order to allow for the effect of temperature on viscosity a shift factor, ar is often used. The Carreau equation then becomes... 

It is not uncommon to encounter emulsions, foams, and suspensions, both in nature and in industry, that contain polymers. If the polymer concentration is high enough, and the dispersed species concentration low enough, the overall viscosity may be better described by the contribution from the polymer solution than that from the dispersed species. One commonly employed equation for describing the viscosity of polymer solutions is the Carreau equation,... 

The viscosity function is often approximated using the Carreau equation which uses three variable parameters ... 

Figure 3.4 Carreau equation for approximating the viscosity function...

If the material to be processed is subject to shear thinning, the linear relationships for the pressure and energy behavior illustrated above no longer apply. With shear thinning, there is a non-linear relationship between the shear rate and shear stress that is reflected in the flow curve (see Chapter 3). As a rule, the zero viscosity and one or two rheological time constants are enough to describe the flow curve with sufficient accuracy. The Carreau equation is often used it contains a dimensionless flow exponent in addition to the zero viscosity and a rheological time constant. 

The apparent viscosity (tja) of the solution can be correlated with shear rate (y) using the Cross (Equation 2.14) or the Carreau (Equation 2.15) equations, respectively. 

As shown in Figure 13.31, the curves describing the variation in the viscosity of polymers filled with homodisperse noninteractive spheres are reminiscent of those of the unfilled polymers, at least up to a solid fraction close to maximum packing (56). The data fit to the Carreau equation (47)... 

A more general model is the Carreau equation (Carreau, 1972 Bird et al., 1987),... 

With a lower power-law index, say, n = 0.2, the apparent viscosity would still be close to the limiting shear viscosity at A of 1.6 s and the shear rate of 0.1 s. For these conditions. Cross equation (17.7) gives 77 = T7o/1.23, and Carreau equation (17.8) gives 77 = 770/I.OI. However, at A of 16 s and the shear rate of 10 s (see above), the apparent viscosity would be much lower, that is, the shear thinning effect would be much more pronounced. Both Cross and Carreau equations give 77 = 77o/14,500, that is, 36 times lower apparent viscosity compared to that at n = 0.5. 

This means that at high shear rates (even at moderate A values), both Cross and Carreau equations are transformed to... 

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 ... 

At high shear rates, the viscosity of a polymer melt behaves like a power law fluid characterized by the Carreau equation ... 

The Carreau equation becomes a power-law equation at high shear rates. 

Although the Carreau equation does offer an improved fit to viscometric data over a wide range of shear rates, it does require four parameters instead of the power law s two. It also makes certain analytical calculations using the viscosity function much more difficult as will be discussed further in later sections. 

Not long ago, an equation with four constants would have been deemed excessively complex for engineering calculations, but the computer does not mind a bit. Actually, in many practical applications, the shear rates don not get high enough to approach the upper-Newtonian region, and a truncated (three-parameter) form of the Carreau equation, with Voo = is adequate. 

Note that the above calculations are facilitated if the flow curve can be written as an explicit function of t, that is, in the form y = y(T)or rj = r (t). Although the power law meets that criterion, neither the Carreau Equation 14.7 nor the modified Cross Equation 14.8 meets the same. For this reason, it is sometimes viewed as being easier to fit data with a polynomial such as Equations 14.9a or A.9b, rather than to use an established constitutive equation to make these calculations (but note the cautions ). 

Compare the truncated (rj o = 0) form of the Carreau Equation 14.7 and the modified Cross Equation 14.8 for fitting these data. 

The film thickness for the ordinary shear-thinning response of lubricants which can be measured is now calculated. There are many generalized Newtonian fluid models that will describe the shear response displayed in Figures 1 and 2 [16]. The Ree-Eyring model utilizes a series in inverse hyperbolic sine to approximate power-law behavior at high shear rate. In others [16] the power-law exponent, , appears explicitly. Today the most widely used model outside of tribology is the Carreau equation [17] that was advanced to describe the results of molecular network theory. 

It is important to note that this model contains no characteristic time. It thus implies that the power-law parameters are independent of shear rate. Of course such a model cannot describe the low-shear-rate portion of the curve, where the viscosity approaches a constant value. Several empirical equations have been proposed to allow for the transition to Newtonian behavior over a range of shear rates. It was noted in the discussion of the Weissenberg number earlier in this chapter that the variation of 77 with y implies the existence of at least one material property with units of time. The reciprocal of the shear rate at which the extrapolation of the power-law line reaches the value of tiq is such a characteristic time. Models that can describe the approach to tIq thus involve a characteristic time. Examples include the Cross equation [64] and the Carreau equation [65], shown below as Eqs. 10.55 and 10.56 respectively. 

Using finite element techniqnes, a mathematical model was developed for the two-dimensional analysis of non-isothermal and transient flow and mixing of a generalised Newtonian fluid with an inert filler. The model could incorporate no-slip, partial-slip or perfect-slip wall conditions using a universally applicable numerical technique. The model was used to simulate the convection of carbon black with flowing rubber in the dispersive section of a tangential rotor (Banbury) mixer. The Carreau equation was used to model the rheological behaviour of the fluid in this example. 31 refs. 

In practical application, to represent the polymer flow curves in a relatively narrow shear rate range (e g., one or two decades of shear rate), the power law equation has sufficient accuracy. To represent the polymer flow curves in a wider shear rate range (e.g., three or four decades), the Carreau equation can be used. In addition to these two equations, other models also have been developed. The interested reader is referred to the works by Tanner (2000), Dealy and Larson (2006), and Brazel and Rosen (2012). 

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