Viscosity

Viscous Shear Behavior

The resistance to flow of a viscous fluid is described by the following relationship between deviatoric stress and strain rate

S = 2 η \dot{e} = η \dot{γ},

where $S$ is the deviatoric stress, $\dot{e}$ is the deviatoric part of the strain rate, $η$ is the viscosity, and $\dot{γ} = 2 \dot{e}$ is the engineering shear strain rate.

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, Abaqus/Explicit supports several models of nonlinear viscosity 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. In addition, in Abaqus/Explicit user subroutine VUVISCOSITY can be used.

In addition to the Newtonian viscous fluid model, Abaqus/Standard supports several models of nonlinear viscosity to describe non-Newtonian fluids for use with fluid pipe elements and fluid pipe connector elements: power law, Herschel-Bulkley, and Bingham plastic.

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.

In Abaqus/Standard this model is used to define the fluid behavior in fluid pipe and fluid pipe connector elements.

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.

In Abaqus/Standard this model is used to define the fluid behavior in fluid pipe and fluid pipe connector elements.

Carreau-Yasuda

The Carreau-Yasuda model describes the shear thinning behavior of polymers and is available only in Abaqus/Explicit. 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})}^{a})}^{\frac{n - 1}{a}},

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 $n$ 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.

Cross

The Cross model is commonly used when it is necessary to describe the low-shear-rate behavior of the viscosity and is available only in Abaqus/Explicit. 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.

Herschel-Bulkley

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

η = {\begin{matrix} η_{0} & if & τ < τ_{0}; \\ \frac{1}{\dot{γ}} (τ_{0} + k ({\dot{γ}}^{n} - {(τ_{0} / η_{0})}^{n})) & if & 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 ( $\dot{γ} \leq τ_{0} / η_{0}$ ), when the stress is below the yield stress, $τ \leq τ_{0}$ . 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.

In Abaqus/Standard this model is used to define the fluid behavior in fluid pipe and fluid pipe connector elements.

Bingham Plastic

The Bingham plastic model is a special case of the Herschel-Bulkley model. Bingham plastics correspond to the case where the flow behavior index in the power law regime, $n$ , is equal to one.

This model is available only in Abaqus/Standard for fluid pipe and fluid pipe connector elements.

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. This model is available only in Abaqus/Explicit. 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, $η_{\infty}$ is the infinite shear viscosity, and $λ$ represents a characteristic time of the measured system.

Ellis-Meter

The Ellis-Meter model, available only in Abaqus/Explicit, 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.

Tabular

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

User-Defined

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

Temperature Dependence of Viscosity (Abaqus/Explicit Only)

The temperature-dependence of the viscosity of many polymer materials of industrial interest obeys a time-temperature shift relationship in the form:

η (\dot{γ}, θ) = a_{T} (θ) η (a_{T} (θ) \dot{γ}, θ_{0}),

where $a_{T} (θ)$ is the shift function and $θ_{0}$ is the reference temperature at which the viscosity versus shear strain rate relationship is known. This concept for temperature dependence is usually referred to as thermorheologically simple (TRS) temperature dependence. In the Newtonian limit for low shear rates, when $\dot{γ} \to 0$ , we have

η_{0} (θ) = lim_{\dot{γ} \to 0} η (\dot{γ} θ) = a_{T} (θ) η_{0} (θ_{0}) .

Thus, the shift function is defined as the ratio of the Newtonian viscosity at the temperature of interest to that of the chosen reference state: $a_{T} (θ) = η_{0} (θ) / η_{0} (θ_{0})$ .

See Thermorheologically Simple Temperature Effects for a description of the different forms of the shift function available in Abaqus.

Material Options

Material shear viscosity in Abaqus/Explicit must be used in combination with an equation of state to define the material's volumetric mechanical behavior (see Equation of State).

Elements

Material shear viscosity can be used with any solid (continuum) elements in Abaqus/Explicit except plane stress elements.

Material shear viscosity can be used only with fluid pipe elements and fluid pipe connector elements in Abaqus/Standard.