Generalized newtonian models

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

Simulation of the Couette flow of silicon rubber - generalized Newtonian model... 

The generalized Newtonian model over-predicted the rotational flow rates and pressure gradients for the channel for most conditions. This over-prediction was caused in part by the utilization of drag flow shape factors (FJ that were too large. Then in order for the sum of the rotational and pressure flows to match the actual flow In the channel, the pressure gradient was forced to be higher than actually required by the process. It has been known for a long time [9] that the power law... 

The generalized Newtonian model has been used successfully for many years for the design of small diameter screws with relatively shallow channels. Its success has brought considerable value to those designers who understood the method and its limitations. An improved method for calculating flows is presented in the next section. 

The numerical and experimental results presented here show that the standard generalized Newtonian model must be used with caution when predicting the pres-... 

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. 

R. B. Bird, Experimental Tests of Generalized Newtonian Models Containing a Zero-Shear Viscosity and a Characteristic Time, Can. J. Chem. Eng. (43) 161,1965. 

For purely viscous fluids, the rheological constitutive equation that relates the stresses x to the velocity gradients is the generalized Newtonian model [5,6,21] and is written as... 

With the aim to theoretically investigate the effect of molecular weight and long chain branching, we have used recently proposed generalized Newtonian model [5] ... 

This generalized Newtonian model has been used together with the film blowing model in the same way as described in [5] for the Newtonian model. In this case, however, the iteration numerical scheme has been required. 

Figure 4 Theoretical investigation of the molecular weight (4a) and long chain branching (4b) on the high stalk bubble formation by the help of the Zatloukal-Vlcek film blowing model and generalized Newtonian model.

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

MODELLING OF STEADY STATE STOKES FLOW OF A GENERALIZED NEWTONIAN FLUID... 

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)... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Non-Newtonian modeling capability Choice of models available, user subroutines. Power law. and difficult to implement user subroutines. Choice of models available. Not available. Power law. Bingham. Generalized power law. and user subroutines. 

Chapter 4 describes in general terms the processing methods which can be used for plastics. All the recent developments in this area have been included and wherever possible the quantitative aspects are stressed. In most cases a simple Newtonian model of each of the processes is developed so that the approach taken to the analysis of plastics processing is not concealed by mathematical complexity. 

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