Time-independent fluid behaviour

In simple shear, the flow behavioin of this class of materials may be described by a constitutive relation of the 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 fimction in equation (1.10) or (1.11), these fluids may be further subdivided into three types  

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Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

Capillary viscometers are the most commonly used instruments for the measurement of viscosity due, in part, to their relative simplicity, low cost and (in the case of long capillaries) accuracy. However, when pressure drives a fluid through a pipe, the velocity is a maximum at the centre the velocity gradient or shear rate y are a maximum at the wall and zero in the centre of the flow. The flow is therefore non-homogeneous and capillary viscometers are restricted to measuring steady shear functions, i.e. steady shear stress-shear rate behaviour for time independent fluids [Macosko 1994]. Due to their inherent similarity to... 

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. 

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. 

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. 

In the simplest case, that of time-independent behaviour, the shear stress depends only on the shear rate but not in the proportional manner of a Newtonian fluid. Various types of time-independent behaviour are shown in Figure 1.19(a), in which the shear stress is plotted against the shear rate on linear axes. The absolute values of shear stress and shear rate are plotted so that irrespective of the sign convention used the curves always lie in the first quadrant. 

The most common type of time-independent non-Newtonian fluid behavioiu observed is pseudoplasticity or shear-thinning, characterised by an apparent viscosity which decreases with increasing shear rate. Both at very low and at very high shear rates, most shear-thinning polymer solutions and melts exhibit Newtonian behaviour, i.e. shear stress-shear rate plots become straight lines. 

Thus, the index m is the slope of the log-log plots of the wall shear stress Xw versus (8V/D) in the laminar region (the limiting condition for laminar flow is discussed in Section 3.3). Plots of x versus (8V/D) thus describe the flow behaviour of time-independent non-Newtonian fluids and may be used directly for scale-up or process design calculations. 

In the same way as there are many equations for predicting friction factor for turbulent Newtonian flow, there are munerous equations for time-independent non-Newtonian fluids most of these are based on dimensional considerations combined with experimental observations [Govier and Aziz, 1982 Heywood and Cheng, 1984]. There is a preponderance of correlations based on the power-law fluid behaviour and additionally some expressions are available for Bingham plastic fluids [Tomita, 1959 Wilson and Thomas, 1985], Here only a selection of widely used and proven methods is presented. 

Bowen [1961], on the other hand, proposed that for turbulent flow of a particular fluid (exhibiting time independent behaviour), the wall shear stress, T, could be expressed as ... 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

In semi-dilute solutions, the Rouse theory fails to predict the relaxation time behaviour of the polymeric fluids. This fact is shown in Fig. 11 where the reduced viscosity is plotted against the product (y-AR). For correctly calculated values of A0 a satisfactory standardisation should be obtained independently of the molar mass and concentration of the sample. 

Newtonian fluids are characterised by pure linear viscous behaviour. When a load is applied they display a linear change in shear over time, and there is a linear relationship between shear rate and stress, i.e. dynamic viscosity is independent of shear rate. When the load is removed, the shear remains completely preserved. 

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