Abstract
We address the flows of dense suspensions of particles within the framework of two-velocity continuum. Thermodynamics of such a continuum is developed by the method suggested in the papers of Landau and Khalatnikov. As an application, we consider the convective settling problem. We capture the Boycott effect and prove that the enhanced sedimentation occurs in a tilted vessel due to vortices. We do not call on additional interphase forces like the Stokes drag, the virtual mass force, the Archimedes force, the Basset-Boussinesq force and etc. Instead, we apply a generalized Fick's law for the particle mass concentration flux vector.
Issue Section:
Multiphase Flows
References
1.
Dmitriev
,
A.
,
Makarov
,
P.
, and
Bouz
,
M. E.
, 2015
, “
A New Regime of Nucleate Boiling in Microsphere Mesostructures: Jumping Pool Boiling
,” Tech. Phys. Lett.
,
41
(3
), pp. 288
–290
.10.1134/S10637850150301902.
Eskin
,
D.
, and
Miller
,
M.
, 2008
, “
A Model of Non-Newtonian Slurry Flow in a Fracture
,” Powder Technol.
,
182
(2
), pp. 313
–322
.10.1016/j.powtec.2007.06.0273.
Boycott
,
A.
, 1920
, “
Sedimentation of Blood Corpuscles
,” Nature
,
104
(2621
), pp. 532
–532
.10.1038/104532b04.
Ungarish
,
M.
, 1993
, Hydrodynamics of Suspensions: Fundamentals of Centrifugal and Gravity Separation
,
Springer-Verlag
,
Berlin/Heidelberg, Germany
.5.
Nakamura
,
H.
, and
Kuroda
,
K.
, 1937
, “
La Cause de L'acceleration de la Vitesse de Sedimentation Des Suspensions Dans Les Recipients Inclines
,” Keijo J. Med.
,
8
, pp. 256
–296
.6.
Ponder
,
E.
, 1925
, “
On Sedimentation and Rouleaux Formation
,” Q. J. Exp. Physiol.
,
15
(3
), pp. 235
–252
.10.1113/expphysiol.1925.sp0003567.
Nevskii
,
Y.
, and
Osiptsov
,
A.
, 2011
, “
Slow Gravitational Convection of Disperse Systems in Domains With Inclined Boundaries
,” Fluid Dyn.
,
46
(2
), pp. 225
–239
.10.1134/S00154628110200508.
Nevskii
,
Y.
, and
Osiptsov
,
A.
, 2008
, “
The Effect of Unsteady and Memory Forces in the Gravity Convection of Suspensions
,” Moscow Univ. Mech. Bull.
,
63
(4
), pp. 85
–88
.10.3103/S00271330080400189.
Einstein
,
A.
, 1906
, “
A New Determination of Molecular Dimensions
,” Ann. Phys.
,
324
(2
), pp. 289
–306
.10.1002/andp.1906324020410.
Batchelor
,
G.
, and
Green
,
J.
, 1972
, “
The Determination of the Bulk Stress in a Suspension of Spherical Particles to Order c2
,” J. Fluid Mech.
,
56
(03
), pp. 401
–427
.10.1017/S002211207200243511.
Eilers
,
H.
, 1941
, “
The Viscosity of the Emulsion of Highly Viscous Substances as Function of Concentration
,” Colloid Polym. Sci.
,
97
, pp. 313
–321
.12.
Mooney
,
M.
, 1951
, “
The Viscosity of a Concentrated Suspension of Spherical Particles
,” Colloid Polym. Sci.
,
6
(2
), pp. 162
–170
.10.1016/0095-8522(51)90036-013.
Quemada
,
D.
, 1978
, “
Rheology of Concentrated Disperse Systems III. General Features of the Propose Non-Newtonian Model. Comparison With Experimental Data
,” Rheol. Acta
,
17
(6
), pp. 643
–653
.10.1007/BF0152203714.
Krieger
,
I. M.
, and
Dougherty
,
T. J.
, 1959
, “
A Mechanism for Non-Newtonian Flow in Suspensions of Rigid Spheres
,” Trans. Soc. Rheol.
,
3
(1
), pp. 137
–152
.10.1122/1.54884815.
Stickel
,
J.
, and
Powell
,
R.
, 2005
, “
Fluid Mechanics and Rheology of Dense Suspensions
,” Annu. Rev. Fluid Mech.
,
37
(1
), pp. 129
–149
.10.1146/annurev.fluid.36.050802.12213216.
Morris
,
J.
, and
Boulay
,
F.
, 1999
, “
Curvilinear Flows of Noncolloidal Suspensions: The Role of Normal Stresses
,” J. Rheol.
,
43
(5
), pp. 1213
–1237
.10.1122/1.55102117.
