12Classification of fluid behaviour
1.2.1 Definition of a Newtonian fluid
Consider a thin layer of a fluid contained between two parallel planes a distance dy apart, as shown in Figure 1.1. Now, if under steady state conditions, the fluid is subjected to a shear by the application of a force F as shown, this will be balanced by an equal and opposite internal frictional force in the fluid. For an incompressible Newtonian fluid in laminar flow, the resulting shear stress is equal to the product of the shear rate and the viscosity of the fluid medium. In this simple case, the shear rate may be expressed as the velocity gradient in the direction perpendicular to that of the shear force, i.e.
Note that the first subscript on both r and p indicates the direction normal to that of shearing force, while the second subscript refers to the direction of the force and the flow. By considering the equilibrium of a fluid layer, it can
readily be seen that at any shear plane there are two equal and opposite shear stressesa positive one on the slower moving fluid and a negative one on the faster moving fluid layer. The negative sign on the right hand side of equation (1.1) indicates that ryx is a measure of the resistance to motion. One can also view the situation from a different standpoint as: for an incompressible fluid of density p, equation (1.1) can be written as:
P dy
The quantity ' pVx' is the momentum in the xdirection per unit volume of the fluid and hence ryx represents the momentum flux in the ydirection and the negative sign indicates that the momentum transfer occurs in the direction of decreasing velocity which is also in line with the Fourier's law of heat transfer and Fick's law of diffusive mass transfer.
The constant of proportionality, a (or the ratio of the shear stress to the rate of shear) which is called the Newtonian viscosity is, by definition, independent of shear rate (yyx) or shear stress (ryx) and depends only on the material and its temperature and pressure. The plot of shear stress (ryx) against shear rate (yyx) for a Newtonian fluid, the socalled 'flow curve' or 'rheogram', is therefore a straight line of slope, a, and passing through the origin; the single constant, a, thus completely characterises the flow behaviour of a Newtonian fluid at a fixed temperature and pressure. Gases, simple organic liquids, solutions of low molecular weight inorganic salts, molten metals and salts are all Newtonian fluids. The shear stressshear rate data shown in Figure 1.2 demonstrate the Newtonian fluid behaviour of a cooking oil and a corn syrup; the values of the viscosity for some substances encountered in everyday life are given in Table 1.1.
Figure 1.1 and equation (1.1) represent the simplest case wherein the velocity vector which has only one component, in the xdirection varies only in the ydirection. Such a flow configuration is known as simple shear flow. For the more complex case of three dimensional flow, it is necessary to set up the appropriate partial differential equations. For instance, the more general case of an incompressible Newtonian fluid may be expressed  for the xplane  as
dx 3
dVy dVz dy
dVy dx dVz dx
Similar sets of equations can be drawn up for the forces acting on the y and zplanes; in each case, there are two (inplane) shearing components and a xx
Table 1.1 Typical viscosity values at room temperature
Substance ß
Air 102
Benzene 0.65
Water 1
Molten sodium chloride (1173 K) 1.01
Ethyl alcohol 1.20
Mercury (293 K) 1.55
Molten lead (673 K) 2.33
Ethylene glycol 20
Olive oil 100
Castor oil 600
100% Glycerine (293 K) 1500
Honey 104
Corn syrup 105
Bitumen 1011
Molten glass 1015
1' 
tzy 
pj 
tzx 
y y 
Figure 1.3 Stress components in three dimensional flow y t y xy xz x z normal component. Figure 1.3 shows the nine stress components schematically in an element of fluid. By considering the equilibrium of a fluid element, it can easily be shown that xyx = rxy; xxz = rzx and xyz = xzy. The normal stresses can be visualised as being made up of two components: isotropic pressure and a contribution due to flow, i.e. Pxx p C rxx Pyy p C ryy Pzz p C rzz where r^, ryy, rzz, contributions arising from flow, are known as deviatoric normal stresses for Newtonian fluids and as extra stresses for nonNewtonian fluids. For an incompressible Newtonian fluid, the isotropic pressure is given by For a Newtonian fluid in simple shearing motion, the deviatoric normal stress components are identically zero, i.e. Thus, the complete definition of a Newtonian fluid is that it not only possesses a constant viscosity but it also satisfies the condition of equation (1.9), or simply that it satisfies the complete NavierStokes equations. Thus, for instance, the socalled constant viscosity Boger fluids [Boger, 1976; Prilutski etal., 1983] which display constant shear viscosity but do not conform to equation (1.9) must be classed as nonNewtonian fluids. 