Chaos in one-dimensional granular flows with oscillating boundaries

The following problem is investigated: N identical, inelastic particles are set in motion along a frictionless horizontal interval bounded on the left by a fixed wall at x=0 and on the right by another wall oscillating according to the equation x = a-α sin ωt, where a > 0 is the initial length of the interval and 0 ≤ α < a and ω > 0 are, respectively, the amplitude and frequency of vibration of the moving wall. For large N the following results, among others, are proved using techniques from dynamical systems theory and also illustrated using a simulation program: (i) If a is sufficiently small, the steady-state granular flow is stationary. (ii) If ω is sufficiently large, the granular flow exhibits period-doubling cascades leading to chaotic motion at certain critical values αc = αc (ω) as ω increases, (iii) Periodic particle motions of any period can be obtained by properly choosing the values of α and ω. An approximate continuum model for the granular flow for N → ∞ in the form of a nonlinear partial differential equation with periodic boundary conditions is also studied. This infinite-dimensional dynamical system is shown to exhibit solutions with properties consistent with those of the large finite particle configurations.