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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.