1.2.2 NonNewtonian fluid behaviourA nonNewtonian fluid is one whose flow curve (shear stress versus shear rate) is nonlinear or does not pass through the origin, i.e. where the apparent viscosity, shear stress divided by shear rate, is not constant at a given temperature and pressure but is dependent on flow conditions such as flow geometry, shear rate, etc. and sometimes even on the kinematic history of the fluid element under consideration. Such materials may be conveniently grouped into three general classes: (1) fluids for which the rate of shear at any point is determined only by the value of the shear stress at that point at that instant; these fluids are variously known as 'time independent', 'purely viscous', 'inelastic' or 'generalised Newtonian fluids', (GNF); (2) more complex fluids for which the relation between shear stress and shear rate depends, in addition, upon the duration of shearing and their kinematic history; they are called 'timedependent fluids', and finally, p =4 (Pxx + Pyy + Pzz) From equations (1.6) and (1.7), it follows that (3) substances exhibiting characteristics of both ideal fluids and elastic solids and showing partial elastic recovery, after deformation; these are categorised as 'viscoelastic fluids'. This classification scheme is arbitrary in that most real materials often exhibit a combination of two or even all three types of nonNewtonian features. Generally, it is, however, possible to identify the dominant nonNewtonian characteristic and to take this as the basis for the subsequent process calculations. Also, as mentioned earlier, it is convenient to define an apparent viscosity of these materials as the ratio of shear stress to shear rate, though the latter ratio is a function of the shear stress or shear rate and/or of time. Each type of nonNewtonian fluid behaviour will now be dealt with in some detail. 1.3 Timeindependent fluid behaviour In simple shear, the flow behaviour of this class of materials may be described by a constitutive relation of the form, or its inverse form, This equation implies that the value of yyx at any point within the sheared fluid is determined only by the current value of shear stress at that point or vice versa. Depending upon the form of the function in equation (1.10) or (1.11), these fluids may be further subdivided into three types: (a) shearthinning or pseudoplastic (b) viscoplastic (c) shearthickening or dilatant Qualitative flow curves on linear scales for these three types of fluid behaviour are shown in Figure 1.4; the linear relation typical of Newtonian fluids is also included. 1.3.1 Shearthinning or pseudoplastic fluidsThe most common type of timeindependent nonNewtonian fluid behaviour observed is pseudoplasticity or shearthinning, characterised by an apparent viscosity which decreases with increasing shear rate. Both at very low and at very high shear rates, most shearthinning polymer solutions and melts exhibit Newtonian behaviour, i.e. shear stressshear rate plots become straight lines, Shear rate Figure 1.4 Types of timeindependent flow behaviour Shear rate Figure 1.4 Types of timeindependent flow behaviour as shown schematically in Figure 1.5, and on a linear scale will pass through origin. The resulting values of the apparent viscosity at very low and high shear rates are known as the zero shear viscosity, and the infinite shear viscosity, respectively. Thus, the apparent viscosity of a shearthinning fluid decreases from to with increasing shear rate. Data encompassing a sufficiently wide range of shear rates to illustrate this complete spectrum of pseudoplastic behaviour are difficult to obtain, and are scarce. A single instrument will not have both the sensitivity required in the low shear rate region and the robustness at high shear rates, so that several instruments are often required to achieve this objective. Figure 1.6 shows the apparent viscosityshear rate behaviour of an aqueous polyacrylamide solution at 293 K over almost seven decades of shear rate. The apparent viscosity of this solution drops from 1400 mPa s to 4.2mPa s, and so it would hardly be justifiable to assign a single average value of viscosity for such a fluid! The values of shear rates marking the onset of the upper and lower limiting viscosities are dependent upon several factors, such as the type and concentration of polymer, its molecular weight distribution and the nature of solvent, etc. Hence, it is difficult to suggest valid generalisations but many materials exhibit their limiting viscosities at shear rates below 10~2 s_1 and above 105 s_1 respectively. Generally, the range of shear rate over which the apparent viscosity is constant (in the zeroshear region) increases as molecular weight of the polymer falls, as its molecular weight distribution becomes narrower, and as polymer concentration (in solution) drops. Similarly, the rate of decrease of apparent viscosity with shear rate also varies from one material to another, as can be seen in Figure 1.7 for three aqueous solutions of chemically different polymers. Figure 1.7 Representative shear stress and apparent viscosity plots for three pseudoplastic polymer solutions Figure 1.7 Representative shear stress and apparent viscosity plots for three pseudoplastic polymer solutions Mathematical models for shearthinning fluid behaviour Many mathematical expressions of varying complexity and form have been proposed in the literature to model shearthinning characteristics; some of these are straightforward attempts at curve fitting, giving empirical relationships for the shear stress (or apparent viscosity)shear rate curves for example, while others have some theoretical basis in statistical mechanics  as an extension of the application of the kinetic theory to the liquid state or the theory of rate processes, etc. Only a selection of the more widely used viscosity models is given here; more complete descriptions of such models are available in many books [Bird et al., 1987; Carreau et al., 1997] and in a review paper [Bird, 1976]. (i) The powerlaw or Ostwald de Waele model The relationship between shear stress and shear rate (plotted on double logarithmic coordinates) for a shearthinning fluid can often be approximated by a straightline over a limited range