TY - JOUR

T1 - Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows

AU - Sturman, R.

AU - Meier, S. W.

AU - Ottino, J. M.

AU - Wiggins, S.

PY - 2008/5/1

Y1 - 2008/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=42949133478&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42949133478&partnerID=8YFLogxK

U2 - 10.1017/S002211200800075X

DO - 10.1017/S002211200800075X

M3 - Article

AN - SCOPUS:42949133478

VL - 602

SP - 129

EP - 174

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -