7.4.5. Viscosity for Non-Newtonian Fluids

7.4.5. Viscosity for Non-Newtonian Fluids

For incompressible Newtonian fluids, the shear stress is proportional to the rate-of-deformation tensor :

(7–31)

where is defined by

(7–32)

and is the viscosity, which is independent of .

For some non-Newtonian fluids, the shear stress can similarly be written in terms of a non-Newtonianviscosity :

(7–33)

In general, is a function of all three invariants of the rate-of-deformation tensor . However, in the

non-Newtonian models available in ANSYS FLUENT, is considered to be a function of the shear rate

only. is related to the second invariant of and is defined as

(7–34)

7.4.5.1. Temperature Dependent Viscosity

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

(7–35)

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

(7–36)

where is the ratio of the activation energy to the thermodynamic constant and is a reference temperature

for which H(T) = 1. , which is the temperature shift, is set to 0 by default, and corresponds to the lowest

7.4.5. Viscosity for Non-Newtonian Fluids http://www.arc.vt.edu/ansys_help/flu_ug/flu_ug_sec_viscosity_non_new...

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temperature that is thermodynamically acceptable. Therefore and are absolute temperatures.

Temperature dependence is only included when the energy equation is enabled. Set the parameter 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

Important:

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 powerlaw described in Power-Law Viscosity Law.

Non-Newtonian model based on single fluid formulation is available for the mixturemodel 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 (for example,[92]).

7.4.5.2. Power Law for Non-Newtonian Viscosity

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

(7–37)

where and are input parameters. is a measure of the average viscosity of the fluid (the consistency

index); is a measure of the deviation of the fluid from Newtonian (the power-law index). The value of

determines the class of the fluid:

→ Newtonian fluid

→ shear-thickening (dilatant fluids)

→ shear-thinning (pseudo-plastics)

7.4.5.2.1. Inputs for the Non-Newtonian Power Law


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To use the non-Newtonian power law, choose non-newtonian-power-law in the drop-down list to the right ofViscosity. The Non-Newtonian Power Law Dialog Box will open, and you can choose between Shear RateDependent and Shear Rate and Temperature Dependent. Enter the Consistency Index , Power-Law

Index , Minimum and Maximum Viscosity Limit , Reference Temperature , and Activation

Energy/R, , which is the ratio of the activation energy to the thermodynamic constant.

7.4.5.3. The Carreau Model for Pseudo-Plastics

The power law model described in Equation 7–37 results in a fluid viscosity that varies with shear rate. For , , and for , , where and 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 piecetogether functions for both Newtonian and shear-thinning ( ) non-Newtonian laws. In the Carreau

model, the viscosity is

(7–38)

and the parameters , , , , and are dependent upon the fluid. is the time constant, is the

power-law index (as described above for the non-Newtonian power law), and are, respectively, the

zero- and infinite-shear viscosities, is the reference temperature, and is the ratio of the activation energy

to thermodynamic constant. Figure 7.16: Variation of Viscosity with Shear Rate According to the CarreauModel shows how viscosity is limited by and at low and high shear rates.

Figure 7.16: Variation of Viscosity with Shear Rate According to the Carreau Model

7.4.5.3.1. Inputs for the Carreau Model

To use the Carreau model, choose carreau in the drop-down list to the right of Viscosity. The Carreau ModelDialog Box will open, and you can choose between Shear Rate Dependent and Shear Rate andTemperature Dependent. Enter the Time Constant , Power-Law Index , Reference Temperature ,

Zero Shear Viscosity , Infinite Shear Viscosity , and Activation Energy/R .


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Figure 7.17: The Carreau Model Dialog Box

7.4.5.4. Cross Model

The Cross model for viscosity is:

(7–39)

where,

= zero-shear-rate viscosity

= natural time (that is, inverse of the shear rate at which the fluid changes from

Newtonian to power-law behavior)

= power-law index

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

7.4.5.4.1. 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 DialogBox will open, and you can choose between Shear Rate Dependent and Shear Rate and TemperatureDependent. Enter the Zero Shear Viscosity , Time Constant , Power-Law Index , Reference

Temperature , and Activation Energy/R, , which is the ratio of the activation energy to the

thermodynamic constant.

7.4.5.5. Herschel-Bulkley Model for Bingham Plastics


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

(7–40)

where is the yield stress:

For , the material remains rigid.

For , 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 lowstrain rates ( ), the “rigid” material acts like a very viscous fluid with viscosity . As the strain rate

increases and the yield stress threshold, , is passed, the fluid behavior is described by a power law.

For

(7–41)

For

(7–42)

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

Figure 7.18: Variation of Shear Stress with Shear Rate According to the Herschel-Bulkley Model shows howshear stress ( ) varies with shear rate ( ) for the Herschel-Bulkley model.

Figure 7.18: Variation of Shear Stress with Shear Rate According to the Herschel-Bulkley Model


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If you choose the Herschel-Bulkley model for Bingham plastics, Equation 7–41 will be used to determine thefluid viscosity.

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

7.4.5.5.1. 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 ShearRate and Temperature Dependent. Enter the Consistency Index , Power-Law Index , Yield Stress

Threshold , Critical Shear Rate , Reference Temperature , and the ratio of the activation energy to

thermodynamic constant , Activation Energy/R.

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7.4.5. Viscosity for Non-Newtonian Fluids