of shear rate (or stress). For this part of the flow curve, an expression of the following form is applicable: so the apparent viscosity for the socalled powerlaw (or Ostwald de Waele) fluid is thus given by: For n < 1, the fluid exhibits shearthinnering properties n = 1, the fluid shows Newtonian behaviour n > 1, the fluid shows shearthickening behaviour In these equations, m and n are two empirical curvefitting parameters and are known as the fluid consistency coefficient and the flow behaviour index respectively. For a shearthinning fluid, the index may have any value between 0 and 1. The smaller the value of n, the greater is the degree of shearthinning. For a shearthickening fluid, the index n will be greater than unity. When n = 1, equations (1.12) and (1.13) reduce to equation (1.1) which describes Newtonian fluid behaviour. Although the powerlaw model offers the simplest representation of shearthinning behaviour, it does have a number of shortcomings. Generally, it applies over only a limited range of shear rates and therefore the fitted values of m and n will depend on the range of shear rates considered. Furthermore, it does not predict the zero and infinite shear viscosities, as shown by dotted lines in Figure 1.5. Finally, it should be noted that the dimensions of the flow consistency coefficient, m, depend on the numerical value of n and therefore the m values must not be compared when the n values differ. On the other hand, the value of m can be viewed as the value of apparent viscosity at the shear rate of unity and will therefore depend on the time unit (e.g. s, minute or hour) employed. Despite these limitations, this is perhaps the most widely used model in the literature dealing with process engineering applications. (ii) The Carreau viscosity equation When there are significant deviations from the powerlaw model at very high and very low shear rates as shown in Figure 1.6, it is necessary to use a model which takes account of the limiting values of viscosities m0 and Based on the molecular network considerations, Carreau [1972] put forward the following viscosity model which incorporates both limiting viscosities m0 and m»: where n (< 1) and X are two curvefitting parameters. This model can describe shearthinning behaviour over wide ranges of shear rates but only at the expense of the added complexity of four parameters. This model predicts Newtonian fluid behaviour m = M0 when either n = 1 or X = 0 or both. (iii) The Ellis fluid model When the deviations from the powerlaw model are significant only at low shear rates, it is perhaps more appropriate to use the Ellis model. The two viscosity equations presented so far are examples of the form of equation (1.11). The threeconstant Ellis model is an illustration of the inverse form, namely, equation (1.10). In simple shear, the apparent viscosity of an Ellis model fluid is given by: In this equation, m0 is the zero shear viscosity and the remaining two constants a(>1) and rT/2 are adjustable parameters. While the index a is a measure of the degree of shearthinning behaviour (the greater the value of a, greater is the extent of shearthinning), rT/2 represents the value of shear stress at which the apparent viscosity has dropped to half its zero shear value. Equation (1.15) predicts Newtonian fluid behaviour in the limit of rT/2 ! 1. This form of equation has advantages in permitting easy calculation of velocity profiles from a known stress distribution, but renders the reverse operation tedious and cumbersome. 1.3.2 Viscoplastic fluid behaviourThis type of fluid behaviour is characterised by the existence of a yield stress (to) which must be exceeded before the fluid will deform or flow. Conversely, such a material will deform elastically (or flow en masse like a rigid body) when the externally applied stress is smaller than the yield stress. Once the magnitude of the external stress has exceeded the value of the yield stress, the flow curve may be linear or nonlinear but will not pass through origin (Figure 1.4). Hence, in the absence of surface tension effects, such a material will not level out under gravity to form an absolutely flat free surface. One can, however, explain this kind of fluid behaviour by postulating that the substance at rest consists of three dimensional structures of sufficient rigidity to resist any external stress less than t0. For stress levels greater than t0, however, the structure breaks down and the substance behaves like a viscous material. A fluid with a linear flow curve for  Tyx  >  t0  is called a Bingham plastic fluid and is characterised by a constant plastic viscosity (the slope of the shear stress versus shear rate curve) and a yield stress. On the other hand, a substance possessing a yield stress as well as a nonlinear flow curve on linear coordinates (for Tyx  > t0), is called a 'yieldpseudoplastic' material. Figure 1.8 illustrates viscoplastic behaviour as observed in a meat extract and in a polymer solution. Figure 1.8 Representative shear stressshear rate data showing viscoplastic behaviour in a meat extract (Bingham Plastic) and in an aqueous carbopol polymer solution (yieldpseudoplastic) Figure 1.8 Representative shear stressshear rate data showing viscoplastic behaviour in a meat extract (Bingham Plastic) and in an aqueous carbopol polymer solution (yieldpseudoplastic) It is interesting to note that a viscoplastic material also displays an apparent viscosity which decreases with increasing shear rate. At very low shear rates, the apparent viscosity is effectively infinite at the instant