Newtonian behaviour

Under certain conditions (e.g. moderate shear rates, at fat contents below 40% and at temperatures above 40°C, at which the fat is liquid and no cold agglutination occurs) milk, skim milk and cream are, in effect, fluids with Newtonian rheological properties. Newtonian behaviour can be described by the equation  

The coefficient of viscosity for whole milk at 20°C, but not affected by cold agglutination of fat globules, is about 2.127mPas. Values for water and milk plasma at 20°C are 1.002 and 1.68 mPas, respectively. Casein, and to a lesser extent fat, are the principal contributors to the viscosity of milk whey proteins and low molecular mass species have less influence. 

The viscosity of milk and Newtonian milk products is influenced by composition, concentration, pH, temperature, thermal history and processing operations. 

The Newtonian coefficient of viscosity at a given temperature for milk, creams and some concentrated milk products is related to the concentration of individual components by Eiler s equation  

The viscosity of milk and creams tends to increase slightly with age, due in part to changes in ionic equilibria. Heating skim milk to an extent that denatures most of the whey proteins increases its viscosity by about 10%. Homogenization of whole milk has little effect on its viscosity. The increase in the volume fraction of fat on homogenization is compensated by a decrease in the volume fractions of casein and whey proteins because some skim milk proteins are adsorbed at the fat-oil interface. Pasteurization has no significant effect on the rheology of whole milk. 

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Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

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. 

In the specific case of polymer melts these almost invariably are of the pseudoplastic type. In such cases the flow behaviour index n is less than 1 the greater the divergence from Newtonian behaviour the lower its value. 

As a complication some sources define a flow index as the reciprocal of that defined above so that some care has to be taken in interpretation. In such cases the values are greater than unity for polymer melts and the greater the value the greater the divergence from Newtonian behaviour.)... 

Equations (5.21), (5.22) and (5.23) are useful for the high strain rates experienced in injection moulding or extrusion but unfortunately they do not predict the low strain rate situation very well where plastic melts tend towards Newtonian behaviour (ie n -) 1). This is illustrated in Fig. 5.7. 

The above correlations may not be valid for non-Newtonian behaviour of biological fluids, nor for the effect of antifoam or the presence of solids. A correlation proposed in the literature as stated in (3.6.4)3 may be true for aerobic non-Newtonian fluid filamentous media of fermentation broth. 

The calculation of heat transfer film coefficients in an air-lift bioreactor is more complex, as small reactors may operate under laminar flow conditions whereas large-scale vessels operate under turbulent flow conditions. It has been found that under laminar flow conditions, the fermentation broths show non-Newtonian behaviour, so the heat transfer coefficient can be evaluated with a modified form of the equation known as the Graetz-Leveque equation 9... 

All gases and most liquids of simple molecular structure exhibit what is termed Newtonian behaviour, and their viscosities are independent of the way in which they are flowing. Temperature may, however, exert a strong influence on viscosity which, for highly viscous liquids, will show a rapid decrease as the temperature is increased. Gases, show the reverse tendency, however, with viscosity rising with increasing temperature, and also with increase of pressure. 

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... 

Many fluids, including some that are encountered very widely both industrially and domestically, exhibit non-Newtonian behaviour and their apparent viscosities may depend on the rate at which they are sheared and on their previous shear history. At any position and time in the fluid, the apparent viscosity pa which is defined as the ratio of the shear stress to the shear rate at that point is given by ... 

An understanding of non-Newtonian behaviour is important to the chemical engineer from two points of view. Frequently, non-Newtonian properties are desirable in that they can confer desirable properties on the material which are essential if it is to fulfil the purpose for which it is required. The example of paint has already been given. Toothpaste should not flow out of the tube until it is squeezed and should stay in place on the brush until it is applied to the teeth. The texture of foodstuffs is largely attributable to rheology. 

Second, it is necessary to take account of non-Newtonian behaviour in the design of process plant and pipelines. Heat and mass transfer coefficients are considerably affected by the behaviour of the fluid, and special attention must be devoted to the selection of appropriate mixing equipment and pumps. 

In this section, some of the important aspects of non-Newtonian behaviour will be quantified, and some of the simpler approximate equations of state will be discussed. An attempt has been made to standardise nomenclature in the British Standard, BS 511 8 1 i. 

In this section, consideration will be given to the equilibrium relationships between shear stress and shear rate for fluids exhibiting non-Newtonian behaviour. Whenever the shear stress or the shear rate is altered, the fluid will gradually move towards its new equilibrium state and, for the present, the period of adjustment between the two equilibrium states will be ignored. 

