skip to main content


Title: A study in three-dimensional chaotic dynamics: Granular flow and transport in a bi-axial spherical tumbler

We study three-dimensional (3D) chaotic dynamics through an analysis of transport in a granular flow in a half-full spherical tumbler rotated sequentially about two orthogonal axes (a bi-axial “blinking” tumbler). The flow is essentially quasi-two-dimensional in any vertical slice of the sphere during rotation about a single axis, and we provide an explicit exact solution to the model in this case. Hence, the cross-sectional flow can be represented by a twist map, allowing us to express the 3D flow as a linked twist map (LTM). We prove that if the rates of rotation about each axis are equal, then (in the absence of stochasticity) particle trajectories are restricted to two-dimensional (2D) surfaces consisting of a portion of a hemispherical shell closed by a “cap''; if the rotation rates are unequal, then particles can leave the surface they start on and traverse a volume of the tumbler. The period-one structures of the governing LTM are examined in detail: analytical expressions are provided for the location of period-one curves, their extent into the bulk of the granular material, and their dependence on the protocol parameters (rates and durations of rotations). Exploiting the restriction of trajectories to 2D surfaces in the case ofmore » equal rotation rates about the axes, a method is proposed for identifying and constructing 3D Kolmogorov--Arnold--Moser (KAM) tubes around the normally elliptic period-one curves. The invariant manifold structure arising from the normally hyperbolic period-one curves is also examined. When the motion is restricted to 2D surfaces, the structure of manifolds of the hyperbolic points in the bulk differs from that corresponding to hyperbolic points in the flowing layer. Each is reminiscent of a template provided by a non-integrable perturbation to a Hamiltonian system, though the governing LTM is not. This highlights the novel 3D chaotic behaviors observed in this model dynamical system.« less
 [1] ;  [2] ;  [2] ;  [3]
  1. Northwestern Univ., Evanston, IL (United States); Princeton Univ., Princeton, NJ (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  2. Northwestern Univ., Evanston, IL (United States)
  3. Univ. of Leeds, Leeds (United Kingdom)
Publication Date:
Report Number(s):
Journal ID: ISSN 1536-0040
Grant/Contract Number:
Accepted Manuscript
Journal Name:
SIAM Journal on Applied Dynamical Systems
Additional Journal Information:
Journal Volume: 13; Journal Issue: 2; Journal ID: ISSN 1536-0040
Society for Industrial and Applied Mathematics
Research Org:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org:
Country of Publication:
United States
42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; Mathematics; chaotic advection; transport; linked twist maps; granular flow
OSTI Identifier: