Viscosity

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

VISCOSITY, DEFINITION=NEWTONIAN (default)

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

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, ${\eta }_{min}$, and/or an upper limit, ${\eta }_{max}$, on the value of the viscosity computed from the power law.

VISCOSITY, DEFINITION=POWER LAW

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

where ${\eta }_0$ is the low shear rate Newtonian viscosity, ${\eta }_{\mathrm{\infty }}$ is the infinite shear viscosity (at high shear strain rates), $\lambda$ is the natural time constant of the fluid ($1/\lambda$ is the critical shear rate at which the fluid changes from Newtonian to power law behavior), and $n$ represents the flow behavior index in the power law regime. The coefficient $a$ is a material parameter. The original Carreau model is recovered when $a$=2.

VISCOSITY, DEFINITION=CARREAU-YASUDA

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

where ${\eta }_0$ is the Newtonian viscosity, ${\eta }_{\mathrm{\infty }}$ is the infinite shear viscosity (usually assumed to be zero for the Cross model), $\lambda$ is the natural time constant of the fluid ($1/\lambda$ 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.

VISCOSITY, DEFINITION=CROSS

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

Here ${\tau }_0$ is the “yield” stress and ${\eta }_0$ is a penalty viscosity to model the “rigid-like” behavior in the very low strain rate regime ($\dot{\gamma }\le {\tau }_0/{\eta }_0$), when the stress is below the yield stress, $\tau \le {\tau }_0$. With increasing strain rates, the viscosity transitions into a power law model once the yield threshold is reached, $\tau \ge {\tau }_0$. 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.

VISCOSITY, DEFINITION=HERSCHEL-BULKLEY

This 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

where ${\eta }_0$ is the Newtonian viscosity, ${\eta }_{\mathrm{\infty }}$ is the infinite shear viscosity, and $\lambda$ represents a characteristic time of the measured system.

VISCOSITY, DEFINITION=POWELL-EYRING

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

where ${\tau }_{1/2}$ is the effective shear stress at which the viscosity is 50% between the Newtonian limit, ${\eta }_0$, and the infinite shear viscosity, ${\eta }_{\mathrm{\infty }}$, and $n$ represents the flow index in the power law regime.

VISCOSITY, DEFINITION=ELLIS-METER

In Abaqus/Explicit the viscosity can be specified directly as a tabular function of shear strain rate and temperature.

VISCOSITY, DEFINITION=TABULAR

In Abaqus/Explicit you can specify a user-defined viscosity in user subroutine VUVISCOSITY (see VUVISCOSITY).

VISCOSITY, DEFINITION=USER