NonNewtonian flows

Rheology. Polymers are often added to change solvent or process flow properties. The addition of polymers almost always causes nonnewtonian flow behavior in the resulting fluid. 

Other Flow Disturbances Other examples of deviations from fully developed, single-phase newtonian flow include nonnewtonian flow, pulsating flow, cavitation, multiphase flow, boundary layer flows, and nonisothermal flows. See Sec. 6. 

Components of the stress tensor (x) for newtonian fluids are given in Table 1.4. This equation may also be used for describing nonnewtonian flow. However, we need relations between the components of T and the various velocity gradients in other words, we have to replace the expressions given in Table 1.4 with other relations appropriate for the nonnewtonian fluid of interest. The expressions for T for some nonnewtonian fluid models are given in Ref. 10. See also Chap. 10. 

Constitutive Equations. For a simple shear flow, Eq. 10.5 describes the dependence of shear stress on shear rate. Equation 10.5 can be extended to an arbitrary nonnewtonian flow ... 

Duct flows of nonnewtonian fluids are described by the governing equations (Eq. 10.24-10.26), by the constitutive equation (Eq. 10.27) with the viscosity defined by one of the models in Table 10.1, or by a linear or nonlinear viscoelastic constitutive equation. To compare the available analytical and experimental results, it is necessary to nondimensionalize the governing equations and the constitutive equations. In the case of newtonian flows, a uniquely defined nondimensional parameter, the Reynolds number, is found. However, a comparable nondimensional parameter for nonnewtonian flow is not uniquely defined because of the different choice of the characteristic viscosity. 

The laminar flow of various kinds of fluids in circular pipes can be easily compared by plotting (DI4) —dPldx) versus 32QI irD ) = SV JD, as shown in Fig, 15.3. This plot (or its equivalent on logarithmic paper) is very widely used in nonnewtonian flow calculations and publications. Its merit can be seen by rewriting Poiseuille s equation (Eq. 6.8) in the form... 

Nonnewtonian Flow. Materials which do not strictly follow newtonian flow behavior are called nonnewtonian fluids. For nonnewtonian materials, it is convenient to define an apparent viscosity, 77, ... 

A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generahzed relationships exist to pi e-dicl the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as... 

Power consumption for impellers in pseudoplastic, Bingham plastic, and dilatant nonnewtonian fluids may be calculated by using the correlating lines of Fig. 18-17 if viscosity is obtained from viscosity-shear rate cuiwes as described here. For a pseudoplastic fluid, viscosity decreases as shear rate increases. A Bingham plastic is similar to a pseudoplastic fluid but requires that a minimum shear stress be exceeded for any flow to occur. For a dilatant fluid, viscosity increases as shear rate increases. 

Pseudoplasticity. Low-concentration solutions of water viscosifiers are usually nonnewtonian fluids(54) and therefore fail to follow the pressure and flow behavior predicted by newtonian models of flow. To... 

The indeterminate character of some Stokes flow problems influence of a nonNewtonian suspending fluid, inertia and deformability. 

It is known that incompressible newtonian fluids at constant temperature can be characterized by two material constants the density p and the viscosity T. The characterization of a purely viscous nonnewtonian fluid using the power law model (or any of the so-called generalized newtonian models) is relatively straightforward. However, the experimental description of an incompressible viscoelastic nonnewtonian fluid is more complicated. Although the density can be measured, the appropriate expression for r poses considerable difficulty. Furthermore there is some uncertainty as to what other properties need to be measured. In general, for viscoelastic fluids it is known that the viscosity is not constant but depends on shear rate, that the normal stress differences are finite and depend on shear rate, and that the stress may also depend on the preshear history. To characterize a nonnewtonian fluid, it is necessary to measure the material functions (apparent viscosity, normal stress differences, etc.) in a relatively simple or standard flow. Standard flow patterns used in characterizing nonnewtonian fluids are the simple shear flow and shear-free flow. 

FIGURE 10.1 Flow curves for newtonian fluid and shear-thinning and shear-thickening nonnewtonian fluids [4]. 

Here t, 4, and 4 2 are three important material functions of a nonnewtonian fluid in steady shear flow. Experimentally, the apparent viscosity is the best known material function. There are numerous viscometers that can be used to measure the viscosity for almost all nonnewtonian fluids. Manipulating the measuring conditions allows the viscosity to be measured over the entire shear rate range. Instruments to measure the first normal stress coefficients are commercially available and provide accurate results for polymer melts and concentrated polymer solutions. The available experimental results on polymer melts show that , is positive and that it approaches zero as y approaches zero. Studies related to the second normal stress coefficient 4 reveal that it is much smaller than 4V and, furthermore, 4 2 is negative. For 2.5 percent polyacrylamide in a 50/50 mixture of water and glycerin, -4 2/4 i is reported to be in the range of 0.0001 to 0.1 [7]. 

The capillary tube viscometer is a flow-through-restriction type. When used carefully, it is capable of accuracies of better than 2 percent over its applicable shear-rate range (300 to 4000 s 1). For laminar flow of a nonnewtonian liquid in a capillary tube, it can be shown [4] that the wall shear stress xw and the shear rate at the wall % are given by... 

Conservation Equations. In the above section, the material functions of nonnewtonian fluids and their measurements were introduced. The material functions are defined under a simple shear flow or a simple shear-free flow condition. The measurements are also performed under or nearly under the same conditions. In most engineering practice the flow is far more complicated, but in general the measured material functions are assumed to hold. Moreover, the conservation principles still apply, that is, the conservation of mass, momentum, and energy principles are still valid. Assuming that the fluid is incompressible and that viscous heating is negligible, the basic conservation equations for newtonian and nonnewtonian fluids under steady flow conditions are given by... 

In shearing flow, y is called the shear rate. Many expressions have been proposed to approximate the actual dependence of the viscosity of the magnitude of the rate-of-deformation tensor. Some of the models used to describe the behavior of purely viscous nonnewtonian fluids are listed in Table 10.1. 

In the presentation of experimental results describing the fluid mechanics of a power-law nonnewtonian fluid flowing through circular tubes, the five different definitions of the Reynolds number shown in Table 10.3 have been used by various investigators ... 

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