Models of non-Newtonian fluids PDF

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Lecture 2: Models for non-Newtonian fluid behaviour and time dependent viscosity

A note on presenting data: Rheologists often plot stress or viscosity as a function of shear rate on log-log plots in order to present the huge range of shear rates that are relevant to the materials behaviour. It is important to note that on such plots, Newtonian, shear thinning and shear thickening fluids can all be represented by straight lines (see figure 1).

Figure 1: Shear stress as a function of shear rate for Newtonian, shear thinning and shear thickening materials. Note how all three can be represented as straight lines due to the log-log nature of the plot.

We can re-plot the flow curves for Newtonian, shear thinning, and shear thickening behaviour in terms of viscosity as a function of shear rate as shown in figure 2. The viscosity is constant (with respect to shear rate) for the Newtonian fluid, decreases for the shear thinning fluid and increases for the shear thickening fluid. However, the curves shown in figure 2 for the shear thinning and shear thickening fluids are only relevant over a limited range of shear rates, for example, if we were to extrapolate the curve for the shear thinning fluid to very small strain rates, the extrapolated data would suggest an infinitely large viscosity, obviously this is not accurate. Similarly, extrapolating to very large strain rates would result in zero viscosity, again this is obviously inaccurate. The viscosity will actually approach asymptotic values as the shear rate becomes either very small or very large, these asymptotic values of viscosity are termed the zero-shear viscosity and the high shear rate plateau viscosity, respectively. Consequently, a plot of viscosity as a function of shear rate for a real shear thinning fluid would appear as shown in figure 3.

Figure 2: Apparent viscosity as a function of shear rate for Newtonian, shear thinning and shear thickening fluids.

Figure 3: Apparent viscosity as a function of shear rate for a shear thinning material.

The first model we will consider was proposed by Malcolm Cross , an ICI Rheologist. Whilst working on dye-stuffs and pigment dispersions Cross noted that the viscosity of many suspension could be described by an equation of the form

This model is known as the Cross model and contains four fitting parameters, the zero shear viscosity, , the high shear plateau viscosity, , K and m. For shear thinning fluids m takes values in the range and indicates the degree of shear thinning. The Cross model describes a Newtonian fluid when and the ‘most shear thinning’ fluids as . One way of measuring viscosity is to use a rheometer. These devices operate in either controlled rate mode, in which a given strain rate is imposed upon a fluid and the resulting stress is measured, or in controlled stress mode in which a given stress is applied to a material and the resulting strain rate measured. Both types of device involve well defined geometries in which uniform strain rates can be achieved. Most commercial rheometers are capable of accurately applying strain rates over a limited range. Whilst this range of shear rates covers several orders of magnitude it is not uncommon for rheological data spanning this range to reveal only a part of the flow curve described by the Cross model, hence it is not always possible to completely define all the parameters in the model. Figure 4 shows viscosity as a function of shear rate for a typical fabric washing liquid, it displays shear thinning behaviour but there appears to be no zero-shear viscosity nor high shear rate plateau viscosity.

Figure 4: Typical viscosity data for a fabric washing liquid

In this case we can assume that

and

, hence the Cross model can be reduced to

or, by a simple change in the parameters to the well know power law model or The power law is one of the most commonly used models of non-Newtonian fluid behaviour. Through suitable choice of the power law index n the model is able to describe shear thinning, shear thickening and Newtonian fluids as shown in table 1.

Fluid Behaviour Power Law Index Shear Thinning 0...