Recent advances on the analytical form of the hydrodynamic force and heat/mass transfer from a particle, bubble, or drop are examined critically. Also some of the recent computational studies, which help strengthen or clarify our knowledge of the complex velocity and temperature fields associated with the momentum and heat/mass transfer processes are also mentioned in a succinct way. Whenever possible, the processes of energy/mass exchange and of momentum exchange from spheres and spheroids are examined simultaneously and any common results and possible analogies between these processes are pointed out. This approach results in a better comprehension of the transport processes, which are very similar in nature, as well as in the better understanding of the theoretical expressions that are currently used to model these processes. Of the various terms that appear in the transient equations, emphasis is given to the history terms, which are lesser known and more difficult to calculate. The origin, form, and method of computation of the history terms are pointed out as well as the effects of various parameters on them. Among the other topics examined here are the differences in the governing and derived equations resulting by finite Reynolds and Peclet numbers; the origin, theoretical validity and accuracy of the semi-empirical expressions; the effects of finite internal viscosity and conductivity of the sphere; the effects of small departures from the spherical shape; the effects of the finite concentration; and the transverse, or lift, components of the force on the sphere.
Issue Section:
Technical Papers
1.
Poisson
, S. A.
, 1831
, “Memoire sur les Mouvements Simultanes d’un Pendule et de L’air Environnemant
,” Mem. Acad. Sci., Paris
,9
, p. 521
–523
.2.
Green
, G.
, 1833
, “Researches on the Vibration of Pendulums in Fluid Media
,” Trans. - R. Soc. Edinbrgh
, 13
, pp. 54
–68
.3.
Stokes
, G. G.
, 1845
, “On the Theories of Internal Friction of the Fluids in Motion
,” Trans. Cambridge Philos. Soc.
, 8
, pp. 287
–319
.4.
Stokes
, G. G.
, 1851
, “On the Effect of the Internal Friction of Fluids on the Motion of a Pendulum
,” Trans. Cambridge Philos. Soc.
, 9
, pp. 8
–106
.5.
Boussinesq
, V. J.
, 1885
, “Sur la Resistance qu’ Oppose un Liquide Inde´fini en Repos…
,” C. R., Acad. Sci.
,100
, pp. 935
–937
.6.
Boussinesq, J., 1885, Applications L’etude des Potentiels, Blanchard, Paris (re-edition 1969).
7.
Basset, A. B., 1888, Treatise on Hydrodynamics, Bell, London.
8.
Basset
, A. B.
, 1888
, “On the Motion of a Sphere in a Viscous Liquid
,” Philos. Trans. R. Soc. London, Ser. A
, 179
, pp. 43
–63
.9.
Whitehead
, A. N.
, 1889
, “Second Approximation to Viscous Fluid Motion. A Sphere Moving Steadily in a Straight Line
,” Q. J. Math.
, 23
, pp. 143
–152
.10.
Oseen
, C. W.
, 1910, “Uber die Stokes’sche Formel und Uber eine verwandte Aufgabe in der Hydrodynamik,” Ark. Mat., Astron. Fys., 6(29).11.
Oseen
, C. W.
, 1913, “Uber den Goltigkeitsbereich der Stokesschen Widerstandsformel,” Ark. Mat., Astron. Fys., 9(19).12.
Faxen
, H.
, 1922
, “Der Widerstand gegen die Bewegung einer starren Kugel in einer zum den Flussigkeit, die zwischen zwei parallelen Ebenen Winden eingeschlossen ist
,” Ann. Phys. (Leipzig)
, 68
, pp. 89
–119
.13.
Tchen, C. M., 1949, “Mean Values and Correlation Problems Connected With the Motion of Small Particles Suspended in a Turbulent Fluid,” doctoral dissertation, Technical University of Delft, Delft, The Netherlands.
14.
Corssin
, S.
, and Lumley
, J. L.
, 1957
, “On the Equation of Motion of a Particle in a Turbulent Fluid
,” Appl. Sci. Res., Sect. A
, 6
, pp. 114
–116
.15.
Sy
, F.
, Taunton
, J. W.
, and Lightfoot
, E. N.
, 1970
, “Transient Creeping Flow Around Spheres
,” AIChE J.
, 16
, pp. 386
–391
.16.
Proudman
, I.
, and Pearson
, J. R. A.
, 1956
, “Expansions at Small Reynolds Numbers for the Flow Past a Sphere and a Circular Cylinder
,” J. Fluid Mech.
, 2
, pp. 237
–262
.17.
Sano
, T.
, 1981
, “Unsteady Flow Past a Sphere at Low Reynolds Number
,” J. Fluid Mech.
, 112
, pp. 433
–441
.18.
Maxey
, M. R.
, and Riley
, J. J.
, 1983
, “Equation of Motion of a Small Rigid Sphere in a Non-Uniform Flow
,” Phys. Fluids
, 26
, pp. 883
–889
.19.
Gatignol
, D.
, 1983
, “The Faxen Formulas for a Rigid Particle in an Unsteady Non-Uniform Stokes Flow
,” J. Mec. Theor. Appl.
, 1
, pp. 143
–154
.20.
Michaelides
, E. E.
, and Feng
, Z.-G.
, 1995
, “The Equation of Motion of a Small Viscous Sphere in an Unsteady Flow With Interface Slip
,” Int. J. Multiphase Flow
, 21
, pp. 315
–321
.21.
Mei
, R.
, Lawrence
, C. J.
, and Adrian
, R. J.
, 1991
, “Unsteady Drag on a Sphere at Finite Reynolds Number With Small Fluctuations in the Free-Stream Velocity
,” J. Fluid Mech.
, 233
, pp. 613
–631
.22.
Mei
, R.
, and Adrian
, R. J.
, 1992
, “Flow Past a Sphere With an Oscillation in the Free-Stream and Unsteady Drag at Finite Reynolds Number
,” J. Fluid Mech.
, 237
, pp. 323
–341
.23.
Lovalenti
, P. M.
, and Brady
, J. F.
, 1993
, “The Hydrodynamic Force on a Rigid Particle Undergoing Arbitrary Time-Dependent Motion at Small Reynolds Numbers
,” J. Fluid Mech.
, 256
, pp. 561
–601
.24.
Hinch
, E. J.
, 1993
, “The Approach to Steady State in Oseen Flows
,” J. Fluid Mech.
, 256
, pp. 601
–603
.25.
Chang
, E. J.
, and Maxey
, M. R.
, 1994
, “Unsteady Flow About a Sphere at Low to Moderate Reynolds Number. Part 1, Oscillatory Motion
,” J. Fluid Mech.
, 277
, pp. 347
–379
.26.
Chang
, E. J.
, and Maxey
, M. R.
, 1995
, “Unsteady Flow About a Sphere at Low to Moderate Reynolds Number. Part 2. Accelerated Motion
,” J. Fluid Mech.
, 303
, pp. 133
–153
.27.
Fourier, J., 1822, Theorie Analytique de la Chaleur, Paris.
28.
Tait, P. G., 1885, Scientific Papers, A. & C. Black, Edinburgh.
29.
Carslaw, H. S., and Jaeger, J. C., 1947, Conduction of Heat in Solids, Oxford University Press, Oxford, UK.
30.
Acrivos
, A.
, and Taylor
, T. E.
, 1962
, “Heat and Mass Transfer from Single Spheres in Stokes Flow
,” Phys. Fluids
, 5
, pp. 387
–394
.31.
Michaelides
, E. E.
, and Feng
, Z. G.
, 1994
, “Heat Transfer From a Rigid Sphere in a Non-Uniform Flow and Temperature Field
,” Int. J. Heat Mass Transf.
, 37
, pp. 2069
–2076
.32.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 1998
, “Transient Heat Transfer From a Particle With Arbitrary Shape and Motion
,” ASME J. Heat Transfer
, 120
, pp. 674
–681
.33.
Lovalenti
, P. M.
, and Brady
, J. F.
, 1993
, “The Force on a Bubble, Drop or Particle in Arbitrary Time-Dependent Motion at Small Reynolds Numbers
,” Phys. Fluids A
, 5
, pp. 2104
–2116
.34.
Leal, L. G., 1992, Laminar Flow and Convective Transport Processes, Butterworth-Heineman, Boston.
35.
Kim, S., and Karila, S. J., 1991, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heineman, Boston.
36.
Sirignano, W. A., 1999, Fluid Dynamics and Transport of Droplets and Sprays, Cambridge University Press, Cambridge, UK.
37.
Leal
, L. G.
, 1980
, “Particle Motions in a Viscous Fluid
,” Annu. Rev. Fluid Mech.
, 12
, pp. 435
–476
.38.
Brady
, J. F.
, and Bossis
, G.
, 1988
, “Stokesian Dynamics
,” Annu. Rev. Fluid Mech.
, 20
, pp. 111
–157
.39.
Feuillebois
, F.
, 1989
, “Some Theoretical Results for the Motion of Solid Spherical Particles on a Viscous Fluid
,” Multiphase Sci. Technol.
,4
, pp. 583
–794
.40.
Sirignano
, W. A.
, 1993
, “Fluid Dynamics of Sprays
,” ASME J. Fluids Eng.
, 115
, pp. 345
–378
.41.
Stock
, D. E.
, 1996
, “Particle Dispersion in Flowing Gases
,” ASME J. Fluids Eng.
, 118
, pp. 4
–17
.42.
Michaelides
, E. E.
, and Feng
, Z.-G.
, 1996
, “Analogies Between the Transient Momentum and Energy Equations of Particles
,” Prog. Energy Combust. Sci.
, 22
, pp. 147
–163
.43.
Michaelides
, E.
, 1997
, “Review—The Transient Equation of Motion for Particles, Bubbles, and Droplets
,” ASME J. Fluids Eng.
, 119
, pp. 233
–247
.44.
Loth
, E.
, 2000
, “Numerical Approaches for the Motion of Dispersed Particles, Droplets, or Bubbles
,” Prog. Energy Combust. Sci.
, 26
, pp. 161
–223
.45.
Koch
, D. L.
, and Hill
, R. J.
, 2001
, “Inertial Effects in Suspension and Porous Media Flows
,” Annu. Rev. Fluid Mech.
, 33
, pp. 619
–647
.46.
Levich, V. G., 1962, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ.
47.
Clift, R., Grace, J. R., and Weber, M. E., 1978, Bubbles, Drops and Particles, Academic Press, New York.
48.
Happel, J., and Brenner, H., 1986, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, Dordrecht, The Netherlands (reprint, orig. publ. 1963).
49.
Govier, G. W., and Aziz, K., 1977, The Flow of Complex Mixtures in Pipes, Kruger Publ., Huntington (reprint).
50.
Soo, S. L., 1990, Multiphase Fluid Dynamics, Science Press, Beijing.
51.
Stock, D. E., Reeks, M. W., Tsuji, Y., Michaelides, E. E., and Gautam, M., 1993, Gas-Particle Flows, FED-Vol. 166, ASME, New York.
52.
Stock, D. E., Reeks, M. W., Tsuji, Y., Michaelides, E. E., and Gautam, M., 1995, Gas-Particle Flows, FED-Vol. 228, ASME, New York.
53.
Stock, D. E., Reeks, M. W., Tsuji, Y., Michaelides, E. E., and Gautam, M., 1997, Gas-Particle Flows–1997, ASME, New York.
54.
Stock, D. E., Reeks, M. W., Tsuji, Y., Michaelides, E. E., and Gautam, M., 1999, Gas-Particle Flows–1999, ASME, New York.
55.
Michaelides
, E. E.
, 1988
, “On the Drag Coefficient and the Correct Integration of the Equation of Motion of Particles in Gases
,” ASME J. Fluids Eng.
, 110
, pp. 339
–342
.56.
Barton
, I. E.
, 1996
, “Exponential-Lagrangian Tracking Schemes Applied to Stokes Law
,” ASME J. Fluids Eng.
, 118
, pp. 85
–89
.57.
Dodemand
, E.
, Prud’homme
, R.
, and Kuentzmann
, P.
, 1995
, “Influence of Unsteady Forces Acting on a Particle in a Suspension Application to the Sound Propagation
,” Int. J. Multiphase Flow
, 21
, pp. 27
–51
.58.
Sangani
, A. S.
, Zhang
, D. Z.
, and Prosperetti
, A.
, 1991
, “The Added Mass, Basset and Viscous Drag Coefficients in Nondilute Bubbly Liquids Undergoing Small-Amplitude Oscillatory Motion
,” Phys. Fluids A
, 3
, pp. 2955
–2970
.59.
Coimbra
, C. F. M.
, and Rangel
, R. H.
, 1997
, “General Solution of the Particle Momentum Equation in Unsteady Stokes Flows
,” J. Fluid Mech.
, 370
, pp. 53
–72
.60.
Coimbra
, C. F. M.
, Edwards
, D. K.
, and Rangel
, R. H.
, 1998
, “Heat Transfer in a Homogenous Suspension Including Radiation and History Effects
,” J. Thermophys. Heat Transfer
, 12
, pp. 304
–312
.61.
Vojir
, D. J.
, and Michaelides
, E. E.
, 1994
, “The Effect of the History Term on the Motion of Rigid Spheres in a Viscous Fluid
,” Int. J. Multiphase Flow
, 20
, pp. 547
–556
.62.
Sazhin
, S. S.
, Goldstein
, V. A.
, and Heikal
, M. R.
, 2001
, “A Transient Formulation of Newton’s Cooling Law for Spherical Bodies
,” ASME J. Heat Transfer
, 123
, pp. 63
–64
.63.
Michaelides
, E. E.
, 1992
, “A Novel Way of Computing the Basset Term in Unsteady Multiphase Flow Computations
,” Phys. Fluids A
, 4
, pp. 1579
–1582
.64.
Maxey
, M. R.
, 1987
, “The Motion of Small Spherical Particles in a Cellular Flow Field
,” Phys. Fluids
, 30
, pp. 1915
–1928
.65.
Reeks
, M. W.
, and McKee
, S.
, 1984
, “The Dispersive Effects of Basset History Forces on Particle Motion in Turbulent Flow
,” Phys. Fluids
, 27
, pp. 1573
–1582
.66.
Lhuillier
, D.
, 2001
, “Internal Variables and the Non-Equilibrium Thermodynamics of Colloidal Suspensions
,” J. Non-Newtonian Fluid Mech.
, 96
, pp. 19
–30
.67.
Theofanous
, T. G.
, Bohrer
, T. H.
, Chang
, M. C.
, and Patel
, P. D.
, 1978
, “Experiments and Universal Growth Relations for Vapor Bubbles with Microlayers
,” ASME J. Heat Transfer
, 100
, pp. 41
–48
.68.
Theofanous
, T. G.
, and Patel
, P. D.
, 1976
, “Universal Relations for Bubble Growth
,” Int. J. Heat Mass Transf.
, 19
, pp. 425
–429
.69.
Magnaudet
, J.
, and Legendre
, D.
, 1998
, “The Viscous Drag Force on a Spherical Bubble With a Time-Dependent Radius
,” Phys. Fluids
, 10
, pp. 550
–554
.70.
Morrison
, F. A.
, and Stewart
, M. B.
, 1976
, “Small Bubble Motion in an Accelerating Fluid
,” ASME J. Appl. Mech.
