Particle accumulation on periodic orbits by repeated free surface collisions
Ernst Hofmanna) and Hendrik C. Kuhlmannb)
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3,A-1040 Vienna, Austria
(Received 8 September 2010; accepted 27 June 2011; published online 27 July 2011)
The motion of small particles suspended in cylindrical thermocapillary liquid bridges is
investigated numerically in order to explain the experimentally observed particle accumulation
structures (PAS) in steady two- and time-dependent three-dimensional flows. Particles moving in
this flow are modeled as perfect tracers in the bulk, which can undergo collisions with the free
surface. By way of free-surface collisions the particles are transferred among different streamlines
which represents the particle trajectories in the bulk. The inter-streamline transfer-process near the
free surface together with the passive transport through the bulk is used to construct an iterative
map that can describe the accumulation process as an attraction to a stable fixed point which
represents PAS. The flow topology of the underlying azimuthally traveling hydrothermal wave
turns out to be of key importance for the existence of PAS. In a frame of reference exactly rotating
with the hydrothermal wave the three-dimensional flow is steady and exhibits co-existing regular
and chaotic streamlines. We find that particles are attracted to accumulation structures if a closed
regular streamline exists in the rotating frame of reference which closely approaches the free
surface locally. Depending on the closed streamline and the particle radius PAS can arise as a
specific trajectory which winds about the closed regular streamline or as the surface of a particular
stream tube containing the closed streamline. VC 2011 American Institute of Physics.
[doi:10.1063/1.3614552]
I. INTRODUCTION
Particle-laden flows are of great importance for natural
phenomena and industrial applications. A fundamental as-
pect is to understand the process of dispersion of the particu-
late phase and its spatial distribution. The clustering of
inertial particles leading to Lagrangian coherent structures
(LCS) is a rapidly emerging field of fluid mechanics and has
recently received considerable attention1 and references
cited therein. LCS are strongly related to topological fluid
mechanics.2 But even in the absence of inertial effects small
particles can accumulate in incompressible flows.
In an experiment on thermocapillary flow in a differen-
tially heated cylindrical liquid bridge Schwabe et al.3
observed that the tracer particles used for flow visualization
in a liquid did not remain randomly distributed in the liquid
volume. Under certain conditions, they accumulate along a
closed thread which moves in the three-dimensional
unsteady flow. Schwabe et al.3 called this phenomenon
dynamic particle accumulation structure (PAS). Dynamic
PAS can take various shapes, depending on the Reynolds
number.4–7 Typically, a closed thread of particles seems to
be wound, once or several times, around a virtual toroid and
rotates azimuthally about the symmetry axis of the toroid [an
axial projection of PAS is shown in Fig. 4(a)]. An experi-
ment under zero gravity conditions confirmed that gravity is
not required for PAS to occur.8 A necessary prerequisite for
dynamic PAS, however, is an underlying flow in form of an
azimuthally traveling hydrothermal wave.9,10 Yet, the funda-
mental mechanism by which PAS comes into existence has
remained obscured.
Particle migration and segregation can be caused by dif-
ferent mechanisms. The migration in shear flow due to iner-
tia-induced lift forces is known as the Segre–Silberberg
effect.11–13 Particle banding has been observed to occur in
rimming flows.14 Jin and Acrivos15 suggested an explanation
of the particle accumulation patterns in terms of a modified
effective viscosity which depends on the particle concentra-
tion. Different from PAS, however, the structures consists of
a quasi-continuous variation of the particle concentration
and do not represent a complete de-mixing. Shinbrot et al.16
reported clustering of very small inertial tracers by exclu-
sively transient effects in volume-conserving flows. Such a
phenomenon can arise when tracers temporarily become
more buoyant than the surrounding fluid due to, e.g., a
change of the particle density caused by external heating via
radiation.
The motion of very small particles suspended in a liquid
depends very much on the underlying flow field. For that rea-
son the flow topology has been an important issue in trans-
port and mixing problems.17 Sapsis and Haller18 have proven
that, under certain conditions, inertial particles cluster on
particular invariant manifolds, which are located close to
certain two-dimensional closed stream surfaces, typically to-
roidal surfaces. They derived existence conditions for clus-
tering in the limit of a small inertia parameter. Since, only a
few elementary types of motion are available in closed-form
solution, the focus has been on the particle motion in Stokes
flow or in inviscid flows where viscosity is taken into
account only for the particle motion. The exact knowledge
of the flow field, as opposed to numerical data on a grid,
a)Electronic mail: [email protected])Electronic mail: [email protected].
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PHYSICS OF FLUIDS 23, 072106 (2011)
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enables high accuracy calculations of streamlines and
trajectories of minute particles. Kroujiline and Stone,19 for
instance, considered regular and chaotic streamlines in a
steady three-dimensional flow inside a sphere which corre-
sponds to an axisymmetric spherical vortex (Hill’s vortex)
superposed by two solid-body rotations, one about the axis
of symmetry of Hill’s vortex and the other one at an oblique
angle. Depending on the oblique rotation rate, the regular
tori of motion for the rotating Hill’s vortex break up and cha-
otic streamlines are generated into which regions of regular
streamlines on tori are embedded.
The basic steady two-dimensional thermocapillary flow
in a liquid bridge is topologically similar to Hill’s vortex
(both are steady and axisymmetric). If the liquid bridge is
rotated very slowly about its axis, all streamlines wind regu-
larly on nested toroidal surfaces. A perturbation of such a
steady axisymmetric basic flow by a hydrothermal wave9,20
may act in a similar way and break up the invariant tori to
create a sea of chaotic streamlines coexisting with the regu-
lar motion. In the axisymmetric rotated Hill’s vortex only
bubbles (qp < qf) can accumulate due to the centripetal
forces on the circle defining the center of the toroidal vortex,
dense particles (qp > qf ) cannot cluster. Provided that the
behavior of particles in Hill’s vortex and in the steady axi-
symmetric flow in a thermocapillary liquid bridge is similar,
a clustering surface should not appear for dense particles.
However, toroidal clustering or 2D-PAS has been observed21
for dense particles. This result suggests that other effects
should be responsible for the clustering. Likewise, the iner-
tial clustering in three-dimensional flows (LCS) is different
from 3D-PAS in thermocapillary liquid bridges. While the
clustering surface in the former is typically toroidal, they are
line-like in the latter case (PAS). Moreover, clustering in an
incompressible flows is only possible if the density of the
particles differ from that of the fluid,22 whereas PAS has also
been found for density-matched particles. The structure of
PAS in thermocapillary flow is nearly the same for a wide
range of particle densities.5 These observations underline the
general trend that while LCS strongly depend on the parti-
cle-to-fluid density ratio 3D-PAS does not. These difference
suggest that other mechanisms are responsible for PAS.
The aim of the present investigation is an explanation of
PAS based on physical arguments. We are interested in the
principle and general mechanisms that lead to the observed
particle clustering along a closed rotating thread. To achieve
this goal certain simplifying assumptions will have to be
made, since the fully nonlinear three-dimensional and time-
dependent flow is only available numerically which limits
certain direct analyses due to error accumulation. In Sec. II
the problem is formulated. The governing equations for the
fluid motion and the motion of small particles are presented
discussing different approximations of the Maxey–Riley
equation.23 Moreover, a model for the interaction of particles
with the boundaries of the domain will be introduced. Sec-
tion III deals with the numerical methods used to compute
the flow and the particle motion. Results for PAS in a three-
dimensional thermocapillary flow are presented in Sec. IV.
