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: ernst.hofmann@tuwien.ac.at.b)Electronic mail: h.kuhlmann@tuwien.ac.at.

1070-6631/2011/23(7)/072106/14/$30.00 VC 2011 American Institute of Physics23, 072106-1

PHYSICS OF FLUIDS 23, 072106 (2011)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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).

072106-7 Particle accumulation on periodic orbits Phys. Fluids 23, 072106 (2011)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

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.

1T. Peacock and J. Dabiri, “Introduction to focus issue: Lagrangian coher-

ent structures,” Chaos 20, 017501 (2010).2Topological Fluid Mechanics, edited by H. K. Moffatt and A. Tsinober,

(Cambridge University Press, Cambridge, UK, 1990).3D. Schwabe, P. Hintz, and S. Frank, “New features of thermocapillary con-

vection in floating zones revealed by tracer particle accumulation struc-

tures (PAS),” Microgravity Sci. Technol. 9, 163 (1996).4S. Tanaka, H. Kawamura, I. Ueno, and D. Schwabe, “Flow structure and

dynamic particle accumulation in thermocapillary convection in a liquid

bridge,” Phys. Fluids 18, 067103 (2006).5D. Schwabe, A. I. Mizev, M. Udhayasankar, and S. Tanaka, “Formation of

dynamic particle accumulation structures in oscillatory thermocapillary

flow in liquid bridges,” Phys. Fluids 19, 072102 (2007).6I. Ueno, Y. Abe, K. Noguchi, and H.Kawamura, “Dynamic particle accu-

mulation structure (PAS) in half-zone liquid bridge–reconstruction of par-

ticle motion by 3-D PTV,” Adv. Space Res. 41, 2145 (2008).7Y. Abe, I. Ueno, and H. Kawamura, “Dynamic particle accumulation

structure due to thermocapillary effect in noncylindrical half-zone liquid

bridge,” Ann. N. Y. Acad. Sci. 1161, 240 (2009).8D. Schwabe, S. Tanaka, A. Mizev, and H. Kawamura, “Particle accumula-

tion structures in time-dependent thermocapillary flow in a liquid bridge

under microgravity conditions,” Microgravity Sci. Technol. 18, 117

(2006).9M. K. Smith and S. H. Davis, “Instabilities of dynamic thermocapillary

liquid layers. Part 1. Convective instabilities,” J. Fluid Mech. 132, 119

(1983).10J. Leypoldt, H. C. Kuhlmann, and H. J. Rath, “Three-dimensional numeri-

cal simulation of thermocapillary flows in cylindrical liquid bridges,” J.

Fluid Mech. 414, 285 (2000).11G. Segre and A. Silberberg, “Radial particle displacements in Poiseuille

flow of suspensions,” Nature (London) 189, 209 (1961).12L. G. Leal, “Particle motions in a viscous fluid,” Annu. Rev. Fluid Mech.

12, 435 (1980).13J.-Ph. Matas, V. Glezer, E. Guazzelli, and J. F. Morris, “Trains of particles

in finite-Reynolds-number pipe flow,” Phys. Fluids 16, 4192 (2004).14M. Tirumkudulu, A. Tripathi, and A. Acrivos, “Particle segregation in

monodisperse sheared suspensions,” Phys. Fluids 11, 507 (1999).15B. Jin and A. Acrivos, “Theory of particle segregation in rimming flows of

suspensions containing neutrally buoyant particles,” Phys. Fluids 16, 641

(2004).

072106-13 Particle accumulation on periodic orbits Phys. Fluids 23, 072106 (2011)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://phf.aip.org/phf/copyright.jsp

16T. Shinbrot, M. M. Alvarez, J. M. Zalc, and F. J. Muzzio, “Attraction of

minute particles to invariant regions of volume preserving flows by transi-

ents,” Phys. Rev. Lett. 86, 1207 (2001).17J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport,

Cambridge Texts in Applied Mathematics (Cambridge University Press,

Cambridge, 1989).18T. Sapsis and G. Haller, “Clustering criterion for inertial particles in two-

dimensional time-periodic and three-dimensional steady flows,” Chaos 20,

017515 (2010).19D. Kroujiline and H. A. Stone, “Chaotic streamlines in steady bounded

three-dimensional Stokes flows,” Physica D 130, 105 (1999).20M. Wanschura, V. S. Shevtsova, H. C. Kuhlmann, and H. J. Rath,

“Convective instability mechanisms in thermocapillary liquid bridges,”

Phys. Fluids 7, 912 (1995).21D. Schwabe and S. Frank, “Particle accumulation structures (pas) in the to-

roidal thermocapillary vortex of a floating-zone model for a step in planet

formation?” Adv. Space Res. 23, 1191 (1999).22G. Haller and T. Sapsis, “Where do inertial particles go in fluid flows?.”

Physica D 237, 573(2008).23M. R. Maxey and J. J. Riley, “Equation of motion for a small rigid sphere

in a nonuniform flow,” Phys. Fluids 26, 883 (1983).24H. C. Kuhlmann and C. Nienhuser,“ in Interfacial Fluid Dynamics

and Transport Processes, Lecture Notes in Physics, Vol. 628, edited by

R. Narayanan and D. Schwabe (Springer, Berlin, Heidelberg, 2003),

pp. 213–239.

25A. Babiano, J. H. E. Cartwright, O. Piro, and A. Provenzale, “Dynamics of

a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian

systems,” Phys. Rev. Lett. 84, 5764 (2000).26H. G. Schuster, Deterministic Chaos: An Introduction (Wiley-VCH, Wein-

heim, 2005).27H. Kawamura, private communication (2004).28Nikolina D. Vassileva, D. van den Ende, F. Mugele, and J. Mellema,

“Capillary forces between spherical particles floating at a liquid–liquid

interface,” Langmuir 21, 11190 (2005).29M. Do-Quang and G. Amberg, “The splash of a solid sphere impacting on

a liquid surface: Numerical simulation of the influence of wetting,” Phys.

Fluids 21, 022102 (2009).30S. Domesi, Numerical analysis of particle dynamics in the thermocapillary

flow of liquid bridges, Ph.D. thesis, (University of Bremen, 2008).31J. Leypoldt, H. C. Kuhlmann, and H. J. Rath, “Stability of hydrothermal-

wave states,” Adv. Space Res. 29, 645 (2002).32T. Dombre, U. Frisch, J. M. Greene, M. Henon, A. Mehr, and A. M. Soward,

“Chaotic streamlines in the ABC flows,” J. Fluid Mech. 167, 353 (1986).33Zh. Kozhoukharova, H. C. Kuhlmann, M. Wanschura, and H. J. Rath,

“Influence of variable viscosity on the onset of hydrothermal waves in

thermocapillary liquid bridges,” Z. Angew. Math. Mech. 79, 535

(1999).34D. E. Melnikov, V. M. Shevtsova, and J. C. Legros, “Numerical simulation

of hydro-thermal waves in liquid bridges with variable viscosity,” Adv.

Space Res. 29, 661 (2002).

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