Flow models 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)... 

Non-Newtonian characteristics are introduced by expressing the wall shear in the capillary tube as an equivalent shear derived from a rheological model such as the power-law model (Equation 1) or the Carreau Model A (Equation 2). Derivations of polymer flow models based upon power-law and Carreau Model A are found in references 6 and 7. Equation 7... 

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). 

If a logarithmic ramp is performed, then the data should not be fit with linear models (unit m.i). These data should be plotted as viscosity versus shear rate on logarithmic axes and the Carreau-Yasuda or Cross models (or subsets) should be used instead. It is unlikely that the zero-shear plateau will be seen in these types of tests. For a complete flow curve, the equilibrium tests described in Basic Protocol 2 should be used. 

Table 11.2 Carreau and Arrhenius model constants for the Coupled Heat Transfer Flow Problem...

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

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. 

Fig. 10.48 Numerical simulation results of nonisothermal flow of HDPE, Melt Flow Index MFI = 0.1 melt obeying the Carreau-Yagoda model for a typical FCM model wedge of e/h — 3 and =15. (a) Velocity (b) shear rate and (c) temperature profiles [Reprinted by permission from E. L. Canedo and L. N. Valsamis, Non Newtonian and Non-isothermal Flow between Non-parallel Plate - Applications to Mixer Design, SPE ANTEC Tech. Papers, 36, 164 (1990).]...

The Carreau model not only described well the flow data of LB gum solutions, but the magnitudes the time constant (Ac) were in good agreement with those of Rouse time (tr) constant derived from solution viscosity data while the Cross (oc) time constants were lower in magnitudes both the Carreau and the Cross time constants followed well power relationships with respect to the concentration (c) of the solutions (Lopes da Silva et al., 1992) ... 

It should be noted that as t becomes large the lowest order term in the coefficient of the K-term is just 60, that is one half the zero-shear-rate value of the primary normal stress function. A similar result was obtained by Bird and Marsh (7) and by Carreau (14) from the slowly varying flow expansions of two continuum models. Hence the time-dependent behavior of the shear stress is related to the steady-state primary normal stress difference in the limit of vanishingly small shear rate. 

The Bird-Carreau model is an integral model which involves taking an integral over the entire deformation history of the material (Bistany and Kokini, 1983). This model can describe non-Newtonian viscosity, shear rate-dependent normal stresses, frequency-dependent complex viscosity, stress relaxation after large deformation shear flow, recoil, and hysteresis loops (Bird and Carreau, 1968). The model parameters are determined by a nonlinear least squares method in fitting four material functions (aj, 2, Ai, and A2). 

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]... 

Khalkhal and Carreau (2011) examined the linear viscoelastic properties as well as the evolution of the stmcture in multiwall carbon nanotube-epoxy suspensions at different concentration under the influence of flow history and temperature. Initially, based on the frequency sweep measurements, the critical concentration in which the storage and loss moduli shows a transition from liquid-like to solid-like behavior at low angular frequencies was found to be about 2 wt%. This transition indicates the formation of a percolated carbon nanotube network. Consequently, 2 wt% was considered as the rheological percolation threshold. The appearance of an apparent yield stress, at about 2 wt% and higher concentration in the steady shear measurements performed from the low shear of 0.01 s to high shear of 100 s confirmed the formation of a percolated network (Fig. 7.9). The authors used the Herschel-Bulkley model to estimate the apparent yield stress. As a result they showed that the apparent yield stress scales with concentration as Xy (Khalkhal and Carreau 2011). 

Chauveteau also studied flow of biopolymers in porous rock. By using well filtered biopolymer solutions, he determined apparent viscosities as a function of Darcy velocity in Fountainbleau sandstone over permeabilities ranging from 3.3 md to 256 md. Polymer retention was low and it was possible to restore the permeability of the rock to its prepolymer value after each polymer flow experiment. Apparent viscosities were fitted with the Carreau Model A. Analysis of the experimental data yields pairs of apparent viscosity and Darcy velocity. Conversion of Darcy velocity to apparent shear rate in the porous rock was done using Equation 14. 

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 ... 

Figure 22 shows the effect of polymer concentration on the flow curves of Statoil polymer in deionized water. At polymer concentrations 2,000 ppm, the apparent viscosity was constant at low shear rates (Newtonian behavior) and decreased at higher shear rates. The Carreau model. Equation 8, predicts the experimental data for this polymer concentration range fairly well. At polymer concentrations > 2,000 ppm, the flow curves showed a shear thinning behavior only. The power-law model. Equation 9, predicts the data fairly well at shear rates > 1 s. ... 

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... 

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... 

Two series of PBTA/PI block copolymers were synthesized in this study and solution processed into molecular composite fibers via dry-jet wet-spinning. The unique rheological properties of liquid-crystalline PBTA homopolymers and PBTA/PI block copolymers were studied with a cone-and-plate rheometer. For block copolymers, the critical concentration decreased with an increase in PBTA content. The flow curves of isotropic and anisotropic solutions could be described via the power-law model and Carreau model, respectively. Copolymer fibers possess tensile strength and modulus located between those of PBTA fibers and PI fibers. Moreover, the tensile strength and modulus of Col fibers increase with an increase in PBTA content. Besides, increasing the draw ratios would give rise to an increase in the mechanical properties of copolymer fibers... 

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. ... 

From a physical standpoint, at small velocities, the polymeric chains of the alginate have a random orientation, increasing the viscosity, while under a sufficient shear they align with the flow, and the viscosity is reduced. Different laws exist for the viscosity of alginate solutions the Carreau-Yasuda law is often used to describe the viscosity of semi-dilute alginate solutions. Similarly, at small velocities, red blood cells form stacks that considerably increase the viscosity. These stacks are dispersed at sufficiently high velocity. Usually blood viscosity is modeled by Cassons law, and an asymptotic value of 4.0 10 Pa.s for the viscosity is obtained when the cells are dispersed. 

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