On the Disordered, Collective and Patterned Movements of Media. Qualitative Analysis of Movements

Abstract

The motion of elements is classified from a topological aspect. We separate three different, disordered, collective and patterned motion forms of media. We will reveal what concrete shapes the different type media take. In case of a limited number of particles, a discrete description is suitable for tracking the mechanic behavior of the medium. In case of the large of a number of particles we apply in some sense continuous mathematical model according to the motion occurring in the medium instead of a discrete description. We postulate a statistical distribution to the disordered motion; we "distribute" the particles for the collective motion in the physical space and apply a differential geometric description. We assign continuous flow lines to the regular patterned motion of the medium with free particles while we assign energy dissipation to the flow image an irregular patterned motion. In the case of deformable solid body built by periodically arranged rigid bodies the state functions with discrete domain of definition are represented by functions with continuous domain of definition. Granular conglomerations seem to be the only such medium, which allow tracking of the state of all granules.
Disordered motion leads to thermodynamic - statistical description. The collective motion of free particles leads to the description of the laminar flow of particles, the collective motion of fixed particles leads to the classical continuum. The patterned motion of free particles leads to the description of vortex or turbulent flow, the patterned motion of fixed particles leads to grid continua.