# Continuity Equation

The continuity equation for the solid-phase *i* can be written as

*e _{i}p_{i} = n_{i}m_{i}* is the mass of phase i per unit volume of mixture,

*e*is its solid volume fraction, and

_{i}*p*is density of phase i. Here у =< c

_{i}_{;}> is the mean velocity of the particle i. The instantaneous velocity c

_{;}is defined as the sum of the average velocity, v

_{;}, and peculiar velocity, С;,

c = v_{;} + С; with < C >= 0

# Momentum Equation

The momentum equation for phase i may be expressed as

# Fluctuating Energy Equation

The fluctuating energy equation for solid-phase i can be expressed as where

*0 _{i}* is the granular temperature or the fluctuating granular energy of the solid-phase I,

In the above governing equations, the relevant variables describing the flow field are the average velocities, the solid volume fractions, and the granular temperatures evaluated at location r of the center of the particle at time t.

# Kinetic Equation

The kinetic equations that characterize the flow of a multiphase system are

where *g _{ip}(r)* is the spatial-pair radial distribution function when the particles i and p are in contact. A solution of Eq. (2.87) near the equilibrium was obtained using the Chapman-Enskog method (Ferziger and Kaper 1972; Chapman and Cowling 1970),

where f^{0} is the Maxwellian velocity distribution

and ф is a perturbation to the Maxwellian velocity distribution. It is a linear function of the first derivative of *n _{i}, d_{i},* and

*v*

_{i}. Note that ф is function of the phase mean velocity, v

_{i}, and not the total flow velocity, because, as mentioned in the introduction, each kind of particle is treated as a separate phase and the interaction is at the interface. The radial distribution function

*g*describes a multisized mixture of hard spheres at contact. Iddir and Arastoopour (2005) modified the Lebowitz (1964) radial distribution function. This approach is in agreement with the results of molecular dynamics (MD) simulation obtained by Alder and Wainwright (1967) at both lower and higher solid volume fractions. This equation can be written as

_{ip}(e_{i}, e_{p})

The expression of *g _{pp}(e_{i}, e_{p})* is obtained by simply interchanging the indices i andp.

For a more detailed explanation for constitutive relation expressions for all solid phases, see Iddir and Arastoopour (2005).