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## 6. Expressions for Describing Steady Shear Non-Newtonian Flow

The expressions shown here are used to characterize the non-Newtonian behavior of fluids under equilibrium, steady shear flow conditions. Many phenomenological and empirical models have been reported in the literature. Only those having a direct and significant implication for suspensions, gels and pastes have been included here. A brief description of each relationship is given with examples of the types of materials to which they typically are applied. In defining the number of parameters associated with a particular model, the term "parameter" in this case refers to adjustable (arbitrary) constants, and therefore excludes measured quantities. Some of these equations have alternative representations other than the one shown. More detailed descriptions and alternative expressions can be found in the sources listed in the bibliography.

### Bingham

The Bingham relation is a two parameter model used for describing viscoplastic fluids exhibiting a yield response. The ideal Bingham material is an elastic solid at low shear stress values and a Newtonian fluid above a critical value called the Bingham yield stress, B. The plastic viscosity region exhibits a linear relationship between shear stress and shear rate, with a constant differential viscosity equal to the plastic viscosity, pl.

### Carreau-Yasuda

A model that describes pseudoplastic flow with asymptotic viscosities at zero ( 0) and infinite ( ) shear rates, and with no yield stress. The parameter is a constant with units of time, where 1/ is the critical shear rate at which viscosity begins to decrease. The power-law slope is (n-1) and the parameter a represents the width of the transition region between 0 and the power-law region. If 0 and are not known independently from experiment, these quantities may be treated as additional adjustable parameters.

### Casson

A two parameter model for describing flow behavior in viscoplastic fluids exhibiting a yield response. The parameter y is the yield stress and pl is the differential high shear ( plastic) viscosity. This equation is of the same form as the Bingham relation, such that the exponent is ½ for a Casson plastic and 1 for a Bingham plastic.

### Cross

Similar in form to the Carreau-Yasuda relation, this model describes pseudoplastic flow with asymptotic viscosities at zero ( 0) and infinite () shear rates, and no yield stress. The parameter is a constant with units of time, and m is a dimensionless constant with a typical range from 2/3 to 1.

### Ellis

A two parameter model, written in terms of shear stress, used to represent a pseudoplastic material exhibiting a power-law relationship between shear stress and shear rate, with a low shear rate asymptotic viscosity. The parameter 2 can be roughly identified as the shear stress value at which has fallen to half its final asymptotic value.

### Herschel-Bulkley

A three parameter model used to describe viscoplastic materials exhibiting a yield response with a power-law relationship between shear stress and shear rate above the yield stress, y. A plot of log ( - y) versus log gives a slope n that differs from unity. The Herschel-Bulkley relation reduces to the equation for a Bingham plastic when n=1.

### Krieger-Dougherty

A model for describing the effect of particle self-crowding on suspension viscosity, where is the particle volume fraction, m is a parameter representing the maximum packing fraction and [] is the intrinsic viscosity. For ideal spherical particles []=2.5 (i.e. the Einstein coefficient). Non-spherical or highly charged particles will exhibit values for [ ] exceeding 2.5. The value of [ ] is also affected by the particle size distribution. The parameter m is a function of particle shape, particle size distribution and shear rate. Both [] and m may be treated as adjustable model parameters.

The aggregate volume fraction (representing the effective volume occupied by particle aggregates, including entrapped fluid) can be determined using this equation if m is fixed at a reasonable value (e.g. 0.64 for random close packing or 0.74 for hexagonal close packing) and [] is set to 2.5. In this case, is the adjustable parameter and is equivalent to the aggregate volume fraction.

### Meter

Expressed in terms of shear stress, used to represent a pseudoplastic material exhibiting a power-law relationship between shear stress and shear rate, with both high ( ) and low (0) shear rate asymptotic viscosity limits. The parameter 2 can be roughly identified as the shear stress value at which has fallen to half its final asymptotic value. The Meter and Carreau-Yasuda models give equivalent representations in terms of shear stress and shear rate, respectively. If 0 and are not known independently from experiment, these quantities may be treated as additional adjustable parameters.

### Powell-Eyring

Derived from the theory of rate processes, this relation is relevant primarily to molecular fluids, but can be used in some cases to describe the viscous behavior of polymer solutions and viscoelastic suspensions over a wide range of shear rates. Here, is the infinite shear viscosity 0 is the zero shear viscosity and the fitting parameter represents a characteristic time of the measured system. If 0 and are not known independently from experiment, these quantities may be treated as additional adjustable parameters.

### power-law [Ostwald-de Waele]

A two parameter model for describing pseudoplastic or shear-thickening behavior in materials that show a negligible yield response and a varying differential viscosity. A log-log plot of versus gives a slope n (the power-law exponent), where n<1 indicates pseudoplastic behavior and n>1 indicates shear-thickening behavior.

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