Neutrally Buoyant Particles and Bailout Embeddings in Three-Dimensional Flows
Julyan H. E. Cartwright
We use the bailout embeddings of three-dimensional volume-preserving maps to study qualitatively the dynamics of small spherical neutrally buoyant impurities suspended in a time-periodic incompressible fluid flow. The accumulation of impurities in tubular vortical structures, the detachment of particles from fluid trajectories near hyperbolic invariant lines, and the formation of nontrivial three-dimensional structures in the distribution of particles are predicted.
bailout_3D.tex
\documentclass[prl,showpacs,superscriptaddress,amsfonts,twocolumn]{revtex4}
\usepackage{bm}
\usepackage{times}
\usepackage[dvips]{graphicx}
\usepackage[T1]{fontenc}
\usepackage{color}

\begin{document}

\title{Neutrally Buoyant Particles and Bailout Embeddings in Three-Dimensional 
Flows}

\author{Julyan H. E. Cartwright} 
\homepage{http://lec.ugr.es/~julyan}
\email{julyan@lec.ugr.es}
\affiliation{Laboratorio de Estudios Cristalogr\'aficos, CSIC, 
E-18071 Granada, Spain}
\author{Marcelo O. Magnasco} 
\homepage{http://asterion.rockefeller.edu/marcelo/}
\email{marcelo@sur.rockefeller.edu}
\affiliation{Mathematical Physics Lab, Rockefeller University, 
Box 212, 1230 York Avenue, NY 10021}
\author{Oreste Piro}
\homepage{http://www.imedea.uib.es/~piro}
\email{piro@imedea.uib.es}
\author{Idan Tuval.} 
\homepage{http://www.imedea.uib.es/~idan} 
\email{idan@imedea.uib.es} 
\affiliation{Institut Mediterrani d'Estudis Avan\c{c}ats, CSIC--UIB,
E-07071 Palma de Mallorca, Spain}

\date{version of \today}

\begin{abstract}
We use the bailout embeddings of three-dimensional volume-preserving maps to
study qualitatively the dynamics of small spherical neutrally buoyant
impurities suspended in a time-periodic incompressible fluid flow. The 
accumulation of impurities in tubular vortical structures, the detachment of 
particles from fluid trajectories near hyperbolic invariant lines, and the 
formation of nontrivial three-dimensional structures in the distribution of 
particles are predicted.
\end{abstract}

\pacs{05.45.Gg, 47.52.+j, 45.20.Jj}

\maketitle

The dynamics of small spherical particles immersed in a fluid flow have received
considerable attention in the past few years from both the theoretical and
experimental points of view. On one hand, these particles are the simplest
models for impurities whose transport in flows is of practical interest to 
understand, and, on the other, their motion is governed by dynamical systems
that display a rich and complex variety of behavior even in the case of the
most minimal approximations.

When the density of the particles does not match that of the fluid, it is
intuitively clear that the trajectories of a particle and a fluid parcel will
in general differ. This has been demonstrated in two-dimensional flows in which
particles with density higher than the basic flow tend to migrate away from the
parts of the flow dominated by rotation --- in the case of chaotic flows, the
KAM (Kolmogorov--Arnold--Moser) islands --- while particles lighter than the
fluid display the opposite tendency \cite{crisanti,tanga,partproc}. 
A more surprising
result, however, is that neutrally buoyant particles may also detach from the
fluid parcel trajectories in the regions in which the flow is dominated by
strain, with the result that they end up settling in the KAM islands
\cite{neutpartprl}. The subtle dynamical mechanism responsable for the latter
phenomenon has suggested a method to target KAM islands in Hamiltonian flows,
and a recent generalization named a bailout embedding permits its extension to
Hamiltonian maps as well \cite{bailout1}. 

