Power law behavior, non-Newtonian

For non-Newtonian fluids the correlations in Figure 35 can be used with generally acceptable accuracy when the process fluid viscosity is replaced by the apparent viscosity. For non-Newtonian fluids having power law behavior, the apparent viscosity can be obtained from shear rate estimated by... 

In both the studies on basalts, the breaks in the slope of the log Pq vs. (1/T,K) curves occur at 20 to 30 volume percent of suspended crystals. The non-Newtonian behavior of these molten silicate suspensions appears to arise from the increasing concentration of suspended crystals in the melt. This suggests that in modeling fluid flow in silicate systems, power law behavior should be considered when the suspended crystal concentration exceeds 20 volume percent. 

The values of the constants were measured to be Tq = 10 Pa, n = 0.630, and m = 0.167 Pa s . This relation is seen to be a combination of the Bingham plastic and power law behavior and is found to fit the measurements to within an accuracy of 1—2%. In Fig. 9.1.3 we have drawn in the velocity distribution for a Bingham plastic fluid using the measured value of the yield stress and a measured value of rja = 0.57 from which we calculate G = 1.4x10 Pam", Mp = 2.28 X 10 Pa s, and 3, = 1.84 m s The agreement between the theory and measurement, although not as excellent as for the yield-power law behavior, is nevertheless seen to be quite good and shows clearly the nature of the non-Newtonian behavior associated with the flow of a colloidal suspension. 

Frequently, non-Newtonian materials show a power-law behavior of the viscosity as a function of the shear rate 7 in which the exponent is a free parameter. An example of this is the following model for the relaxation time (Apelian, 1988)... 

Figure 4-2 shows typical flow curves of Newtonian and non-Newtonian fluids. Figure 4-3 illustrates shear viscosity as a function of shear rate for a pseudoplastic fluid, showing regions of low-shear limiting viscosity (Yo), high-shear limiting viscosity (Yoo) and power law behavior. 

Hydraulic fracturing fluids are solutions of high-molecular-weight polymers whose rheological behavior is non-Newtonian. To describe the flow behavior of these fluids, it is customary to characterize the fluid by the Power Law parameters of Consistency Index (K) and Behavior Index (n). These parameters are obtained experimentally by subjecting the fluid to a series of different shear rates (y) and measuring the resultant shear stresses (t). The slope and Intercept of a log shear rate vs log shear stress plot yield the Behavior Index (n) and Consistency Index (Kv), respectively. Consistency Indices are corrected for the coaxial cylinder viscometers by ... 

The rheological properties change behavior, relative to more dilute solutions, above cp = 0.2, where non-Newtonian behavior is then exhibited. The power law dependence of rj on cp is in harmony with the Zimm rather than the Rouse model, which suggests that hydrodynamic interactions between these polymers, in a mean field sense, are important. Electrical properties also begin to deviate for Nafion solutions above cp = 0.2, and mechanical percolation is essentially the same for the sodium and acid forms. 

In addition to temperature, the viscosity of these mixtures can change dramatically over time, or even with applied shear. Liquids or solutions whose viscosity changes with time or shear rate are said to be non-Newtonian, that is, viscosity can no longer be considered a proportionality constant between the shear stress and the shear rate. In solutions containing large molecules and suspensions contain nonattracting aniso-metric particles, flow can orient the molecules or particles. This orientation reduces the resistance to shear, and the stress required to increase the shear rate diminishes with increasing shear rate. This behavior is often described by an empirical power law equation that is simply a variation of Eq. (4.3), and the fluid is said to be a power law fluid ... 

Both polymeric and some biological reactors often contain non-Newtonian liquids in which viscosity is a function of shear rate. Basically, three types of non-Newtonian liquids are encountered power-law fluids, which consist of pseudoplastic and dilatant fluids viscoplastic (Bingham plastic) fluids and viscoelastic fluids with time-dependent viscosity. Viscoelastic fluids are encountered in bread dough and fluids containing long-chain polymers such as polyamide and polyacrylonitrite that exhibit coelastic flow behavior. These... 

The major characteristic of a polymeric reactor that is different from most other types of reactors discussed earlier is the viscous and often non-Newtonian behavior of the fluid. Shear-dependent rheological properties cause difficulties in the estimation of the design parameters, particularly when the viscosity is also time-dependent. While significant literature on the design parameters for a mechanically agitated vessel containing power-law fluid is available, similar information for viscoelastic fluid is lacking. 

A single figure for t] is not appropriate for non-Newtonian substances, and it common practice to plot flow curves of such malerials in terms of (apparent viscosity) against corresponding values of y. Many equations have been proposed to describe non-Newtonian behavior. Generally, however, the mathematics involved is not worth the effort except for the simplest problems. It is most efficient to read the required viscosity values from experimental rjn-y plots. These relations can usually be described over limited shear rate ranges by power law expressions of the form ... 

In contrast to a cone and plate geometry to be discussed next, the shear rate of non-Newtonian foods cannot be determined from a simple expression involving the angular velocity and often one must use a suitable relationship between rotational speed and shear stress to correct for non-Newtonian behavior. More complex equations are needed to describe the flow of non-Newtonian fluids in concentric cylinder geometry. For example, for fluids that can be described by the power law model, an expression presented by Krieger and Elrod (Van Wazer et al., 1963) has been used extensively in the literature ... 

However, they may not indicate the true bulk viscosity of a suspension that forms a thin layer of the continuous phase (e.g., serum of tomato juice) around the immersed probe or when the probe is covered by a higher viscosity gel due to fouling. Vibrational viscometers are suitable for measuring viscosities of Newtonian fluids, but not the shear-dependent rheological behavior of a non-Newtonian fluid (e.g., to calculate values of the power law parameters). 

The homogeneous non-Newtonian capillary tube-power law model has a number of limitations. The models assume a power law relationship for the emulsion, and any deviations from this rheological behavior will lead to errors. The power law constants n and K are obtained by using viscometry, and their validity in porous media is questionable. No transient permeability reduction (assumption 4) is predicted, even though experimental evidence suggests otherwise. This model is seen to have validity only for high-quality emulsions that approach steady state quickly and have small droplet-size to pore-size ratios. 

In this chapter, we focus on the first stage of clogging and investigate the impaction of non-Newtonian power-law fluids on thin fibers. We aim to obtain the threshold radius of impacting droplets in different impact velocities. Effect of shear-thinning and shear-thickening behavior of droplets is evaluated and compared with corresponding Newtonian fluids. For this purpose, volume of fluid method is used and open source OpenFOAM software is applied for simulations. 

The flow behavior of molten chocolate can also be affected by changes in processing conditions. These may lead to different values, because of the effect of non-Newtonian flow at the walls of the processing equipment. The wall shear rate is often characterized by the Nusselt number equation, which contains the so-called <5 factor. The S factor is the ratio of the wall shear rate for a non-Newtonian fluid to that of a Newtonian fluid at the same flow rate. For power law fluids, this factor can be calculated ... 

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