48 research outputs found
Slow axial drift in three-dimensional granular tumbler flow
Models of monodisperse particle flow in partially filled three-dimensional
tumblers often assume that flow along the axis of rotation is negligible. We
test this assumption, for spherical and double cone tumblers, using experiments
and discrete element method simulations. Cross sections through the particle
bed of a spherical tumbler show that, after a few rotations, a colored band of
particles initially perpendicular to the axis of rotation deforms: particles
near the surface drift toward the pole, while particles deeper in the flowing
layer drift toward the equator. Tracking of mm-sized surface particles in
tumblers with diameters of 8-14 cm shows particle axial displacements of one to
two particle diameters, corresponding to axial drift that is 1-3% of the
tumbler diameter, per pass through the flowing layer. The surface axial drift
in both double cone and spherical tumblers is zero at the equator, increases
moving away from the equator, and then decreases near the poles. Comparing
results for the two tumbler geometries shows that wall slope causes axial
drift, while drift speed increases with equatorial diameter. The dependence of
axial drift on axial position for each tumbler geometry is similar when both
are normalized by their respective maximum values
Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations
We present a computational study of finite-time mixing of a line segment by
cutting and shuffling. A family of one-dimensional interval exchange
transformations is constructed as a model system in which to study these types
of mixing processes. Illustrative examples of the mixing behaviors, including
pathological cases that violate the assumptions of the known governing theorems
and lead to poor mixing, are shown. Since the mathematical theory applies as
the number of iterations of the map goes to infinity, we introduce practical
measures of mixing (the percent unmixed and the number of intermaterial
interfaces) that can be computed over given (finite) numbers of iterations. We
find that good mixing can be achieved after a finite number of iterations of a
one-dimensional cutting and shuffling map, even though such a map cannot be
considered chaotic in the usual sense and/or it may not fulfill the conditions
of the ergodic theorems for interval exchange transformations. Specifically,
good shuffling can occur with only six or seven intervals of roughly the same
length, as long as the rearrangement order is an irreducible permutation. This
study has implications for a number of mixing processes in which
discontinuities arise either by construction or due to the underlying physics.Comment: 21 pages, 10 figures, ws-ijbc class; accepted for publication in
International Journal of Bifurcation and Chao
Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua
We compare and contrast two types of deformations inspired by mixing
applications -- one from the mixing of fluids (stretching and folding), the
other from the mixing of granular matter (cutting and shuffling). The
connection between mechanics and dynamical systems is discussed in the context
of the kinematics of deformation, emphasizing the equivalence between stretches
and Lyapunov exponents. The stretching and folding motion exemplified by the
baker's map is shown to give rise to a dynamical system with a positive
Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting
and shuffling does not stretch. When an interval exchange transformation is
used as the basis for cutting and shuffling, we establish that all of the map's
Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per
unit volume, is shown to be exponentially fast when there is stretching and
folding, but linear when there is only cutting and shuffling. We also discuss
how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The
following article appeared in the American Journal of Physics and may be
found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright
2011 American Association of Physics Teachers. This article may be downloaded
for personal use only. Any other use requires prior permission of the author
and the AAP