Rate and shear

Therefore, the viscosity can be determined from the torque and angular velocity. However, the viscosity is usually calculated from the shear rate and shear stress, which can be obtained from the Margules equation. The shear rate is given by equation 27, where r is any given radius. 

The shear rate and shear stress can be calculated for any radius r from these equations. In most cases the radius used is R because the shear stress and shear rate of interest are at the inner, torque-sensing cylinder. Thus equations 27 and 28 become... 

From equation 3.123 when n > 1. pa increases with increase of shear rate, and shear-thickening behaviour is described ... 

In order to overcome the shortcomings of the power-law model, several alternative forms of equation between shear rate and shear stress have been proposed. These are all more complex involving three or more parameters. Reference should be made to specialist works on non-Newtonian flow 14-171 for details of these Constitutive Equations. 

As the shear rate and shear stress are radially dependent, each radial position provides a viscosity data point 77(7) at a different shear rate ... 

The flow behaviour of aqueous coating dispersions, because of their high pigment and binder content, is often complex. They have viscosities which are not independent of the shear rate and are therefore non-Newtonian. Shear thickening (when the viscosity of the dispersion increases with shear rate) and shear thinning or pseudoplastic behaviour (when the viscosity decreases with shear rate), may... 

Figure 2.2 Relationships between shear rate and shear stress for Newtonian and non-Newtonian fluids.

To avoid the apparent complications with absolute rheologic measurement techniques, a number of investigators (4,5). have used relative measurement systems to make rheologic measurements. The major difference between the relative and absolute measurement techniques is that the fluid mechanics in the relative systems are complex. The constitutive equations needed to find the fundamental rheologic variables cannot be readily solved. Relative measurement systems require the use of Newtonian and non-Newtonian calibrations fluids with known properties to relate torque and rotational speed to the shear rate and shear stress (6). 

Newtonian and non-Newtonian calibration fluids were used to determine the necessary calibration constants for the impeller method. It has been previously determined that the impeller method is only valid for a Reynolds number (Re) <10. Impeller rotational speed and torque data from Newtonian calibration fluids of known viscosity were employed to determine the Newtonian calibration constant, c. Cone-and plate-viscometer data from non-Newtonian calibration fluids were used to determine a viscosity vs shear rate relationship. Impeller rotational speed and torque data of the non-Newtonian calibration fluids combined with a determined viscosity vs shear rate correlation were utilized to calculate the shear rate constant, k. The impeller method calibration constants allow the calculation of viscosity, shear rate, and shear stress data of non-Newtonian suspensions. Metz et al. (2) have thoroughly discussed the equations utilized in the impeller method. 

Fig. E2.5b Schematic representation velocity shear rate and shear stress profiles of a Newtonian fluid between parallel plates.

Figure E2.5b depicts the shear rate and shear stress profiles normalized by the pure drag flow values for a number of pressure-to-drag flow ratios.

The extensional melt behavior was assessed with the new SER Universal Testing Platform from X-pansion Instruments, described by Sentmanat (53,54) and shown in Fig. 12. 25. The obtained tensile stress of the two resins at 170°C and extensional rate of 20 s-1 are shown on Fig. 12.26. It is evident that Resin E has a higher modulus and higher tensile stress values, at a given strain below yield, than Resin C. From this, and the experimental data discussed previously, we see that the values of the critical shear rate and shear stress for the onset of sharkskin fracture are inversely proportional to the magnitude of the tensile stress of the resins. This suggests that the rapid increase... 

If the material to be processed is subject to shear thinning, the linear relationships for the pressure and energy behavior illustrated above no longer apply. With shear thinning, there is a non-linear relationship between the shear rate and shear stress that is reflected in the flow curve (see Chapter 3). As a rule, the zero viscosity and one or two rheological time constants are enough to describe the flow curve with sufficient accuracy. The Carreau equation is often used it contains a dimensionless flow exponent in addition to the zero viscosity and a rheological time constant. 

Many of the comments in the previous chapter about the selection of grade, additives and mixing before moulding apply equally in preparation for extrusion. It is important of course that the material should be appropriate for the purpose, uniform, dry, and free from contamination. It should be tested for flow and while many tests have been devised for this it is convenient to classify them as either for low or high rates of shear. The main terms used in such testing ( viscosity , shear rate , shear strain , etc.) are defined in words and expressed as formulae in ISO 472, and it is not necessary to repeat them here. Viscosity may be regarded as the resistance to flow or the internal friction in a polymer melt and often will be measured by means of a capillary rheometer, in which shear flow occurs with flow of this type—one of the most important with polymer melts—when shearing force is applied one layer of melt flows over another in a sense that could be described as the relationship between two variables—shear rate and shear stress.1 In the capillary rheometer the relationship between the measurements is true only if certain assumptions are made, the most important of which are ... 

An article by Karam (1) gives typical data to illustrate the difference between shear rate and shear stress. Table II is extracted from cross plots of their data, showing the shear rate required with different continuous phase viscosities and one dispersed phase viscosity to break up a second fluid of the same size droplet. This shows that the shear stress in grams per centimeter squared is the basic parameter and the viscosity and shear rate are inversely proportional to give the required shear stress. 

The effect of particle shape on the filtration rate and shear strength of quartz and dolomite mineral filter cakes... 

Viscosity, is the internal friction of a fluid or its tendency to resist flow. It is denoted by the symbol t] for Newtonian fluids, whose viscosity does not depend on the shear rate, and for non-Newtonian fluids to indicate shear rate dependence by Depending on the flow system and choice of shear rate and shear stress, there are several equations to calculate. Here, it is defined by the equation ... 

Yoo, B., Rao, M. A., and Steffe, J. F. 1995. Yield stress of food suspensions with the vane method at controlled shear rate and shear stress. J. Texture Stud. 26 1-10. 

An outline of the steps involved to derive the equations for shear rate and shear stress for fully developed flow in a tube is given in Appendix 3-B. The shear stress (ctw) is given by Equation (3.34) and the shear rate by Equation 3.35, where the subscript w is used to emphasize that the values obtained are those at the pipe wall. 

One can now construct diagrams of shear rate versus shear stress for non-Newtonian foods. Vitali and Rao (1982) obtained volumetric flow versus pressure drop data given in Table 3-3, that were then converted to shear rate and shear stress values shown in Figure 3-18 and also listed in the table. 

Other types of viscometers, sueh as an oseillation viscometer, that are useful for eharaeterizing Newtonian foods are also available. However, their use for characterizing non-Newtonian fluid foods can be justified only if the complex flow fields can be analyzed and expressions are derived for the shear rate and shear stress. 

Time-Dependent Non-Newtonian Fluids. Time-dependent non-Newtonian fluids are characterized by the property that their viscosities are a function of both shear rate and shear history. 

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