Viscosity Options

Newtonian fluids are characterized by a viscosity that only depends on temperature, $η (θ)$ . In the more general case of non-Newtonian fluids the viscosity is a function of the temperature and shear strain rate:

η = η (\dot{γ}, θ),

where $\dot{γ} = \sqrt{\frac{1}{2} \dot{γ} : \dot{γ}}$ is the equivalent shear strain rate. In terms of the equivalent shear stress, $τ = \sqrt{\frac{1}{2} S : S}$ , we have:

τ = η \dot{γ} .

Non-Newtonian fluids can be classified as shear-thinning (or pseudoplastic), when the apparent viscosity decreases with increasing shear rate, and shear-thickening (or dilatant), when the viscosity increases with strain rate.

In addition to the Newtonian viscous fluid model, several models of nonlinear viscosity are available to describe non-Newtonian fluids: power law, Carreau-Yasuda, Cross, Herschel-Bulkley, Powell-Eyring, and Ellis-Meter. Other functional forms of the viscosity can also be specified in tabular format.


Input Data	Description
Newtonian	Viscosity that depends only on temperature.
Carreau Yasuda	Describes the shear thinning behavior of polymers.
Cross	Commonly used when it is necessary to describe the low-shear-rate behavior of the viscosity.
Ellis Meter	Expresses the viscosity in terms of the effective shear stress.
Herschel Bulkley	Describes the behavior of viscoplastic fluids, such as Bingham plastics, that exhibit a yield response.
Powell Eyring	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.
Power Law	Commonly used to describe the viscosity of non-Newtonian fluids.
Tabular	Direct specification of viscosity as a tabular function of shear strain rate and temperature. In flow analyses, only shear strain rate dependence is supported.
Sutherland's Law	Computes viscosity as a function of temperature.

Newtonian

The Newtonian model is useful to model viscous laminar flow governed by the Navier-Poisson law of a Newtonian fluid, $τ = η \dot{γ} .$ . Newtonian fluids are characterized by a viscosity that depends only on temperature, $η (θ)$ . You need to specify the viscosity as a tabular function of temperature when you define the Newtonian viscous deviatoric behavior.


Input Data	Description
Viscosity	Dynamic viscosity.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Carreau Yasuda

The Carreau-Yasuda model describes the shear thinning behavior of polymers. This model often provides a better fit than the power law model for both high and low shear strain rates. The viscosity is expressed as

η = η_{\infty} + (η_{0} - η_{\infty}) {(1 + {(λ \dot{γ})}^{α})}^{\frac{n - 1}{α}},

where $η_{0}$ is the low shear rate Newtonian viscosity, $η_{\infty}$ is the infinite shear viscosity (at high shear strain rates), $λ$ is the natural time constant of the fluid ( $1 / λ$ is the critical shear rate at which the fluid changes from Newtonian to power law behavior), and $η$ represents the flow behavior index in the power law regime. The coefficient $α$ is a material parameter. The original Carreau model is recovered when $α$ =2.


Input Data	Description
Viscosity at Low Shear Rates	$η_{0}$
Viscosity at Large Shear Rates	$η_{\infty}$
Time Constant	$λ$
Flow Behavior Index	$η$
Const A	$α$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Cross

The Cross model is commonly used when it is necessary to describe the low-shear-rate behavior of the viscosity. The viscosity is expressed as

η = η_{\infty} + \frac{(η_{0} - η_{\infty})}{1 + {(λ \dot{γ})}^{1 - n}},

where $η_{0}$ is the Newtonian viscosity, $η_{\infty}$ is the infinite shear viscosity (usually assumed to be zero for the Cross model), $λ$ is the natural time constant of the fluid ( $1 / λ$ is the critical shear rate at which the fluid changes from Newtonian to power-law behavior), and $n$ is the flow behavior index in the power law regime.


