Geometrical and physical models of abrasion
We extend the geometrical theory presented in [5] for collisional and frictional particle abrasion to include an independent physical equation for the evolution of mass and volume. We introduce volume weight functions as multipliers of the geometric equations and use these mutipliers to enforce physical volume evolution in the unified equations. The latter predict, in accordance with Sternberg's Law, exponential decay for volume evolution. We describe both the PDE versions, which are generalisations of Bloore's equations and their heuristic ODE approximations, called the box equations. The latter are suitable for tracking the collective abrasion of large particle populations. The mutual abrasion of identical particles, called the self-dual ows, play a key role in explaining geological scenarios. We give stability criteria for the self-dual ows in terms of the parameters of the physical volume evolution models and show that under reasonable assumptions these criteria can be met by physical systems. We also study a natural generalisation, the unidirectional Bloore equation, covering the case of unidirectional abrasion. We have previously shown that his equation admits travelling front solutions with circular profiles. More generally, in three dimensions, they are so-called linear or special Weingarten surfaces.