Baumgarten
,
A.
, and
Kamrin
,
K.
, 2019
, “
A General Fluid-Sediment Mixture Model and Constitutive Theory Validated in Many Flow Regimes
,” J. Fluid Mech.
,
861
, pp. 721
–764
.10.1017/jfm.2018.91418.
Khalatnikov
,
I.
, 1989
, Introduction to the Theory of Superfluidity
,
Addison-Wesley
,
New York
.19.
Landau
,
L.
, and
Lifshits
,
E.
, 1987
, Fluid Mechabics. Course of Theoretical Physics
,
Pergamon Press
, Oxford, UK
.20.
DeGroot
,
S.
, and
Mazur
,
P.
, 1962
, Non Equilibrium Thermodynamics
,
North-Holland Publishing Company
,
Amsterdam, The Netherlands
.21.
Blokhin
,
A.
, and
Dorovskii
,
V.
, 1995
, Mathematical Modelling in the Theory of Multivelocity Continuum
,
Nova Science Publishers
,
New York
.22.
Dorovskii
,
V.
, and
Perepechko
,
Y.
, 1996
, “
The Hydrodynamic Model of Solution in Cracking-Porous Media
,” Russ. Geol. Geophys.
,
9
, pp. 123
–134
.23.
Shelukhin
,
V.
, 2014
, “
A Poroelastic Medium Saturated by a Two-Phase Capillary Fluid
,” Continuum Mech. Thermodyn.
,
26
(5
), pp. 619
–638
.10.1007/s00161-013-0321-x24.
Shelukhin
,
V.
, 2018
, “
Thermodynamics of Two-Phase Granular Fluids
,” J. Non-Newtonian Fluid Mech.
,
262
, pp. 25
–37
.10.1016/j.jnnfm.2018.02.00425.
Anderson
,
T.
, and
Jackson
,
R.
, 1967
, “
Fluid Mechanical Description of Fluidized Beds. Equations of Motion
,” Ind. Eng. Chem. Fundam.
,
6
(4
), pp. 527
–539
.10.1021/i160024a00726.
Jackson
,
R.
, 1997
, “
Locally Averaged Equations of Motion for a Mixture of Identical Particles and a Newtonian Fluid
,” Chem. Eng. Sci.
,
52
(15
), pp. 2457
–2469
.10.1016/S0009-2509(97)00065-127.
Buyevich
,
Y.
, and
Shchelchkova
,
I.
, 1979
, “
Flow of Dense Suspensions
,” Prog. Aerosp. Sci.
,
18
, pp. 121
–150
.10.1016/0376-0421(77)90004-528.
Zhang
,
D.
, and
Prosperetti
,
A.
, 1997
, “
Momentum and Energy Equations for Disperse Two-Phase Flows and Their Closure for Dilute Suspensions
,” Int. J. Multiphase Flow
,
23
(3
), pp. 425
–453
.10.1016/S0301-9322(96)00080-829.
Miller
,
R.
,
Singh
,
J.
, and
Morris
,
J.
, 2009
, “
Suspensions Flow Modeling for General Geometries
,” Chem. Eng. Sci.
,
64
(22
), pp. 4597
–4610
.10.1016/j.ces.2009.04.03330.
Brady
,
J.
, and
Morris
,
J.
, 1997
, “
Microstructure of Strongly Sheared Suspensions and Its Impact on Rheology and Diffusion
,” J. Fluid Mech.
,
348
, pp. 103
–139
.10.1017/S002211209700632031.
Lhuillier
,
D.
, 2009
, “
Migration of Rigid Particles in Non-Brownian Viscous Suspensions
,” Phys. Fluids
,
21
(2
), p. 023302
.10.1063/1.307967232.
Nott
,
P.
, and
Brady
,
J.
, 1994
, “
Pressure-Driven Flow of Suspensions: Simulation and Theory
,” J. Fluid Mech.
,
275
, pp. 157
–199
.10.1017/S002211209400232633.
Crowe
,
C.
,
Schwarzkopf
,
J.
,
Sommerfield
,
M.
, and
Tsuji
,
Y.
, 2011
, Multiphase Flows With Droplets and Particles
,
CRC Press
,
Boca Raton, FL
.34.
Shelukhin
,
V.
, and
Neverov
,
V.
, 2016
, “
Thermodynamics of Micropolar Bingham Fluids
,” J. Non-Newtonian Fluid Mech.
,
236
, pp. 83
–90
.10.1016/j.jnnfm.2016.08.00535.
Leighton
,
D.
, and
Acrivos
,
F.
, 1987
, “
The Shear-Induced Migration of Particles in Concentrated Suspensions
,” J. Fluid Mech.
