American Institute of Mathematical Sciences

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January  2009, 8(1): 275-293. doi: 10.3934/cpaa.2009.8.275

Inelastic Collapse in a Corner

Ming Gao 1, , Jonathan J. Wylie 2, and  Qiang Zhang 1,

1. 

Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, Hong Kong
2. 

Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China

Received  March 2008 Revised  July 2008 Published  October 2008

We consider the interaction of a rigid, frictionless, inelastic particle with a rigid boundary that has a corner. Typically, two possible final outcomes can occur: the particle escapes from the corner after experiencing a certain number of collisions with the corner, 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 a surprising phenomenon that if the corner has the form of a cusp, 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.
Keywords: Inelastic collapse, corner.
Mathematics Subject Classification: Primary: 70E18; Secondary: 70F25.
Citation: Ming Gao, Jonathan J. Wylie, Qiang Zhang. Inelastic Collapse in a Corner. Communications on Pure & Applied Analysis, 2009, 8 (1) : 275-293. doi: 10.3934/cpaa.2009.8.275
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