1,571 research outputs found
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
Homogenization induced by chaotic mixing and diffusion in an oscillatory chemical reaction
A model for an imperfectly mixed batch reactor with the chlorine dioxide-iodine-malonic acid (CDIMA) reaction, with the mixing being modelled by chaotic advection, is considered. The reactor is assumed to be operating in oscillatory mode and the way in which an initial spatial perturbation becomes homogenized is examined. When the kinetics are such that the only stable homogeneous state is oscillatory then the perturbation is always entrained into these oscillations. The rate at which this occurs is relatively insensitive to the chemical effects, measured by the Damkohler number, and is comparable to the rate of homogenization of a passive contaminant. When both steady and oscillatory states are stable, spatially homogeneous states, two possibilities can occur. For the smaller Damkohler numbers, a localized perturbation at the steady state is homogenized within the background oscillations. For larger Damkohler numbers, regions of both oscillatory and steady behavior can co-exist for relatively long times before the system collapses to having the steady state everywhere. An interpretation of this behavior is provided by the one-dimensional Lagrangian filament model, which is analyzed in detail
Surgery on breast cancer in pregnancy
Pregnancy-associated breast cancer (PABC) is defined as breast cancer develops either during or within 1 year after pregnancy, it is a rare disease arising in 1:3,000 to 1:10,000 pregnant women. Prognosis of this tumor is influenced by local or systemic treatment, which might be conditioned by gestational age and limited by the concern on potential adverse impact on fetus. The aim of this literature review is to analyze the main topics regarding surgical treatment of patients diagnosed with breast cancer in pregnancy: anesthesia and maternal-fetal monitoring, type of breast surgery, immediate breast reconstruction after radical surgery and management of the axilla. Some important topics remain controversial since the relative rarity of PABC precludes the feasibility of large studies leading to a lack of literature data. Multi-institutional collaboration is warranted to collect women with PABC, in order to best define surgical treatment in view of associated maternal and fetal risks
Slow decay of concentration variance due to no-slip walls in chaotic mixing
Chaotic mixing in a closed vessel is studied experimentally and numerically
in different 2-D flow configurations. For a purely hyperbolic phase space, it
is well-known that concentration fluctuations converge to an eigenmode of the
advection-diffusion operator and decay exponentially with time. We illustrate
how the unstable manifold of hyperbolic periodic points dominates the resulting
persistent pattern. We show for different physical viscous flows that, in the
case of a fully chaotic Poincare section, parabolic periodic points at the
walls lead to slower (algebraic) decay. A persistent pattern, the backbone of
which is the unstable manifold of parabolic points, can be observed. However,
slow stretching at the wall forbids the rapid propagation of stretched
filaments throughout the whole domain, and hence delays the formation of an
eigenmode until it is no longer experimentally observable. Inspired by the
baker's map, we introduce a 1-D model with a parabolic point that gives a good
account of the slow decay observed in experiments. We derive a universal decay
law for such systems parametrized by the rate at which a particle approaches
the no-slip wall.Comment: 17 pages, 12 figure
Phase separation in a chaotic flow
The phase separation between two immiscible liquids advected by a
bidimensional velocity field is investigated numerically by solving the
corresponding Cahn-Hilliard equation. We study how the spinodal decomposition
process depends on the presence -or absence- of Lagrangian chaos. A fully
chaotic flow, in particular, limits the growth of domains and for unequal
volume fractions of the liquids, a characteristic exponential distribution of
droplet sizes is obtained. The limiting domain size results from a balance
between chaotic mixing and spinodal decomposition, measured in terms of
Lyapunov exponent and diffusivity constant, respectively.Comment: Minor changes - Version accepted for publication - Physical Review
Letter
Walls Inhibit Chaotic Mixing
We report on experiments of chaotic mixing in a closed vessel, in which a
highly viscous fluid is stirred by a moving rod. We analyze quantitatively how
the concentration field of a low-diffusivity dye relaxes towards homogeneity,
and we observe a slow algebraic decay of the inhomogeneity, at odds with the
exponential decay predicted by most previous studies. Visual observations
reveal the dominant role of the vessel wall, which strongly influences the
concentration field in the entire domain and causes the anomalous scaling. A
simplified 1D model supports our experimental results. Quantitative analysis of
the concentration pattern leads to scalings for the distributions and the
variance of the concentration field consistent with experimental and numerical
results.Comment: 4 pages, 3 figure
Experimental evidence of chaotic advection in a convective flow
Lagrangian chaos is experimentally investigated in a convective flow by means
of Particle Tracking Velocimetry. The Fnite Size Lyapunov Exponent analysis is
applied to quantify dispersion properties at different scales. In the range of
parameters of the experiment, Lagrangian motion is found to be chaotic.
Moreover, the Lyapunov depends on the Rayleigh number as . A
simple dimensional argument for explaining the observed power law scaling is
proposed.Comment: 7 pages, 3 figur
Moving walls accelerate mixing
Mixing in viscous fluids is challenging, but chaotic advection in principle
allows efficient mixing. In the best possible scenario,the decay rate of the
concentration profile of a passive scalar should be exponential in time. In
practice, several authors have found that the no-slip boundary condition at the
walls of a vessel can slow down mixing considerably, turning an exponential
decay into a power law. This slowdown affects the whole mixing region, and not
just the vicinity of the wall. The reason is that when the chaotic mixing
region extends to the wall, a separatrix connects to it. The approach to the
wall along that separatrix is polynomial in time and dominates the long-time
decay. However, if the walls are moved or rotated, closed orbits appear,
separated from the central mixing region by a hyperbolic fixed point with a
homoclinic orbit. The long-time approach to the fixed point is exponential, so
an overall exponential decay is recovered, albeit with a thin unmixed region
near the wall.Comment: 17 pages, 13 figures. PDFLaTeX with RevTeX 4-1 styl
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