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References
Adamson AW (1967) Physical Chemistry of Surfaces. Second Edition, Interscience Publishers (John Wiley & Sons), New York
Ahmadi G (1987) On the mechanics of incompressible multiphase suspensions. Adv Water Res 10:32-43
Ahmadi G, Ma D (1990) A thermodynamical formulation for dispersed multiphase turbulent flows-I Basic theory. Int J Multiphase Flow 16:323-340
Albråten PJ (1982) The Dynamics of Two-Phase Flow. PhD thesis, Chalmers University of Technology, Göteborg, Sweden
Aleinov I, Puckett EG (1995) Computing Surface Tension with High-Order Kernels. In: Oshima, K., (ed) Proceedings of the 6th International Symposium on Computational Fluid Dynamics, pp. 13-18, Lake Tahoe, CA, USA.
Amsden AA, Harlow FH (1970) A Simplified MAC Technique for Incompressible Fluid Flow Calculations. J Comput Phys 6:322-325
Andersson TB, Jackson R (1967) A Fluid Mechanical Descriptin of Fluidized Beds: Equations of Motion. Ind Engng Chem Fundam 6(4):527-539
Ashgriz N, Poo JY (1991) FLAIR: Flux Line-Segment Model for Advection and Interface Reconstruction. J Comput Phys 93:449-468.
Aris R (1962) Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover, Inc., New York
Banerjee S, Chan AMC (1980) Separated Flow Models-I: Analysis of the Averaged and Local Instantaneous Formulations. Int J Multiphase Flow 6:1-24
Banerjee S (1999) Multifield Formulations. In: Modelling and Computation of Multiphase Flows, Short Course, Zurich, Switzerland, March 8-12, 14B:1-49.
Barkhudarov MR, Chin SB (1994) Stability of a numerical algorithm for gas bubble modelling. Int. J Numer Meth Fluids 19:415-437
Batchelor GK (1970) An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge
Beckermann C, Viskanta R (1993) Mathematical modeling of transport phenomena during alloy solidification. Appl Mech Rev 46(1):1-27
Bedford A, Drumheller DS (1983) Theories of Immiscible and Structured Mixtures. Int J Engng Sci 21(8):863-960
Bendiksen K, Malnes D, Moe R and Nuland S (1991) The dynamic two-fluid model OLGA: Theory and applications. SPE Production Engineers, pp. 171-180
Bennon WD, Incropera FP (1987) A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems-II. Application to solidification in a regular cavity. Int J Heat Transfer 30(10): 2171-2187
Bertola F (2003) Modelling of Bubble Columns by Computational Dynamics. PhD thesis, Politecnico Di Torino, Torino
Bertola F, Grundseth J, Hagesaether L, Dorao C, Luo H, Hjarbo KW, Svendsen HF, Vanni M, Baldi G, Jakobsen HA (2005) Numerical Analysis and Experimental Validation of Bubble Size Distribution in Two-Phase Bubble Column Reactors. Multiphase Science & Technology, 17(1-2):123-145
Beux F, Banerjee S (1996) Numerical Simulation of Three-Dimensional Two-Phase Flows by Means of a Level Set Method. ECCOMAS 96 Proceedings, John Wiley.
Biesheuvel A, Wijngaarden L van (1984) Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J Fluid Mech 168:301-318
Bourè JA (1978) Constitutive Equations for Two-Phase Flows. In: Two-phase flows and heat transfer with application to nuclear reactor design problems, chap 9, von Karman Inst. Book, Hemisphere, New York
Bourè JA (1979) On the form of the pressure terms in the momentum and energy equations of the two-phase flow models. Int J Multiphase Flow 5:159-164
Bourè JA, Delhaye JM (1982) General Equations and Two-Phase Flow Modeling. In: Hetsroni G (ed) Handbook of Multiphase Systems, Section 1.2, pp. (1-36) - (1-95), McGraw-Hill, New York
Brackbill JU, Kothe DB, Zemach C (1992) A Continuum Method for Modeling SurfaceTension. J Comput Phys 100:335-354.