, 97
, pp. 399
–402
.71.
Thorncroft
, G. E.
, Klausner
, J. F.
, and Mei
, R.
, 2001
, “Bubble Forces and Detachment Models
,” Multiph. Sci. Technol.
,13
(3–4
), pp. 35
–76
.72.
Ormieres
, D.
, and Provancal
, M.
, 1999
, “Transition to Turbulence in the Wake of a Sphere
,” Phys. Rev. Lett.
, 83
, pp. 80
–83
.73.
Ghidersa
, B.
, and Dusek
, J.
, 2000
, “Breaking of Axisymmetry and Onset of Unsteadiness in the Wake of a Sphere
,” J. Fluid Mech.
, 423
, pp. 33
–69
.74.
Mebarek
, M.
, Bouchet
, G.
, Gidhersa
, B.
, and Dusek
, J.
, 2002, “Hydrodynamical Forces Acting on a Rigid Fixed Sphere Placed in a Uniform Flow,” Int. J. Numer. Methods Fluids, in print.75.
Maxworthy
, T.
, 1965
, “Accurate Measurements of a Sphere Drag at Low Reynolds Numbers
,” J. Fluid Mech.
, 23
, pp. 369
–372
.76.
Schiller
, L.
, and Nauman
, A.
, 1933
, “Uber die grundlegende Berechnung bei der Schwekraftaufbereitung
,” Ver. Deutch. Ing.
,44
, pp. 318
–320
.77.
Whitaker
, S.
, 1972
, “Forced Convection Heat Transfer Correlations for Flow in Pipes Past Flat Plates, Single Cylinders, Single Spheres, and for Flow in Packed Beds and Tubes Bundles
,” AIChE J.
, 18
, pp. 361
–371
.78.
Incropera, F. P., and DeWitt, C. P., 1985, Fundamentals of Heat and Mass Transfer, John Wiley and Sons, New York.
79.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 2001
, “Heat and Mass Transfer Coefficients of Viscous Spheres
,” Int. J. Heat Mass Transf.
, 44
, pp. 4445
–4454
.80.
Feuillebois, F., and Lasek, A., 1980, “Boundary Layer on a Sphere Accelerating From Rest,” Boundary and Interior Layers—Computational and Asymptotic Methods, J. J. H. Miller, ed., Boole Press, Dublin.
81.
Mei
, R.
, 1994
, “Flow due to an Oscillating Sphere and an Expression for Unsteady Drag on the Sphere at Finite Reynolds Numbers
,” J. Fluid Mech.
, 270
, pp. 133
–174
.82.
Lawrence
, C. J.
, and Mei
, R.
, 1995
, “Long-Time Behavior of the Drag on a Body in Impulsive Motion
,” J. Fluid Mech.
, 283
, pp. 307
–332
.83.
Lovalenti
, P. M.
, and Brady
, J. F.
, 1995
, “Force on a Body in Response to an Abrupt Change in Velocity at Small but Finite Reynolds Number
,” J. Fluid Mech.
, 293
, pp. 35
–46
.84.
Choudhury
, P. N.
, and Drake
, D. G.
, 1971
, “Unsteady Heat Transfer from a Sphere in a Low Reynolds Number Flow
,” Q. J. Mech. Appl. Math.
, 24
, pp. 23
–36
.85.
Konoplin
, N.
, and Sparrow
, E. M.
, 1972
, “Unsteady Heat Transfer and Temperature for Stokesian Flow About a Sphere
,” ASME J. Heat Transfer
, 94
, pp. 266
–272
.86.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 1996
, “Unsteady Heat Transfer from a Spherical Particle at Finite Peclet Numbers
,” ASME J. Fluids Eng.
, 118
, pp. 96
–102
.87.
Pozrikidis
, C.
, 1997
, “Unsteady Heat or Mass Transport from a Suspended Particle at Low Peclet Numbers
,” J. Fluid Mech.
, 289
, pp. 652
–688
.88.
Lawrence
, C. J.
, and Weinbaum
, S.
, 1986
, “The Force on an Axisymmetric Body in Linearized Time-Dependent Motion: A New Memory Term
,” J. Fluid Mech.
, 171
, pp. 209
–218
.89.
Lawrence
, C. J.
, and Weinbaum
, S.
, 1988
, “The Unsteady Force on a Body at Low Reynolds Number: The Axisymmetric Motion of a Spheroid
,” J. Fluid Mech.
, 189
, pp. 463
–498
.90.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 1997
, “Transient Heat and Mass Transfer From a Spheroid
,” AIChE J.
, 43
, pp. 609
–616
.91.
Konoplin
, N.
, 1971
, “Gravitationally Induced Acceleration of Spheres in a Creeping Flow
,” AIChE J.
, 17
, pp. 1502
–1503
.92.
Chaplin
, J. R.
, 1999
, “History Forces and the Unsteady Wake of a Cylinder
,” J. Fluid Mech.
, 393
, pp. 99
–121
.93.
Julien, P. Y., 1995, Erosion and Sedimentation, Cambridge Press, Cambridge, UK.
94.
Loseno, C., and Easson, W. J., “Free Falling of Irregular Particles,” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
95.
Odar
, F.
, and Hamilton
, W. S.
, 1964
, “Forces on a Sphere Accelerating in a Viscous Fluid
,” J. Fluid Mech.
, 18
, pp. 302
–303
.96.
Odar
, F.
, 1966
, “Verification of the Proposed Equation for Calculation of the Forces on a Sphere Accelerating in a Viscous Fluid
,” J. Fluid Mech.
, 25
, pp. 591
–592
.97.
Al-taweel
, A. M.
, and Carley
, J. F.
, 1971
, “Dynamics of Single Spheres in Pulsated Flowing Liquids: Part I. Experimental Methods and Results
,” AIChE Symp. Sr.
, 67
(116
), pp. 114
–123
.98.
Al-taweel
, A. M.
, and Carley
, J. F.
, 1972
, “Dynamics of Single Spheres in Pulsated Flowing Liquids: Part II. Modeling and Interpretation of Results
,” AIChE Symp. Ser.
, 67
(116
), pp. 124
–131
.99.
Schoneborn
, P. R.
, 1975
, “The Interaction Between a Single Sphere and an Oscillating Fluid
,” Int. J. Multiphase Flow
, 2
, pp. 307
–317
.100.
Karanfilian
, S. K.
, and Kotas
, T. J.
, 1978
, “Drag on a Sphere in Unsteady Motion in a Liquid at Rest
,” J. Fluid Mech.
, 87
, pp. 85
–96
.101.
Temkin
, S.
, and Mehta
, H. K.
, 1982
, “Droplet Drag in an Accelerating and Decelerating Flow
,” J. Fluid Mech.
, 116
, pp. 297
–313
.102.
Tsuji
, Y.
, Kato
, N.
, and Tanaka
, T.
, 1991
, “Experiments on the Unsteady Drag and Wake of a Sphere at High Reynolds Numbers
,” Int. J. Multiphase Flow
, 17
, pp. 343
–354
.103.
Darwin
, C.
, 1953
, “A Note on Hydrodynamics
,” Proc. Roy. Soc.
, 49
, pp. 342
–353
.104.
Bataille, J., Lance, M., and Marie, J. L., 1990, “Bubbly Turbulent Shear Flows,” J. Kim, U. Rohatgi, and M. Hashemi, eds., FED-Vol. 99, ASME, New York, pp. 1–7.
105.
Rivero
, M.
, Magnaudet
, J.
, and Fabre
, J.
, 1991
, “Quelques Resultats Nouveaux Concernants les Forces Exerce´es sur une Inclusion Spherique par Ecoulement Accelere
,” C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers
, 312
, ser. II, pp. 1499
–1506
.106.
