For non-Newtonian fluids

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

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... 

The power law, r =, is widely used as a model for non-Newtonian fluids. It holds for many solutions and can describe Newtonian,... 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

For non-Newtonian fluids in slow flow, friclion loss across a square-woven or fuU-twill-woven screen can be estimated by considering the screen as a set of parallel tubes, each of diameter equal to the average minimal opening between achacent wires, and length twice the diameter, without entrance effects (Carley and Smith, Polym. Eng. Set., 18, 408-415 [1978]). For screen stacks, the losses of individual screens should be summed. 

Using the Perez and Sandall [21] eorrelation for non-Newtonian fluid... 

Comparing this with the equation for Newtonian fluids shows that the oxygen transfer coefficient for non-Newtonian fluids is less sensitive to power input changes. Thus, more power input is required to reach the same mass transfer coefficient value than in a Newtonian fluid. 

Yoo, S, S.i Ph.D. Thesis, University of Illinois, Chicago (1974). Heat transfer and friction factors for non-Newtonian fluids in circular tubes. 

The ratio of extensional viscosity r e to shear viscosity r s is known as the Trouton ratio, which is three for Newtonian fluids in uniaxial extension and larger than three for non-Newtonian fluids. For a viscoelastic fluid such as a polymer in solution, the uniaxial extensional viscosity characterizes the resistance of the fluid... 

Shah, S.N. "Proppant Settling Correlations for Non-Newtonian Fluids Under Static and Dynamic Conditions," SPE paper 9330, 1980 Annual Fall Technical Conference and Exhibition of the SPE of AIME, Dallas, September 21 24. 

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)]. 

You must determine the horsepower required to pump a coal slurry through an 18 in. diameter pipeline, 300 mi long, at a rate of 5 million tons/yr. The slurry can be described by the Bingham plastic model, with a yield stress of 75 dyn/cm2, a limiting viscosity of 40 cP, and a density of 1.4 g/cm3. For non-Newtonian fluids, the flow is not sensitive to the wall roughness. 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

A procedure analogous to the one followed can be used for non-Newtonian fluids that follow the power law or Bingham plastic models (Darby and Melson, 1982). 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

With the development of modern computation techniques, more and more numerical simulations occur in the literature to predict the velocity profiles, pressure distribution, and the temperature distribution inside the extruder. Rotem and Shinnar [31] obtained numerical solutions for one-dimensional isothermal power law fluid flows. Griffith [25], Zamodits and Pearson [32], and Fenner [26] derived numerical solutions for two-dimensional fully developed, nonisothermal, and non-Newtonian flow in an infinitely wide rectangular screw channel. Karwe and Jaluria [33] completed a numerical solution for non-Newtonian fluids in a curved channel. The characteristic curves of the screw and residence time distributions were obtained. 

Karwe, M. V. and Jaluria, Y., Numerical Simulation of Fluid Flow and Heat Transfer in a Single-Screw Extruder for Non-Newtonian Fluids, Numer. Heat Transfer, Part A, 17, 167 (1990)... 

Based on this relationship, a Reynolds number can be derived and estimated for non-Newtonian fluids from ... 

Substitution of the generalized Reynolds number (Section IIB) for the simple Newtonian Reynolds number has been shown (06) to enable approximate prediction of agitator power consumption for non-Newtonian fluids at low Reynolds numbers. The conventional Newtonian power number-Reynolds number charts which have been drawn up by Rushton et al. (R9) were shown to be applicable in the laminar region. This laminar region, however, appeared to extend to Reynolds numbers of 20 to 25, as compared with critical values of 8 to 10 for Newtonian liquids. Above Reynolds numbers of about 70 the conventional Newtonian curve again appeared to be followed. 

For non-Newtonian fluids the best that one can do at the present time is to make use of various empirical models for non-Newtonian flow. For example, for the incompressible Bingham plastic one can use in place of Eq. (28) the expression... 

The characterisation of the viscosity is difficult for non-Newtonian fluids because the viscosity changes as a result of the flow process, which increases the shear rate. This is further complicated for two-phase fluids because the presence of bubbles will also affect the viscosity. The simpler methods to obtain G for high viscosity fluids make the simplifying assumptions that the fluid viscosity is equal to the liquid viscosity and that the fluid is Newtonian. 

Non-Newtonian fluids are generally those for which the viscosity is not constant even at constant temperature and pressure. The viscosity depends on the shear rate or, more accurately, on the previous kinematic history of the fluid. The linear relationship between the shear stress and the shear rate, noted in Equation (1), is no longer sufficient. Strictly speaking, the coefficient of viscosity is meaningful only for Newtonian fluids, in which case it is the slope of a plot of stress versus rate of shear, as shown in Figure 4.2. For non-Newtonian fluids, such a plot is generally nonlinear, so the slope varies from point to point. In actual practice, the data... 

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