Brenner H (1979) A Micromechanical Derivation of the Differential Equation of Interfacial Statics. Journal of Colloid and Interface Science 68 (3):422-439
Buyevich YA (1971) Statistical hydrodynamics of disperse systems. Part 1. Physical background and general equations. J Fluid Mech 49(3):489-507
Buyevich YA, Shchelchkova IN (1978) Flow of Dense Suspensions. Prog Aerospace Sci 18:121-150
Carmo MP do (1976) Differential Geometry of Curves and Surfaces. Pretice-Hall, Inc, Englewood Cliffs
Celik I (1993) Numerical Uncertainty in Fluid Flow Calculations: Needs for Future Research. ASME J Fluids Engineering 115:194-195.
Chandrasekhar S (1981) Hydrodynamic and hydromagnetic stability. Dover Publications, New York.
Chang YC, Hou TY, Merriman B, Osher S (1996) A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows. J Comput Phys 124:449-464.
Chorin AJ (1968) Numerical solution of the Navier-Stokes equations. Math Comput 22:745-762.
Clift R, Grace JR, Weber ME (1978) Bubble, Drops, and Particles. Academic Press, New York
Colebrook CF (1939) Turbulent Flow in Pipes, with particular reference to the Transition Region between Smooth and Rough Pipe Laws. J Inst Civ Eng 12(4):133-156
Crowe CT (1982) Review: numerical models for dilute gas-particle flows. J Fluids Engng 104:297-303
Crowe CT, Troutt TR, Chung JN (1996) Numerical Models for Two-Phase Turbulent Flows. Annu Rev Fluid Mech 28:11-43.
Crowe CT, Sommerfeld M, Tsuji Y (1998) Multiphase Flows with Droplets and Particles. CRC Press, Boca Raton.
Danckwerts PV (1953) Continuous flow systems: Distribution of Residence Times. Chem Eng Sci 2(1):1-18
Danov KD, Gurkov TD, Dimitrova T, Ivanov IB, Smith D (1997) Hydrodynamic Theory for Spontaneously Growing Dimple in Emulsion Films with Surfactant Mass Transfer. Journal of Colloid and Interface Science 188:313-324
Deemer AR, Slattery JC (1978) Balance Equations and Structural Models for Phase Interfaces. Int J Multiphase Flow 4:171-192
Delhaye JM (1974) Jump Conditions and Entropy Sources in Two-Phase Systems: Local Instant Formulation. Int J Multiphase Flow 1:395-409
Delhaye JM, Achard JL (1977) On the averaging operators introduced in two-phase flow. In: Banerjee S, Weaver JR (eds) Transient Two-phase Flow. Proc. CSNI Specialists Meeting, Toronto, 3.-4. august
Delhaye JM (1977) Instantaneous space-averaged equations. In: Kakac S, Mayinger F Two-Phase Flows and Heat Transfer. Vol. 1, pp. 81-90, Hemisphere, Washington, DC
Delhaye JM (1977) Local time-averaged equations. In: Kakac S, Mayinger F Two-Phase Flows and Heat Transfer. Vol. 1, pp. 91-100, Hemisphere, Washington, DC
Delhaye JM (1977) Space/time and Time/space-averaged equations. In: Kakac S, Mayinger F Two-Phase Flows and Heat Transfer. Vol. 1, pp. 101-114, Hemisphere, Washington, DC
Delhaye JM, Achard JL (1978) On the use of averaging operators in two phase flow modeling: Thermal and Aspects of Nuclear Reactor Safty, 1: Light Water Reactors. ASME Winter Meeting
Delhaye JM (1981) Basic Equations for Two-Phase Flow Modeling. In: Bergles AE, Collier JG, Delhaye JM, Hewitt GF, Mayinger F (eds) Two-Phase Flow and Heat Transfer in the Power and Process Industries. Hempsherer Publishing, Washington
Delnoij E, Kuipers JAM, van Swaaij WPM (1997) Computational fluid dynamics applied to gas-liquid contactors. Chem Eng Sci 52 (21/22):3623-3638.
Delnoij E (1999) Fluid Dynamics of Gas-Liquid Bubble Columns. PhD thesis, University of Twente, The Netherlands, Twente.