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter /z, the viscosity of the fluid. No single parameter model will describe non-Newtonian behaviour and models involving two or even more parameters only approximate to the characteristics of real fluids, and can be used only over a limited range of shear rates. 

Thus, by selecting an appropriate value of n, both shear-thinning and shear-thickening behaviour can be represented, with n = 1 representing Newtonian behaviour which essentially marks the transition from shear-thinning to shear-thickening characteristics. 

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... 

A fluid which exhibits non-Newtonian behaviour is flowing in a pipe of diameter 70 mm and the pressure drop over a 2 m length of pipe is 4 x 104 N/m2. A pitot lube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. Calculate the power-law index n and consistency coefficient K. 

The Non-Newtonian behaviour, i.e. the decrease of the viscosity as a function of the shear rate, becomes increasingly important when the polymer concentration and molecular weight... 

These flow features are of importance in a great number of technical processes, especially for high process velocities when extremely high shear rates can be observed. For polymeric systems this can lead to a so-called non-Newtonian behaviour, i.e. the rheological material functions become dependent on the shear or elongational rate. 

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

For a Newtonian fluid, the shear stress is proportional to the shear rate, the constant of proportionality being the coefficient of viscosity. The viscosity is a property of the material and, at a given temperature and pressure, is constant. Non-Newtonian fluids exhibit departures from this type of behaviour. The relationship between the shear stress and the shear rate can be determined using a viscometer as described in Chapter 3. There are three main categories of departure from Newtonian behaviour behaviour that is independent of time but the fluid exhibits an apparent viscosity that varies as the shear rate is changed behaviour in which the apparent viscosity changes with time even if the shear rate is kept constant and a type of behaviour that is intermediate between purely liquid-like and purely solid-like. These are known as time-independent, time-dependent, and viscoelastic behaviour respectively. Many materials display a combination of these types of behaviour. 

Clearly, shear thinning behaviour corresponds to nshear thickening behaviour to n> 1. The special case, n = 1, is that of Newtonian behaviour and in this case the consistency coefficient K is identical to the viscosity fx. Values of n for shear thinning fluids often extend to 0.5 but less commonly can be as low as 0.3 or even 0.2, while values of n for shear thickening behaviour usually extend to 1.2 or 1.3. 

Under conditions of steady fully developed flow, molten polymers are shear thinning over many orders of magnitude of the shear rate. Like many other materials, they exhibit Newtonian behaviour at very low shear rates however, they also have Newtonian behaviour at very high shear rates as shown in Figure 1.20. The term pseudoplastic is used to describe this type of behaviour. Unfortunately, the same term is frequently used for shear thinning behaviour, that is the falling viscosity part of the full curve for a pseudoplastic material. The whole flow curve can be represented by the Cross model [Cross (1965)] ... 

The final main category of non-Newtonian behaviour is viscoelasticity. As the name implies, viscoelastic fluids exhibit a combination of ordinary liquid-like (viscous) and solid-like (elastic) behaviour. The most important viscoelastic fluids are molten polymers but other materials containing macromolecules or long flexible particles, such as fibre suspensions, are viscoelastic. An everyday example of purely viscous and viscoelastic behaviour can be seen with different types of soup. When a thin , watery soup is stirred in a bowl and the stirring then stopped, the soup continues to flow round the bowl and gradually comes to rest. This is an example of purely viscous behaviour. In contrast, with certain thick soups, on cessation of stirring the soup rapidly slows down and then recoils slightly. 

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. 

Figure 3.10 shows a friction factor - Reynolds number chart for Bingham plastics at various values of the Hedstrom number. The turbulent flow line is that for Newtonian behaviour and is followed by some Bingham plastics with low values of the yield stress [Thomas (1962)]. 

For Molecular weight determination by viscometry we do not need absolute h value, viscosity measurements may be carried out in simple Ostwald Viscometer. Because of (the non-Newtonian behaviour of most macromolecular solutions at high velocity gradients in the capillary, the viscometer dimensions are chosen in such a manner that the viscosity gradient is the smallest possible. 

The melt flow index is a useful indication of the molar mass, since it is a reciprocal measure of the melt viscosity p. p depends very strongly on 77 ( ) (doubling of results in a 10.6 times higher 77 ). This relation is valid for the zero-shear viscosity the melt index is measured at a shear stress where the non-Newtonian behaviour, and thus the width of the molar mass distribution, is already playing a part (see MT 5.3.2). The melt index is a functional measure for the molar mass, because for a producer of end products the processability is often of primary importance. 

As a result of the non-Newtonian behaviour both expressions for the pressure flows bxpirj and cxplrj) are no longer valid. The curve for the die is now curled upward since the apparent viscosity decreases with increasing shear stress. Also the shape of the screw characteristic changes. 

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