Despite its obvious importance from the point of view of applications, the case
in which the base flow is three-dimensional has been much less investigated. A
probable reason for this is the fact that very few simple three-dimensional
incompressible flow models exist. The few that are simple are not realistic ---
e.g., the ABC (Arnold--Beltrami--Childress) flow \cite{arnold65} --- and the few
that are realistic are far more complex. In the study of the Lagrangian
structure of three-dimensional incompressible time-periodic flows, where this
difficulty was already present, the alternative approach of qualitatively
modeling the flows by iterated three-dimensional volume-preserving maps was
successful in predicting fundamental structures that were later found in more
realistic theoretical flows and in experiments 
\cite{feingold88.2,pirofein,3dpaper,spherespaper}. However, no similar approach
has been followed to describe the motion of impurities in this kind of flows.

This Letter has the dual purpose of extending the idea of bailout embedding
to a class of dynamical systems outside the Hamiltonian domain, namely 
three-dimensional volume-preserving maps, and by means of this embedding
investigating the generic structures that one might expect in the dynamics of
neutrally buoyant particles suspended in incompressible three-dimensional
time-periodic flows.

Let us first recall the equation of motion for a small, neutrally buoyant,
spherical tracer in an incompressible fluid (the Maxey--Riley equation)
\cite{maxey,michaelides}. Under assumptions allowing us to retain only the
Bernoulli, Stokes drag, and Taylor added mass contributions to the force
exerted by the fluid on the sphere, the equation of motion for the particle at
position $x$ is
\begin{equation} \label{part_in_fluid} 
\frac{d}{dt}(\dot{\bm x}-{\bm u}({\bm x}))=-(\lambda +{\bm\nabla\bm
u})\cdot(\dot{\bm x}-{\bm u}({\bm x}))  
,\end{equation}
where $\dot{\bm x}$ represents the velocity of the particle, $\bm {u}$ that of
the fluid, $\lambda$ a number inversely proportional to the Stokes number of
the particle, and $\bm\nabla\bm u$ the Jacobian derivative matrix of the flow.

\begin{figure*}[ht]
\begin{center}
\includegraphics*[width=0.9\textwidth]{fig1.eps}
\end{center}
\caption{\label{baildraw} 
For a homogeneous distribution of initial conditions
we plot only the last 1000 steps of the map evolution for different values of
$\lambda$: (a) \protect\( \lambda =2,\protect \) (b) \protect\( \lambda
=1,\protect \) (c) \protect\( \lambda =0.6,\protect \) (d) \protect\( \lambda
=0.5\protect \). The images represent the [0,2$\pi$] cube in the phase
space.}
\end{figure*}

The difference between the particle velocity and the velocity of the
surrounding fluid is exponentially damped with negative damping term
$-(\lambda+\bm\nabla\bm u)$. In the case in which the flow gradients reach the
magnitude of the viscous drag coefficient, there is the possibility that
the Jacobian matrix $\bm\nabla\bm u$ may acquire an eigenvalue of positive real
part in excess of the drag coefficient. In these instances, the trajectories of
the impurities, instead of converging exponentially onto those defined by
$\dot{\bm x}=\bm u$, may detach from them.

For incompressible two-dimensional
flows, since the Jacobian matrix is traceless, the two eigenvalues must add up
to zero, which implies that they are either both purely imaginary or both purely
real, mutually equal in absolute value and opposite in sign. The result is that
the particles can abandon the fluid trajectories in the neighborhood of the
saddle points and other unstable orbits, where the Jacobian eigenvalues are
real, and eventually overcome the Stokes drag, to finally end up in a regular
region of the flow on a KAM torus dominated by the imaginary eigenvalues. From
a more physical point of view, this effect implies that the particles tend to
stay away of the regions of strongest strain.

In contrast to the two-dimensional case, in time-dependent three-dimensional
flows the incompressibility condition only implies that the sum of the three
independent eigenvalues must be zero. This less restrictive condition allows
for many more combinations.  Triplets of real eigenvalues, two positive and one
negative or vice versa, as well as one real eigenvalue of any sign together
with a complex-conjugate pair whose real part is of the opposite sign, are
possible. Accordingly, chaotic trajectories may have one or two positive
Lyapunov numbers, and a richer range of dynamical situations may be expected.

