8.4.5 Viscosity for Non-Newtonian Fluids

For incompressible Newtonian fluids, the shear stress is proportional to the rate-of-deformation tensor $\overline{\overline{D}}$:

where $\overline{\overline{D}}$ is defined by

and $\mu$ is the viscosity, which is independent of $\overline{\overline{D}}$.

For some non-Newtonian fluids, the shear stress can similarly be written in terms of a non-Newtonian viscosity $\eta$:

In general, $\eta$ is a function of all three invariants of the rate-of-deformation tensor $\overline{\overline{D}}$. However, in the non-Newtonian models available in ANSYS FLUENT, $\eta$ is considered to be a function of the shear rate $\dot{\gamma}$ only. $\dot{\gamma}$ is related to the second invariant of $\overline{\overline{D}}$ and is defined as

Temperature Dependent Viscosity

If the flow is non-isothermal, then the temperature dependence on the viscosity can be included along with the shear rate dependence. In this case, the total viscosity consists of two parts and is calculated as

where H(T) is the temperature dependence, known as the Arrhenius law.

where $\alpha$ is the ratio of the activation energy to the thermodynamic constant and $T_{\alpha}$ is a reference temperature for which H(T) = 1. $T_0$, which is the temperature shift, is set to 0 by default, and corresponds to the lowest temperature that is thermodynamically acceptable. Therefore $T_{\alpha}$ and $T_0$ are absolute temperatures. Temperature dependence is only included when the energy equation is enabled. Set the parameter $\alpha$ to 0 when you want temperature dependence to be ignored, even when the energy equation is solved.

ANSYS FLUENT provides four options for modeling non-Newtonian flows:

• power law

• Carreau model for pseudo-plastics

• Cross model

• Herschel-Bulkley model for Bingham plastics

Note that the models listed above are not available when modeling turbulent flow.

Note that the non-Newtonian power law described below is different from the power law described in Section  8.4.2.

Note:    Non-Newtonian model for single phase is available for the mixture model and it is recommended that this should be attached to the primary phase.

Appropriate values for the input parameters for these models can be found in the literature (e.g., [ 84]).

Power Law for Non-Newtonian Viscosity

If you choose non-newtonian-power-law in the drop-down list to the right of Viscosity, non-Newtonian flow will be modeled according to the following power law for the non-Newtonian viscosity:

where $k$ and $n$ are input parameters. $k$ is a measure of the average viscosity of the fluid (the consistency index); $n$ is a measure of the deviation of the fluid from Newtonian (the power-law index). The value of $n$ determines the class of the fluid:

$n = 1$	$\rightarrow$	Newtonian fluid
$n > 1$	$\rightarrow$	shear-thickening (dilatant fluids)
$n < 1$	$\rightarrow$	shear-thinning (pseudo-plastics)

Inputs for the Non-Newtonian Power Law

To use the non-Newtonian power law, choose non-newtonian-power-law in the drop-down list to the right of Viscosity. The Non-Newtonian Power Law dialog box will open, and you can choose between Shear Rate Dependent and Shear Rate and Temperature Dependent. Enter the Consistency Index $k$, Power-Law Index $n$, Minimum and Maximum Viscosity Limit, Reference Temperature $T_{alpha}$, and Activation Energy/R, which is the ratio of the activation energy to the thermodynamic constant $\alpha$.

The Carreau Model for Pseudo-Plastics

The power law model described in Equation  8.4-20 results in a fluid viscosity that varies with shear rate. For $\dot{\gamma} \rightarrow 0$, $\eta \rightarrow \eta_0$, and for $\dot{\gamma} \rightarrow \infty$, $\eta \rightarrow \eta_\infty$, where $\eta_0$ and $\eta_\infty$ are, respectively, the upper and lower limiting values of the fluid viscosity.

