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Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows

R. STURMANa1, S. W. MEIERa2, J. M. OTTINOa2a3 and S. WIGGINSa4 a1 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK r.sturman@maths.leeds.ac.uk a2 Department of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208, USA smeier@chem-eng.northwestern.edu a3 The Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, IL 60208, USA jm-ottino@northwestern.edu a4 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK s.wiggins@bristol.ac.uk

Article author query sturman r  meier sw  ottino jm  wiggins s

Abstract

We study the mixing properties of two systems: (i) a half-filled quasi-two-dimensional circular drum whose rotation rate is switched between two values and which can be analysed in terms of the existing mathematical formalism of linked twist maps; and (ii) a half-filled three-dimensional spherical tumbler rotated about two orthogonal axes bisecting the equator and with a rotational protocol switching between two rates on each axis, a system which we call a three-dimensional linked twist map, and for which there is no existing mathematical formalism. The mathematics of the three-dimensional case is considerably more involved. Moreover, as opposed to the two-dimensional case where the mathematical foundations are firm, most of the necessary mathematical results for the case of three-dimensional linked twist maps remain to be developed though some analytical results, some expressible as theorems, are possible and are presented in this work. Companion experiments in two-dimensional and three-dimensional systems are presented to demonstrate the validity of the flow used to construct the maps. In the quasi-two-dimensional circular drum, bidisperse (size-varying or density-varying) mixtures segregate to form lobes of small or dense particles that coincide with the locations of islands in computational Poincaré sections generated from the flow model. In the 3d spherical tumbler, patterns formed by tracer particles reveal the dynamics predicted by the flow model.

(Received September 01 2007)

(Revised January 21 2008)