Carreau fluid model

Using the concept of a shear thinning fluid, Shuler and Advani [8] investigated the use of a Carreau fluid model to fit the squeeze flow data for a clay/nylon and an APC-2 composite material. Their results were calculated numerically assuming a full-stick flow condition and seem to match their experimental data quite well. In particular, they studied the... 

Rodrigue et al. (1994) Cylinders, bars, and irregular-shaped rock particles Carreau fluid model Extensive results on drag in pseudoplastic, viscoelastic, and Boger fluids, Re< 50... 

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 Cross and the temperature-dependent Cross-WLF model (42) is an often used GNF-type model accounting for, like the Ellis and Carreau fluids for the viscosity at both low and high shear rates,... 

Plot these data in the form of t — / and yu, — / on logarithmic coordinates. Evaluate the power-law parameters for this fluid. Does the use of the Ellis fluid (equation 1.15) or of the truncated Carreau fluid (equation 1.14) model offer any improvement over the power-law model in representing these data What are the mean and maximum % deviations from the data for these three models ... 

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]... 

Figure 3 illustrates the relationship between steady shear viscosity and shear rate for PBTA homopolymer solutions in NMP/4% LiCl with various concentrations. This figure clearly revels the shear-thinning effect for isotropic (C C r) solutions and anisotropic (C > Ccj-) solutions with the most shear rate region. Meanwhile, a Newtonian plateau appears in a low shear rate region for anisotropic solutions, especially for C = 6 wt% and C = 6.5 wt%. Furthermore, the experimental data could be fitted with theoretical non-Newtonian fluid model. Among which, power-law model was applied for isotropic solutions and Carreau model (22) for anisotropic solutions, as shown below ... 

If shear thinning is the main phenomenon to be described, the simplest model is the general viscous fluid. Section 2.4. It has no time dependence, nor can it predict any normal stresses or extensional thickening (however, recaU eq. 2.4.24). Nevertheless, it should generally be the next step after a Newtonian solution to a complex process flow. The power law. Cross or Carreau-type models are available on all large-scale fluid mechanics computation codes. As discussed in Section 2.7, they accurately predict pressure drops in flow through channels, forces on rollers and blades, and torques on mixing blades. 

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. 

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)... 

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

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)]. 

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). 

Evaluation of GNF Fluid Constants from Viscometric Data Using the flow curve of Chevron/Philips 1409 MI = 50 LDPE in Appendix A, calculate the parameters of the Power Law, Cross and Carreau models. 

We can generalize to include fluids for non-constant viscosity to obtain further dimensionless characteristic values. Two examples are given in Fig. 6.12, and a numerical example for an extruder with a product whose viscosity can be described by the Carreau model is given in Chapter 6. 

A typical viscosity characteristic of many non-Newtonian fluids (e.g., polymeric fluids, flocculated suspensions, colloids, foams, gels, etc.) is illustrated by the curves labeled structural viscosity in Figures 5.2 and 5.3. These flnids exhibit Newtonian behavior at very low and very high shear rates, with shear thinning or pseudoplastic behavior at intermediate shear rates. This can often be attributed to a reversible structure or network that forms in the rest or eqnilibrinm state. When the material is sheared, the structure breaks down, resnlting in a shear-dependent (shear thinning) behavior. This type of behavior is exhibited by flnids as diverse as polymer solutions, blood, latex emulsions, paint, mud (sediment), etc. An example of a useful model that represents this type of behavior is the Carreau model ... 

Grmela, M., and Carreau, P. J., Conformation tensor rheological models, J. Non-Newtonian Fluid Mech., 23, 271-294 (1987). 

Rheological models have also been developed to describe fluid behavior over the shear rate range which include Newtonian behavior at low and high shear rates. The Carreau Model Pi has been found to fit polymer data satisfactorily. Equation 2 is the Carreau Model A. In Equation 2, i is the Newtonian viscosity in the low shear region, x is the Newtonian viscosity in the high shear regions, and i is the shear rate. The parameter n is the power-law exponent and Tj- is a characteristic time constant. All parameters are determined by fitting experimental data. 

Feme J, Ausias G, Heuzey MC, Carreau PJ (2009) Modeling fiber interactions in semiconcentrated fiber suspensions. J Rheol 53 49—72 Ferrari A, Dumbser M, Toto EF, Armanini A (2009) A new 3D parallel SPH scheme for free surface flows. Comput Fluids 38 1203—1217 Ferry JD (1980) Viscoelastic properties of polymers. Wiley, New York... 

Note that equation 18.10 in chapter 18 essentially applies to settling at low particle concentrations (below 0.5% by volume) in Newtonian liquids, which have a constant viscosity. In principle, it can be also used in non-Newtonian fluids where viscosity /x then becomes the apparent viscosity but, depending on the type of the non-Newtonian behaviour (= model), its determination may require an iterative procedure. Not only is such behaviour shear-dependent (i.e. the apparent viscosity depends on how fast is the particle settling) but it may also be time-dependent and the model may contain a zero shear viscosity as a parameter. Ref. 4 reviews the state of the art to 1993 and research is still in progress, for example, on particle settling in the Carreau model fluids (e.g. polymeric liquids). ... 

Machac, 1., Siska, B. and Machacova, L., Terminal falling velocity of spherical particles moving through a Carreau model fluid , Chem. Eng. Proc., accepted for publication in early 2000... 

Prior to 1993 the results of theoretical and experimental smdies on the flow of non-Newtonian fluids past a sphere have been reviewed by Chabra". Since then a number of research smdies have been published, most notably by Machac and co-workers at the University of Pardubice. They investigated experimentally the drag coefficients and settling velocities of spherical particles in power law and Herschel-BuUdey model fluids, in Carreau model fluids (spherical in ref. 13 and non-spherical in ref. 14) and also the effect of the wall in a rectangular ceU, for power law fluids. ... 

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