Auton
, T. R.
, Hunt
, J. R. C.
, and Prud’homme
, M.
, 1988
, “The Force Exerted on a Body in Inviscid Unsteady Non-Uniform Rotational Flow
,” J. Fluid Mech.
, 197
, pp. 241
–257
.107.
Kim
, I.
, Elghobashi
, S.
, and Sirignano
, W. A.
, 1998
, “On the Equation for Spherical-Particle Motion: Effect of Reynolds and Acceleration Numbers
,” J. Fluid Mech.
, 367
, pp. 221
–253
.108.
Bagchi, P., and Balachandar, S., 2001, “On the Effect of Nonuniformity and the Generalization of the Equation of Motion of a Particle,” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
109.
Kaviani, M., 1994, Principles of Convective Heat Transfer, Springer-Verlag, New York.
110.
Matsumoto
, Y.
, and Takagi
, S.
, 1998
, “Numerical Simulations of Multiphase Flows
,” Computational Fluid Dyn. Rev.
,4
, pp. 994
–1010
.111.
LeClair
, B. P.
, Hamielec
, A. E.
, and Pruppracher
, H. R.
, 1970
, “A Numerical Study of the Drag on a Sphere at Intermediate Reynolds and Peclet Numbers
,” J. Atmos. Sci.
, 27
, pp. 308
–319
.112.
Rivkind
, V. Y.
, Ryskin
, G. M.
, and Ishbein
, G. A.
, 1976
, “Flow Around a Spherical Drop in a Fluid Medium at Intermediate Reynolds Numbers
,” Appl. Math. Mech.
, 40
, pp. 687
–691
.113.
Abramzon
, B.
, and Elata
, C.
, 1984
, “Heat Transfer from a Single Sphere in Stokes Flow
,” Int. J. Heat Mass Transf.
, 27
, pp. 687
–695
.114.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 2000
, “A Numerical Study on the Transient Heat Transfer From a Sphere at High Reynolds and Peclet Numbers
,” Int. J. Heat Mass Transf.
, 43
, pp. 219
–229
.115.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 2000
, “Mass and Heat Transfer From Fluid Spheres at Low Reynolds Numbers
,” Powder Technol.
, 112
, pp. 63
–69
.116.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 2001
, “Drag Coefficients of Viscous Spheres at Intermediate and High Reynolds Numbers
,” ASME J. Fluids Eng.
, 123
, pp. 841
–849
.117.
Winnikow
, S.
, and Chao
, B. T.
, 1966
, “Droplet Motion in Purified Systems
,” Phys. Fluids
, 9
, pp. 50
–61
.118.
Magnaudet
, J.
, Rivero
, M.
, and Fabre
, J.
, 1995
, “Accelerated Flows Past a Rigid Sphere or a Spherical Bubble. Part 1. Steady Straining Flow
,” J. Fluid Mech.
, 284
, pp. 97
–135
.119.
Mei
, R.
, and Lawrence
, C. J.
, 1996
, “The Flow Field due to a Body in Impulsive Motion
,” J. Fluid Mech.
, 325
, pp. 79
–111
.120.
Kim
, I.
, Elghobashi
, S.
, and Sirignano
, W. A.
, 1997
, “Unsteady Flow Interactions Between a Pair of Advected Vortex Tubes and a Rigid Sphere
,” J. Multiphase Flow
, 23
, pp. 1
–23
.121.
Masoudi
, M.
, and Sirignano
, W. A.
, 2000
, “Collision of a Vortex with a Vaporizing Droplet
,” Int. J. Multiphase Flow
, 26
, pp. 1925
–1949
.122.
Kim
, I.
, Elghobashi
, S.
, and Sirignano
, W. A.
, 1993
, “Three-Dimensional Flow Over Two Spheres Placed Side by Side
,” J. Fluid Mech.
, 246
, pp. 465
–488
.123.
Chiang
, H.
, and Kleinstreuer
, C.
, 1993
, “Numerical Analysis of Variable-Fluid-Property Effects on the Convective Heat and Mass Transfer of Fuel Droplets
,” Combust. Flame
, 92
, pp. 459
–464
.124.
Feng
, J.
, and Joseph
, D. D.
, 1995
, “The Unsteady Motion of Solid Bodies in Creeping Flows
,” J. Fluid Mech.
, 303
, pp. 83
–102
.125.
Cheng
, H.
, and Papanicolaou
, G.
, 1997
, “Flow Past Periodic Arrays of Spheres at Low Reynolds Number
,” J. Fluid Mech.
, 335
, pp. 189
–212
.126.
Chiang
, H.
, and Kleinstreuer
, C.
, 1992
, “Transient Heat and Mass Transfer of Interacting Vaporizing Droplets in a Linear Array
,” Int. J. Heat Mass Transf.
, 35
, pp. 2675
–2682
.127.
Pozrikidis
, C.
, 2000
, “Conductive Mass Transport From a Semi-Infinite Lattice of Particles
,” Int. J. Heat Mass Transf.
, 43
, pp. 493
–502
.128.
Balachandar
, S.
, and Ha
, M. Y.
, 2001
, “Unsteady Heat Transfer From a Sphere in a Uniform Cross-Flow
,” Phys. Fluids
, 13
(12
), pp. 3714
–3728
.129.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 1998
, “Motion of a Permeable Sphere at Finite but Small Reynolds Numbers
,” Phys. Fluids
, 10
, pp. 1375
–1383
.130.
Graham
, D. J.
, and Moyeed
, R. A.
, 2002
, “How Many Particles for My Lagrangian Simulations?
” Powder Technol.
, 114
, pp. 254
–259
.131.
Hjelmfelt
, A. T.
, and Mockros
, L. F.
, 1966
, “Motion of Discrete Particles in a Turbulent Fluid
,” Appl. Sci. Res.
, 16
, pp. 149
–161
.132.
Druzhinin
, O. A.
, and Ostrovsky
, L. A.
, 1994
, “The Influence of Basset Force on Particle Dynamics in Two Dimensional Flows
,” Physica D
, 76
, pp. 34
–43
.133.
Domgin, J.-F., Huilier, D. G. F., Karl, J.-J., Gardin, P., and Burnage, H., 1998, “Experimental and Numerical Study of Rigid Particles, Droplets, and Bubbles Motion in Quiescent and Turbulent Flows—Influence of the History Force,” Third International Conference on Multiphase Flow, ICMF-98, Lyon, France.
134.
Launey, K., and Huillier, D., 1999, “On the Influence of the Non-Stationary Forces on the Particles Dispersion,” Proceedings of the ASME, ASME, New York, FEDSM99-7874, San Fransisco, CA.
135.
Abbad, M., and Souhar, M., 2001, “Experimentation on the History Force Acting on Rigid or Fluid Particles at Low Reynolds Number in an Oscillating Frame,” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
136.
Thomas
, P. J.
, 1997
, “A Numerical Study of the Influence of the Basset Force on the Statistics of LDV Velocity Data Sampled in a Flow Region With a Large Spatial Velocity Gradient
,” Exp. Fluids
, 23
, pp. 48
–53
.137.
Gay
, M.
, and Michaelides
, E. E.
, 2002, “Effect of the History Term on the Transient Energy Equation for a Sphere,” Int. J. Heat Mass Transf., in print.138.
Michaelides, E. E., and Feng, Z.-G., 2002, “History Terms in the Heat and Mass Transfer Equations of Particles,” Proc. 12th Intern. Conf. on Heat Transfer, Elsevier, Amsterdam, Paper No. 991.
139.
White, F. M., 1985, Viscous Fluid Flow, McGraw-Hill, New York.
140.
Carlson
, D. J.
, and Hoglund
, R. F.
, 1964
, “Particle Drag and Heat Transfer in Rocket Nozzles
,” AIAA J.
, 2
, pp. 1980
–1984
.141.
Ranz
, W. E.
, and Marshall
, W. R.
, 1952
, “Evaporation From Drops
,” Chem. Eng. Prog.
, 48
, pp. 141
–146
.142.
Polyanin
, A. D.
, 1984
, “An Asymptotic Analysis of Some Nonlinear Boundary Value Problems of Convective Mass and Heat Transfer of Reacting Particles With the Flow
,” Int. J. Heat Mass Transf.
, 27
, pp. 163
–189
.143.
Yuge
, T.
, 1960
, “Experiments on Heat Transfer from Spheres Including Combined Natural and Forced Convection
,” ASME J. Heat Transfer
, 82
, pp. 214
–220
.144.
Bankoff
, S. G.
, and Mason
, J. P.
, 1962
, “Heat Transfer from the Surface of a Steam Bubble in Turbulent Subcooled Liquid Stream
,” AIChE J.
, 8
, pp. 30
–33
.145.
Gilbert
, H.
, and Angellino
, H.
, 1974
, “Transfere de Matiere entre une Sphere Soumise a des Vibrations et un Liquide en Mouvement
,” Int. J. Heat Mass Transf.
, 17
, pp. 625
–632
.146.
Hayward
, G. L.
, and Pei
, D. C. T.
, 1978
, “Local Heat Transfer from a Single Sphere to a Turbulent Air Stream
,” Int. J. Heat Mass Transf.
, 21
, pp. 35
–41
.147.
Chen
, J. L. S.
, 1974
, “Growth of the Boundary Layer on a Spherical Gas Bubble
,” ASME J. Appl. Mech.
, 41
, pp. 873
–878
.148.
Lhuillier
, D.
, 1982
, “Forces d’Inertie sur une Bulle en Expansion se Deplacant dans un Fluide
,” C. R. Acad. Sci., Paris
,295
, pp. 95
–106
.149.
Drew
, D. A.
, and Lahey
, R. T.
, 1987
, “The Virtual Mass and Lift Force on a Sphere in Rotating and Straining Inviscid Flow
,” Int. J. Multiphase Flow
, 13
, pp. 113
–121
.150.
Drew
, D. A.
, and Lahey
, R. T.
, 1990
, “Some Supplemental Analysis on the Virtual Mass and Lift Force on a Sphere in a Rotating and Straining Inviscid Flow
,” Int. J. Multiphase Flow
, 16
, pp. 1127
–1130
.151.
Mei
, R.
, Klausner
, J. F.
, and Lawrence
, C. J.
, 1994
, “A Note on the History Force on a Spherical Bubble at Finite Reynolds Number
,” Phys. Fluids
, 6
, pp. 418
–420
.152.
Park
, W. C.
, Klausner
, J. F.
, and Mei
, R.
, 1995
, “Unsteady Forces on Spherical Bubbles
,” Exp. Fluids
, 19
, pp. 167
–172
.153.
Chisnell
, R. F.
, 1987
, “The Unsteady Motion of a Drop Moving Vertically Under Gravity
,” J. Fluid Mech.
, 176
, pp. 434
–464
.154.
Sy
, F.
, and Lightfoot, 1971
, “Transient Creeping Flow Around Fluid Spheres
,” AIChE J.
, 17
, pp. 177
–181
.155.
Galindo
, V.
, and Gerbeth
, G.
, 1993
, “A Note on the Force on an Accelerating Spherical Drop at Low Reynolds Numbers
,” Phys. Fluids
, 5
, pp. 3290
–3292
.156.
Feng
, Z. G.
, Michaelides
, E. E.
, and Scibilia
, M.-F.
, 1996
, “The Energy Equation of a Sphere in an Unsteady and Non-Uniform Temperature Field
,” Rev. Gen. Therm.
, 35
, pp. 5
–13
.157.
Yang
, S.-M.
, and Leal
, L. G.
, 1991
, “A Note on Memory-Integral Contributions to the Force of an Accelerating Spherical Drop at Low Reynolds Number
,” Phys. Fluids A
, 3
, pp. 1822
–1824
.158.
Hadamard
, J. S.
, 1911
, “Mouvement Permanent Lent d’une Sphere Liquide et Visqueuse dans un Liquide Visqueux
,” C. R. Acad. Sci., Paris
,152
, pp. 1735
–1752
.159.
Rybczynski
, W.
, 1911
, “On the Translatory Motion of a Fluid Sphere in a Viscous Medium
,” Bull. Acad. Sci., Cracow, Series A
,p.
40
40
.160.
Harper
, J. F.
, and Moore
, D. W.
, 1968
, “The Motion of a Spherical Liquid Drop at High Reynolds Number
,” J. Fluid Mech.
, 32
, pp. 367
–391
.161.
Ryskin
, G.
, and Leal
, L. G.
, 1983
, “Numerical Simulation of a Free Boundary Problem in Fluid Mechanics—Part III: Bubble Deformation in an Axysymmetric Straining Flow
,” J. Fluid Mech.
, 148
, pp. 37
–54
.162.
Fan, L.-S., and Tsuchiya, K., 1990, Bubble Wake Dynamics in Liquids and Liquid Solid Suspensions, Butterworth-Heinemann, Stoneham, MA.
163.
Takagi
, S.
, Prosperetti
, A.
, and Matsumoto
, Y.
, 1994
, “Drag Coefficient of a Gas Bubble in an Axisymmetric Shear Flow
,” Phys. Fluids
, 6
(9
), pp. 3186
–3188
.164.
Tomiyama, A., Miyoshi, K., Tamai, H., Zun, I., and Sakaguchi, T., 1998, “A Bubble Tracking Method for the Prediction of Spatial-Evolution of Bubble Flow in a Vertical Pipe,” Proceedings of the ICMF-98, Lyon, France.
165.
Ford
, B.
, and Loth
, E.
, 1998
, “Forces on Ellipsoidal Bubbles in a Turbulent Shear Layer
,” Phys. Fluids
, 10
(1
), pp. 178
–188
.166.
Fujiwara, A., Tokuhiro, A., Hishida, K., and Maeda, M., 1998, “Investigation of Oscillatory Bubble Motion Using a Dual Shadow Technique and Its Surrounding Flow Field by LIF-PIV,” Proceedings of the 3th International Conference on Multiphase Flow, ICMF-98, Lyon, France.
167.
Fujiwara, A., Tokuhiro, A., Hishida, K., and Maeda, M., 2001, “Flow Structure Around Rising Bubble Measured by PIV/LIF (Effect of Shear Rate and Bubble Size),” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
168.
Tokuhiro
, A.
, Maekawa
, M.
, Iizuka
, K.
, Hishida
, K.
, and Maeda
, M.
, 1998
, “Turbulent Flow Past a Bubble and an Ellipsoid Using Shadow-Image and PIV Techniques
,” Int. J. Multiphase Flow
, 24
, pp. 1383
–1406
.169.
Sirignano
, W. A.
, and Mehring
, C.
, 2000
, “Review of Theory of Distortion and Disintegration of Liquid Streams
,” Prog. Energy Combust. Sci.
, 26
, pp. 609
–655
.170.
Garner, F. H., and Lihou, D. A., 1965, DECHEMA–Monograph, 55, pp. 155–178.
171.
LeClair
, B. P.
, and Hamielec
, A. E.
, 1972
, “A Theoretical and Experimental Study of the Internal Circulation in Water Drops Falling at Terminal Velocity in Air
,” J. Atmos. Sci.
, 29
(2
), pp. 728
–740
.172.
Rivkind
, V. Y.
, and Ryskin
, G. M.
