Non-Newtonian behaviour

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. 

The term viscosity has no meaning for a non-Newtonian fluid unless it is related to a particular shear rate y. An apparent viscosity fia can be defined as follows (using the negative sign convention for stress)  

From such a flow curve, the apparent viscosity can be calculated at any 

Some very concentrated suspensions are dilatant. If, in such a suspension, the particles are closely packed, then when the suspension is sheared the particles have to adopt a greater spacing in order to move past neighbouring particles and as a result the suspension expands, ie it dilates. Dilatant materials tend to be shear thickening but it does not follow that shear thickening behaviour is necessarily due to dilatancy. Consequently, dilatancy should not be used as a synonym for shear thickening behaviour. 

It should be noted that for shear thinning and shear thickening behaviour the shear stress-shear rate curve passes through the origin. This type of behaviour is often approximated by the power law and such materials are called power law fluids . Using the negative sign convention for stress components, the power law is usually written as 

In addition to the general decrease in viscosity with increasing temperature, heating milk can also influence its rheology by heat-induced denatura-tion of cryoglobulins and/or other whey proteins. Concentration of milk, e.g. by ultrafiltration, prior to heating results in a greater increase in i/ pp than in milk heated before concentration. 

The addition of hydrocolloids (e.g. carrageenans, pectins or car-boxymethyl cellulose) as thickening agents will greatly increase the apparent viscosity of the product. The production of extracellular polysaccharides by certain bacteria will also increase the viscosity of milk products. 

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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. 

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... 

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. 

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.)...

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. 

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. 

The presence of fillers in viscous polymer melts not only increases their viscosity but also infiuences their shear rate dependency, especially with non-spherical particles (fibrous or fiake-like) which become oriented in the fiow field. As Fig. 6 shows, particle orientation increases the non-Newtonian behaviour which commences at a lower rate of shear than for unfilled melt. 

The experimental evidence concerning the effects of LCB on the non-Newtonian behaviour of polyethylene melts is not as extensive as might be wished. Guillet and co-workers (167) studied fractions of both linear and branched polyethylenes and found that, for a given low shear-rate viscosity. 

Mendelson (169) studied the effect of LCB on the flow properties of polyethylene melts, using two LDPE samples of closely similar M and Mw plus two blends of these. Both zero-shear viscosity and melt elasticity (elastic storage modulus and recoverable shear strain) decreased with increasing LCB, in this series. Non-Newtonian behaviour was studied and the shear rate at which the viscosity falls to 95% of the zero shear-rate value is given this increases with LCB from 0.3 sec"1 for the least branched to 20 sec"1 for the most branched (the text says that shear sensitivity increases with branching, but the numerical data show that it is this shear rate that increases). This comparison, unlike that made by Guillet, is at constant Mw, not at constant low shear-rate viscosity. 

Ham and co-workers (173) compared a LDPE with three HDPE samples, the MJMn ratios of which bracketted that of the LDPE their non-Newtonian behaviour, as measured by the exponent in a power-law relationship between stress and shear rate, also bracketted that of the LDPE. However, the LDPE had considerably higher melt elasticity than the HDPE, which was ascribed to LCB. 

Wolff C (1982) Non-Newtonian behaviour of associations of macromolecules in dilute solutions Adv Coll Interf Sci 17 263... 

As we already know, viscosity is affected by four factors. These factors can be the reason that the flow behaviour is non-Newtonian for those masses %/v is not constant. Figures 9.18, 9.19 and 9.20 are graphic representations of Newtonian flow behaviour and some forms of non-Newtonian behaviour. 

The model is based on the idea that the glassy phase is composed of two layers - a normal glassy phase layer behaving in a Newtonian way, embedded into an over-condensed layer with non-Newtonian behaviour. Thus, for stresses lower than the critical stress, the creep is controlled by the normal glassy phase (n= 1), and when the stress exceeds a critical value, the squeeze of this phase makes the two over-condensed layers come into contact, thus the material creeps in a non-Newtonian way (n = 0.5). The creep rate is written ... 

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