
For incompressible Newtonian fluids, the shear stress is proportional to the rateofdeformation tensor :
(8.414) 
where is defined by
(8.415) 
and is the viscosity, which is independent of .
For some nonNewtonian fluids, the shear stress can similarly be written in terms of a nonNewtonian viscosity :
(8.416) 
In general, is a function of all three invariants of the rateofdeformation tensor . However, in the nonNewtonian 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
Temperature Dependent Viscosity
If the flow is nonisothermal, 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
(8.418) 
where H(T) is the temperature dependence, known as the Arrhenius law.
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 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 nonNewtonian flows:

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


Note that the nonNewtonian power law described below is different from the power law described in Section
8.4.2.

Appropriate values for the input parameters for these models can be found in the literature (e.g., [ 84]).
Power Law for NonNewtonian Viscosity
If you choose nonnewtonianpowerlaw in the dropdown list to the right of Viscosity, nonNewtonian flow will be modeled according to the following power law for the nonNewtonian viscosity:
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 powerlaw index). The value of determines the class of the fluid:
Newtonian fluid  
shearthickening (dilatant fluids)  
shearthinning (pseudoplastics) 
Inputs for the NonNewtonian Power Law
To use the nonNewtonian power law, choose nonnewtonianpowerlaw in the dropdown list to the right of Viscosity. The NonNewtonian Power Law dialog box will open, and you can choose between Shear Rate Dependent and Shear Rate and Temperature Dependent. Enter the Consistency Index , PowerLaw Index , Minimum and Maximum Viscosity Limit, Reference Temperature , and Activation Energy/R, which is the ratio of the activation energy to the thermodynamic constant .
The Carreau Model for PseudoPlastics
The power law model described in Equation 8.420 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 curvefit to piece together functions for both Newtonian and shearthinning ( ) nonNewtonian laws. In the Carreau model, the viscosity is
and the parameters , , , , and are dependent upon the fluid. is the time constant, is the powerlaw index (as described above for the nonNewtonian power law), and are, respectively, the zero and infiniteshear viscosities, is the reference temperature, and is the ratio of the activation energy to thermodynamic constant. Figure 8.4.1 shows how viscosity is limited by and at low and high shear rates.
Inputs for the Carreau Model
To use the Carreau model, choose carreau in the dropdown 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 , PowerLaw Index , Reference Temperature , Zero Shear Viscosity , Infinite Shear Viscosity , and Activation Energy/R .
Cross Model
The Cross model for viscosity is:
where  =  zeroshearrate viscosity  
=  natural time (i.e., inverse of the shear rate at which the fluid changes  
from Newtonian to powerlaw behavior)  
=  powerlaw index 
The Cross model is commonly used to describe the lowshearrate behavior of the viscosity.
Inputs for the Cross Model
To use the Cross model, choose cross in the dropdown 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 , Time Constant , PowerLaw Index , Reference Temperature , and Activation Energy/R, which is the ratio of the activation energy to the thermodynamic constant .
HerschelBulkley 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 nonzero shear stress when the strain rate is zero:
(8.423) 
where is the yield stress:
The HerschelBulkley model combines the effects of Bingham and powerlaw behavior in a fluid. For low strain 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
For
(8.425) 
where is the consistency factor, and is the powerlaw index.
Figure 8.4.3 shows how shear stress ( ) varies with shear rate ( ) for the HerschelBulkley model.
If you choose the HerschelBulkley model for Bingham plastics, Equation 8.424 will be used to determine the fluid viscosity.
The HerschelBulkley 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 noflow regime, the HerschelBulkley model can also exhibit a shearthinning or shearthickening behavior depending on the value of .
Inputs for the HerschelBulkley Model
To use the HerschelBulkley model, choose herschelbulkley in the dropdown list to the right of Viscosity. The HerschelBulkley dialog box will open, and you can choose between Shear Rate Dependent and Shear Rate and Temperature Dependent. Enter the Consistency Index , PowerLaw Index , Yield Stress Threshold , Critical Shear Rate , Reference Temperature , and the ratio of the activation energy to thermodynamic constant , Activation Energy/R.