, 1976
, “Flow Structure in Motion of a Spherical Drop at Intermediate Reynolds Numbers
,” Fluid Mech.-Sov. Res.
, 11
(1
), pp. 5
–12
.173.
Oliver
, D. L.
, and Chung
, J. N.
, 1987
, “Flow About a Fluid Sphere at Low to Moderate Reynolds Numbers
,” J. Fluid Mech.
, 177
, pp. 1
–18
.174.
Tomiyama, A., 2001, “Reconsideration of Three Fundamental Problems in Modeling Bubbly Flows,” 39th European Two-Phase Flow Group Meeting, Aveiro, Portugal.
175.
Wu
, M.
, and Gharib
, M.
, 2002
, “Experimental Studies on the Shape and Path of Small Air Bubbles Rising in Clean Water
,” Phys. Fluids
, 14
, pp. L49–L52
L49–L52
.176.
Prosperetti, A., 2002, “Some Problems Associated With the Formation and Flow of Bubbles,” 14th Congress of Theoretical and Applied Mechanics, Blacksburg, VA.
177.
Bowman
, C. W.
, Ward
, D. M.
, Johnson
, A. I.
, and Trans
, O.
, 1961
, “Mass Transfer From Fluid and Solid Spheres at Low Reynolds Numbers
,” Can. J. Chem. Eng.
, 39
, pp. 9
–13
.178.
Abramzon
, B.
, and Sirignano
, W. A.
, 1989
, “Droplet Vaporization for Spray Combustion Calculations
,” Int. J. Heat Mass Transf.
, 32
, pp. 1605
–1618
.179.
Chigier
, N. A.
, 1976
, “The Atomization and Burning of Liquid Fuel Sprays
,” Prog. Energy Combust. Sci.
, 2
, pp. 97
–114
.180.
Chigier, N. A., 1981, Energy, Combustion and Environment, McGraw-Hill, New York.
181.
Chigier
, N. A.
, and McCreath
, C. G.
, 1974
, “Combustion of Droplets in Sprays
,” Acta Astronaut.
, 1
, pp. 687
–710
.182.
Crespo
, A.
, and Linan
, A.
, 1975
, “Unsteady Effects in Droplet Evaporation and Combustion
,” Combust. Sci. Technol.
, 1
, pp. 9
–18
.183.
Law
, C. K.
, 1982
, “Recent Advances in Droplet Vaporization and Combustion
,” Prog. Energy Combust. Sci.
, 8
, pp. 171
–201
.184.
Williams, A., 1985, Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, Benjamin-Cummings, Menlo Park, CA.
185.
Saffman
, P. G.
, 1965
, “The Lift on a Small Sphere in a Slow Shear Flow
,” J. Fluid Mech.
, 22
, pp. 385
–398
.186.
Saffman
, P. G.
, 1968
, “The Lift on a Small Sphere in a Slow Shear Flow-Corrigendum
,” J. Fluid Mech.
, 31
, pp. 624
–625
.187.
Childress
, S.
, 1964
, “The Slow Motion of a Sphere in a Rotating Viscous Fluid
,” J. Fluid Mech.
, 20
, pp. 305
–314
.188.
Fan
, F. G.
, and Ahmadi
, G.
, 2000
, “Wall Deposition of Small Ellipsoids From Turbulent Air Flows—A Brownian Dynamics Simulation
,” J. Aerosol. Sci.
, 31
(10
), pp. 1205
–1229
.189.
Crowe
, C.
, Chung
, J. N.
, and Troutt
, T. R.
, 1988
, “Particle Mixing in Free Shear Flows
,” Prog. Energy Combust. Sci.
, 14
, pp. 171
–194
.190.
Bohnet
, M.
, Gottschalk
, O.
, and Morweiser
, M.
, 1997
, “Modern Design of Aerocyclones
,” Adv. Powder Technol.
,8
(2
), pp. 137
–161
.191.
Auton
, T. R.
, 1987
, “The Lift Force on a Spherical Body in a Rotational Flow
,” J. Fluid Mech.
, 183
, pp. 199
–218
.192.
Tsuji
, Y.
, Morikawa
, Y.
, and Mizuno
, O.
, 1985
, “Experimental Measurements of the Magnus Force on a Rotating Sphere at Low Reynolds Numbers Bubble in an Axisymmetric Shear Flow
,” ASME J. Fluids Eng.
, 107
, pp. 484
–498
.193.
McLaughlin
, J. B.
, 1991
, “Inertial Migration of a Small Sphere in Linear Shear Flows
,” J. Fluid Mech.
, 224
, pp. 261
–274
.194.
Cherukat
, P.
, McLaughlin
, J. B.
, and Graham
, A. L.
, 1994
, “The Inertial Lift on a Rigid Sphere Translating in a Linear Shear Flow Field
,” Int. J. Multiphase Flow
, 20
, pp. 339
–353
.195.
Sridhar
, G.
, and Katz
, J.
, 1995
, “Drag and Lift Forces on Microscopic Bubbles Entrained by a Vortex
,” Phys. Fluids
, 7
(2
), pp. 389
–399
.196.
Mei
, R.
, and Klausner
, J. F.
, 1994
, “Shear Lift Force on Spherical Bubbles
,” Int. J. Heat Mass Transf.
, 15
(1
), pp. 62
–65
.197.
Magnaudet, J., and Legendre, D., 1998, “Some Aspects of the Lift Force on a Spherical Bubble,” In Fascination of Fluid Dynamics, A. Biesheuvel and G. F. van Heijst, eds., Kluwer Academic, Dordrecht, The Netherlands, pp. 441–461.
198.
Legendre
, D.
, and Magnaudet
, J.
, 1997
, “A Note on the Lift Force on a Spherical Bubble or Drop in a Low Reynolds Number Linear Shear Flow
,” Phys. Fluids
, 9
, pp. 3572
–3574
.199.
Tomiyama, A., Tamai, H., Zun, I., and Hosokawa, S., “Transverse Migration of Single Bubbles in Simple Shear Flows,” Proceedings 2nd Int. Symposium on Two-Phase Flow Modeling and Experimentation, 2, pp. 941–948.
200.
Oesterle
, B.
, and Dihn
, Bui
, 1998
, “Experiments on the Lift of a Spinning Sphere in the Range of Intermediate Reynolds Numbers
,” Exp. Fluids
, 25
, pp. 16
–22
.201.
Rubinow
, S. I.
, and Keller
, J. B.
, 1961
, “The Transverse Force on a Spinning Sphere Moving in a Viscous Fluid
,” J. Fluid Mech.
, 11
, pp. 447
–459
.202.
Dennis
, S. C. R.
, Singh
, S. N.
, and Ingham
, D. B.
, 1980
, “The Steady Flow Due to a Rotating Sphere at Low and Moderate Reynolds Numbers
,” J. Fluid Mech.
, 101
, pp. 257
–279
.203.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 2002
, “Inter-Particle Forces and Lift on a Particle Attached to a Solid Boundary in Suspension Flow
,” Phys. Fluids
, 14
, pp. 49
–60
.204.
Aggarwal, S. K., Chen, G., Yapo, J. B., Grinstein, F. F., and Kailasanath, K., 1992, “Numerical Simulation of Particle Dynamics in Planar Shear Layers,” AIAA Paper No. 92-0107.
205.
Moursali
, E.
, Marie
, J. L.
, and Bataille
, J.
, 1995
, “An Upward Turbulent Bubbly Boundary Layer Along a Vertical Flat Plate
,” Int. J. Multiphase Flow
, 21
, pp. 107
–117
.206.
Climent
, E.
, and Magnaudet
, J.