Dobran F (1983) On the formulation of conservation, balance and constitutive equations for multiphase flows. In: Vezirogly TN (ed) Proc 3rd Multiphase Flow and Heat Transfer Symposium.
Drew DA (1971) Averaged Field Equations for Two-Phase Media. Stud Appl Math 50(2):133-166
Drew DA, Lahey RT Jr (1979) Application of general constitutive principles to the derivation of multidimensional two-phase flow equations. Int J Multiphase Flow. 5:243-264
Drew DA (1983) Mathematical Modeling of Two-Phase Flow. Ann Rev Fluid Mech 15:261-291
Drew DA (1992) Analytical Modeling of Multiphase Flows: Modern Developments and Advances. In: Lahey RT jr (ed) Boiling Heat Transfer, pp. 31-83, Elsevier Science Publishers BV, Amsterdam
Drew DA, Lahey RT Jr (1993) Analytical Modeling of Multiphase Flow. In: Roco MC (ed) Particulate Two-Phase Flow, Chapt. 16, pp. 509-566, Butterworth-Heinemann, Boston
Drew DA, Wallis GB (1994) Fundamentals of Two-Phase Flow Modeling. Multiphase Science and Technology 8:1-67
Drew DA, Passman SL (1999) Theory of Multicomponent Fluids. Springer, New York
Drew DA (2005) Probability and repeatibility: One particle diffusion. Nuclear Engineering and Design 235:1117-1128
Duckworth OW, Cygan RT, Martin ST (2004) Linear Free Energy Relationships between Dissolution Rates and Molecular Modeling Energies of Rhombohedral Carbonates. Langmuir 20:2938-2946
Duducovič MP (1999) Trends in Catalytic Reaction Engineering. Catal Today 48 (1-4):5-15
Edwards CH Jr, Penny DE (1982) Calculus and Analytic Geometry. Prentice-Hall Inc, Englewood Cliffs, New Jersey
Edwards DA, Brenner H, Wasan DT (1991) Interfacial Transport Processes and Rheology. Butterworth-Heinemann, Boston
Elghobashi SE, Abou-Arab TW (1983) A two-equation turbulence model for two-phase flows. Phys Fluids 26(4):931-938.
Elghobashi SE, Truesdell GC (1993) On the two-way interaction between homogeneous turbulence and dispersed solid particles. I: Turbulence modification. Phys Fluids A5 (7):1790-1801
Elghobashi SE (1994) On predicting particle laden turbulent flows. Appl Sci Res 52:309-329.
Enwald H, Peirano E, Almstedt AE (1996) Eulerian Two-Phase Flow Theory Applied to Fluidization. Int J Multiphase Flow 22:21-66, Suppl.
Ervin EA, Tryggvason G (1997) The rise of bubbles in a vertical shear flow. ASME J Fluid Engineering 119:443-449
Esmaeeli A, Ervin E, Tryggvason G (1994) Numerical simulations of rising bubbles. In: Blake JR, Boulton-Stone JM, Thomas NH (eds) Bubble Dynamics and Interfacial Phenomena. Kluwer Academic Publishers, Dordrecht
Esmaeeli A, Tryggvason G (1996) An Inverse Energy Cascade in Two-Dimensional Low Reynolds Number Bubbly Flows. J Fluid Mech 314:315-330
Esmaeeli A, Tryggvason G (2004) Computations of film boiling. Part I: numerical method. Int J Heat Mass Transfer 47:5451-5461
Esmaeeli A, Tryggvason G (2004) A front tracking method for computations of boiling in complex geometries. Int J Multiphase Flow 30:1037-1050
Fan F-S, Tsuchiya K (1990) Bubble Wake Dynamics in Liquids and Solid-Liquid Suspensions. Butterworth-Heinemann, Boston
Fan F-S, Zhu C (1998) Principles of Gas-Solid Flows. Cambridge University Press, Cambridge
Favre A (1965) Equations des gaz turbulents compressibles. J Mechanique 4(3):361-390
Favre A (1969) Statistical Equations of Turbulent Gases. Problems of Hydrodynamics and Continuum Mechanics. SIAM, pp. 231-266, Philadelphia (PA)
Ferziger JH, Kaper HG (1972) Mathematical Theory of Transport Processes in Gases. North-Holland Publishing Company, Amsterdam
Fogler Scott H (2006) Elements of Chemical Reaction Engineering. Fourth Edition, Prentice-Hall International, Inc, New Jersey
Freitas J (1993) Editorial. Transactions of the ASME, Journal of Fluids Engineering. New York: American Society of Mechanical Engineers 115:339-340.