Based on an analysis of the flow topology a physical model
is presented which can explain the demixing of density-
matched particles. The results are summarized and discussed
in Sec. VI.
II. FORMULATION OF THE PROBLEM
A. Fluid flow
We consider the incompressible flow in a liquid bridge
under zero gravity conditions (Fig. 1). The liquid with den-
sity qf and kinematic viscosity � is suspended between two
parallel, coaxial rigid disks of dimensional radius �R sepa-
rated by a distance d and kept at a constant temperature dif-
ference DT. The liquid is kept in place by its surface tension
r. The fluid motion is governed by the Navier–Stokes, conti-
nuity and energy equations
@u
@tþ u � ru ¼ �rpþr2u; (1a)
r � u ¼ 0; (1b)
@T
@tþ u � rT ¼ 1
Prr2T; (1c)
where uðr;u; z; tÞ ¼ uer þ veu þ wez is the velocity field in
cylindrical coordinates ðr;u; zÞ with unit vectors ðer; eu; ezÞ,p is the pressure field and T is the temperature field. The
Prandtl number is Pr ¼ �=j with j being the thermal diffu-
sivity of the liquid. Equations (1) have been scaled using the
scales d, d2=�, �=d, qf�2=d2 and DT for length, time, veloc-
ity, pressure, and temperature, respectively.
We consider the asymptotic limit of large surface ten-
sion r such that all dynamic surface deformations are absent
and the shape of the free surface is given by its static equilib-
rium shape.24 This condition can be cast more precisely into
a vanishing capillary number Ca ¼ cDT=r! 0 where
c ¼ �@r=@T is the negative surface-tension coefficient. For
simplicity, we consider a liquid volume of V ¼ p �R2d and
contact lines pinned to the edges of the supporting disks such
that the liquid shape is exactly upright cylindrical. This
assumption is justified, because PAS is relatively insensitive
to the precise shape of the free surface.7
Due to the absence of gravity the flow is driven by ther-
mocapillary forces only. Neglecting the viscosity of the am-
bient gas and assuming adiabatic free surface conditions the
unknown field variables must satisfy the free-surface bound-
ary conditions at r ¼ R ¼ 1=C, with C ¼ d= �R being the
aspect ratio,
FIG. 1. Sketch of the liquid bridge.
072106-2 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
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S � er þ Re I� ererð Þ � rT ¼ 0; (2a)
er � rT ¼ 0: (2b)
Here, S ¼ ruþ ðruÞT is the viscous part of the dimension-
less stress tensor and I the identity. A measure for the
strength of the flow is the thermocapillary Reynolds number
Re ¼ cDTd
qf�2: (3)
The remaining boundary conditions on the rigid disks at
z ¼ 61=2 are no-slip and constant temperatures, hence
u ¼ 0 and T ¼ 61
2: (4)
It remains to solve the transport equations (1) subject to the
boundary conditions (2) and (4), respectively, inside the flow
domain V ¼ fx j r � R; �1=2 � z � 1=2g.
B. Particle motion
1. Inertial particle
According to Schwabe et al.,5 particle–particle interac-
tion does not play any major role in the formation of PAS. In
addition, we assume one-way coupling for the motion of a
small spherical particle suspended in the liquid bridge. This
is consistent with the assumption that the particles are suffi-
ciently small and dilute, such that the Maxey–Riley equa-
tion23 represents a good model for the motion of the
particles. We employ the model of Babiano et al.25 which
represents a simplified version of the Maxey–Riley equation.
Taking into account the pressure gradient, the added mass
and the Stokes drag, the equation of motion for the particle
in the absence of gravity reads
€y ¼ 1
.þ 12
� .St
_y� uð Þ þ 3
2
Du
Dt
� �: (5)
Here, yðtÞ is the position of the particle’s center of mass,
u ¼ uðx ¼ yðtÞ; tÞ is the fluid velocity at the current particle
position, . ¼ qp=qf is the particle-to-fluid density ratio and
D=Dt ¼ @=@tþ u � r is the substantial derivative with
respect to the fluid motion. The magnitude of the Stokes
drag is measured by the Stokes number St ¼ 2.�a2=ð9�sfÞ,where �a is the (dimensional) radius of the particle (assumed
to be spherical) and sf the characteristic time of the flow. In
Eq. (5) we use the dimensionless velocity field u from Eq.
(1). Therefore, we have to employ the same viscous diffusion
time scale sf ¼ d2=� as for the fluid motion and obtain the
Stokes number in the form
St ¼ 2.�a2
9d2: (6)
Equation (5) holds true in the combined limit of a small
dimensionless particle radius a ¼ �a=d � 1 and a small parti-
cle Reynolds number Rep ¼ �aj _y� uj=� � 1.
Schwabe et al.5 found PAS experimentally for a wide
range of the Stokes numbers. Their experiments were limited
to Stexp � 10�3. This limit was obtained using �a ¼ 25 lm,
. ¼ 1:8 and a time scale sf ¼ 0:2 s estimated from what they
called the time of the action of the cold spot. Using the
material data of Schwabe et al.5 we find the relation between
their and the present Stokes number as Stexp
¼ St d2=ð� 0:2 sÞ ¼ St� 36. Thus, we shall consider
St �< 10�5 in the following.
A very important experimental observation is the exis-
tence of PAS for density-matched ð. ¼ 1Þ particles. Even
more, PAS formation is most rapid for density-matching.5
This observation suggests to investigate the dynamics of
density-matched particles. Under zero gravity and for a
steady flow Eq. (5) reduces to
€y ¼ � 2
3St_y� uð Þ þ u � ru: (7)
The steady flow assumption will be justified in Sec. III B.
2. Point tracer
For particles with St ¼ Oð10�5Þ an initially large parti-
cle–flow velocity mismatch will decay exponentially fast
due to the large Stokes drag. Therefore, it is justified to use
particle–flow velocity-matching as the initial condition
_yðt0Þ ¼ uðx ¼ yðt0Þ; t0Þ with t0 ¼ 0. Furthermore, this initial
condition together with . ¼ 1 yields a vanishing Stokes drag
for all times and Eq. (7) reduces to €y ¼ u � ru, which is the
equation of motion for a fluid element in a steady flow, thus
_y ¼ u: (8)
Hence, particle trajectories and streamlines are in excellent
agreement for density-matched and initially velocity-
matched particles with St� 1. Such particles will behave as
ideal passive tracers, from now on called point tracers. Point
tracers can move in the full domain V occupied by the
liquid.
Due to the incompressibility of the flow, point tracers
cannot form dynamic PAS in form of a closed thread.
Because otherwise PAS would represent an attracting
streamline and attractors cannot appear in incompressible
flows.26 For that reason the point tracer is not an appropriate
model to explain PAS, hence we introduce the model of a fi-
nite-size tracer.
