Following the recent discovery of new three-dimensional particle attractors driven by joint (fluid) thermovibrational and (particle) inertial effects in closed cavities with various shapes and symmetries [M. Lappa, Phys. Fluids 26(9), 093301 (2014); ibid. 31(7), 073303 (2019)], the present analysis continues this line of inquiry by probing influential factors hitherto not considered; among them, the role of the steady component of thermovibrational convection, i.e., the time-averaged velocity field that is developed by the fluid due to the non-linear nature of the overarching balance equations. It is shown how this apparently innocuous problem opens up a vast parameter space, which includes several variables, comprising (but not limited to) the frequency of vibrations, the so-called “Gershuni number,” the size of particles (Stokes number), and their relative density with respect to the surrounding fluid (density ratio). A variety of new particle structures (2D and 3D) are uncovered and a complete analysis of their morphology is presented. The results reveal an increase in the multiplicity of solutions brought in by the counter-intuitive triadic relationship among particle inertial effects and the instantaneous and time-averaged convective thermovibrational phenomena. Finally, a universal formula is provided that is able to predict correctly the time required for the formation of all the observed structures.
REFERENCES
1.
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
).2.
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
).3.
D. E.
Melnikov
, D. O.
Pushkin
, and V. M.
Shevtsova
, “Synchronization of finite-size particles by a traveling wave in a cylindrical flow
,” Phys. Fluids
25
(9
), 092108
(2013
).4.
M.
Gotoda
, D. E.
Melnikov
, I.
Ueno
, and V.
Shevtsova
, “Experimental study on dynamics of coherent structures formed by inertial solid, particles in three-dimensional periodic flows
,” Chaos
26
(7
), 073106
(2016
).5.
R. N.
Parthasarathy
and G. M.
Faeth
, “Turbulent dispersion of particles in self-generated homogeneous turbulence
,” J. Fluid Mech.
220
, 515
–537
(1990
).6.
K.
Ehara
, C.
Hagwood
, and K. J.
Coakley
, “Novel method to classify aerosol particles according to their mass-to-charge ratio-Aerosol particle mass analyser
,” J. Aerosol Sci.
27
, 217
–234
(1996
).7.
M.
Lappa
, “On the nature of fluid-dynamics
, in Understanding the Nature of Science
, Science, Evolution and Creationism, edited by P.
Lindholm
(Nova Science Publishers, Inc.
, 2019
), BISAC: SCI034000, ISBN: 978-1-53616-016-1, Chap. 1, pp. 1
–64
, https://novapublishers.com/shop/understanding-the-nature-of-science/.8.
A.
Bracco
, P. H.
Chavanis
, A.
Provenzale
, and E. A.
Spiegel
, “Particle aggregation in a turbulent Keplerian flow
,” Phys. Fluids
11
(8
), 2280
–2287
(1999
).9.
M.
Lappa
, “On the nature, formation and diversity of particulate coherent structures in microgravity conditions and their relevance to materials science and problems of astrophysical interest
,” Geophys. Astrophys. Fluid Dyn.
110
(4
), 348
–386
(2016
).10.
T.
Haszpra
and T.
Tél
, “Volcanic ash in the free atmosphere: A dynamical systems approach
,” J. Phys.: Conf. Ser.
333
, 012008
(2011
).11.
V. P.
Srivastava
, “Particle-fluid suspension model of blood flow through stenotic vessels with applications
,” Int. J. Bio-Med. Comput.
38
(2
), 141
–154
(1995
).12.
M. Z.
Saghir
and A.
Mohamed
, “Effectiveness in incorporating Brownian and thermophoresis effects in modelling convective flow of water-Al2O3 nanoparticles
,” Int. J. Numer. Methods Heat Fluid Flow
28
(1
), 47
–63
(2018
).13.
D.
Aquaro
, “Erosion due to the impact of solid particles of materials resistant at high temperature
,” Meccanica
41
, 539
–551
(2006
).14.
S.
Dasgupta
, R.
Jackson
, and S.
Sundaresan
, “Gas-particle flow in vertical pipes with high mass loading of particles
,” Powder Technol.
96
, 6
–23
(1998
).15.
C. S. R.
Rao
, “Fluidized-bed combustion technology—A review
,” Combust. Sci. Technol.
16
(3-6
), 215
–227
(1977
).16.
J. Z.
Zhao
, L.
Ratke
, and B.
Feuerbacher
, “Microstructure evolution of immiscible alloys during cooling through the miscibility gap
,” Model. Simul. Mater. Sci. Eng.
6
, 123
–139
(1998
).17.
F.
Balboa-Usabiaga
and R.