Input Data	Description
Viscosity at Low Shear Rates	$η_{0}$
Viscosity at Large Shear Rates	$η_{\infty}$
Time Constant	$λ$
Flow Behavior Index	$n$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Ellis Meter

The Ellis-Meter model expresses the viscosity in terms of the effective shear stress, $τ = \sqrt{\frac{1}{2} S : S}$ , as:

η = η_{\infty} + \frac{(η_{0} - η_{\infty})}{1 + {(τ / τ_{1 / 2})}^{(1 - n) / n}},

where $τ_{1 / 2}$ is the effective shear stress at which the viscosity is 50% between the Newtonian limit, $η_{0}$ , and the infinite shear viscosity, $η_{\infty}$ , and $n$ represents the flow index in the power law regime.


Input Data	Description
Viscosity at Low Shear Rates	$η_{0}$
Viscosity at Large Shear Rates	$η_{\infty}$
Effective Shear Stress	$τ_{1 / 2}$
Flow Behavior Index	$n$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Herschel Bulkley

The Herschel-Bulkley model can be used to describe the behavior of viscoplastic fluids, such as Bingham plastics, that exhibit a yield response. The viscosity is expressed as

η = {\begin{matrix} η_{0} & i f & τ < τ_{0}; \\ \frac{1}{\dot{γ}} (τ_{0} + k ({\dot{γ}}^{n} - {(τ_{0} / η_{0})}^{n})) & i f & τ \geq τ_{0} . \end{matrix}

Here $τ_{0}$ is the "yield" stress and $η_{0}$ is a penalty viscosity to model the "rigid-like" behavior in the very low strain rate regime ( $\dot{γ} = τ_{0} / η_{0}$ ), when the stress is below the yield stress, $τ < τ_{0}$ . With increasing strain rates, the viscosity transitions into a power law model once the yield threshold is reached, $τ \geq τ_{0}$ . The parameters $k$ and $n$ are the flow consistency and the flow behavior indexes in the power law regime, respectively. Bingham plastics correspond to the case $n$ =1


Input Data	Description
Viscosity at Low Shear Rates	$η_{0}$
Yield Shear Stress	$τ_{0}$
Consistency Index	$k$
Flow Behavior Index	$n$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Powell Eyring

The Powell-Eyring model, which is derived from the theory of rate processes, 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. The viscosity is expressed as

η = η_{\infty} + (η_{0} - η_{\infty}) \frac{\sinh^{- 1} (λ \dot{γ})}{λ \dot{γ}},

where $η_{0}$ is the Newtonian viscosity, $η_{0}$ is the infinite shear viscosity, and $λ$ represents a characteristic time of the measured system.


Input Data	Description
Viscosity at Low Shear Rates	$η_{0}$
Viscosity at Large Shear Rates	$η_{\infty}$
Time Constant	$λ$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Power Law

The power law model is commonly used to describe the viscosity of non-Newtonian fluids. The viscosity is expressed as

η = k {\dot{γ}}^{n - 1}; η_{\min} \leq η \leq η_{\max},

where

k

is the flow consistency index and

n

is the flow behavior index. When

n < 1

, the fluid is shear-thinning (or pseudoplastic): the apparent viscosity decreases with increasing shear rate. When

n > 1

, the fluid is shear-thickening (or dilatant); and when

n = 1

, the fluid is Newtonian. Optionally, you can place a lower limit,

η_{\min}

, and/or an upper limit,

η_{\max}

, on the value of the viscosity computed from the power law.


Input Data	Description
Consistency Index	$k$
Flow Behavior Index	$n$
Maximum Viscosity	$η_{\max}$
Minimum Viscosity	$η_{\min}$
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Tabular

For explicit analyses the viscosity can be specified directly as a tabular function of shear strain rate and temperature. In flow analyses only shear strain rate dependence is supported.


Input Data	Description
Viscosity	Dynamic viscosity.
Shear Strain	Shear strain rate.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Sutherland's Law

Sutherland's law describes the relationship between the dynamic viscosity and the absolute temperature of a fluid. It is expressed as:

μ = μ_{r e f} {(\frac{T}{T_{r e f}})}^{3 / 2} \frac{T_{r e f} + S}{T + S}


Input Data	Description
Reference Viscosity	Dynamic viscosity, $μ_{r e f}$ .
Reference Temperature	Reference temperature, $T_{r e f}$ .
Sutherland Temperature	Sutherland temperature, $S$ .