3. Finite-size tracer
Due to the absence of inertial effects in Eq. (8) and the
absence of attractors ðr � u ¼ 0Þ the only remaining possi-
bility for PAS formation is a particle transfer from one
streamline to another until a stable configuration, namely
PAS, is reached. Experiments4,5 indicate that PAS touches
the free surface, while PAS does not contact the rigid top
and bottom disks. Such particle–free-surface interactions
are likely to take place frequently, because the streamlines
are very dense in the vicinity of the free surface due to the
cylindrical geometry and the thermocapillary surface
forces.
To account for these particle–free-surface interactions,
we devise an interaction model that takes into account the
072106-3 Particle accumulation on periodic orbits Phys. Fluids 23, 072106 (2011)
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finite size of the tracer. Within this model the tracer is
assumed to be a sphere of radius a. Due to the non-zero ra-
dius, the center of the tracer corresponding to y cannot pene-
trate into a layer of thickness a covering all boundaries of
the flow domain V. Hence the reduced domain for the tracer
motion is V� ¼ fx jr � R�; z�� � z � z�þg, where
R� ¼ R� a; (9a)
z�6 ¼ 61
2 a: (9b)
Even if the particle is treated as a point tracer in the bulk by
employing Eq. (8), i.e., by disregarding the effects of Stokes
number and particle size, the restriction of the tracer motion
to the domain V�, owing to its finite size, will cause impor-
tant consequences.
The particle model defined by Eq. (8) together with the
restricted domain of motion V� shall be called finite-size-tracer model. We will show that PAS follows naturally from
this model by a sequence of particle–free-surface interac-
tions. A similar particle model has also been suggested by
Kawamura.27
C. Particle-transfer model
1. Particle–boundary interaction model
Finite-size tracers will collide with the boundary of
the restricted domain V� if they move on streamlines inter-
secting R� or z�6. Realistic models for particle–free-surface
interactions must take into account the wetting properties
(contact angle) and surface deflections, see e.g., Refs. 28
and 29. The numerical effort for the implementation of
these models is very high. Thus, we only implement a
simple but practical approach which allows to explicitly
compute trajectories of many individual tracers, as it was
already done by Domesi.30 Particle collisions with the
solid heated disks do not occur for particles on PAS.
Thus, the particle–wall collision model seems to be only
of minor importance. Particles on PAS, however, approach
the free surface very closely and a particle–free-surface
interaction model may be important.
For the following considerations we have chosen the so-
called partially elastic reflection model. If a particle hits a
boundary, say at r ¼ R�, the radial momentum is annihilated
and forced to zero as long as the radial component of the ve-
locity field is positive at the particle’s center of mass y, thus
uðyÞ > 0, i.e., as long as the flow is directed outward. As a
result, the particle will be subject to a transfer process as
shown in the Sec. II C 2.
The partially elastic reflection model seems to be justi-
fied, in particular for particle–free-surface collisions, since a
small particle cannot pierce out of a wetting liquid due to the
high capillary pressure that would be associated with a sur-
face bulging on the small scale of the particle radius a. The
model is also consistent with the assumption of an asymp-
totically large mean surface tension ðCa! 0Þ made for the
flow. In addition the model, implies a perfect wettability of
the particle by the liquid.
2. Basic particle transfer process
Based on the finite-size tracer motion together with the
partially elastic reflection model, the expected particle
motion in the steady axisymmetric thermocapillary flow of a
liquid bridge is illustrated in Fig. 2.
If the tracer, moving along a streamline, gets in contact
with the free surface it will slide along the free surface as
long as the radial component of the flow is positive, i.e.,
u 0. During sliding, the center of the spherical tracer
moves on a cylinder of radius R�. By this process the tracer
is continuously transferred from one streamline to another. It
is released to the bulk again at a release point P� on R� for
which u¼ 0 [Fig. 2 (a)]. Thereafter, it will perfectly follow
the streamline through P�.All finite-size tracers initially located in the light gray-
shaded region in Fig. 2(a) of the liquid bridge, a subset of
V�, will undergo a free surface collision, slide and detach
from the free surface at P�. Thus, all those tracers will end
up on the streamline being tangent to R� in P�. This stream-
line represents the 2D-PAS-trajectory LPAS for finite-size
tracers. In a real thermocapillary flow, the radial velocity utypically changes its sign as indicated in Fig. 2(b), where
three points satisfy uðr ¼ R�Þ ¼ 0. The release point P� then
lies on the innermost corresponding streamline.
FIG. 2. (Color online) (a) Sketch of the transfer of finite-size tracers to a
single streamline (full line) tangent to R� in P� (dot). (b) In thermocapillary
flows typically three streamlines are tangent to R�. The boundaries of the
light gray area indicate V� and the dark area delineates VnV�.
072106-4 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
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In three dimensions the three points ðP�;P1;P2Þ that sat-
isfy uðr ¼ R�Þ ¼ 0 become curves on the cylindrical surface
R�, which we call from now on C0, C1, and C2. The most im-
portant curve for the following investigations will be the
release line C0. In the particular case of steady axisymmetric
flow all curves Ci are circles on R� and the axial coordinate
of all release points is constant, thus zCiðuÞ ¼ const:
III. NUMERICAL METHODS
A. Flow field
The primary thermocapillary flow in a liquid bridge is
stationary and two-dimensional as long as the Reynolds
number is below the critical value Rec which marks the
onset of three-dimensional flow. For supercritical driving
ðRe > RecÞ and sufficiently large Prandtl numbers ðPr �> 1Þ,hydrothermal waves exist as secondary flows.20 Here, we
are interested in the time-asymptotic state of a traveling
hydrothermal wave, because experiments have shown that
the existence of a pure traveling hydrothermal wave is a
necessary condition for PAS to occur. For the model
described in Sec. II A, Leypoldt et al.31 have shown that
traveling hydrothermal waves are stable immediately above
the onset of instability if Pr �< 8, whereas standing hydro-
thermal waves are stable immediately above the onset if
Pr �> 8. Therefore, we consider Pr ¼ 4 which ensures the
existence of traveling hydrothermal waves at moderate
supercritical driving.
To compute the velocity field uðx; tÞ of the hydrothermal
wave we employ the code of Leypoldt et al.10 It is based on
finite volumes in r and z and on a pseudospectral Fourier
method in u. The resolution is set to Nr � Nz � Nu
¼ 100� 66� 32 with grid stretching in r and z. The travel-
ing-wave state is obtained by simulation after imposing ini-
tial wave-like perturbations onto the unstable two-
dimensional basic steady state solution. The relaxation to the
traveling-wave state is terminated at t1 after the transient
relative peak-to-peak oscillations of the Nusselt numbers on
the hot and on the cold wall are both less than 10�3.
In the inertial frame of reference K, a pure traveling
hydrothermal wave has the form
uðx; tÞ ¼X1
n¼0; n mod m
eunðr; zÞ ei nu�xntð Þ þ c:c: (10)
with complex Fourier amplitudes eun including the relative
phases. All Fourier components are harmonics of the funda-
mental mode m and propagate at the same azimuthal phase
velocity xn=n (no dispersion),10 such that the flow field
rotates like a rigid body with the constant angular velocity
X ¼ ðxn=nÞez ¼ Xez. By a coordinate transformation into a
rotating frame of reference K0, rotating with X, the hydro-
thermal wave becomes a stationary three-dimensional flow.