Instead of investigating all these in terms of a given fully fledged
three-dimensional time-periodic model flow, we follow a qualitative
approach based on iterated maps that roughly reproduces the properties of the
impurity dynamics in a generic flow of this kind. In order to construct the map
we first note that the dynamical system governing the behavior of neutrally
buoyant particles is composed of some dynamics within another larger set of
dynamics. Equation~(\ref{part_in_fluid}) can be seen as an equation for a
variable $\bm \delta=(\dot{\bm x}-\bm u)$ which in turn will define the
equation of motion $(\dot{\bm x}=\bm u)$ of a fluid element whenever the
solution of the former be zero. In this sense we may say that the fluid parcel
dynamics is embedded in the particle dynamics. In reference to the fact that
some of the embedding trajectories abandon some of those of the embedded
dynamics, the generalization of this process is dubbed a bailout embedding
\cite{bailout1}.

It is rather easy to construct this type of embedding for map dynamics. 
Given a map
\begin{equation}\label{passive_map}
\bm X_{n+1}=\bm T(\bm X_{n})
,\end{equation}
a general bailout embedding is given by
\begin{equation}\label{bailout_map}
\bm X_{n+2}-\bm T(\bm X_{n+1})=\bm K(\bm X_{n})\cdot (\bm X_{n+1}-\bm T(\bm
X_{n}))
,\end{equation}
where $\bm K(\bm X)$ is the so-called bailout function whose properties
determine which trajectories of the embedded map will be eventually
abandoned by the embedding.
The particular choice --- naturally imposed by the particle dynamics ---
of the gradient as the bailout function in a flow
translates in the map setting to
\begin{equation}\label{bail_function}
\bm K(x)=e^{-\lambda }\cdot\bm \nabla \bm T
.\end{equation}

Bailout embeddings have been used to investigate targeting of KAM tori in
Hamiltonian systems as well as to explore generic properties of the
distribution of small particles immersed in incompressible two-dimensional
fluid flows \cite{bailout1}, which are also of a Hamiltonian nature. Following
a similar approach here leads us to consider the bailout embedding of a
class of non-Hamiltonian systems: three-dimensional volume-preserving maps.
In particular, we chose to represent qualitatively chaotic three-dimensional
incompressible base flows periodic in time by ABC maps, a family
\begin{equation}
\label{Tabc}
\bm T=\bm T_{ABC}: (x_{n},y_{n},z_{n})\longrightarrow(x_{n+1},y_{n+1},z_{n+1})
,\end{equation}
where
\begin{eqnarray}
\label{Tabc_explicita}
x_{n+1} &=&x_{n}+A\sin z_{n}  +C\cos y_{n}\,(\bmod 2\pi), 
\nonumber \\ 
y_{n+1} &=&y_{n}+B\sin x_{n+1}+A\cos z_{n}\,(\bmod 2\pi),
\\ 
z_{n+1} &=&z_{n}+C\sin y_{n+1}+B\cos x_{n+1}\,(\bmod 2\pi),
\nonumber 
\end{eqnarray}
that displays all the basic features of interest of the  evolution of fluid
flows. Depending on the parameter values, this map possesses two
quasi-integrable behaviors: the one-action type, in which a KAM-type theorem
exists, and with it invariant surfaces shaped as tubes or sheets; and the
two-action type displaying the phenomenon of resonance-induced diffusion
leading to global transport throughout phase space \cite{feingold88.2}.

Let us now study the dynamics defined by Eqs~(\ref{bailout_map}),
(\ref{bail_function}) and (\ref{Tabc}). We first concentrate on the cases in
which the flow is dominated by one-action behavior. In these we find an
interesting generalization of the behavior already found in two dimensions.
Particle trajectories are expelled from the chaotic regions to finally settle
in the regular KAM tubes. As an example, we take values of $A$, $B$, and $C$
that lead to almost ergodic behavior of the fluid map: a single fluid
trajectory  almost completely covers the phase space. However, from randomly
distributed initial conditions, the particle trajectories inevitably visit some
hyperbolic regions where they detach from the corresponding fluid trajectory.
In this fashion they find their way inside the invariant elliptic structures
where they can finally relax back onto a safe fluid trajectory. In
Fig.~\ref{baildraw} we show how a homogeneous distribution of one hundred
particles in the fluid flow, after a large number of stabilization iterations,
finally settles inside the tubular KAM structures for different values of the
parameter $\lambda$. When the value of $\lambda$ decreases, more random
trajectories follow this evolution; more particles fall into the
invariant tubes.

\begin{figure}[tb]
\begin{center}
\includegraphics*[width=0.8\columnwidth]{fig2.eps}
\end{center}
\caption{\label{2acciones}
(a) The evolution of one component of the difference $\bm \delta_{n}$ between
the particles and the fluid flow velocities as a function of the number of
iterations. (b) An $xy$ slice of the phase space with those points where $\bm
\delta _{n}$ is greater than $\bm \delta_{0}$. The square is the region 
[0,2$\pi$]$\times$[0,2$\pi$].}
\end{figure}

In the two-action case, the eigenvalues of the Jacobian are very small on large
portions of the motion, so that separation may only occur sporadically during
the short time intervals in which the fluid parcel crosses the fast motion
resonances \cite{feingold88.2}. Most of the time, particles and fluid parcels
follow exponentially convergent trajectories causing the separations to be then
practically unobservable except for very small values of $\lambda$. Most
probably, once the particles converge to the fluid dynamics they remain
attached. However, by adding a small amount of white noise, we can continually
force the impurity to fluctuate around the flow trajectory \cite{bailout2}. 
From the application point of view, this noise may be considered to represent
the effect of small scale turbulance, thermal fluctuations, etc, but here we
will use it only as a dynamical device. With this, the particles arrive in the
neighborhood of the resonances with a non-negligible velocity mismatch with the
fluid that is considerably amplified during the transit across the resonance.
The measure of this mismatch is then a good detector of the proximity of the
resonance. Let us specifically study the following stochastic iterative system:
\begin{equation}
\label{bail_map}
\bm X_{n+2}-\bm T(\bm X_{n+1})=e^{-\lambda }\bm \nabla \bm T\cdot (\bm
X_{n+1}-\bm T(\bm X_{n}))+\bm \xi_{n} 
.\end{equation}
The noise term $\xi_{n}$ satisfies
\begin{eqnarray}
\label{est_ruido}
\langle {\bm{\xi}_{n}} \rangle &=& 0, \nonumber \\
\langle {\bm{\xi}_{n}\,\bm{\xi}_{m}} \rangle &=&
\varepsilon(1-e^{-2\gamma})\,\delta_{mn}\,{\mathrm I}
.\end{eqnarray}

We can recast Eq.~(\ref{bail_map}) into
\begin{eqnarray} 
\label{bail_map2}
\bm x_{n+1}&=&\bm T(\bm x_{n})+\bm \delta _{n} \nonumber \\
\bm \delta _{n+1}&=&e^{-\lambda }\bm \nabla \bm T\cdot \bm \delta _{n}+\bm
\xi _{n} 
\end{eqnarray}
if we define the velocity separation between the fluid and the particle
as $\bm \delta _{n}=(\bm x_{n+1}-\bm T(\bm x_{n}))$.

\begin{figure}[tb]
\begin{center}
\includegraphics*[width=0.8\columnwidth]{fig3.eps}
\end{center} 
\caption{\label{temp}
The temperature amplitude for the one-action, two-action and most chaotic cases
((a), (c), and (e) respectively), together with the corresponding slices of the
impurity dynamics (histogram) in the phase space ((b), (d), and (f)). All
images are the [0,2$\pi$]$\times$[0,2$\pi$] region in the $xy$ axis, for a
slice in the $z$ direction corresponding to the values $z\in$[0,0.49].} 
\end{figure}

Let us illustrate the behavior referred above by studying the most ergodic
two-action case in which all the fluid trajectories intersect the resonant
lines. In Fig.~\ref{2acciones}a we show how $\bm \delta _{n}$ grows strongly at
some points. These points correspond to the crossings of the resonant lines. In
Fig.~\ref{2acciones}b we plot these points in an $xy$ slice of the
three-dimensional cube, choosing those points where the value of $\bm
\delta_{n}$ is greater than a certain minimum value $\bm \delta_{0}$. As shown,
we recover the resonant structure previously noted \cite{feingold88.2,pirofein}.

This is the most primitive way to obtain useful information from the the noisy
particle dynamics. A shrewder analysis \cite{bailout2}
shows how the variance of the separation $\bm\delta_{n}$ between particles and
fluid trajectories and the variance of the noise $\xi_{n}$ are related by a
function that only depends on the particular point of the phase space that we
look at, in a sort of temperature amplitude for the fluctuations of $\bm\delta$,
\begin{equation}
\label{temperatura}
\mathcal{T}(x)=\frac{\left\langle \bm \delta^{2}\right\rangle }
{\left\langle \bm \xi^{2}\right\rangle }=\sum \left( e^{-j\lambda }\cdot 
\prod \bm \nabla \bm T\left|_{T^{-k}(x)}\right. \right)^{2}
.\end{equation}
This amplitude takes different values at different points of the flow. At those
points that the particle dynamics tries to avoid, its value increases, so
the particle prefers to escape the hot regions and to fall into the
cold ones. 

In Fig.~\ref{temp} we show this phenomenon. First we analyze a one-action
situation showing the temperature amplitude, as well as the impurity dynamics
(Figs~\ref{temp}a and \ref{temp}b respectively). Again we use slices of the
three-dimensional cube to show the situation more clearly. Fig.~\ref{temp}a
shows the temperature in a scaled color code (lighter is hotter). 
Fig.~\ref{temp}b shows a
histogram of visits that a single particle pays to each bin of the space. The
agreement between the higher temperature regions and the less visited ones is
evident. Next we plot the same pictures but in the two-action case studied
before (Figs~\ref{temp}c and \ref{temp}d). Finally, we apply this analysis to a
generic chaotic case where we do not have any information about the phase space
structure. We show in Fig.~\ref{temp}e how the invariant manifolds are very
twisted, and in Fig.~\ref{temp}f how the particles, even so, try to find the
coldest regions of the flow.

In order to show that the above described behavior is not an artefact of our
mapping-based approach, we have performed analogous simulations using a
continuous-time model as a base flow. Specifically we have considered neutrally
buoyant particles immersed in a modified version of the ABC flow, in which each
component of the velocity vector field is sinusoidally modulated with a
relative phase shift of $2\pi/3$ and where $x$, $y$, and $z$ are to be 
considered ($\bmod 2\pi$)
\begin{eqnarray} \label{ABC-time-dependent}
\frac{dx}{dt}&=&(1+\sin\ 2\pi t)\cdot(A\sin z+C\cos y), \\
\frac{dy}{dt}&=&(1+\sin\ 2\pi (t+\frac{1}{3}))\cdot(B\sin x+A\cos z), 
\nonumber \\  
\frac{dz}{dt}&=&(1+\sin\ 2\pi (t+\frac{2}{3}))\cdot(C\sin y+B\cos x). 
\nonumber  
\end{eqnarray}

