Behavior of intruders in a granular flow

Abstract

In this thesis, we investigate the behavior of intruders in a 2-dimensional granular system composed of rigid, frictionless, inelastic disks. The system is driven by adding energy at the boundary which is held at constant temperature in a thermodynamic sense. We consider an intruder with the shape of a circular segment. In the simulations, we fix one end point of the intruder in the center of the system. If all the collisions in the system are elastic, the rotation of the intruder behaves like a Brownian motion with zero drift. However, in the inelastic case the intruder will rotate towards a preferred direction. We give an explanation of this phenomenon and also present a theoretical analysis of the force exerted on the intruder in the dilute case. The result from the simulation validates our theory. We also consider an intruder with a “_” shape. Similar phenomena are observed, and a theoretical analysis is also performed in the dilute case. Then we study the behavior of two intruders in the granular flow and find that in the elastic case the positions of the two intruders are uniformly distributed in the system. However, in the inelastic case the two intruders have a clear trend to stay together. We investigate the “attractive force” between the two intruders and find that the “attractive force” comes from two sources: a) Each intruder exerts a force towards the center of the system because of the temperature gradient caused by inter-particle collisions. b) There exists an interaction between the two intruders caused by the inelastic particleintruder collisions. We present a theoretical analysis of the interaction between the two intruders in the dilute case. The theoretical prediction is in excellent agreement with the result from the simulation in the dilute case. In the less dilute case, the theory can still give qualitative predictions about the interaction between the two intruders. In the latter part of the thesis, we study the interaction of a rigid, frictionless, in elastic particle with a rigid boundary that has a corner. Typically, two possible final outcomes can occur in such a system: the particle escapes from the corner after experiencing a certain number of collisions with the boundary, or the particle experiences an inelastic collapse in which an infinite number of collisions can occur in a finite time interval. For the former case, we determine the number of collisions that the particle will experience with the boundary before escaping the corner. For the latter case, we determine the conditions for which inelastic collapse can occur. For a corner composed of two straight walls, we derive simple analytic solutions and show that for a given coefficient of restitution, there is a critical corner angle above which inelastic collapse cannot occur. We show that as the corner angle tends to the critical corner angle from below, the process of inelastic collapse takes infinitely long. We also show that if the corner has the form of a cusp, then inelastic collapse can occur without the velocity of the particle becoming zero. Surprisingly, this means that the particle can have an infinite number of collisions with the boundary in a finite time interval without losing all of its energy, and eventually escapes from the corner.