Thus, we only need a snapshot of the fully developed asymp-
totic hydrothermal-wave state uðx; t1Þ for all particle calcu-
lations in the rotating frame of reference. This liberates us
from numerically simulating the flow in addition to the parti-
cle motion and saves an enormous amount of computing
time. The flow field in the rotating frame of reference K0 is
u0ðx0Þ ¼ uðx; t1Þ � X� x: (11)
To obtain a perfect traveling wave, according to Eq.
(10), the numerical solution is filtered. During filtering every
velocity component is Fourier transformed with respect to uand inverse Fourier transformed using Eq. (10). Hence, all
small non-harmonic contributions caused by aliasing are
eliminated and the filtered flow field becomes strictly peri-
odic with period 2p=m.
B. Particle motion
To obtain the equation of motion for the particle in the
rotating frame of reference K0, we transform Eq. (5) into K0,rotating with X, and obtain
€y0 ¼ 1
.þ 12
� .Stð _y0 �u0Þþ3
2u0 �r0u0
� �
�2X� _y0 � 3
2.þ1u0
� ��X�ðX� y0Þ 1� 3
2.þ1
� �;
(12)
where y0ðtÞ is the particle’s center of mass in the rotating
frame of reference K0 and u0 ¼ u0 x0 ¼ y0ðtÞð Þ the fluid veloc-
ity of the hydrothermal wave at the current particle position
in K0. The second and third term of the equation represents
Coriolis and centrifugal accelerations, respectively. With the
choice of density-matching and particle–flow velocity-
matching as initial condition we find again, in a one-to-one
analogy to Eq. (8),
_y0 ¼ u0: (13)
After having computed the asymptotic flow field, Eq. (13)
is integrated using the solver ode15s of MATLAB. The
ODE-solver’s tolerances are both set to AbsTol ¼RelTol ¼ 10�6. A further decrease of the tolerances does
not affect the results. The particle–boundary interaction is
detected using the built-in function events. For the inte-
gration of Eq. (13) the flow field is required at an arbitrary
point of the volume. This is accomplished by linear
interpolation.
C. Simulation parameters
For a representative supercritical simulation we assume
zero gravity, an adiabatic non-deformable free surface,
Pr ¼ 4, C ¼ 0:66, and Re ¼ 1800. For the values Pr and Cselected the critical Reynolds number10,20 is Rec � 1080 and
hydrothermal waves for Re > Rec have a fundamental wave
number of m¼ 3.
We consider density-matched, spherical particles with
radius a ¼ 0:015 corresponding to a Stokes number
St ¼ 5� 10�5. Furthermore, we employ the model of a fi-
nite-size tracer and integrate Eq. (13) within the restricted
domain V�. Particle–free-surface collisions are treated as
partially elastic reflections. Note that from now on all con-
siderations refer to the rotating frame of reference K0.
072106-5 Particle accumulation on periodic orbits Phys. Fluids 23, 072106 (2011)
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D. Initial conditions
In a direct approach to PAS in three-dimensions one
would typically simulate a sufficient number of particles ini-
tially regularly or randomly distributed in the restricted do-
main V�. Guided by the result for two-dimensional PAS and
the experimental finding that PAS touches the free surface
we consider, instead, particular initial conditions for the fi-
nite-size tracers.
For the three-dimensional flow of a hydrothermal wave
the curves C0, C1, and C2 are no longer circles, as for axi-
symmetric flows. They form closed wavy curves on r ¼ R�
as shown in Fig. 3.
The tracers are then introduced to the liquid equidis-
tantly distributed along a circle sector at ðr00; z00Þ ¼ ðR�; 0:4Þ.The height z00 is chosen, such that the radial flow velocity is
guaranteed to be positive for all tracers, i.e.
z00 ¼ 0:4 > max�z0C0ðu0Þ
�. Hence, all tracers introduced will
initially slide along R�. Due to the azimuthal periodicity of
the flow, the tracers are only introduced in the interval
½0; 2p=m�, equidistantly distributed with Du00 ¼ 2p=360. The
plane u0 ¼ 0 is defined by the angle at which the free surface
temperature of the fundamental harmonic m¼ 3 takes its
maximum at the midplane z0 ¼ 0 on the given discrete mesh.
The dashed line in Fig. 3 indicates the initial positions.
IV. RESULTS
A. Three-dimensional PAS
After t¼ 8 the majority of the tracers, i.e. 106 out of
120, have been attracted to the PAS trajectory LPAS shown in
Fig. 4(b). The spatio-temporal structure of LPAS agrees quali-
tatively with the experimental observations.4–6
Figure 5 shows a bird’s-eye view of the liquid bridge
with a representative trajectory of a finite-size tracer on the
PAS trajectory LPAS. The collision and release points are
shown as circles. One can clearly see that all three release
points P� lie on the wavy release line C0. The distance
between a collision and a release point is the short sliding
section on R�.While the hydrothermal wave travels clockwise with the
constant angular velocity X ¼ �10:142 ez in the laboratory
frame K, indicated by the arrow in in Fig. 4(b), the averaged
angular velocity of every PAS tracer in K0 is �x0PAS � 14:5 e0z.
Hence, the net movement of all PAS tracers in the laboratory
frame K, given by �xPAS ¼ �x0PAS þ X > 0, is anticlockwise
opposing the azimuthal direction of propagation of the
hydrothermal wave. This result is consistent with the experi-
ments of Ueno et al.,6 and the analysis of Leypoldt et al.10
Quite generally we find that if the tracer–free-surface
interaction takes place in one of three particular azimuthal
sectors the tracer rapidly converges to PAS. This is demon-
strated by the Poincare map for the intersection of tracer tra-
jectories with the plane z0 ¼ 0. Figure 6 shows the mapping
of the azimuthal angles u0n for consecutive intersection
points. The figure shows the dynamic evolution of all tracers
with an initial angle lying in the azimuthal interval
u00 2 ½30 ; 83
�. The two other equivalent intervals, which
we call sectors are located periodically, due to the period
m¼ 3. The three sectors and the free-surface collision points
are shown in Fig. 7. We conclude that a finite-size tracer is
unconditionally transferred to PAS if and only if the tracer is
released from C0 in one of the three sectors.
The remaining 14 tracers are found (for t¼ 8) in a more
or less regular secondary structure shown together with LPAS
in Fig. 7. This secondary structure is approximately a thread
which is nearly periodic with period 4p. The tracers on the
FIG. 3. Unrolled cylinder surface at R�. The curves from C0 to C2 are solu-
tions of uðr ¼ R�Þ ¼ 0 for the hydrothermal wave obtained for C ¼ 0:66,
Pr ¼ 4, Re ¼ 1800 and finite-size tracers with a ¼ 0:015. The dashed line at
z0 ¼ 0:4 indicates the initial positions of the tracers.
FIG. 5. (Color online) Bird’s-eye view of the liquid bridge and LPAS. The
collision and release points (P�), both shown as dots, are displayed together
with the wavy release line C0. The dashed lines represent the plane u0 ¼ 0
and the marker symbol indicates the reference point x0: ðr0;u0; z0Þ¼ ðR; 0; 0Þ.
FIG. 4. (a) Axial view of PAS with period m¼ 3 in an experiment (Ref. 5)
and (b) numerical result for LPAS (full line) for finite-size tracers with
C ¼ 0:66, Pr ¼ 4, Re ¼ 1800, and a ¼ 0:015. The flow field is shown in the
laboratory frame K for z¼ 0. The direction of rotation of the pattern (in K)
is indicated by arrows.
