For the first time evidence is provided that one-dimensional objects formed by the accumulation of tracer particles can emerge in flows of thermogravitational nature (in the region of the space of parameters, in which the so-called OS (oscillatory solution) flow of the Busse balloon represents the dominant secondary mode of convection). Such structures appear as seemingly rigid filaments, rotating without changing their shape. The most interesting (heretofore unseen) feature of such a class of physical attractors is their variety. Indeed, distinct shapes are found for a fixed value of the Rayleigh number depending on parameters accounting for particle inertia and viscous drag. The fascinating “sea” of existing potential paths, their multiplicity and tortuosity are explained according to the granularity of the loci in the physical space where conditions for phase locking between the traveling thermofluid-dynamic disturbance and the “turnover time” of particles in the basic toroidal flow are satisfied. It is shown, in particular, how the observed wealth of geometric objects and related topological features can be linked to a general overarching attractor representing an intrinsic (particle-independent) property of the base velocity field.

1.
P.
Bergè
,
Y.
Pomeau
, and
C.
Vidal
,
Order Within Chaos-Towards a Deterministic Approach to Turbulence
(
John Wiley
,
New York, USA
,
1984
).
2.
W.
Pesch
, “
Complex spatiotemporal convection patterns
,”
Chaos
6
(
3
),
348
(
1996
).
3.
F.H.
Busse
and
R.M.
Clever
, “
Asymmetric squares as an attracting set in Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
81
,
341
(
1998
).
4.
H.
Riecke
and
S.
Madruga
, “
Geometric diagnostics of complex patterns: Spiral defect chaos
,”
Chaos
16
,
013125
(
2006
);
[PubMed]
D. A.
Egolf
,
I. V.
Melnikov
,
W.
Pesch
, and
R. E.
Ecke
, “
Mechanisms of extensive spatiotemporal chaos in Rayleigh–Bénard convection
,”
Nature
404
,
733
(
2000
);
[PubMed]
B. A.
Malomed
and
A. A.
Nepomnyashchy
, “
Coexistence of patterns on a ramp of overcriticality
,”
Phys. Lett. A
244
(
1–3
),
92
(
1998
).
5.
F. H.
Busse
and
R. M.
Clever
, “
Instabilities of convection rolls in a fluid of moderate Prandtl number
,”
J. Fluid Mech.
91
,
319
(
1979
).
6.
R. M.
Clever
and
F. H.
Busse
, “
Three-dimensional knot convection in a layer heated from below
,”
J. Fluid Mech.
198
,
345
(
1989
);
R. M.
Clever
and
F. H.
Busse
, “
Steady and oscillatory bimodal convection
,”
J. Fluid Mech.
271
,
103
(
1994
).
7.
M.
Lappa
,
Thermal Convection: Patterns, Evolution and Stability
(
John Wiley & Sons
,
Chichester, England
,
2010
).
8.
C.
Bizon
,
J.
Werne
,
A. A.
Predtechensky
,
K.
Julien
,
W. D.
McCormick
,
J. B.
Swift
, and
H. L.
Swinney
, “
Plume dynamics in quasi-2D turbulent convection
,”
Chaos
7
(
1
),
107
(
1997
).
9.
A. P.
Vincent
and
D. A.
Yuen
, “
Plumes and waves in two-dimensional turbulent thermal convection
,”
Phys. Rev. E
60
(
3
),
2957
(
1999
).
10.
S.
Childress
, “
Eulerian mean flow from an instability of convective plumes
,”
Chaos
10
(
1
),
28
(
2000
).
11.
M.
Lappa
, “
Some considerations about the symmetry and evolution of chaotic Rayleigh–Bénard convection: The flywheel mechanism and the “wind” of turbulence
,”
C. R. Méc.
339
,
563
(
2011
).
12.
N.
Raju
and
E.
Meiburg
, “
The accumulation and dispersion of heavy particles in forced two-dimensional mixing layers. Part 2: The effect of gravity
,”
Phys. Fluids
7
,
1241
(
1995
).
13.
M. R.
Maxey
,
B. K.
Patel
,
E. J.
Chang
, and
L.-P.
Wang
, “
Simulations of dispersed turbulent multiphase flow
,”
Fluid Dyn. Res.
20
(
1–6
),
143
(
1997
).
14.
E.
Balkovsky
,
G.
Falkovich
, and
A.
Fouxon
, “
Intermittent distribution of inertial particles in turbulent flows
,”
Phys. Rev. Lett.
86
,
2790
(
2001
).
15.
C.
Pasquero
,
A.
Provenzale
, and
E. A.
Spiegel
, “
Suspension and fall of heavy particles in random two-dimensional flow
,”
Phys. Rev. Lett.
91
,
054502
(
2003
).
16.
A.
Esmaeeli
, “
Phase distribution of bubbly flows under terrestrial and microgravity conditions
,”
Fluid Dyn. Mater. Process.
1
(
1
),
63
(
2005
);
M.
Lappa
, “
Coalescence and non-coalescence phenomena in multi-material problems and dispersed multiphase flows: Part 1, a critical review of theories
,”
Fluid Dyn. Mater. Process.
1
,
201
(
2005
);
M.
Lappa
, “
Coalescence and non-coalescence phenomena in multi-material problems and dispersed multiphase flows: Part 2, a critical review of CFD approaches
,”
Fluid Dyn. Mater. Process.
1
,
213
(
2005
).
17.
R. D.
Vilela
and
A. E.
Motter
, “
Can aerosols be trapped in open flows?
,”
Phys. Rev. Lett.
99
,
264101
(
2007
).
18.
D.
Di Carlo
,
J. F.
Edd
,
K. J.
Humphry
,
H.A.
Stone
, and
M.
Toner
, “
Particle segregation and dynamics in confined flows
,”
Phys. Rev. Lett.
102
,
094503
(
2009
).
19.
E. W.
Saw
,
R. A.
Shaw
,
S.
Ayyalasomayajula
,
P. Y.
Chuang
, and
A.
Gylfason
, “
Inertial clustering of particles in high-Reynolds-number turbulence
,”
Phys. Rev. Lett.
100
,
214501
(
2008
).
20.
D.
Schwabe
,
P.
Hintz
, and
S.
Frank
, “
New features of thermocapillary convection in floating zones revealed by tracer particle accumulation structures
,”
Microgravity Sci. Tech.
9
,
163
(
1996
).
21.
I.
Ueno
,
S.
Tanaka
, and
H.
Kawamura
, “
Oscillatory and chaotic thermocapillary convection in a half-zone liquid bridge
,”
Phys. Fluids
15
(
2
),
408
(
2003
);
I.
Ueno
,
S.
Tanaka
, and
H.
Kawamura
, “
Various flow patterns in thermocapillary convection in half-zone liquid bridge of high Prandtl number fluid
,”
Adv. Space Res.
32
(
2
),
143
(
2003
).
22.
D.
Schwabe
,
A. I.
Mizev
,
S.
Tanaka
, and
H.
Kawamura
, “
Particle accumulation structures in time-dependent thermocapillary flow in a liquid bridge under microgravity
,”
Microgravity Sci. Tech.
18
(
3–4
),
117
(
2006
);
S.
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
).
23.
D.
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
(
7
),
072102
(
2007
).
24.
M.
Lappa
, “
Assessment of the role of axial vorticity in the formation of particle accumulation structures in supercritical Marangoni and hybrid thermocapillary-rotation-driven flows
,”
Phys. Fluids
25
,
012101
(
2013
).
25.
E.
Hofmann
and
H. C.
Kuhlmann
, “
Particle accumulation on periodic orbits by repeated free surface collisions
,”
Phys. Fluids
23
,
072106
(
2011
).
26.
M. R.
Maxey
and
J. J.
Riley
, “
Equation of motion for a small rigid sphere in a nonuniform flow
,”
Phys. Fluids
26
,
883
(
1983
).
27.
M. R.
Maxey
, “
The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields
,”
J. Fluid Mech.
174
,
441
465
(
1987
);
G.
Haller
and
T.
Sapsis
, “
Where do inertial particles go in fluid flows?
,”
Physica D
237
(
5
),
573
(
2008
).
28.
T.
Sapsis
and
G.
Haller
, “
Clustering criterion for inertial particles in two-dimensional time-periodic and three-dimensional steady flows
,”
Chaos
20
,
017515
(
2010
).
29.
D.
Pushkin
,
D.
Melnikov
, and
V.
Shevtsova
, “
Ordering of small particles in one-dimensional coherent structures by time-periodic flows
,”
Phys. Rev. Lett.
106
,
234501
(
2011
).
30.
D.
Schwabe
and
A. I.
Mizev
, “
Particles of different density in thermocapillary liquid bridges under the action of travelling and standing hydrothermal waves
,”
Eur. Phys. J. Special Topics
192
,
13
(
2011
).
31.
F. H.
Busse
and
R. M.
Clever
, “
An asymptotic model of two-dimensional convection in the limit of low Prandtl number
,”
J. Fluid Mech.
102
,
75
(
1981
).
32.
R. M.
Clever
and
F. H.
Busse
, “
Low Prandtl number convection in a layer heated from below
,”
J. Fluid Mech.
102
,
61
(
1981
);
R. M.
Clever
and
F. H.
Busse
, “
Nonlinear oscillatory convection
,”
J. Fluid Mech.
176
,
403
(
1987
).
33.
V.
Croquette
, “
Convective pattern dynamics at low Prandtl number: Part II
,”
Contemp. Phys.
30
,
153
(
1989
).
34.
M.
Lappa
,
Fluids, Materials and Microgravity: Numerical Techniques and Insights into the Physics
(
Elsevier Science
,
Oxford, England
,
2004
).
35.
M.
Lappa
, “
Three-dimensional numerical simulation of Marangoni flow instabilities in floating zones laterally heated by an equatorial ring
,”
Phys. Fluids
15
,
776
(
2003
);
M.
Lappa
, “
Combined effect of volume and gravity on the three-dimensional flow instability in non-cylindrical floating zones heated by an equatorial ring
,”
Phys. Fluids
16
(
2
),
331
(
2004
);
M.
Lappa
,
R.
Savino
, and
R.
Monti
, “
Influence of buoyancy forces on Marangoni flow instabilities in liquid bridges
,”
Int. J. Numer. Methods Heat Fluid Flow
10
(
7
),
721
(
2000
).
36.
D.
Melnikov
,
D.
Pushkin
, and
V.
Shevtsova
, “
Accumulation of particles in time-dependent thermocapillary flow in a liquid bridge. Modeling of experiments
,”
Eur. Phys. J. Special Topics
192
,
29
(
2011
).
37.
K. A.
Atkinson
,
An Introduction to Numerical Analysis
, 2nd ed. (
John Wiley & Sons
,
New York
,
1989
).
38.
M.
Lappa
,
Rotating Thermal Flows in Natural and Industrial Processes
(
John Wiley & Sons
,
Chichester, England
,
2012
).
39.
B.
Hof
,
G. J.
Lucas
, and
T.
Mullin
, “
Flow state multiplicity in convection
,”
Phys. Fluids
11
,
2815
(
1999
).
40.
A. Yu.
Gelfgat
,
P. Z.
Bar-Yoseph
, and
A. L.
Yarin
, “
Stability of multiple steady states of convection in laterally heated cavities
,”
J. Fluid Mech.
388
,
315
(
1999
).
41.
Y.
Hu
,
R.
Ecke
, and
G.
Ahlers
, “
Transition to spiral-defect chaos in low Prandtl number convection
,”
Phys. Rev. Lett.
74
,
391
(
1995
).
42.
R. V.
Cakmur
,
D. A.
Egolf
,
B. B.
Plapp
, and
E.
Bodenschatz
, “
Bistability and competition of spatiotemporal chaotic and fixed point attractors in Rayleigh-Bénard convection
,”
Phys. Rev. Lett.
79
(
10
),
1853
(
1997
).
43.
M.
Lappa
, “
Thermal convection and related instabilities in models of crystal growth from the melt on earth and in microgravity: Past history and current status
,”
Cryst. Res. Technol.
40
(
6
),
531
(
2005
);
M.
Lappa
and
R.
Savino
, “
3D analysis of crystal/melt interface shape and Marangoni flow instability in solidifying liquid bridges
,”
J. Comput. Phys.
180
(
2
),
751
(
2002
).
44.
M.
Lappa
, “
Secondary and oscillatory gravitational instabilities in canonical three-dimensional models of crystal growth from the melt, Part 1: Rayleigh-Bènard systems
,”
C. R. Mec.
335
(
5–6
),
253
(
2007
);
M.
Lappa
, “
Part 2: Lateral heating and the Hadley circulation
,”
C. R. Mec.
335
(
5–6
),
261
(
2007
).
45.
M.
Lappa
,
D.
Castagnolo
, and
L.
Carotenuto
, “
Sensitivity of the non-linear dynamics of Lysozyme ‘Liesegang Rings’ to small asymmetries
,”
Physica A
314/1-4
,
623
(
2002
).
46.
L.
Carotenuto
,
C.
Piccolo
,
D.
Castagnolo
,
M.
Lappa
, and
J. M.
Garcìa-Ruiz
, “
Experimental observations and numerical modelling of diffusion-driven crystallisation processes
,”
Acta Crystallogr.
58
,
1628
(
2002
);
M.
Lappa
, “
A theoretical and numerical multiscale framework for the analysis of pattern formation in protein crystal engineering
,”
Int. J. Multiscale Comp. Eng.
9
(
2
),
149
(
2011
).
You do not currently have access to this content.