Delgado-Buscalioni
, “Particle hydrodynamics: From molecular to colloidal fluids
,” in Particle-Based Methods II: Fundamentals and Applications
(Springer Nature
, 2011
), pp. 152
–163
.18.
J. K.
Eaton
and J. R.
Fessler
, “Preferential concentration of particles by turbulence
,” Int. J. Multiphase Flow
20
, 169
–209
(1994
).19.
R.
Jayaram
, H. I.
Andersson
, L.
Zhao
, and H. I.
Andersson
, “Clustering of inertial spheres in evolving Taylor-Green vortex flow
,” Phys. Fluids
32
, 043306
(2020
).20.
W.
Yuan
, L.
Zhao
, H. I.
Andersson
, and J.
Deng
, “Three-dimensional Voronoï analysis of preferential concentration of spheroidal particles in wall turbulence
,” Phys. Fluids
30
, 063304
(2018
).21.
K.
Luo
, Z.
Wang
, D.
Li
, J.
Tan
, and J.
Fan
, “Fully resolved simulations of turbulence modulation by high-inertia particles in an isotropic turbulent flow
,” Phys. Fluids
29
, 113301
(2017
).22.
G.
Haller
and T.
Sapsis
, “Where do inertial particles go in fluid flows?
,” Physica D
237
(5
), 573
–583
(2008
).23.
S. G.
Love
and D. R.
Pettit
, “Fast repeteable clumping of solid particles in microgravity
,” Lunar Planet. Sci.
XXXV
, 1119
(2004
).24.
S. S.
Tabakova
and Z. D.
Zapryanov
, “On the hydrodynamic interaction of two spheres oscillating in a viscous fluid. II. Three dimensional case
,” J. Appl. Math. Phys. (ZAMP)
33
, 487
–502
(1982
).25.
R.
Wunenburger
, V.
Carrier
, and Y.
Garrabos
, “Periodic order induced by horizontal vibrations in a two-dimensional assembly of heavy beads in water
,” Phys. Fluids
14
(7
), 2350
–2359
(2002
).26.
C. C.
Thomas
and J. P.
Gollub
, “Structures and chaotic fluctuations of granular clusters in a vibrated fluid layer
,” Phys. Rev. E
70
, 061305
(2004
).27.
A. A.
Ivanova
, V. G.
Kozlov
, and A. F.
Kuzaev
, “Vibrational lift force acting on a body in a fluid near a solid surface
,” Dokl. RAN
402
(4
), 1
–4
(2005
)A. A.
Ivanova
, V. G.
Kozlov
, and A. F.
Kuzaev
[Dokl. Phys.
50
(6
), 311
–314
(2005
)].28.
V. G.
Kozlov
, A. A.
Ivanova
, and P.
Evesque
, “Block stratification of sedimenting granular matter in a vessel due to vertical vibration
,” Fluid Dyn. Mater. Process.
2
(3
), 203
–210
(2006
).29.
D. V.
Lyubimov
, T. P.
Lyubimova
, and A. V.
Straube
, “Accumulation of solid particles in convective flows
,” Microgravity Sci. Technol.
16
(1
), 210
–214
(2005
).30.
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. Spec. Top.
192
, 29
–39
(2011
).31.
D. E.
Melnikov
, T.
Watanabe
, T.
Matsugase
, I.
Ueno
, and V.
Shevtsova
, “Experimental study on formation of particle accumulation structures by a thermocapillary flow in a deformable liquid column
,” Microgravity Sci. Technol.
26
, 365
–374
(2014
).32.
D. E.
Melnikov
and V.
Shevtsova
, “Different types of Lagrangian coherent structures formed by solid particles in three-dimensional time-periodic flows
,” Eur. Phys. J.: Spec. Top.
226
(6
), 1239
–1251
(2017
).33.
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
(1
), 012101
(2013
).34.
M.
Lappa
, “On the existence and multiplicity of one-dimensional solid particle attractors in time-dependent Rayleigh-Bénard convection
,” Chaos
23
(1
), 013105
(2013
).35.
M.
Lappa
, “On the variety of particle accumulation structures under the effect of g-jitters
,” J. Fluid Mech.
726
, 160
–195
(2013
).36.
M.
Lappa
, “Stationary solid particle attractors in standing waves
,” Phys. Fluids
26
(1
), 013305
(2014
).37.
M.
Lappa
, “The patterning behaviour and accumulation of spherical particles in a vibrated non-isothermal liquid
,” Phys. Fluids
26
(9
), 093301
(2014
).38.
M.
Lappa
, “Towards new contact-less techniques for the control of inertial particles dispersed in a fluid
,” in 12th International Conference on Thermal Engineering, February 23–26, 2019
(PDPU
, Gandhinagar, Gujarat, India
, 2019
).39.
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
).40.
I.
Ueno
, Y.
Abe
, K.
Noguchi
, and H.
Kawamura
, “Dynamic particle accumulation structure (PAS) in half-zone liquid bridge - reconstruction of particle motion by 3-D PTV
,” Adv. Space Res.
41
(12
), 2145
–2149
(2008
).41.
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. Spec. Top.
192
, 13
–27
(2011
).42.
M.
Gotoda
, T.
Sano
, T.
Kaneko
, and I.
Ueno
, “Evaluation of existence region and formation time of particle accumulation structure (PAS) in half-zone liquid bridge
,” Eur. Phys. J. Spec. Top.
224
, 299
(2015
).43.
A.
Toyama
, M.
Gotoda
, T.
Kaneko
, and I.
Ueno
, “Existence conditions and formation process of second type of spiral loop particle accumulation structure (SL-2 PAS) in half-zone liquid bridge
,” Microgravity Sci. Technol.
29
, 263
–274
(2017
).44.
G. Z.
Gershuni
and D. V.
Lyubimov
, Thermal Vibrational Convection
(Wiley
, England
, 1998
).45.
A.
Mialdun
, I. I.
Ryzhkov
, D. E.
Melnikov
, and V.
Shevtsova
, “Experimental evidence of thermal vibrational convection in a nonuniformly heated fluid in a reduced gravity environment
,” Phys. Rev. Lett.
101
, 084501
(2008
).46.
T. P.
Lyubimova
, A. V.
Perminov
, and M. G.
Kazimardanov
, “Stability of quasi-equilibrium states and supercritical regimes of thermal vibrational convection of a Williamson fluid in zero gravity conditions
,” Int. J. Heat Mass Transfer
129
, 406
–414
(2019
).47.
S.
Bouarab
, F.
Mokhtari
, S.
Kaddeche
, D.
Henry
, V.
Botton
, and A.
Medelfef
, “Theoretical and numerical study on high frequency vibrational convection: Influence of the vibration direction on the flow structure
,” Phys. Fluids
31
(4
), 043605
(2019
).48.
V.
Shevtsova
, T.
Lyubimova
, Z.
Saghir
, D.
Melnikov
, Y.
Gaponenko
, V.
Sechenyh
, J. C.
Legros
, and A.
Mialdun
, “IVIDIL: On-board g-jitters and diffusion controlled phenomena
,” J. Phys.: Conf. Ser.
327
, 012031
(2011
).49.
V.
Shevtsova
, A.
Mialdun
, D.
Melnikov
, I.
Ryzhkov
, Y.
Gaponenko
, Z.
Saghir
, T.
Lyubimova
, and J. C.
Legros
, “The IVIDIL experiment onboard the ISS: Thermodiffusion in the presence of controlled vibrations
,” C. R. Mécaniq.
339
(5
), 310
–317
(2011
).50.
B.
Maryshev
, T.
Lyubimova
, and D.
Lyubimov
, “Two-dimensional thermal convection in porous enclosure subjected to the horizontal seepage and gravity modulation
,” Phys. Fluids
25
, 084105
(2013
).51.
M.
Lappa
, “Control of convection patterning and intensity in shallow cavities by harmonic vibrations
,” Microgravity Sci. Technol.
28
(1
), 29
–39
(2016
).52.
A.
Vorobev
and T.
Lyubimova
, “Vibrational convection in a heterogeneous binary mixture. Part 1. Time-averaged equations
,” J. Fluid Mech.
870
, 543
–562
(2019
).53.
M.
Lappa
, “Numerical study into the morphology and formation mechanisms of three-dimensional particle structures in vibrated cylindrical cavities with various heating conditions
,” Phys. Rev. Fluids
1
(6
), 064203
(2016
).54.
M.
Lappa
, “On the multiplicity and symmetry of particle attractors in confined non-isothermal fluids subjected to inclined vibrations
,” Int. J. Multiphase Flow
93
, 71
–83
(2017
).55.
M.
Lappa
, “On the formation and morphology of coherent particulate structures in non-isothermal enclosures subjected to rotating g-jitters
,” Phys. Fluids
31
(7
), 073303
(2019
).56.
M.
Lappa
, “Time reversibility and non-deterministic behaviour in oscillatorily sheared suspensions of non-interacting particles at high Reynolds numbers
,” Comput. Fluids
184
, 78
–90
(2019
).57.
M.
Lappa
, Thermal Convection: Patterns, Evolution and Stability
(John Wiley and Sons, Ltd.
, Chichester, England
, 2009
), 700
pages.58.
M.
Lappa
, “On the transport, segregation, and dispersion of heavy and light particles interacting with rising thermal plumes
,” Phys. Fluids
30
(3
), 033302
(2018
).59.
