Driven inelastic-particle systems with drag.
We study steady states of the motion of a large number of particles in a closed box that are excited by a vibrating boundary and experience a linear drag force from the interstitial fluid. The dissipation in such systems arises from two main sources: Inelasticity in particle collisions and the effects of interstitial fluid on the particles. In many applications, order of magnitude estimates suggest that the dissipation due to interstitial fluid effects may greatly exceed that due to inelasticity and one is naturally led to neglect inelastic effects. In this study, we show that, if one adopts a linear drag force and inelastic effects are neglected, a steady state only exists when the vibration speed of the boundary is below a critical value. For vibration speeds above this critical value, no steady state exists since the kinetic energy of the particles grows without bound. We show that, for vibration speeds above the critical value, inelastic effects must be included to obtain a steady state even if order of magnitude estimates suggest they are negligible. Numerical simulations confirm these theoretical predictions. We also show that inclusion of apparently small nonlinear drag terms can also play a similar role in preventing the kinetic energy of the particles growing without bound.
Wylie
Jonathan J
JJ
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong.
Zhang
Qiang
Q
Li
Yun
Y
Hengyi
Xu
X