Particle accumulation on periodic orbits by repeated free surface collisions Ernst Hofmann a) and Hendrik C. Kuhlmann b) 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. V C 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 attention 1 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 Acrivos 15 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 Haller 18 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]. b) Electronic mail: [email protected]. 1070-6631/2011/23(7)/072106/14/$30.00 V C 2011 American Institute of Physics 23, 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
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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
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.