Mathematical Modelling of Particulate Flows

In this chapter, the equations governing the dynamics of particulate flows are presented and discussed. We focus here on the notion of ‘particle resolution scale’, in terms of whether individual particle dynamics are resolved or not. The concept of particle resolution scale is fundamental for obtaining insights into mechanisms ranging from the particle scale processes up to the geophysical flow scales. Since it is not feasible to simultaneously resolve all of these scales, we presently discuss micro- and meso-scale models of relevance for macro-geophysical applications. In this chapter, the particle-resolved methods are shown first, which are based on the Eulerian description of the carrier flow and the Lagrangian description of the motion of individual particles. In order to investigate the meso-scale processes of geophysical flows, it is necessary to work with equations averaged over scales much larger than the particle scale, enabling Eulerian description of both an equivalent fluid phase and an equivalent particle phase (Euler-Euler). The Euler-Euler approach requires closures, as part of the dynamics and mechanics are not resolved, which include fluid-particle interaction forces, subgrid turbulence and granular rheology. Such problem closures are discussed, though not exhaustively, along the book where necessary. There are other approaches that focus on resolution of different scales that may be found in the literature and some being also used in other chapters of this book, and will be briefly explained throughout the work as needed. Some of them result from simplifications of the physical processes involved in the fluid-particle system. As for instance, depending on the size, the relative density and the solid volume fraction, the dynamics of the particle phase can be either coupled or uncoupled (one-way coupling approach) from that of the fluid. This can lead to single-phase methods augmented with a particle concentration transport equation, useful to describe systems that extend over very large scales (kilometers and more).