The Energy-Minimized Multiscale EMMS Model

There are thus three scales of interaction between the fluid and the particles  

According to the above physical model, the following equations have been derived  

The above six equations for continuity and force balance do not, however, afford a complete description of a heterogeneous particle-fluid system in which a dense phase and a dilute phase coexist. An additional constraint needs to be identified to account for the stability of the system. 

This constraint is to be provided through the concept of minimal energy. According to this concept, particles in a vertical flow system tend toward certain dynamic array which results in minimal energy. 

Consequently, modeling of a two-phase flow system is subject to both the constraints of the hydrodynamic equations and the constraint of minimizing N. Such modeling is a nonlinear optimization problem. Numerical solution on a computer of this mathematical system yields the eight parameters  

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