,
181
(1
), pp. 415
–439
.10.1017/S002211208700215536.
Nott
,
P.
,
Guazzelli
,
E.
, and
Pouliquen
,
O.
, 2011
, “
The Suspension Balance Model Revisited
,” Phys. Fluids
,
23
(4
), p. 043304
.10.1063/1.357092137.
Schwarze
,
R.
,
Gladkyy
,
A.
,
Uhlig
,
F.
, and
Luding
,
S.
, 2013
, “
Rheology of Weakly Wetted Granular Materials: A Comparison of Experimental and Numerical Data
,” Granular Matter
,
15
(4
), pp. 455
–465
.10.1007/s10035-013-0430-z38.
Boyer
,
F.
,
Guazzelli
,
E.
, and
Pouliquen
,
O.
, 2011
, “
Unifying Suspension and Granular Rheology
,” Phys. Rev. Lett.
,
107
(18
), p. 188301
.10.1103/PhysRevLett.107.18830139.
Pietsch
,
W.
, 2002
, Agglomeration Processes: Phenomena, Technologies, Equipment
,
Wiley-VCH
,
Weinheim, Germany
.40.
Guo
,
Y.
,
Wu
,
C.
, and
Thornton
,
C.
, 2011
, “
The Effects of Air and Particle Density Difference Segregation of Powder Mixtures During Die Filling
,” Chem. Eng. Sci.
,
66
(4
), pp. 661
–673
.10.1016/j.ces.2010.11.01741.
Guo
,
Y.
,
Wu
,
C.
,
Kafui
,
K.
, and
Thornton
,
C.
, 2011
, “
3ddem/Cfd Analysis of Size-Induced Segregation During Die Filling
,” Powder Technol.
,
206
(1–2
), pp. 177
–188
.10.1016/j.powtec.2010.05.02942.
Beeley
,
P.
, 2001
, Foundry Technology
,
Butterworth-Heinemann
,
Oxford, UK
.43.
Guazzelli
,
E.
, and
Hinch
,
J.
, 2011
, “
Fluctuations and Instability in Sedimentation
,” Annu. Rev. Fluid Mech.
,
43
(1
), pp. 97
–116
.10.1146/annurev-fluid-122109-16073644.
Marchetti
,
B.
,
Bergougnoux
,
L.
, and
Guazzelli
,
E.
, 2021
, “
Falling Clouds of Particles in Vortical Flows
,” J. Fluid Mech.
,
908
, p. A30
.10.1017/jfm.2020.88345.
Gillissen
,
J.
,
Ness
,
C.
,
Peterson
,
J.
,
Wilson
,
H.
, and
Cates
,
M.
, 2019
, “
Constitutive Model for Time-Dependent Flows of Shear-Thickening Suspensions
,” Phys. Rev. Lett.
,
123
(21
), p. 214504
.10.1103/PhysRevLett.123.21450446.
Gillissen
,
J.
,
Ness
,
C.
,
Peterson
,
J.
,
Wilson
,
H.
, and
Cates
,
M.
, 2020
, “
Constitutive Model for Shear-Thickening Suspensions: Predictions for Steady Shear With Superposed Transverse Oscillations
,” J. Rheol.
,
64
(2
), pp. 353
–365
.10.1122/1.512965747.
Acrivos
,
A.
, and
Herbolzheimer
,
E.
, 1979
, “
Enhanced Sedimentation in Settling Tanks With Inclined Walls
,” J. Fluid Mech.
,
92
(3
), pp. 435
–457
.10.1017/S002211207900072048.
Leontovich
,
M.
, 1985
, Introduction to Thermodynamics. Statistical Physics
,
Nauka, Moscow, Russia
.49.
Ishii
,
M.
, and
Mishima
,
K.
, 1984
, “
Two-Fluid Model and Hydrodynamic Constitutive Relations
,” Nucl. Eng. Des.
,
82
(2–3
), pp. 107
–126
.10.1016/0029-5493(84)90207-350.
Kinosita
,
K.
, 1949
, “
Sedimentation in Tilted Vessels
,” J. Colloid Interface Sci.
,
4
(5
), pp. 525
–536
.10.1016/0095-8522(49)90049-551.
Hill
,
W. D.
,
Rothfus
,
R. R.
, and
Li
,
K.
, 1977
, “
Boundary-Enhanced Sedimentation Due to Settling Convection
,” Int. J. Multiph. Flow
,
3
(6
), pp. 561
–583
.10.1016/0301-9322(77)90030-152.
McCaffery
,
S.
,
Elliott
,
L.
, and
Ingham
,
D.
, 1999
, Enhanced Sedimentation in Inclined Fracture Channels
,
Computational Mechanics Publications
,
Southampton, Russia
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