immediately before the substance yields and begins to flow. It is thus possible to regard these materials as possessing a particular class of shearthinning behaviour. Strictly speaking, it is virtually impossible to ascertain whether any real material has a true yield stress or not, but nevertheless the concept of a yield stress has proved to be convenient in practice because some materials closely approximate to this type of flow behaviour, e.g. see [Barnes and Walters, 1985; Astarita, 1990; Schurz, 1990 and Evans, 1992]. The answer to the question whether a fluid has a yield stress or not seems to be related to the choice of a time scale of observation. Common examples of viscoplastic fluid behaviour include particulate suspensions, emulsions, foodstuffs, blood and drilling muds, etc. [Barnes, 1999] Mathematical models for viscoplastic behaviourOver the years, many empirical expressions have been proposed as a result of straightforward curve fitting exercises. A model based on sound theory is yet to emerge. Three commonly used models for viscoplastic fluids are briefly described here. (i) The Bingham plastic model This is the simplest equation describing the flow behaviour of a fluid with a yield stress and, in steady one dimensional shear, it is written as: Often, the two model parameters, rf and , are treated as curve fitting constants irrespective of whether or not the fluid possesses a true yield stress. (ii) The HerschelBulkley fluid model A simple generalisation of the Bingham plastic model to embrace the nonlinear flow curve (for I ryx I > I rf I ) is the three constant HerschelBulkley fluid model. In one dimensional steady shearing motion, it is written as: Tyx = ToH C m(Kyx)" for I rx I > I r^ I (1.17) Note that here too, the dimensions of m depend upon the value of n. With the use of the third parameter, this model provides a somewhat better fit to some experimental data. (iii) The Casson fluid model Many foodstuffs and biological materials, especially blood, are well described by this two constant model as: ( I Tyx I )1/2 = ( I Tq I )1/2 C (Mc I )V I )1/2 for I Tyx I > I Tq I (1.18) This model has often been used for describing the steady shear stressshear rate behaviour of blood, yoghurt, tomato puree, molten chocolate, etc. The flow behaviour of some particulate suspensions also closely approximates to this type of behaviour. The comparative performance of these three as well as several other models for viscoplastic behaviour has been thoroughly evaluated in an extensive review paper by Bird et al. [1983]. 1.3.3 Shearthickening or dilatant fluid behaviourDilatant fluids are similar to pseudoplastic systems in that they show no yield stress but their apparent viscosity increases with increasing shear rate; thus these fluids are also called shearthickening. This type of fluid behaviour was originally observed in concentrated suspensions and a possible explanation for their dilatant behaviour is as follows: At rest, the voidage is minimum and the liquid present is sufficient to fill the void space. At low shear rates, the liquid lubricates the motion of each particle past others and the resulting stresses are consequently small. At high shear rates, on the other hand, the material expands or dilates slightly (as also observed in the transport of sand dunes) so that there is no longer sufficient liquid to fill the increased void space and prevent direct solidsolid contacts which result in increased friction and higher shear stresses. This mechanism causes the apparent viscosity to rise rapidly with increasing rate of shear. The term dilatant has also been used for all other fluids which exhibit increasing apparent viscosity with increasing rate of shear. Many of these, such as starch pastes, are not true suspensions and show no dilation on shearing. The above explanation therefore is not applicable but nevertheless such materials are still commonly referred to as dilatant fluids. Of the timeindependent fluids, this subclass has received very little attention; consequently very few reliable data are available. Until recently, dilatant fluid behaviour was considered to be much less widespread in the chemical and processing industries. However, with the recent growing interest in the handling and processing of systems with high solids loadings, it is no longer so, as is evidenced by the number of recent review articles on this subject [Barnes etal., 1987; Barnes, 1989; Boersma et al., 1990; Goddard and Bashir, 1990]. Typical examples of materials exhibiting dilatant behaviour include concentrated suspensions of china clay, titanium dioxide [Metzner and Whitlock, 1958] and of corn flour in water. Figure 1.9 shows the dilatant behaviour of dispersions of polyvinylchloride in dioctylphthalate [Boersma etal., 1990]. The limited information reported so far suggests that the apparent viscosityshear rate data often result in linear plots on double logarithmic coordinates over a limited shear rate range and the flow behaviour may be represented by the powerlaw model, equation (1.13), with the flow behaviour index, n, greater than one, i.e. One can readily see that for n > 1, equation (1.13) predicts increasing viscosity with increasing shear rate. The dilatant behaviour may be observed in moderately concentrated suspensions at high shear rates, and yet, the same suspension may exhibit pseudoplastic behaviour at lower shear rates, as shown in Figure 1.9; it is not yet possible to ascertain whether these materials also display limiting apparent viscosities. 
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