Objective fluxes in a multi-scale continuum description of sparse medium dynamics

We discuss reasons justifying a multi-scale continuum description of sparse media, which do not admit a choice of a representative volume element remaining permanent in time with constant mass. We suggest the choice of objective derivatives for time-varying terms in the balance equations pertinent to the scheme that we analyze, to avoid some problems connected with SO(3)-based changes in observers, which emerge within the setting of the standard theory of gases when we start from it to get continuum schemes. The scheme discussed here can be reduced to versions of the averaged regularizations of the Navier-Stokes equations. In our approach we have primarily in mind the continuum description of bodies like macro-molecular fluids or granular ones, even if what we propose can be significant in a broader setting.