Flows of suspended grains: mixing a sparse phase with simple and complex continua

We propose a multi-scale view on the motion of grains inside a continuum body undergoing strain. We consider the whole compound as a mixture and attribute to every material point the statistical properties of a grain cluster in a control volume open to mass exchange with the surrounding medium. Description is entirely Eulerian and accounts for anisotropy in grain (particle) distribution within the continuum, and its effects on the global motion. Essentially, what we propose here is a framework useful for construct- ing specific models of body classes like suspensions with various nature, landslides, dune motion under wind action, some transient states during formation of particulate composites, general colloids. As re- gards foundational aspects of mixture theory—which we extend here—besides showing once again that two of the five Truesdell’s classical axioms emerge from a unique assumption, the so-called rule of total external power, we show also that another Truesdell’s axiom requires to be refined when we describe at least one component as a complex material. We also underline the role of two Reynolds tensors, one at macroscopic scale, associated with fluctuations of each single component, the other pertaining to grain fluctuations. Our main arguments rely on invariance requirements with respect to classes of changes in observers.