, 1998
, “Modifications d’une Couche de Melange Verticale Induites par la Presence de Bulles
,” Academie des Sciences Paris
326
(11b
), pp. 627
–634
.207.
Narayanan
, C.
, Lakehal
, D.
, and Yadigaroglu
, G.
, 2002
, “Linear Stability Analysis of Particle-Laden Mixing Layers Using Lagrangian Particle Tracking
,” Powder Technol.
, 125
, pp. 122
–130
.208.
Marie
, J. L.
, Moursali
, E.
, and Tran-Cong
, S.
, 1997
, “Similarity Law and Turbulence Intensity Profiles in a Bubbly Boundary Layer at Low Void Fraction
,” Int. J. Multiphase Flow
, 23
, pp. 227
–247
.209.
Joia
, I. A.
, Ushijima
, T.
, and Perkins
, R. J.
, 1997
, “Numerical Study of Bubble and Particle Motion in Turbulent Boundary Layer Using Proper Orthogonal Decomposition
,” Appl. Sci. Res.
, 57
, pp. 263
–277
.210.
Tran-Cong, S., Marie, J. L., and Perkins, R. J., 2001, “Bubble Migration in a Turbulent Boundary Layer—The Influence of Coherent Structures,” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
211.
Druzhinin
, O. A.
, and Elghobashi
, S. E.
, 2001
, “Direct Numerical Simulation of a Three-Dimensional Spatially Developing Bubble-Laden Mixing Layer with Two-Way Coupling
,” J. Fluid Mech.
, 429
, pp. 23
–61
.212.
Felton
, K.
, and Loth
, E.
, 2002
, “Diffusion of Spherical Bubbles in a Turbulent Boundary Layer
,” Int. J. Multiphase Flow
, 28
, pp. 69
–92
.213.
Bohlin, T., 1960, “Terminal Velocities of Solid Spheres in Cylindrical Enclosures,” Transactions of the Royal Institute of Technology, Stockholm, Report No. 155.
214.
Haberman, W. L., and Sayre, R. M., 1958, “Motion of Rigid and Fluid Spheres in Stationary and Moving Liquids Inside Cylindrical Tubes,” Report No. 1143, David Taylor Model Basin, U.S. Navy, Washington, DC.
215.
Paine
, P. L.
, and Scherr
, P.
, 1975
, “Drag Coefficients for the Movement of Rigid Spheres Through Liquid-Filled Cylindrical Pores
,” Biophys. J.
, 15
, pp. 1087
–1091
.216.
Iwaoka
, M.
, and Ishii
, T.
, 1979
, “Experimental Wall Correction Factors of Single Solid Spheres in Circular Cylinders
,” J. Chem. Eng. Jpn.
, 12
, pp. 239
–242
.217.
Miyamura
, A.
, Iwasaki
, S.
, and Ishii
, T.
, 1981
, “Experimental Wall Correction Factors of Single Solid Spheres in Triangular and Square Cylinders, and Parallel Plates
,” Int. J. Multiphase Flow
, 7
, pp. 41
–46
.218.
Masmoudi
, K.
, Lecoq
, N.
, Anthore
, R.
, Bostel
, F.
, and Feuillebois
, F.
, 2002
, “Accurate Measurement of Hydrodynamic Interactions Between a Particle and Walls
,” Exp. Fluids
, 32
, pp. 55
–65
.219.
Fayon
, A. M.
, and Happel
, J.
, “Effect of a Cylindrical Boundary on a Fixed Rigid Sphere in a Moving Viscous Fluid
,” AIChE J.
, 6
, pp. 55
–58
.220.
Feng
, Z.-G.
, and Michaelides
, E. E.
, 2002
, “Hydrodynamic Force on Spheres in Cylindrical and Prismatic Enclosures
,” Int. J. Multiphase Flow
, 28
, pp. 479
–496
.221.
Poiseuille
, J. L. M.
, 1841
, “Recherches sur le Mouvement du Sang dans les Vein Capillaires
,” Mem. Acad. Roy. Sci.
,7
, pp. 105
–175
.222.
Wang
, H.
, and Skalak
, R.
, 1969
, “Viscous Flow in a Cylindrical Tube Containing a Line of Spherical Particles
,” J. Fluid Mech.
, 38
, pp. 75
–96
.223.
Westborg
, H.
, and Hassager
, O.
, 1989
, “Creeping Motion of Long Bubbles and Drops in Capillary Tubes
,” J. Colloid Interface Sci.
, 133
, pp. 135
–147
.224.
Xu
, Q.
, and Michaelides
, E. E.
, 1996
, “A Numerical Study of the Flow Over Ellipsoidal Objects Inside a Cylindrical Tube
,” Int. J. Numer. Methods Fluids
, 22
, pp. 1075
–1088
.225.
Brenner
, H.
, and Happel
, J.
, 1958
, “Slow Viscous Flow Past a Sphere in a Cylindrical Tube
,” J. Fluid Mech.
, 4
, pp. 195
–213
.226.
Cox
, R. G.
, and Brenner
, H.
, 1967
, “The Slow Motion of a Sphere through a Viscous Fluid Towards a Plane Surface
,” Chem. Eng. Sci.
, 22
, pp. 1753
–1777
.227.
Cox
, R. G.
, and Hsu
, S. K.
, 1977
, “The Lateral Migration of Solid Spheres in a Laminar Flow Near a Plane
,” Int. J. Multiphase Flow
, 3
, pp. 201
–222
.228.
Vasseur
, P.
, and Cox
, R. G.
, 1976
, “The Lateral Migration of Spherical Particles in Two-Dimensional Shear Flow
,” J. Fluid Mech.
, 78
, pp. 385
–413
.229.
Cherukat
, P.
, and McLaughlin
, J. B.
, 1990
, “Wall-Induced Lift on a Sphere
,” Int. J. Multiphase Flow
, 16
(5
), pp. 899
–907
.230.
Segre
, G.
, and Silberberg
, A.
, 1962
, “Behavior of Macroscopic Rigid Spheres in Poiseuille Flow
,” J. Fluid Mech.
, 14
, pp. 115
–157
.231.
Inamuro
, T.
, Maeba
, K.
, and Ogino
, F.
, 2000
, “Flow Between Parallel Walls Containing the Lines of Neutrally Buoyant Circular Cylinders
,” Int. J. Multiphase Flow
, 26
, pp. 1981
–2004
.232.
Mortazavi
, S.
, and Tryggvason
, G.
, 2000
, “A Numerical Study of the Motion of Drops in Poiseuille Flow. Part 1: Lateral Migration of One Drop
,” J. Fluid Mech.
, 411
, pp. 325
–350
.233.
Joseph
, D. D.
, Ocando
, D.
, and Huang
, P. Y.
, 2002, “Slip Velocity and Lift,” J. Fluid Mech., in print.234.
Patankar
, N.
, Ko
, T.
, Choi
, H. G.
, and Joseph
, D. D.
, 2001
, “A Correlation for the Lift-Off of Many Particles in Plane Poiseuille Flows of Newtonian Fluids
,” J. Fluid Mech.
, 445
, pp. 55
–76
.235.
Feuillebois
, F.
, and Lasek
, A.
, 1978
, “On the Rotational Historic Term in Non-Stationary Shear Flow
,” Q. J. Mech. Appl. Math.
, 31
, pp. 435
–443
.236.
Brenner
, H.
, 1961
, “The Slow Motion of a Sphere Through a Viscous Fluid Toward a Plane Surface
,” Chem. Eng. Sci.
, 16
, pp. 242
–251
.237.
Kalio, G. A., 1993, “Random Walk Modeling of Particle Deposition,” Gas-Solid Flows–1993, Stock et al., eds., ASME FED Vol. 166, ASME, New York, pp. 161–167.
238.
Portela
, L. M.
