Generalized Newtonian behaviour

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. 

Figure 1.2 Comparison of the rheological behaviour of Newtonian and typical generalized Newtonian fluids...

Typical rheograms representing the behaviour of various types of generalized Newtonian fluids are shown in Figure 1.2. 

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... 

Liquids of complex structure, such a polymer solutions and melts, and pseudo-homogeneous suspensions of fine particles, will generally exhibit non-Newtonian behaviour, with their apparent viscosities depending on the rate at which they are sheared, and the time for which they have been subjected to shear. They may also exhibit significant elastic... 

Because concentrated flocculated suspensions generally have high apparent viscosities at the shear rates existing in pipelines, they are frequently transported under laminar flow conditions. Pressure drops are then readily calculated from their rheology, as described in Chapter 3. When the flow is turbulent, the pressure drop is difficult to predict accurately and will generally be somewhat less than that calculated assuming Newtonian behaviour. As the Reynolds number becomes greater, the effects of non-Newtonian behaviour become... 

Use of this generalized Reynolds number was suggested by Metzner and Reed (1955). For Newtonian behaviour, K = n and n = 1 so that the generalized Reynolds number reduces to the normal Reynolds number. 

In practice, most polymers vary by a factor between 1.5 and 5 over a temperature difference of 30 °C. With non-Newtonian behaviour it is important how the temperature dependence, e.g. the value of the energy of activation E, is defined, namely for constant shear stress or for constant shear rate. Constant stress T gives, in general, a more constant value for E, since the log q - log r curves shifts, upon temperature change, in the vertical direction (just as with a change in M), whereas, with constant y the pattern is more deformed (see Figure 5.12). 

Under actual melt-spinning conditions, the situation is far more complicated as the velocity always increases (D decreases), while the fluid generally does not show Newtonian behaviour. Moreover, the temperature decreases, so that the physical properties change in the course of the process. 

Rheological studies on cement pastes and concrete have been reviewed in a book (T42) and several shorter articles (H40,L40,S76). The results in the literature show wide variations, many of which reflect the large effects of seemingly minor differences in experimental technique. Since cement pastes do not show Newtonian behaviour, methods giving only a single parameter are inadequate. Viscometers in which the material is studied in shear between a shallow cone and a plate, or between concentric cylinders, have generally been used. 

The viscosity is defined as the shear force per unit area necessary to achieve a velocity gradient of unity. Equation 5.2 applies to the majority of fluids, and they are generally known as Newtonian fluids, or fluids that display Newtonian behaviour. There are exceptions, and some fluids (usually liquids) do not conform to Equation 5.2, and these are generally classified as non-Newtonian fluids although within this grouping there is a sub classification with distinctly different "viscosity" behaviour for the fluids within the different groups. 

The scaling up or down of mixing systems with Newtonian rheological characteristics is comprehensively covered in the literature and in this book. However, the majority of emulsions with medium phase volume (30-55%) exhibit non-Newtonian behaviour. Literature is bereft of methods to scale up non-Newtonian systems. An approach is presented in Appendix 2 based on a generalized scale-up equation for agitated systems where ... 

Figure 13.2 Generalized flow curve for non-Newtonian behaviour. (A) shear-thinning (pseudoplastic) behaviour with low shear Newtonian plateau (B) shear-thinniiig behaviour with high shear Newtonian plateau i oo (C) dilatant (shear thidceniiig) behaviour...

For all fluids, the nature of the flow is governed by the relative importance of the viscous and the inertial forces. For Newtonian fluids, the balance between these forces is characterised by the value of the Reynolds munber. The generally accepted value of the Reynolds number above which stable laminar flow no longer occms is 2100 for Newtonian fluids. For time-independent fluids, the critical value of the Reynolds number depends upon the type and the degree of non-Newtonian behaviour. For power-law fluids (n = n ), the criterion of Ryan and Johnson [1959] can be used. 

Diluted solutions of high molecular weight polymers display generally in a laminar simple shear floW a shearthinning effect which is a peculiar case of non Newtonian behaviour CLohmander (I), Wolff (2), Kotaka et al. (3) Yang (4)]. 

Polymer solutions virtually always show non-Newtonian flow behaviour at sufficiently high polymer concentrations. However, before discussing this type of fluid in more detail, an outline of how non-Newtonian behaviour fits into the general area of fluid mechanics is presented. This is done by considering the total stress tensor and the equation of momentum conservation. 

Experimentally, Newton s law is generally followed only by gases and by simple liquids in laminar flow. It is only at a very low deformation rate and shear stress that solutions of macromolecules, molten polymers, emulsions and suspensions, display approximately Newtonian behaviour. 

In this Section we describe in detail the shear flow examples discussed by Leslie [163] for nematic liquid crystals. We first make some comments on Newtonian and non-Newtonian behaviour of fluids in Section 5.5.1 before going on to derive the general explicit governing equations for shear flow, equations (5.121) and (5.122), in Section 5.5.2. We then specialise in Sections 5.5.3 and 5.5.4 to the specific problems of shear flow near a boundary and shear flow between parallel plates. Section 5.5.5 discusses some scaling properties for nematics. 

The non-Newtonian behaviour of nematic Uquid crystals will become evident from the form of the stress tensor given via equation (4.128) and the viscous stress derived in, for example, equations (5.100) to (5.103). We remark that, in general, it is necessary but not sufficient for the graph of (7 to be a straight line through the origin for a fiuid to be Newtonian however, it is sufficient for any one of the above four requirements to be violated for a fluid to be regarded as non-Newtonian [86, pp.9 13]. 

Chapter 4 describes in general terms the processing methods which can be used for plastics 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. Chapter 5 deals with the aspects of the flow behaviour of polymer melts which are relevant to the processing methods. The models are developed for both Newtonian and Non-Newtonian (Power Law) fluids so that the results can be directly compared. 

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