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8.4.5 Viscosity for Non-Newtonian Fluids

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


 \overline{\overline{\tau}} = \mu \overline{\overline{D}} (8.4-14)

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


 \overline{\overline{D}} = \left( \frac{\partial u_j}{\partial x_i} +\frac{\partial u_i}{\partial x_j} \right) (8.4-15)

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 $\eta$:


 \overline{\overline{\tau}} = \eta \left(\overline{\overline{D}}\right) \overline{\overline{D}} (8.4-16)

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 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


 \dot{\gamma} = \sqrt{\frac{1}{2}\overline{\overline{D}}:\overline{\overline{D}}} (8.4-17)

FLUENT provides four options for modeling non-Newtonian flows:

figure   

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., [ 368]).



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:


 \eta = k \dot{\gamma}^{n-1} e^{T_0/T} (8.4-18)

FLUENT allows you to place upper and lower limits on the power law function, yielding the following equation:


 \eta_{\rm min} < \eta = k \dot{\gamma}^{n-1}e^{T_0/T} < \eta_{\rm max} (8.4-19)

where $k$, $n$, $T_0$, $\eta_{\rm min}$, and $\eta_{\rm max}$ 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), as described below; $T_0$ is the reference temperature; and $\eta_{\rm min}$ and $\eta_{\rm max}$ are, respectively, the lower and upper limits of the power law. If the viscosity computed from the power law is less than $\eta_{\rm min}$, the value of $\eta_{\rm min}$ will be used instead. Similarly, if the computed viscosity is greater than $\eta_{\rm max}$, the value of $\eta_{\rm max}$ will be used instead. Figure  8.4.1 shows how viscosity is limited by $\eta_{\rm min}$ and $\eta_{\rm max}$ at low and high shear rates.

Figure 8.4.1: Variation of Viscosity with Shear Rate According to the Non-Newtonian Power Law
figure

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 panel will open, and you can enter the Consistency Index $k$, Power-Law Index $n$, Reference Temperature $T_0$, Minimum Viscosity Limit $\eta_{\rm min}$, and Maximum Viscosity Limit $\eta_{\rm max}$. For temperature-independent viscosity, the value of $T_0$ should be set to zero. If the energy equation is not being solved, FLUENT uses a default value of $T$=273 K in Equation  8.4-18.



The Carreau Model for Pseudo-Plastics


The power law model described in Equation  8.4-18 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


 \eta = \eta_\infty + (\eta_0 - \eta_\infty) [1 + (H(T)\dot{\gamma} \lambda)^2]^{(n-1)/2} (8.4-20)

where

 H(T) = exp \left[ \alpha \left( \frac{1}{T - T_0} - \frac{1}{T_{\alpha}- T_0} \right) \right] (8.4-21)

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), $T_0$ is set to zero by default, while $T_{\alpha}$ is the reference temperature at which $H(T) = 1$. $T$ and $T_{\alpha}$ are the absolute temperatures and $\eta_0$ and $\eta_\infty$ are, respectively, the zero- and infinite-shear viscosities, and $\alpha$ is the activation energy. Figure  8.4.2 shows how viscosity is limited by $\eta_0$ and $\eta_\infty$ at low and high shear rates.

Figure 8.4.2: Variation of Viscosity with Shear Rate According to the Carreau Model
figure

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 panel will open, and you can 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 $\alpha$.

Figure 8.4.3: The Carreau Model Panel
figure



Cross Model


The Cross model for viscosity is


 \eta = \frac{\eta_0}{1 + \left( \lambda \dot{\gamma} \right) ^{1-n} } (8.4-22)


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 when it is necessary 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 panel will open, and you can enter the Zero Shear Viscosity $\eta_0$, Time Constant $\lambda$, and Power-Law Index $n$ .



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:


 \overline{\overline{\tau}} = \overline{\overline{\tau}}_0 + \eta \overline{\overline{D}} (8.4-23)

where $\tau_0$ is the yield stress:

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.


 \eta = \frac{\tau_0 + k [\dot{\gamma}^{n} -(\tau_0/\mu_0)^n]}{ \dot{\gamma}} (8.4-24)

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

Figure  8.4.4 shows how shear stress ( $\tau$) varies with shear rate ( $\dot{\gamma}$) for the Herschel-Bulkley model.

Figure 8.4.4: Variation of Shear Stress with Shear Rate According to the Herschel-Bulkley Model
figure

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 panel will open, and you can enter the Consistency Index $k$, Power-Law Index $n$, Yield Stress Threshold $\tau_0$, and Yielding Viscosity $\mu_0$.


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