Carreau

Carreau, P. J., 1968. PhD thesis. Department of Chemical Engineering, University of Wisconsin, Wisconsin. 

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)... 

Step 4 - update the value of viscosity (r/) using an appropriate rheological equation (e.g. temperature-dependent form of the Carreau model given as Equation (5.4)). 

Carreau proposed an alternative equation to overcome this problem. It has the form... 

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

Example 5.1 Applying the Carreau model to polypropylene, the following constants are known at 190°C. 

Carraghenmoos, n. carrageen moss, Irish moss. Carreau, n. check, square carrieren, v.t. checker, cartesisch, a. Cartesian. 

Jolicoeur M, Chavarie C, Carreau PJ, Archambault J (1992) Biotechnol Bioeng 39 511... 

Carreau PJ, De Kee DCR, Chhabra RP (1997) Rheology of polymeric systems. Principles and applications. Hanser, Munich, p 35... 

Carreau, P. J. DeKee, D. C. R. Chhabra, R. P. Rheology ofPolymeric Systems Principles and Applications. Hanser Gardner Publications, Cincinnati. 

The Carreau model (Carreau, 1972) is very useful for describing the viscosity of structural fluids ... 

Figure 3-8 Viscosity data and Carreau model fit of polyacrylamide solutions. (From Darby and Pivsa-Art, 1991.)...

If r]oo <3C ( /. j/o), the Carreau model reduces to a three-parameter model ( 0,k, and p) that is equivalent to a power law model with a low shear limiting viscosity, also known as the Ellis model ... 

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

The Yasuda model (Yasuda et al., 1981) is a modification of the Carreau model with one additional parameter a (a total of five parameters) ... 

For non-Newtonian fluids, any model parameter with the dimensions or physical significance of viscosity (e.g., the power law consistency, m, or the Carreau parameters r,]co and j/0) will depend on temperature in a manner similar to the viscosity of a Newtonian fluid [e.g., Eq. (3-34)]. 

The complete expression for tVDe is given by Darby and Pivsa-Art (1991) as a function of the viscoelastic fluid properties of the fluid (i.e., the Carreau parameters /y0, X, and p). This expression is... 

ARe>s is the Reynolds number based on the solvent properties, /zs is the solvent viscosity, D is the pipe diameter, F is the velocity in the pipe, and A is the fluid time constant (from the Carreau model fit of the viscosity curve). 

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... 

Devineau, J. and C. Amiard Triquet 1985. Patterns of bioaccumulation of an essential trace element (zinc) and a pollutant metal (cadmium) in larvae of the prawn Palaemon serratus. Mar. Biol. 86 139-143. Dib, A., J.P Clavel, and J.P. Carreau. 1989. Effects of gamma-linolenic acid supplementation on lipid composition of liver microsomal membranes. I. Pregnant rats fed a zinc-deficient diet and those fed a balanced one. Jour. Clin. Biochem. Nutr. 6 95-102. 

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