Flow Curves Definitions and Equations

Most characterisation of non-linear responses of materials with De 1 have concerned the application of a shear rate and the shear stress has been monitored. The ratio at any particular rate has defined the apparent viscosity. When these values are plotted against one another we produce flow curves. The reason for the popularity of this approach is partly historic and is related to the type of characterisation tool that was available when rheology was developing as a subject. As a consequence there are many expressions relating shear stress, viscosity and shear rate. There is also a plethora of interpretations for meaning behind the parameters in the modelling equations. There are a number that are commonly used as phenomenological descriptions of the flow behaviour. 

A plastic material is one which displays a yield stress. As the shear rate is reduced the apparent viscosity progressively increases, diverging and never reaching a constant value. If a stress is applied to a plastic material no flow will be observed until the stress exceeds the yield stress. Expressions describing plastic materials must include at least two terms, a yield stress and a limiting high shear viscous term. Expressions usually describe shear stress in terms of shear rate  

The term /pj is the plastic viscosity and aB 0c and tHb are the Bingham, Casson and the Herchel-Bulkley yield stresses. 

It is important when using the term yield stress to distinguish between an extrapolated value, sometimes called the dynamic yield stress and a true or static yield stress . The latter can only be observed for plastic solids whilst the former is readily obtained with pseudoplastic liquids. In practical terms this can be critical in evaluating the performance of a material. 

There is an expression that does not truly fit either class of behaviour, for power law fluids which can be expressed in terms of stress, rate or apparent viscosity with relative ease. They can describe shear thickening or thinning depending upon the sign of the power law index n  

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