The Sisko model

Many real flows take place for structured liquids at shear rates where the viscosity is just coming out of the power-law region of the flow curve and flattening off towards z/. This situation is easily dealt with by simply adding a Newtonian contribution to the power-law description of the viscosity, giving 

This is called the Sisko equation, and it is very good at describing the flow behaviour of most emulsions and suspensions in the practical everyday shear rate range of 0.1 to 1000 s-i. 

Exercise In figure 19, which models would you attempt to fit to each the flow ciirves shown  

Let US suppose we have a set of shear-stress/shear-rate data (cr,y) which we want to fit to an appropriate equation. First we plot the data on a linear basis to see if it fits the Bingham equation. If it does, then aU we need to do is to perform a linear regression on the data, using a simple spread-sheet software package, or the software usually provided nowadays with the viscometer/rheometer we have used. Then the values of cto and % can be used to predict flows of the liquid in other geometries, see chapter 10. 

If the linear plot shows curvature, the data should be plotted on a logarithmic basis. From this plot we can see if we have a reasonable straight line, or a straight-line and some curvature. If a fair straight line is seen, then the data can be submitted for power-law regression and the k and n extracted and used for prediction. Last, if there is considerable curvature, then the Sisko model, or if necessary the complete Cross equation is indicated. However, here we have a problem, because the equation is nonlinear and simple regression analysis is inadequate. However, most modem viscometers have suitable software for this purpose, otherwise other specialised commercial mathematical software. 

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The power law model can be extended by including the yield value r — Tq = / 7 , which is called the Herschel-BulMey model, or by adding the Newtonian limiting viscosity,. The latter is done in the Sisko model, 77 +. These two models, along with the Newtonian, Bingham, and Casson... 

If r/0 (i], 1] ) and (ky)2 >S> 1, the Carreau model reduces to the equivalent of a power law model with a high shear limiting viscosity, called the Sisko model ... 

If the value of p is set equal to 1/2 in the Sisko model, the result is equivalent to the Bingham plastic model ... 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

The power law does not describe the regions of the viscosity curve near y = 0 and y —> oo. To this end, the Ellis model at low shear rates and the Sisko model at high shear rates can be used (2). The models are given by... 

Other practical applications for the Sisko model have been reported for yogurt and polymer liquid crystals. ... 

If the liquid flowing in a pipe/tube can be described by the Sisko model (q =kj n-i +r ), then the pressure drop is given (to a very good approximation) by... 

The Bingham model is the most non-Newtonian example of the Sisko model, which is itself a simplification of the Cross (or Carreau model) under the appropriate conditions [3]. 

Similarly, the two models could be reduced by neglecting the lower Newtonian region. This has been done by Sisko for the Cross model in order to account for the rheological properties of grease in bearings ... 

In Chapter 3, Section 3-4-2-2, the Sisko, Cross, Meter, and Bird rheological models were presented. Chilton and Stainsby (1998) stressed the limitations of these models and the shear rates at which they are valid. 

If we make various simplifying assumptions, it is not difficult to show that the Qoss equation can be reduced to Sisko, power-law and Newtonian behaviour, see below. There is another Cross-like model which uses the stress rather than the shear rate as the independent variable, it has been called the Ellis or sometimes the Meter model, and for some specific values of the exponent, it has been gives other names for an exponent of unity it has been called the Williamson or Dougherty and Krieger model, while for an exponent of two it has been called the Phihppoff model, etc. 

For most structured liquids at high shear rates, rjo rjoo and Ky l,and then it is easy to show that the Cross model simplifies to the Sisko equation... 

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

Basic Protocol 2 is for time-dependent non-Newtonian fluids. This type of test is typically only compatible with rheometers that have steady-state conditions built into the control software. This test is known as an equilibrium flow test and may be performed as a function of shear rate or shear stress. If controlled shear stress is used, the zero-shear viscosity may be seen as a clear plateau in the data. If controlled shear rate is used, this zone may not be clearly delineated. Logarithmic plots of viscosity versus shear rate are typically presented, and the Cross or Carreau-Yasuda models are used to fit the data. If a partial flow curve is generated, then subset models such as the Williamson, Sisko, or Power Law models are used (unithi.i). 

Better fits to the portions of the flow curve that are near y Q and y oo are the Ellis (Equations 6.26a and 6.26b) and Sisko (Equation 6.27) models, respectively ... 

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