072106-6 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
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secondary structure move with the averaged angular velocity
�x0 anticlockwise in K0 where j �x0j < jXj. Hence, the net
movement of all secondary-structure tracers is clockwise in
K in contrast to the tracers of LPAS.
B. Flow topology
Two-dimensional PAS of finite-size tracers is identical
to a closed streamline. In the three-dimensional hydrother-
mal-wave state LPAS is similar to a closed streamline in the
rotating frame of reference K0, because LPAS is identical to
streamline segments in the bulk which are only disrupted by
very small segments of sliding motion on R� (Fig. 5). It is
natural, therefore, to inquire about the existence of exactly
closed streamlines in the hydrothermal-wave state in K0. To
that end, we cover the ðr0; z0Þ -plane at u0 ¼ 0 (Fig. 5) by
equidistant grid points and grid spacing Dr0 ¼ Dz0 ¼ 0:01.
For every grid point we determine the corresponding stream-
line by integration of Eq. (13) within the full domain V of
the fluid flow.
The streamlines emerging from ðr0i; z0jÞð0Þ
at u00 ¼ 0 are
computed up to u01 ¼ 2p=3 where they end at ðr0i; z0jÞð1Þ
.
Rather than integrating the streamlines for one full azimuthal
revolution, we account for the periodicity of the flow and
evaluate the data at u01 ¼ 2p=3. This keeps the accumulation
of numerical errors to a minimum. From the data we obtain
the discrete offset functions dri ¼ r0ð1Þi � r0
ð0Þi and
dzj ¼ z0ð1Þj � z0
ð0Þj . Interpolating the zeros of the offset func-
tions dri and dzj we identify streamlines with azimuthal peri-
odicity of 2p=3 by the simultaneous zeros dr ¼ dz ¼ 0.
These streamlines are closed, because they are also 2p peri-
odic. The result is shown in Fig. 8(a).
Closed streamlines exist in the rotating frame of refer-
ence. Four of them are unambiguously identified and indi-
cated by dots in Fig. 8(a). The corresponding streamlines,
projected to z0 ¼ 0, are shown below in Fig. 8(b). In the
Poincare section at u0 ¼ 0, shown in Fig. 9, these closed
streamlines appear as fixed points, i.e., the intersection points
of the respective periodic orbits.
Three of the periodic orbits are clearly surrounded by
quasi-periodic orbits, i.e., streamlines that spiral on nested
closed stream tubes representing invariant tori of the flow
field in the rotating frame of reference. These streamlines are
shown in Fig. 8(b) as full curves. For each of these closed
orbits a critical torus exists. It is the largest invariant torus
before break up, marking the border to the chaotic sea. The
remaining orbit, shown as dashed curve in Fig 8(b), seems to
lie within the chaotic sea. However, it is most likely that this
streamline is again encapsulated by invariant stream tubes
which are too small to be resolved numerically.
The streamlines on any of the closed stream tubes
(invariant tori) are open in general and wind around the to-
roidal tube in an incommensurate fashion such that the
stream tube is densely covered by a single streamline. This is
FIG. 7. (Color online) Axial view of all 120 trajectories for the period
t 2 ½7; 8� within the rotating frame of reference K0 showing LPAS and the sec-
ondary structure. The gray areas indicate the sectors and the circles represent
free-surface collision points.
FIG. 8. (Color) (a) Interpolated isolines dr ¼ 0 (black) and dz ¼ 0 (red) for
Pr ¼ 4, Re ¼ 1800 and C ¼ 0:66. The color indicates the absolute value of
the offset d ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidr2 þ dz2p
. The red and green markers indicate closed
streamlines. (b) Closed streamlines corresponding to the dots in (a): dashed
(red dot), full (green dots) and the streamline related to LPAS (thick).
FIG. 6. (Color online) Poincare return map u0n ! u0nþ1 for the dynamics of
finite-size tracers near one of the three ðm ¼ 3Þ fixed points of the periodic
orbit representing PAS in the plane z0 ¼ 0. Shown are the trajectories with
initial angles of the interval u00 ¼ ½30 ; 83
� (first sector).
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illustrated schematically in Fig. 10. Again, this topology
applies to the rotating frame of reference.
The closed streamline corresponding to the green dot at
ðr0; z0Þ � ð0:62; 0:12Þ in Fig. 8(a), drawn as thick curve in
Fig. 8(b) and enclosed by the green stream tubes in Fig. 9
will be denoted LC in the following. By comparing Figs. 7
and 8(b), LC seems to be identical to the PAS trajectory
LPAS. We shall show below that LPAS and LC are similar in
general but identical in a certain limit. We find, moreover,
that the secondary structure (see Fig. 7) is similar to the
dashed streamline in Fig. 8(b).
Figure 11 shows 60 streamlines forming the maximum
reconstructible invariant stream tube around the closed
streamline LC which is also shown as the closed curve in the
center of the stream tube. The streamlines are computed from
u0 ¼ 0 to u0 ¼ 2p=3. The start and end points are indicated as
blue curves. To illustrate the three-dimensional movement of
a single passive tracer (fluid element), one streamline is high-
lighted in red.
V. MECHANISM OF PAS FORMATION
Based on the finite-size-tracer model we propose a
mechanism for PAS formation. The finite-size-tracer model
precludes bulk-flow effects from being responsible for PAS.
We thus inquire into the role of surface collisions. The fact
that PAS has only been found, to date, in thermocapillary
systems and that all PAS seem to touch the free surface also
suggest the importance of particle–free-surface collisions.
To understand the PAS mechanism one has to clearly
distinguish between (a) the asymptotic trajectory LPAS of fi-
nite-size tracers forming PAS which is determined by Eq.
(13) within the reduced domain V� (with surface interaction)
and (b) the closed streamline LC in the vicinity of LPAS which
is determined by Eq. (13) within the full domain V (point
tracer without free surface interaction). If PAS is observed,
these two orbits are very close to each other. But both trajec-
tories are in general distinct.
Consider a finite-size tracer which collides with the free
surface. After the first contact with the free surface the tracer
will slide on R� for a short distance. After sliding, it leaves
the surface at the release point P01 which lies on the wavy
release line C0. Due to the passive motion of the tracer in the
bulk the tracer trajectory is identical with the streamline
emerging from P01. Let this streamline be denoted L1. The
tracer crosses the bulk on L1, hits the free surface again,
slides along R� and detaches from the free surface at the
release point P02. The corresponding streamline L2 will, in
general, be different from L1. Hence, the particle–free-sur-
face interaction has transferred the tracer from streamline L1
to streamline L2. This sequence of collision and transport
through the bulk will occur repeatedly. During each particle–
free-surface interaction the finite-size tracer will be trans-
ferred from one streamline to another. For a given flow to-
pology and under certain conditions this process may lead to
PAS (see e.g. Fig. 6) and the release points P0n will converge
to the PAS release point: limn!1 P0n ¼ P�.In the following, we shall consider the convergence to
PAS using certain simplifying but generic assumptions about
the local flow topology near the free surface (see Fig. 13).
