Granular media at multiple scales: mathematical analysis, modelling and computation

Abstract

There are many challenges in modelling granular media, in particular due to hard particle interactions such as collisions. Modelling and simulating at a microscopic level produces very accurate results, but simulations are generally restricted to relatively small systems of particles. It is also difficult to construct a simple continuum model which accurately describes all the properties of granular media. In this thesis, we consider a number of the problems associated with modelling granular media. We first look at the microscopic dynamics of individual particles and how to derive physically appropriate interactions between them, and discuss Event-Driven Particle Dynamics (EDPD) as an accurate and efficient way to model a system of hard, spherical particles. We then present a novel derivation of the weak form of the Liouville equation which can model systems where particles interact instantaneously (e.g. via inelastic collisions). From here we construct the BBGKY hierarchy and use moment closure methods to construct a new, accurate continuum model for granular media, based on Dynamical Density Functional Theory (DDFT). We then use EDPD to construct approximations for the radial correlation function which accounts for friction, packing fraction and inelasticity. This is then included in the DDFT in simulated examples.