Granular flow theories governing

The two-fluid granular flow model is formulated applying the classical Eulerian continuum concept for the continuous phase, while the governing equations of the particle phase are developed in accordance with the principles of kinetic theory. In this theory it is postulated that the particulate system can be represented considering a collection of identical, smooth, rigid spheres, adapting a Boltzmann type of equation. This microscopic balance describes the rate of change of the distribution function with respect to position and time. 

For reactive flows the governing equations used by Lindborg et al [92] resemble those in sect 3.4.3, but the solid phase momentum equation contains several additional terms derived from kinetic theory and a frictional stress closure for slow quasi-static flow conditions based on concepts developed in soil mechanics. Moreover, to close the kinetic theory model the granular temperature is calculated from a separate transport equation. To avoid misconception the model equations are given below (in which the averaging symbols are disregarded for convenience) ... 

It is therefore common to assume that the state variables that describe the rapid deformation response of granular materials would border on the parameters that describe the behavior of fluids and Coulomb type dissipation of energy (static). In view of the above it is common to find that the theories governing granular flow are formulated around the assumption of a continuum similar in some regard to viscous fluids however, the equilibrium states of the theories are not states of hydrostatic pressure, as would be in the case of fluids, but are rather states that are specified by the Mohr-Coulomb criterion (Cowin, 1974). The advantage of continuum formulations over alternative particulate (stochastic) formulations is that use of continuum is more capable of generating predictive results. Mathematically one... 

Chapter 2 contains a summary of the basic concepts of kinetic theory of dilute and dense gases. This theory serves as basis for the development of the continuum scale conservation equations by averaging the governing equations determining the discrete molecular scale phenomena. This method is an alternative to, or rather both a verification and an extension of, the continuum approach described in Chap. 1. These kinetic theory concepts also determine the basis for a group of models used describing granular flows, further outlined in Chap. 4. 

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