Spatial averaging in the mechanics of heterogeneous and dispersed systems

Spatial averaging of the equations describing two single-phase media is separately considered in this paper regarding the volumes occupied by either phase with allowance for the boundary conditions on phase interfaces. The equations obtained are specialized to describe monodispersed mixtures within a “cellular” scheme. It is shown that it is necessary to consider the average values both over the whole cells and over those intersected by the boundary of a selected mixture volume.The problem of motion in the cell is formulated. Fictitious parameters are introduced at “infinity” for the carrier phase to solve the problem. These parameters do not coincide with the average values for this phase. A closed system of equations is derived for two extreme cases: an ideal incompressible carrier fluid and an extremely viscous incompressible carrier fluid. These are correlative with the inertial and viscous motions in the cell.Various effects are discussed in this paper. These include the radial motion of bubbles, the oriented rotation of dispersed particles (the symmetry and asymmetry of stress tensor), viscosity, phase transitions and the finite volume content of dispersed particles. Some aspects of earlier studies are critically analyzed.