The Carreau model attempts to describe a wide range of fluids by the establishment of a curve-fit to piece together functions for both Newtonian and shear-thinning ( $n < 1$) non-Newtonian laws. In the Carreau model, the viscosity is

and the parameters $n$, $\lambda$, $T_{\alpha}$, $\eta_0$, and $\eta_\infty$ are dependent upon the fluid. $\lambda$ is the time constant, $n$ is the power-law index (as described above for the non-Newtonian power law), $\eta_0$ and $\eta_\infty$ are, respectively, the zero- and infinite-shear viscosities, $T_{\alpha}$ is the reference temperature, and $\alpha$ is the ratio of the activation energy to thermodynamic constant. Figure  8.4.1 shows how viscosity is limited by $\eta_0$ and $\eta_\infty$ at low and high shear rates.

Figure 8.4.1: Variation of Viscosity with Shear Rate According to the Carreau Model

Inputs for the Carreau Model

To use the Carreau model, choose carreau in the drop-down list to the right of Viscosity. The Carreau Model dialog box will open, and you can choose between Shear Rate Dependent and Shear Rate and Temperature Dependent. Enter the Time Constant $\lambda$, Power-Law Index $n$, Reference Temperature $T_\alpha$, Zero Shear Viscosity $\eta_0$, Infinite Shear Viscosity $\eta_{\infty}$, and Activation Energy/R $\alpha$.

Figure 8.4.2: The Carreau Model Dialog Box

Cross Model

The Cross model for viscosity is:

where	$\eta_0$	=	zero-shear-rate viscosity
	$\lambda$	=	natural time (i.e., inverse of the shear rate at which the fluid changes
			from Newtonian to power-law behavior)
	$n$	=	power-law index

The Cross model is commonly used to describe the low-shear-rate behavior of the viscosity.

Inputs for the Cross Model

To use the Cross model, choose cross in the drop-down list to the right of Viscosity. The Cross Model dialog box will open, and you can choose between Shear Rate Dependent and Shear Rate and Temperature Dependent. Enter the Zero Shear Viscosity $\eta_0$, Time Constant $\lambda$, Power-Law Index $n$, Reference Temperature $T_{\alpha}$, and Activation Energy/R, which is the ratio of the activation energy to the thermodynamic constant $\alpha$.

Herschel-Bulkley Model for Bingham Plastics

The power law model described above is valid for fluids for which the shear stress is zero when the strain rate is zero. Bingham plastics are characterized by a non-zero shear stress when the strain rate is zero:

where $\tau_0$ is the yield stress:

• For $\tau < \tau_0$, the material remains rigid.

• For $\tau > \tau_0$, the material flows as a power-law fluid.

The Herschel-Bulkley model combines the effects of Bingham and power-law behavior in a fluid. For low strain rates ( $\dot{\gamma} < \tau_0/\mu_0$), the "rigid'' material acts like a very viscous fluid with viscosity $\mu_0$. As the strain rate increases and the yield stress threshold, $\tau_0$, is passed, the fluid behavior is described by a power law.

For $\dot{\gamma} > \dot{\gamma_c}$

For $\dot{\gamma} < \dot{\gamma_c}$

where $k$ is the consistency factor, and $n$ is the power-law index.

Figure  8.4.3 shows how shear stress ( $\tau$) varies with shear rate ( $\dot{\gamma}$) for the Herschel-Bulkley model.

Figure 8.4.3: Variation of Shear Stress with Shear Rate According to the Herschel-Bulkley Model

If you choose the Herschel-Bulkley model for Bingham plastics, Equation  8.4-24 will be used to determine the fluid viscosity.

The Herschel-Bulkley model is commonly used to describe materials such as concrete, mud, dough, and toothpaste, for which a constant viscosity after a critical shear stress is a reasonable assumption. In addition to the transition behavior between a flow and no-flow regime, the Herschel-Bulkley model can also exhibit a shear-thinning or shear-thickening behavior depending on the value of $n$.

Inputs for the Herschel-Bulkley Model

To use the Herschel-Bulkley model, choose herschel-bulkley in the drop-down list to the right of Viscosity. The Herschel-Bulkley dialog box will open, and you can choose between Shear Rate Dependent and Shear Rate and Temperature Dependent. Enter the Consistency Index $k$, Power-Law Index $n$, Yield Stress Threshold $\tau_0$, Critical Shear Rate $\dot{\gamma_c}$, Reference Temperature $T_{\alpha}$, and the ratio of the activation energy to thermodynamic constant $\alpha$, Activation Energy/R.