, Cota
, P.
, and Oliemans
, R. V. A.
, 2002
, “Numerical Study of the Near-Wall Behavior of Particles in Turbulent Pipe Flows
,” Powder Technol.
, 125
, pp. 149
–157
.239.
Crowe, C. T., Sommerfeld, M., and Tsuji, Y., 1998, Multiphase Flows with Droplets and Particles, CRC Press, Boca Raton, FL.
240.
Sommerfeld
, M.
, and Huber
, N.
, 1999
, “Experimental Analysis and Modelling of Particle-Wall Collisions
,” Int. J. Multiphase Flow
, 25
, pp. 1457
–1489
.241.
Sommerfeld, M., 2000, “Theoretical and Experimental Modelling of Particulate Flow,” Von Karman Institute for Fluid Dynamics Series, B. Rhode, ed., Von Karman Institute, Saint Genese, Belgium.
242.
Yamamoto
, Y.
, Potthoff
, M.
, Tanaka
, T.
, Kajishima
, T.
, and Tsuji
, Y.
, 2001
, “Large-Eddy Simulation of Turbulent Gas-Particle Flow in a Vertical Channel: Effect of Considering Inter-Particle Collisions
,” J. Fluid Mech.
, 442
, pp. 303
–334
.243.
Michaelides, E. E., ed., 2001, Proceedings of the 4th International Conference on Multiphase Flow, New Orleans, LA.
244.
ICMF-98 1998, “Proceedings of the 3rd International Conference on Multiphase Flow,” Lyon, France.
245.
De Vries, A. W. G., Biesheuvel, A., and Van Wijngaarden, L., 2001, “Notes on the Path and Wake of a Gas Bubble Rising in Pure Water,” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
246.
Lyubimov, D. V., Lyubimova, T. P., Meradji, S., and Roux, B., 2001, “Unsteady Motion and Deformation of a Gaseous Bubble in a Vibrating Liquid in Zero Gravity,” Proceedings of the 4th International Conference on Multiphase Flow, E. E. Michaelides, ed., New Orleans, LA.
247.
Richardson
, J. F.
, and Zaki
, W. N.
, 1954
, “The Fall Velocities of Spheres in Viscous Fluids
,” Trans. Inst. Chem. Eng.
, 32
, pp. 35
–41
.248.
Rowe
, P. N.
, 1961
, “The Drag Coefficient of a Sphere
,” Trans. Inst. Chem. Eng.
, 39
, pp. 175
–181
.249.
Aidi
, M.
, Feuillebois
, F.
, Lasek
, A.
, Anthore
, R.
, Petipas
, C.
, and Auvray
, X.
, 1989
, “Mesure de la Vitese de Sedimentation d’ une Suspension par Absorption de Rayons X
,” Rev. Phys. Appl.
, 24
, pp. 1077
–1084
.250.
Batchelor
, G. K.
, and Wen
, C. S.
, 1983
, “Sedimentation in a Dilute Polydisperse System of Interacting Spheres. Part 2. Numerical Results
,” J. Fluid Mech.
, 137
, pp. 467
–468
.251.
Batchelor
, G. K.
, and Wen
, C. S.
, 1982
, “Sedimentation in a Dilute Polydisperse System of Interacting Spheres—Part II: Numerical Results
,” J. Fluid Mech.
, 124
, pp. 495
–511
.252.
Di Felice
, R.
, 1994
, “The Voidage Function for Fluid-Particle Interaction Systems
,” Int. J. Multiphase Flow
, 20
, pp. 153
–162
.253.
Ladd
, A. J. C.
, 1994
, “Numerical Simulations of Particulate Suspensions via a Discretized Boltzmann Equation—Part I: Theoretical Foundation
,” J. Fluid Mech.
, 271
, pp. 285
–310
.254.
Ladd
, A. J. C.
, 1994
, “Numerical Simulations of Particulate Suspensions via a Discretized Boltzmann Equation—Part II: Numerical Results
,” J. Fluid Mech.
, 271
, pp. 311
–332
.255.
Aidun
, C. K.
, Lu
, Y.-N.
, and Ding
, E.-J.
, 1998
, “Direct Analysis of Particulate Suspensions With Inertia Using the Discrete Boltzmann Equation
,” J. Fluid Mech.
, 373
, pp. 287
–303
.256.
Kaneda
, Y.
, 1986
, “The Drag on a Sparse Random Array of Fixed Spheres in Flow at Small but Finite Reynolds Number
,” J. Fluid Mech.
, 167
, pp. 455
–463
.257.
Koch
, D. L.
, and Ladd
, A. J. C.
, 1997
, “Moderate Reynolds Number Flows Through Periodic and Random Arrays of Aligned Cylinders
,” J. Fluid Mech.
, 349
, pp. 31
–66
.258.
Rojas
, S.
, and Koplik
, J.
, 1998
, “Non-Linear Flow in Porous Media
,” Phys. Rev. E
, E58
(4
), pp. 4776
–4782
.259.
Koch
, D. L.
, and Sangani
, A. S.
, 1999
, “Particle Pressure and Marginal Stability Limits for a Homogeneous Monodisperse Gas Fluidized Bed: Kinetic Theory and Numerical Calculations
,” J. Fluid Mech.
, 400
, pp. 229
–263
.260.
Ergun
, S.
, 1952
, “Fluid Flow Through Packed Columns
,” Chem. Eng. Prog.
, 48
, pp. 93
–98
.261.
Zuber
, N.
, 1964
, “On the Dispersed Two-Phase Flow in the Laminar Flow Regime
,” Chem. Eng. Sci.
, 19
, pp. 897
–917
.262.
van Wijngaarden
, L.
, 1976
, “Hydrodynamic Interaction Between Bubbles in a Liquid
,” J. Fluid Mech.
, 77
, pp. 27
–44
.263.
Spelt, P. D. M., and Sangani, A. S., 1998, “Properties and Averaged Equations for Flows of Bubbly Liquids,” In Fascination of Fluid Dynamics, A. Biesheuvel, and G. F. van Heijst, eds., Kluwer Academic, Dordrecht, pp. 337–386.
264.
Sangani
, A. S.
, and Didwania
, A. K.
, 1993
, “Dynamic Simulation of Flows of Bubbly Liquids at Large Reynolds Numbers
,” J. Fluid Mech.
, 250
, pp. 307
–337
.265.
Duineveld, P. C., 1998, “Bouncing and Coalescence of Bubble Pairs,” In Fascination of Fluid Dynamics, A. Biesheuvel, and G. F. van Heijst, eds., Kluwer Academic, Dordrecht, pp. 409–439.
266.
Neale
, G.
, Epstein
, N.
, and Nader
, W.
, 1973
, “Creeping Flow Relative to Permeable Spheres
,” Chem. Eng. Sci.
, 28
, pp. 1865
–1874
.267.
Beavers
, G. S.
, and Joseph
, D. D.
, 1967
, “Boundary Conditions at a Naturally Permeable Wall
,” J. Fluid Mech.
, 30
, pp. 197
–207
.268.
Saffman
, P. G.
, 1971
, “On the Boundary Condition at the Surface of a Porous Medium
,” Stud. Appl. Math.
, 50
, pp. 93
–101
.269.
Jones
, I. P.
, 1973
, “Low Reynolds Number Flow Past a Porous Spherical Shell
,” Proc. Cambridge Philos. Soc.
, 73
, pp. 231
–238
.270.
Joseph
, D. D.
, and Tao
, L. N.
, 1964
, “The Effect of Permeability on the Slow Motion of a Porous Sphere in a Viscous Liquid
,” ZAMP
, 44
, pp. 361
–364
.Copyright © 2003
by ASME
You do not currently have access to this content.