To derive the models to be presented in the subsequent sec-
tions we assume that the closed streamline LC is tangent or
nearly tangent to the cylindrical surface R�. Let QC be the
point of LC with the largest distance from the axis r¼ 0. We
then define the collision plane as the plane perpendicular to
LC in point QC. This is illustrated in Fig. 12 for the case
when QC lies within V�.Based on the close proximity of LPAS and LC we con-
sider, moreover, slender stream tubes about LC such that part
of the volume of the stream tubes extends beyond R�. Let
Vtube be the volume occupied by the largest-diameter stream
tube considered. The part of Vtube that is located outside of
R�, i.e., VtubenV�, is assumed to have a maximum linear
extension b such that it is small compared to the length scale
of the flow, i.e. b� d. This assumption is justified by the
slenderness of the invariant stream tubes and the proximity
of LC and R�.Under these conditions, finite-size tracers on invariant
stream tubes can collide with the free surface only on the
small area Vtube \ R� (see e.g., the elliptical shaped blue area
in Fig. 13) and in the vicinity of the collision plane which is
also contained in Vtube. If a finite-size tracer is released from
the wavy release line C0 on R� in the vicinity of the collision
plane, it will move in a winding fashion on an invariant
stream tube through the bulk of the fluid and may collide
again with the surface in Vtube \ R� and in the vicinity of the
collision plane. Since the sliding distance on R� is very short,
the winding about the closed streamline can be neglected for
the sliding fraction of the trajectory. Moreover, the short
sliding justifies that the collision point, say P†n, and the
FIG. 10. (Color online) Sketch of the flow topology of an incompressible
flow in the vicinity of a closed streamline LC (center). LC is encapsulated by
an infinite number of invariant stream tubes. Two of them are shown. On the
outer stream tube a streamline (arrows) is shown to illustrate the winding
about the closed one.
FIG. 9. (Color) Poincare section at u0 ¼ 0 showing intersection points of
regular streamlines which lie on tori (black, green, and blue symbols) and of
chaotic streamlines (red symbols).
072106-8 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
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release point P0n of a single collision process can be projected
onto the point Pn in the collision plane. The dynamics of the
finite-size tracer can now be described entirely by a two-
dimensional map M : R2 ! R2 defined by Pn ! Pnþ1.
It remains to determine the map M. This map is com-
posed of two parts: One part reflects the motion of a point
tracer through the bulk. This motion is restricted to invariant
stream tubes. Hence, the image ~Pn of Pn must lie on the
same cross section of the invariant stream tube as Pn. The
other part of the map M concerns the collision. If the image~Pn lies outside of R� a collision of a finite-size tracer must
have taken place. In that case the collision part of the map is
just the orthogonal projection of ~Pn onto R�. Thus, if the
cross sections of the invariant stream tubes were known, the
map M would be completely defined by specifying the accu-
mulated effect of the winding about LC of a point tracer
released at an arbitrary point Pn from the collision plane as it
travels through the bulk. The cumulative effect of the wind-
ing in the bulk is given by the cumulative winding angle, in
the following just called winding angle h.
The construction of approximations to the map M is our
primary interest. It contains all the information about the
motion of finite-size tracers near LC. Note that the details of
the flow all along the invariant stream tubes are not of con-
cern: the particle dynamics can be completely described by
the shape of the cross sections of the invariant stream tubes
in the collision plane and the winding angle. In the absence
of precise knowledge about the shape of the invariant stream
tubes, and thus about the true map M we shall make reasona-
ble assumptions regarding the shape of the cross sections of
the invariant stream tubes and the winding angle in the colli-
sion plane. Since, we consider slender stream tubes, the
winding angle will be approximated by a constant value
which corresponds to the lowest-order term of a Taylor se-
ries expansion of h around QC.
A. Stream tubes of circular cross section
As a first qualitative approximation, we shall assume
circular cross sections of the nested stream tubes around LC
in the collision plane. This assumption about the shape of the
stream tubes will be relaxed further below to stream tubes
which have elliptic cross sections in the region of the parti-
cle–free-surface interaction.
1. Tangent case
If the particle size is such that its radius a ¼ R� R�
equals the minimum distance between LC and the free sur-
face, the closed streamline LC is exactly tangent to the sur-
face R�. In that case all regular stream tubes around LC
intersect with R�. This is illustrated in Fig. 13 showing a
three-dimensional sketch of the collision process with nested
stream tubes of circular cross sections. We consider the as-
ymptotic case of a small distance from the point QC. In that
case the surface r ¼ R� is represented by the ðx; tÞ -plane and
the collision plane results as the ðx; yÞ-plane. The corre-
sponding unit vectors of this local coordinate system are the
tangential unit vector et ¼ u0ðQCÞ=ju0ðQCÞj, the unit vector
of y corresponding to the radial direction, thus ey ¼ e0r, and
ex which is perpendicular to both, ey and et. A tracer
FIG. 11. (Color)(a) Side view of the maximum reconstructible stream tube for u0 ¼ ½0; 2p=3� around the closed streamline LC which is also shown as closed
curve. The red streamline illustrates the motion of a fluid element along this invariant stream tube for the estimation of the winding angle as g � p (introduced
below). The horizontal lines indicate the top and bottom disks. (b) Side view as indicated in (a).
FIG. 13. (Color) Three-dimensional sketch of the collision process and par-
ticle transfer process if LC is tangent to R�, here represented by the ðx; tÞ-plane. The blue and green horizontal surfaces in the ðx; tÞ -plane indicate
cuts through the two stream tubes made by R�. The vertical ðx; yÞ -plane rep-
resents the collision plane which cuts the invariant stream tubes at their
apex. The sliding distance P†1 � P01 is short in reality. More details are given
in the text.FIG. 12. (Color online) Definition of the collision plane.
072106-9 Particle accumulation on periodic orbits Phys. Fluids 23, 072106 (2011)
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approaching R� on the outer (blue) stream tube will collide
with R� at P†1. Thereafter, it slides on R� (black arrow in Fig.
13) until it is released to the bulk at P01. Point P01 however,
lies on a stream tube (green) fully contained in the larger
outer (blue) stream tube. The finite-size tracer cannot reach~P1 beyond R�, which could only be reached by a point tracer.
The finite tracer size thus leads to a displacement of the
tracer from an outer stream tube to an inner stream tube.
Since the length of the sliding phase (black arrow) is negligi-
bly small compared to the length of the closed streamline LC,
the winding of the streamline about LC can be neglected for
the collision process. Hence, the normal projection of a colli-
sion point P†n and its corresponding release point P0n onto the
collision plane will coincide both with Pn. Therefore, it is
justified to reduce the discussion from now on to the colli-
sion plane.
To find the map M, we analyze the situation given in Fig.
13 for successive collisions within the collision plane, as
shown in Fig. 14. A tracer that is released to the bulk from
y¼ 0 (corresponding to R�) on some stream tube at P0 will
move on that stream tube and experiences a certain winding
about the closed streamline LC as it travels through the bulk.
Due to the winding, the tracer will return to the free surface at
a different angle h 6¼ 0. Again, let h be this constant winding
angle between successive returns to the free surface. Then a
point tracer would return to the collision plane at point ~P1. A
finite-size tracer, however, will collide with the free surface
and its center of mass will not cross R�. After being released
from R� at point P1 the process (motion through the bulk and
collision) repeats itself. By successive collisions, any tracer
on a closed stream tube will finally be transferred to LPAS
which is identical to LC in this tangent case.
