Function of shear rate

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)... 

Fig. 1. Melt viscosity as a function of shear rate for (—) linear BPA polycarbonate and (-) branched polycarbonate. To convert Pa-s to poise, multiply...

For a Hquid under shear the rate of deformation or shear rate is a function of the shearing stress. The original exposition of this relationship is Newton s law, which states that the ratio of the stress to the shear rate is a constant, ie, the viscosity. Under Newton s law, viscosity is independent of shear rate. This is tme for ideal or Newtonian Hquids, but the viscosities of many Hquids, particularly a number of those of interest to industry, are not independent of shear rate. These non-Newtonian Hquids may be classified according to their viscosity behavior as a function of shear rate. Many exhibit shear thinning, whereas others give shear thickening. Some Hquids at rest appear to behave like soHds until the shear stress exceeds a certain value, called the yield stress, after which they flow readily. 

Orthokinetic flocculation is induced by the motion of the Hquid obtained, for example, by paddle stirring or any other means that produces shear within the suspension. Orthokinetic flocculation leads to exponential growth which is a function of shear rate and particle concentration. Large-scale one-pass clarifiers used in water installations employ orthokinetic flocculators before introducing the suspension into the settling tank (see Water,... 

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

Figure 9 Shear viscosity as a function of shear rate at the wall at 200°C, (O) Resin A, ( ) Resin B, (O) Resin C, (A) Resin D, (A) Resin E. (Refer to Table 2 for symbols code.) Source Ref. 56.

The plot of the extrudate swelling ratios with ACM content are shown in Fig. 21 as a function of shear rate... 

The plot of the extrudate swelling ratios with AU content are shown in Fig. 27 as a function of shear rate for both types of blends. The die-swell gradually decreases with the addition of AU in the blend up to the... 

This equation has the advantage of having two adjustable parameters, [17] and , which fit the data for both low- and high-shear rates, as shear rate. Since the viscosity of the suspension decreases the faster it is sheared, 4>m is evidently higher for higher shear rates. In practice, higher loadings also impart a... 

The empirical frictional factor (T fric) is independent of shear rate but increases in poor solvent this permits to account for the dependence of the scission rate constant on solvent quality. The entanglement part (r enl), as given by Graessley s theory which considers the effect of entanglement and disentanglement processes, is a complex function of shear rate ... 

The fluid may be either shear-thinning or, less often, shear-thickening, and in either case the shear- stress and the apparent viscosity fia are functions of shear rate, or ... 

Optical and electro-optical behavior of side-chain liquid crystalline polymers are described 350-351>. The effect of flexible siloxane spacers on the phase properties and electric field effects were determined. Rheological properties of siloxane containing liquid crystalline side-chain polymers were studied as a function of shear rate and temperature 352). The effect of cooling rate on the alignment of a siloxane based side-chain liquid crystalline copolymer was investigated 353). It was shown that the dielectric relaxation behavior of the polymers varied in a systematic manner with the rate at which the material was cooled from its isotropic phase. 

Fig. 8—Effective viscosity of confined hexadecane measured on SFA as a function of shear rate and film thickness, from which it is seen that the shear thinning gradually disappears as the film thickness increases and the viscosity finally has approached the bulk values at h=122 nm.

The polymer behaves as a generalized Newtonian fluid, where viscosity is an arbitrary function of shear rate and temperature. 

Shear rate x exposure time Fig. 29. Activity of catalase as a function of shear rate x time [42]... 

Caustic Waterflooding. In caustic waterflooding, the interfacial rheologic properties of a model crude oil-water system were studied in the presence of sodium hydroxide. The interfacial viscosity, the non-Newtonian flow behavior, and the activation energy of viscous flow were determined as a function of shear rate, alkali concentration, and aging time. The interfacial viscosity drastically... 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, q0. Flow curves (plots of log q vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

Fig. 19. Shear stress and first normal stress difference plotted as a function of shear rate for different molar masses, and b at different concentrations of polystyrene in toluene... 

Fig. 2.8.16 Director orientation, 0, as a function of shear rate for both flow aligning (solid squares) and tumbling (open squares 325 K, solid circles 328 K and open circles 333 K) nematic polymers. (From Siebert et al. [10].)...

Utilization of a microfabricated rf coil and gradient set for viscosity measurements has recently been demonstrated [49]. Shown in Figure 4.7.9 is the apparent viscosity of aqueous CMC (carboxymethyl cellulose, sodium salt) solutions with different concentrations and polymer molecular weights as a function of shear rate. These viscosity measurements were made using a microfabricated rf coil and a tube with id = 1.02 mm. The shear stress gradient, established with the flow rate of 1.99 0.03 pL s-1 was sufficient to observe shear thinning behavior of the fluids. 

Newtonian flow, and their viscosity is not constant but changes as a function of shear rate and/or time. The rheological properties of such systems cannot be defined simply in terms of one value. These non-Newtonian phenomena are either time-independent or time-dependent. In the first case, the systems can be classified as pseudoplastic, plastic, or dilatant, in the second case as thixotropic or rheopective. 

Fig. 3 Newtonian and non-Newtonian behaviours as a function of shear rate (a) flow profile (b) viscosity profile. (From Ref. 65.)...

Figure 14 gives limiting viscosity numbers for hydrolyzed copolymer 11 as a function of shear rate. Since limiting viscosity number is a function of molecular size, these data show that solution pseudoplasticity occurs because of compaction of the solvated polymer with increasing shear. 

Effective viscosity as a function of shear rate for 0.15 g/dL of copolymer 5 in distilled water is given in Figure 15 The Ostwald-DeWaele exponent for copolymer solutions is greater than that of matching hydrolyzed copolymer solutions at a given concentration. Thus, copolymer molecules are less compactable in solution than are their hydrolyzed derivatives, and pseudoplasticity of polymer solutions increases upon hydrolysis. 

Theoretical collision efficiencies as a function of shear rate and particle size were based on theory derived for spherical particles in simple shear (20,21). Effective floe size radii (22) were calculated according to... 

Just as the variation of vx with r was unknown, so is the variation of y. However, if the fluid is time-independent and homogeneous, the shear stress is a function of shear rate only. The inverse is that the shear rate y is a function of shear stress rrx only and the variation of rrx with r is known ... 

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