M. R.
Maxey
and J. J.
Riley
, “Equation of motion for a small rigid sphere in a nonuniform flow
,” Phys. Fluids
26
, 883
–889
(1983
).60.
H. C.
Kuhlmann
et al, “The JEREMI-Project on thermocapillary convection in liquid bridges. Part A: Overview of particle accumulation structures
,” Fluid Dyn. Mater. Process.
10
(1
), 1
–36
(2014
).61.
R.
Monti
, R.
Savino
, and M.
Lappa
, “Microgravity sensitivity of typical fluid physics experiment
,” paper presented at the 17th Microgravity Measurements Group Meeting
, Cleveland, Ohio
, 1998
, published in the Meeting Proceedings in NASA CP-1998-208414, pp. 1
–15
.62.
R.
Monti
, R.
Savino
, and M.
Lappa
, “On the convective disturbances induced by g-jitter on the space station
,” Acta Astronaut.
48
(5-12
), 603
–615
(2001
).63.
R.
Savino
and M.
Lappa
, “Assessment of thermovibrational theory: Application to g-jitter on the space station
,” J. Spacecr. Rockets
40
(2
), 201
–210
(2003
).64.
G. Z.
Gershuni
, E. M.
Zhukhovitskii
, and Yu. S.
Yurkov
, “Vibrational thermal convection in a rectangular cavity
,” Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza
4
, 94
–99
(1982
).65.
G. Z.
Gershuni
and E. M.
Zhukhovitskii
, “Vibrational thermal convection in zero gravity
,” Fluid Mech. Sov. Res.
15
(1
), 63
–84
(1986
).66.
R.
Clift
, J. R.
Grace
, and M. E.
Weber
, Bubbles, Drops, and Particles
(Academic Press
, New York
, 1978
).67.
P. M.
Gresho
and R. L.
Sani
, “On pressure boundary conditions for the incompressible Navier-Stokes equations
,” Int. J. Numer. Methods Fluids
7
, 1111
–1145
(1987
).68.
G. E.
Karniadakis
, M.
Israeli
, and S. A.
Orszag
, “High-order splitting methods for the incompressible Navier-Stokes equations
,” J. Comput. Phys.
97
, 414
–443
(1991
).69.
N. A.
Petersson
, “Stability of pressure boundary conditions for Stokes and Navier-Stokes equations
,” J. Comput. Phys.
172
, 40
–70
(2001
).70.
O. A.
Ladyzhenskaya
, The Mathematical Theory of Viscous Incompressible Flow
, 2nd ed. (Gordon and Breach
, New York; London
, 1969
).71.
D. L.
Brown
, R.
Cortez
, and M. L.
Minion
, “Accurate projection methods for the incompressible Navier-Stokes equations
,” J. Comput. Phys.
168
(2
), 464
–499
(2001
).72.
S.
Armfield
and R.
Street
, “An analysis and comparison of the time accuracy of fractional-step methods for the Navier-Stokes equations on staggered grids
,” Int. J. Numer. Methods Fluids
38
(3
), 255
–282
(2002
).73.
J. L.
Guermond
, P.
Minev
, and J.
Shen
, “An overview of projection methods for incompressible flows
,” Comput. Methods, Comput. Methods Appl. Mech. Eng.
195
, 6011
–6045
(2006
).74.
S.
Patankar
, Numerical Heat Transfer and Fluid Flow
, Hemisphere Series on Computational Methods in Mechanics and Thermal Science (Taylor and Francis
, 1980
).75.
M.
Lappa
, “Strategies for parallelizing the three-dimensional Navier-Stokes equations on the Cray T3E
,” Sci. Supercomput. CINECA
11
, 326
–340
(1997
).76.
M.
Lappa
, “A mathematical and numerical framework for the simulation of oscillatory buoyancy and Marangoni convection in rectangular cavities with variable cross section
,” in Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics
, Computational Methods in Applied Sciences, edited by G.
Alexander
(Springer Mathematical Series
, 2019
), ISBN: 978-3-319-91493-0, Vol. 50, Chap. 12, pp. 419
–458
.77.
H.
Khallouf
, G. Z.
Gershuni
, and A.
Mojtabi
, “Numerical study of two-dimensional thermovibrational convection in rectangular cavities
,” Numer. Heat Transfer Part A
27
, 297
–305
(1995
).78.
M.
Lappa
, “Some considerations about the symmetry and evolution of chaotic Rayleigh-Bénard convection: The flywheel mechanism and the “wind” of turbulence
,” C. R. Mécanique
339
, 563
–572
(2011
).© 2020 Author(s).
2020
Author(s)
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