2. Intersecting case
Next we consider the more general case that the closed
streamline LC intersects with R�. By definition of the colli-
sion plane, the distance A represents the maximum distance
between QC and R�, as shown in Fig. 15. With coordinates
centered in QC,
xnþ1 ¼ xn cos h� yn sin h; (14a)
ynþ1 ¼ min A; xn sin hþ yn cos h½ � (14b)
maps the coordinates Pn ¼ ðxn; ynÞ (see Fig. 15) of a tracer
released from y¼A, or a non-colliding tracer with y < A, to
those of the next release point Pnþ1 ¼ ðxnþ1; ynþ1Þ after one
return to the free surface. This iterative map models the colli-
sion process as a projection of the point ~Pn to the current
release point Pn on R� if ynþ1 > A. In case ynþ1 < A no colli-
sion occurs for the current return to the free surface.
The map (14) has the trivial periodic fixed points
xnþ1 ¼ xnð�1Þk for h ¼ kp, k 2 Z where the location of the
periodic points depend on the initial release-point coordinate
x1. For A � 0, i.e., if LC is tangent to or intersects with R�,the map also has a non-trivial, stable fixed point
x�; y�ð Þ ¼ � A
tan h=2ð Þ ;A� �
: (15)
All tracers on a closed stream tube about LC and initially on
R� will converge, after repeated collisions, to Eq. (15) corre-
sponding to the release point P� of LPAS. An example is
shown in Fig. 17(a). For certain intermediate ranges of the
winding angle h, convergence is very rapid and merely a few
collisions are required to transfer the finite-size tracer to
PAS. For winding angles near h ¼ kp convergence slows
down. In the intersecting case LPAS originating from P� does
no longer coincide with the closed streamline LC originating
from QC.
3. Non-intersecting case
Finally, we consider the case when the closed streamline
approaches R� up to the distance A > 0, see Fig. 16. For
A > 0 only the trivial periodic fixed points exists. However,
for h 6¼ kp, k 2 Z, all initial conditions are rapidly attracted
to the stream tube around QC which is tangent to R�. Par-
ticles then accumulate on the invariant torus represented by
this stream tube. We call this accumulation structure tubularPAS. The convergence to the stream tube with radius r¼A is
illustrated in Fig. 17(b). While the x and y coordinates scatter
FIG. 14. (Color online) Schematic representation of the mapping in the col-
lision plane. A finite-size tracer encounters free surface collisions if the
closed streamline LC is tangent to R� in QC ¼ P�. The white area indicates
the liquid and the gray shaded area indicates the out-of-reach region for fi-
nite-size tracers.
FIG. 15. (Color online) Schematic representation of the mapping of a finite-
size tracer encountering free surface collisions. The iterated release point
Pnþ1 is already close to the fixed point P�.
072106-10 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
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in the range ½�A;A�, the distance from the origin dN shows a
rapid convergence to A.
B. Stream tubes of elliptic cross section
The invariant stream tubes of the hydrothermal wave in
the rotating frame of reference do not have circular cross
sections in the collision plane. Fig. 18 shows a Poincare sec-
tion of the maximum reconstructible stream tube, which is
already very close to the critical stream tube. The Poincare
section is taken at z0 ¼ 0:312 which is the vertical position of
the maximum distance between LC and the axis r¼ 0. For
simplicity, the plane for the Poincare section has been
selected normal to the z-axis instead of being normal to LC.
The figure unambiguously shows that the stream tubes get
radially squeezed near the free surface.
Thus, a better model can be expected by assuming an
elliptic shape of the cross sections of the invariant stream
tubes in the collision plane. For the winding bulk motion
about LC we use again the lowest-order approximation of a
constant winding angle h between two successive returns to
the free surface. If we define the axes ratio as E ¼ a=b,
where a and b are the major and the minor semi axes in x-
and y-direction, respectively, the iterated map in the collision
plane takes the form
xnþ1 ¼ xn cos h� ynE sin h; (16a)
ynþ1 ¼ min A;xn
Esin hþ yn cos h
h i: (16b)
The axes ratio E enters as an additional parameter. For
E¼ 1, Eq. (14) is recovered. Again, Eq. (16) has trivial peri-
odic fixed points xn ¼ xnð�1Þk for h ¼ kp, k 2 Z. The non-
trivial stable fixed point, in analogy to Eq. (15) and in case
LC intersects with R�ðA < 0Þ, is given by
x�; y�ð Þ ¼ � AE
tan h=2ð Þ ;A� �
: (17)
The winding angle h must not to be confused with the polar
angle of the ellipse g, as shown in Fig. 19. The relation
between these angles is
tan h ¼ E tan g: (18)
The representation of Eq. (16) using h has been used for its
compact form and to enable a comparison with Eq. (14) in
the limit of E! 1.
If the closed streamline intersects with R�, we find rapid
convergence to the fixed point Eq. (17) in analogy to the cir-
cular case. If the closed streamline does not intersect with R�
FIG. 16. (Color online) Same as Fig. 14, but for a closed stream tube tan-
gent to R� enclosing the closed streamline LC.
FIG. 18. (Color online) Poincare section at z0 ¼ 0:312 of the maximum
reconstructible stream tube for Pr ¼ 4, Re ¼ 1800, and C ¼ 0:66. The axes
ratio is E � 50. The circular arcs indicate R and R�. Since the piercing point
of the closed streamline (dot) lies between R and R� we have an intersecting
case.
FIG. 17. (Color online) Coordinates xN=Rc (circles) and yN=Rc (squares) af-
ter N¼ 20 iterations of Eq. (14) using the initial vector x0=Rc ¼ ð0:8;AÞ. (a)
A=Rc ¼ �0:3. The line indicates the exact value of the fixed point coordi-
nate x�=Rc according to Eq. (15). (b) A=Rc ¼ 0:3. The diamonds indicate the
distance from the origin dN ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2
N þ y2N
p=Rc.
072106-11 Particle accumulation on periodic orbits Phys. Fluids 23, 072106 (2011)
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we find attraction to a toroid of elliptic cross section in the
ðx; yÞ-plane. In that case and in the limit n!1, the iterated
release point ðxn; ynÞ wanders on an ellipse of E¼ 50 and
b ¼ A about the apex QC of the closed streamline.
C. Real stream tubes
The above elliptical model has been constructed to eluci-
date the principle mechanisms of PAS formation. It cannot
reproduce all details of the real flow. A closer look at the real
flow reveals its complexity and the remaining deviations of
the PAS formation process from our model. Figure 20 shows
again the cross section (normal to z) of the maximum recon-
structible stream tube at z0 ¼ 0:312 as in Fig. 18. For clarity,
the figure is stretched in y-direction and rotated such that the
intersection point QC of LC, indicated by the central (red
online) dot, is located at x¼ 0. This representation illustrates
the deviation in the collision plane of the maximum recon-
structible stream tube from an ellipse assumed in the model:
the bending due to the cylindrical free surface and the asym-
metry with respect to the plane x¼ 0 which is due to the fore-
aft asymmetry of the traveling hydrothermal wave.
