### 5.6.1. Solids Stress Tensor

Additional stresses due to inter-particle collisions are modeled using a collisional solids stress tensor in the solid phase momentum equation only:

| (5–88)

Here, denotes solids pressure, denotes solids shear viscosity, and denotes solids bulk viscosity. There are two important classes of models for these quantities:

#### 5.6.1.1. Empirical Constitutive Equations

There exist wide classes of models where the constitutive elements of the solids stress are specified using empirical constitutive relations. See, for example, (Enwald et al. [97]). In many of these, the solids pressure, shear and bulk viscosities are simple functions of the solid phase volume fraction.

#### 5.6.1.2. Kinetic Theory Models for the Solids Stress Tensor

These are a class of models, based on the kinetic theory of gases, generalized to take into account inelastic particle collisions. In these models, the constitutive elements of the solids stress are functions of the solid phase granular temperature, defined to be proportional to the mean square of fluctuating solid phase velocity due to inter-particle collisions:

| (5–89)

In the most general kinetic theory models, the granular temperature is determined from a transport equation. However, in many circumstances, it is possible to ignore the transport terms, and determine granular temperature from the resulting algebraic equation.

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