While a detailed analysis of the dynamical aspects of this flow is beyond
the scope of this Letter, we anticipate that it shows structures similar
to those of the ABC maps, i.e, a complex array of KAM sheets and tubes
surrounded by chaotic volumes. Neutrally buoyant particles evolved
according to the true (simplified) Maxey--Riley equations,
Eq.~(\ref{part_in_fluid}), based on this flow, show exactly
the same tendency to accumulate inside KAM tubes as in the map case. This is
depicted in  Fig.~\ref{ABCtimedependent}, where ten particles, initially
distributed at random in the cubic cell, are shown to end up in the interior of
two of the tubes mentioned above.

\begin{figure}[tb]
\begin{center}
\includegraphics*[width=0.8\columnwidth]{fig4.eps}
\end{center} 
\caption{\label{ABCtimedependent}
Stroboscopic sampling ($\Delta t$=1) of the position of 10 particles initially 
distributed at random in a flow described by Eq.~(\ref{ABC-time-dependent})
with $A=2$, $B=0.4$, and $C=1.2$. The dots represent the positions of these
particles at the strobing periods 1000 to 2000.} 
\end{figure} 

This confirmation demonstrates the validity of the qualitative approach based 
on the bailout embedding of three-dimensional volume-preserving maps to
describe the dynamics of neutrally buoyant particles immersed in
three-dimensional time-periodic incompressible flows. This application of the
bailout concept is the first to be reported for a non-Hamiltonian dynamical
system. Our approach can be pursued with two different goals in mind: on one
hand, it contributes to the understanding of the physical behavior of
impurities, and on the other hand, it provides a mathematical device to learn
about the dynamical structures of the base flow in situations where these are
very difficult to elucidate directly. Notice that both bodies of information
are very important to improve our presently scarce knowledge of the transport
properties of three-dimensional fluid flows.

This research was performed by IT as part of his doctoral thesis under the
supervision of OP, together with additional input from JHEC and MOM. JHEC
acknowledges the financial support of the Spanish CSIC, Plan Nacional del
Espacio contract PNE-007/2000-C, MOM acknowledges the support of the Meyer
Foundation, and OP and IT acknowledge the Spanish Ministerio de Ciencia y
Tecnologia, Proyecto CONOCE, contract BFM2000-1108 and Proyecto IMAGEN,
contract REN2001-0802-C02-01.

\bibliography{database}

\end{document}

bailout_3D.bbl
\begin{thebibliography}{13}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

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  \bibinfo{editor}{\bibfnamefont{D.}~\bibnamefont{Reguera}},
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\bibitem[{\citenamefont{Feingold et~al.}(1988)\citenamefont{Feingold, Kadanoff,
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\bibitem[{\citenamefont{Piro and Feingold}(1988)}]{pirofein}
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\bibitem[{\citenamefont{Cartwright et~al.}(1994)\citenamefont{Cartwright,
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\bibitem[{\citenamefont{Cartwright et~al.}(1996)\citenamefont{Cartwright,
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  \bibinfo{author}{\bibfnamefont{J.~J.} \bibnamefont{Riley}},
  \bibinfo{journal}{Phys. Fluids} \textbf{\bibinfo{volume}{26}},
  \bibinfo{pages}{883} (\bibinfo{year}{1983}).

\bibitem[{\citenamefont{Michaelides}(1997)}]{michaelides}
\bibinfo{author}{\bibfnamefont{E.~E.} \bibnamefont{Michaelides}},
  \bibinfo{journal}{J. Fluids Eng.} \textbf{\bibinfo{volume}{119}},
  \bibinfo{pages}{233} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Cartwright
  et~al.}(2002{\natexlab{b}})\citenamefont{Cartwright, Magnasco, and
  Piro}}]{bailout2}
\bibinfo{author}{\bibfnamefont{J.~H.~E.} \bibnamefont{Cartwright}},
  \bibinfo{author}{\bibfnamefont{M.~O.} \bibnamefont{Magnasco}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Piro}},
  \bibinfo{journal}{Chaos} \textbf{\bibinfo{volume}{submitted}}
  (\bibinfo{year}{2002}{\natexlab{b}}).

\end{thebibliography}