To compare the motion of point tracers in the real flow
with those predicted by our model we release point tracers
from the circle through QC. Such a release would correspond
to the tangential case when the release line passes through
QC and should thus yield an estimate of the value of h and
indicate whether this winding angle is approximately con-
stant as assumed in our model. The initial conditions for the
motion of the point tracers are indicated by open dots in Fig.
20. Streamlines are then computed up to their first intersec-
tions with z0 ¼ 0:312 (return to the free surface). These
return points are indicated by full dots and rotated by 2p=3
to compare their locations relative to their respective initial
points.
The streamlines originating from points 1, 2, etc., return
to the plane z0 ¼ 0:312 at points 10, 20, etc., respectively. The
remaining doublets of initial and final points can be identi-
fied by the sequence of pairs of points which results if the
two sets of points are correlated following the directions of
the two arrows in Fig. 20. On a first sight, the polar windingangle seems to be close to g � p for all streamlines (stream
tubes). To a first approximation one would thus expect
g � p � h. From Eq. (16) this value would yield an oscilla-
tory behavior and not the experimentally and numerically
observed rapid convergence to PAS. A closer inspection
shows that all streamlines originating from x > 0 on the cen-
tral curve through QC end above the central curve at x < 0.
Hence, they experience a more or less constant polar wind-
ing angle g which is slightly less than p. All streamlines
emerging from x < 0 end, even more pronounced, below the
central curve, corresponding as well to a polar winding angle
slightly below p. This small systematic deviation from p is
important in view of Eq. (18): The winding angle h depends
sensitively on g in the vicinity of p if the axes ratio E is
large. Thus, if E� 1 and the winding angle g is slightly
above or below p, then one finds for the cumulative winding
angle h � const: � p=2 at which the iteration Eq. (16) con-
verges most rapidly. This is another strong indication for the
robustness of PAS and will favor PAS at even higher Reyn-
olds numbers for which even larger axes ratios can be
expected.
VI. DISCUSSION AND CONCLUSIONS
We have developed a simple model for PAS. The model
is aimed at incorporating the physical key effects that cause
PAS. It is based on the following experimental observations.
1. In all experiments reported, PAS is touching the free sur-
face. Thus, the particle–free-surface interaction must be
taken into account.
2. PAS has been found for a wide range of particle sizes and
densities, including neutral density.5 This observation
suggests that the mechanism for PAS does not qualita-
tively depend on the particle size and density. Moreover,
the PAS trajectory is very similar to a streamline.4 This
indicates that inertia effects are very small.
3. PAS has not been observed in systems other than thermo-
capillary flows in which the streamlines are very crowded
near the free surface. Hence, streamlines and transported
particles can closely approach the free surface.
FIG. 19. (Color online) Sketch to illustrate the winding angle h and the po-
lar angle g.
FIG. 20. (Color online) Poincare section at z0 ¼ 0:312 of the maximum
reconstructible stream tube for Pr ¼ 4, Re ¼ 1800, and C ¼ 0:66 (closed
curve, blue online). Shown is the mapping of point tracers (open dots)
released on the cylindrical surface (inner arc) through QC (dot at x¼ 0, red
online) and their first return to this plane (full dots). The free surface at R is
indicated by the outer arc.
072106-12 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp
These observations suggest that PAS is made possible
by particle–free-surface interactions.
To investigate this hypothesis, we have exploited the
density matched case. From there we have developed the
finite-size tracer model. In the rotating frame of reference,
in which the propagating hydrothermal wave is stationary,
we unraveled the flow topology and discovered a closed
streamline which approaches the free surface very closely.
Finite-size tracers moving on streamlines near the closed
streamline are likely to undergo free-surface collisions.
Taking into account the flow topology in the vicinity of
the closed streamline and using a sliding model for the
particle–free-surface interaction of finite-size tracers, we
suggest a model for the transfer of particles between dif-
ferent streamlines by way of particle–free-surface colli-
sions. Each collision transfers the particle from a closed
(invariant) stream tube to another, smaller closed stream
tube which is fully embedded in the original stream tube.
Repeated collisions can be described as an iterative map
for the release point of the particle from the free surface.
This map is defined in the introduced and so called colli-
sion plane. If the flow in the vicinity of the closed stream-
line is regular, the iterated map exhibits either a nontrivial
fixed point (corresponding to line-like PAS) or a certain
distance from the closed streamline remains constant (cor-
responding to tubular PAS). The results obtained from the
iterated map strongly suggest that PAS is indeed governed
by the same qualitative mechanisms.
The only properties of the bulk flow that enter the itera-
tive map for the particle–free-surface collision are the shapes
of the invariant stream tubes and the cumulative winding
angle h about the closed streamline LC between two succes-
sive returns of the particle to the free surface. For simplicity,
we have assumed elliptic cross sections of the stream tubes
in the collision plane, which are topologically similar to the
cross sections of the real stream tubes, and a constant wind-
ing angle for all nested invariant stream tubes. Within our
model, a tracer is transferred to an inner stream tube upon
collision and remains on the original stream tube if no colli-
sion occurs.
Some further remarks are in order. Gravity is not
included in the model and is not an essential factor for the
mechanism of PAS. This has been demonstrated by the
semi-quantitative agreement between PAS on ground and in
zero gravity.8 Gravity forces do, however, modify PAS indi-
rectly by perturbing the motion of non-density-matched
tracer particles such that the trajectories are no longer identi-
cal with the streamlines. Moreover, buoyancy modifies the
underlying flow and shape of the liquid bridge and will thus
change the invariant stream tubes. These factors, if suffi-
ciently strong, may even lead to a complete suppression of
PAS. A corresponding analysis cannot be easily done for in-
ertial particles, since the flow field could no longer be
directly exploited to gain information on the particle trajec-
tories. An analysis may be possible, however, along the lines
of Sapsis and Haller.18 Similar as in the present case they
showed that particles cluster on a toroidal surface which lies
entirely inside of the toroidal stream tube of the regular
motion (KAM torus) of ABC flow for A2 ¼ 1, B2 ¼ 2=3, and
C2 ¼ 1=3 which is non-integrable exhibiting coexisting reg-
ular and chaotic streamlines32 as in the present flow field in
the rotating frame. In the absence of any theory, however,
room is left for further numerical simulations of PAS with
inertial particles. On the one hand, it may be expected that
particle–free-surface collisions are promoted, in zero gravity,
for particles heavier than the liquid due to the centrifugal
forces. On the other hand, strong inertial effects could move
particles too far away from their initial stream tube and the
convergence of the iterative map might be degraded. This
may explain the global minimum of the PAS formation time
for . ¼ 1 found in the experiments of Schwabe et al.5 An
additional complication is posed by the slight variation of
the liquid density due to the thermal expansion33,34 which is
frequently disregarded in numerical simulations, but which
cannot be avoided in experiments.
Finally, it should be mentioned that a sufficiently high
accuracy of the numerically computed flow field is indispen-
sable. Otherwise, error accumulation can perturb the particle
motion on the closed stream tubes and may fake a PAS-like
phenomenon which would not be present otherwise or even
prevent PAS from appearing in the numerical simulation.
H.C.K. gratefully acknowledges support from BMVIT
through the Austrian Space Application Programme under
Grant No. 819714.
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072106-14 E. Hofmann and H. C. Kuhlmann Phys. Fluids 23, 072106 (2011)
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp