The solids stress tensor contains shear and bulk viscosities arising from particle momentum exchange due to translation and collision. A frictional component of viscosity can also be included to account for the viscous-plastic transition that occurs when particles of a solid phase reach the maximum solid volume fraction.
The collisional and kinetic parts, and the optional frictional part, are added to give the solids shear viscosity:
 | (14–362) |
The third term in Equation 14–362 is only included when the solids fraction is greater than the friction packing limit.
Ansys Fluent provides two expressions for the kinetic part.
The default expression is from Syamlal et al. [640] :
 | (14–364) |
The following optional expression from Gidaspow et al. [205] is also available:
 | (14–365) |
The solids bulk viscosity accounts for the resistance of the granular particles to compression and expansion. It has the following form from Lun et al. [397]:
 | (14–366) |
Note that the bulk viscosity is set to a constant value of zero, by default. It is also possible to select the Lun et al. expression or use a user-defined function.
In dense flow at low shear, where the secondary volume fraction for a solid phase nears the packing limit, the generation of stress is mainly due to friction between particles. The solids shear viscosity computed by Ansys Fluent does not, by default, account for the friction between particles.
To model the frictional viscosity, Ansys Fluent provides the following models:
Schaeffer [575]
constant
user-defined
Schaeffer is given by:
 | (14–367) |
where 
is the frictional pressure, 
is the angle of internal friction, and 
is the second invariant of the deviatoric stress tensor. It is also possible to specify a constant or user-defined frictional viscosity.
In granular flows with high solids volume fraction, instantaneous collisions are less important. The application of kinetic theory to granular flows is no longer relevant since particles are in contact and the resulting frictional stresses need to be taken into account. Ansys Fluent extends the formulation of the frictional viscosity and employs other models, as well as providing new hooks for UDFs. See the Fluent Customization Manual for details.
The frictional stresses are usually written in Newtonian form:
 | (14–368) |
The frictional stress is added to the stress predicted by the kinetic theory when the solids volume fraction exceeds a critical value. This critical value is referred to as the friction packing limit in Ansys Fluent. Then
 | (14–369) |
 | (14–370) |
The derivation of the frictional pressure is mainly semi-empirical, while the frictional viscosity can be derived from the first principles.
For frictional pressure, Ansys Fluent offers the following models:
The Johnson and Jackson model for frictional pressure is defined as:
 | (14–371) |
where 
is a function of the volume fraction computed as:
 | (14–372) |
In Equation 14–371, 
is the friction packing limit, 
is the packing limit, the coefficient 
= 2, and 
= 5 [483].
The second model that is employed is Syamlal et al. [640]. Comparing the two models results in the frictional normal stress differing by orders of magnitude.
The radial distribution function is an important parameter in the description of the solids pressure resulting from granular kinetic theory. If we use the models of Lun et al. [397] or Gidaspow [204] the radial function tends to infinity as the volume fraction tends to the packing limit. It would then be possible to use this pressure directly in the calculation of the frictional viscosity, as it has the desired effect. This approach is also available in Ansys Fluent as the based-ktgf option.
Important: The introduction of the frictional viscosity helps in the description of frictional flows, however a complete description would require the introduction of more physics to capture the elastic regime with the calculation of the yield stress and the use of the flow-rule. These effects can be added by you via UDFs to model static regime. Small time steps are required to get good convergence behavior.