The Equations of Motion for Granular Flows

The equations of motion for granular flows have been derived by adopting the kinetic theory of dense gases. This approach involves a statistical-mechanical treatment of transport phenomena rather than the kinematic treatment more commonly employed to derive these relationships for fluids. The motivation for going to the formal approach (i.e., dense gas theory) is that the stress field consists of static, translational, and collisional components and the net effect of these can be better handled by statistical mechanics because of its capability for keeping track of collisional trajectories. However, when the static and collisional contributions are removed, the equations of motion derived from dense gas theory should (and do) reduce to the same form as the continuity and momentum equations derived using the traditional continuum fluid dynamics approach. In fact, the difference between the derivation of the granular flow equations by the kinetic approach described above and the conventional approach via the Navier Stokes equations is that, in the latter, the material properties, such as viscosity, are determined by experiment while in the former the fluid properties are mathematically deduced by statistical mechanics of interparticle collision. 

Equations of motion and the pertinent constitutive equations for the flow of granular materials have been developed by Lun et al. (1984) using the hard sphere kinetic theory of dense gas approach. In this derivation a fixed control volume was considered in which a discrete number of smooth but inelastic particles are undergoing deformation. The resulting system of equations was given as follows  

In addition to mass conservation. Equation (4.3), and momentum conservation. Equation (4.4), a third relationship that is required to describe the flow is some form of energy conservation equation. The total energy per unit mass of the granular material, E, may be broken into three components Oohnson and Jackson, 1987)  

The kinetic energy, E., associated with the local average velocity u 

The pseudo-thermal energy, Epj, associated with deviations of the motion of individual particles from the local average Epj can be represented by the kinetic energy definition of temperature as 

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