Rheology Cross equation

Lai et al. [100] proposed the use of the Dow Rheology Index (DRI) as an indicator for comparing branching level in industrial polymers. For a linear polymer molecule, like unbranched polyethylene, the viscosity of the polymer as a function of the applied shear rate is given by the Cross equation [84,100],... 

At the percolation limit, the rheological behavior of the suspension changes from Newtonian to either the Cross equation with a low shear limit viscosity or the Bingham plastic equation with an apparent yield... 

FIGURE 12S Schematic of the low shear viscosity of TiOj as a function of pH. Near the zero point of charge (ZPC) the rheology is non-Newtonian for dilute suspensions, conforming to the Cross equation, which suggests that aggregation is responsible for this increase in viscosity. Away from the ZPC, the rheology is Newtonian for dilute suspensions. 

If we are to use a direct analogy of suspension rheology to the Cross equation derived for polymer solutions, we should consider that the... 

In this chapter, we have presented the rheological behavior of homopolymers, placing emphasis on the relationships between the molecular parameters and rheological behavior. We have presented a temperature-independent correlation for steady-state shear viscosity, namely, plots of log ri T, Y) r](jiT) versus log or log j.y, where Tq is a temperature-dependent empirical constant appearing in the Cross equation and a-Y is a shift factor that can be determined from the Arrhenius relation for crystalline polymers in the molten state or from the WLF relation for glassy polymers at temperatures between and + 100 °C. 

It is often sufficient for the technologist to know the difference between the pressures, which is required to extrude the gas-liquid medium through the channel. This problem can be solved, at least for flows in straight pipes with an unchanging cross section, without resorting to rheological equations of such two-phase media. This idea is based on two concessions ... 

Cross, M. M. J. Colloid Sci. 20 (1965) 417. Rheology of non-Newtonian fluids a new flow equation for pseudoplastic systems. 

Reynolds apparatus for tracing flow patterns 59 Rheogram 105, 197 Rheological equation (Cross) 110 Rheology 105, 195... 

Cross, M.M., Rheology of Non-Newtonian Fluids A New Flow Equation for Pseudoplastic Systems, /. Colloid ScL, 20, 417 (1965)... 

The melt flow under isothermal conditions, when it is described by the rheological equation for the Newtonian or power law liquid, has been studied in detail63 66). The flow of the non-Newtonian liquid in the channels of non-round cross section for the liquid obeying the Sutterby equation have also been studied 67). In particular, the flow in the channels of rectangular and trigonal cross section was studied. In the analysis of the non-isothermal flow, attention should be paid to the analysis 68) of pseudo-plastic Bingham media. 

If the capillary has a slit cross-section of thickness h, which is very small compared with the slit width w, then the basic rheological equations are as follows for shear stress and shear rate taken to be very near the wall ... 

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

Both the Carreau and the Cross models can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given in Equation (2.16) was employed in the study of the rheological behavior of glass-filled polymers (Poslinski et al., 1988) ... 

Equations describing velocity profiles can be used, among other applications, to study the effect of different rheological models on the distribution of velocities and to understand the concept of residence-time distribution across the cross-section of a pipe or a channel. 

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