Froment GF, Bischoff KB (1990) Chemical Reactor Analysis and Design. John Wiley and Sons, New York
Ganesan S, Poirier DR (1990) Conservation of Mass and Momentum for the Flow of Interdendritic Liquid during Solidification. Metallurgical Transactions B 21B:173-181
Ganesan V, Brenner H (2000) A diffuse interface model of two-phase flow in porous media. Proc R Soc Lond A 456:731-803
Gibbs JW (1928) The Collected Works of J. Willard Gibbs. Longmans, Green & Co, New York
Gidaspow D (1974) Introduction to Modeling of Two-Phase Flow. Round Table Discussion (RT-1-2). Proc 5th Int Heat Transfer Conf Vol VII, p. 163.
Gidaspow D (1994) Multiphase Flow and Fluidization-Continuum and Kinetic Theory Descriptions. Academic Press, Harcourt Brace & Company, Publishers, Boston
Gosman AD, Pun WM, Runchal AK, Spalding DB, Wolfshtein M (1969) Heat and Mass Transfer in Recirculating Flows. Academic Press, London and New York
Gosman AD, Lekakou C, Polits S, Issa RI, Looney MK (1992) Multidimensional Modeling of Turbulent Two-Phase Flows in Stirred Vessels. AIChE J 38(12):1946-1956
Gotaas C, Havelka P, Roth N, Hase M, Weigand B, Jakobsen HA, Svendsen HF (2004) Influence of viscosity on droplet-droplet collision behaviour: Experimental and numerical results. CHISA 2004, Prague, Czech Republic, August 22-26.
Gray WG (1975) A Derivation of the Equations for Multi-Phase Transport. Chem Eng Sci 30:229-233
Gray WG, Lee PCY (1977) On the Theorems for Local Volume Averaging of Multiphase Systems. Int J Multiphase Flow 3:333-340
Gray WG (1983) Local Volume Averaging of Multiphase Systems Using A Non-Constant Averaging Volume. Int J Multiphase Flow 9(6):755-761
Gueyffier D, Li J, Nadim A, Scardovelli R, Zaleski S (1999) Volume-of Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows. J Comput Phys 152:423-456.
Hagesæther L, Jakobsen HA, Svendsen HF (1999) Theoretical analysis of fluid particle collisions in turbulent flow. Chem Eng Sci 54:4749-4755.
Heinbockel JH (2001) Introduction to Tensor Calculus and Continuum Mechanics. Trafford Publishing, Canada (ISBN 1553691334)
Han J, Tryggvason G (1999) Secondary breakup of axisymmetric liquid drops. I. Acceleration by a constant body force. Physics of Fluids 11(12):3650-3667
Harlow FH, Welch JE (1965) Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. Physics and Fluids 8:2182-2189
Harlow FH, Amsden AA (1975) Numerical Calculation of Multiphase Fluid Flow. J Comput physics 17:19-52
Hassanizadeh M, Gray WG (1979) General conservation equations for multi-phase systems: 1. Averaging procedure. Advances in Water Resources 2:131-144
Hassanizadeh M, Gray WG (1979) General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations. Advances in Water Resources 2:191-203
Hassanizadeh M, Gray WG (1980) General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Advances in Water Resources 3:25-40
Hassanizadeh M, Gray WG (1987) High Velocity Flow in Porous Media. Transport in Porous Media 2:521-531
Hassanizadeh M, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resources 13(4):169-186
Hidy GM, Broch JR (1970) The Dynamics of Aerocolloidal Systems. Pergamon, Oxford
Hinch EJ (1977) An average-equation approach to particle interactions in a fluid suspension. J Fluid Mech 83(4):695-720
Hinze JO (1975) Turbulence. Second Edition, McGraw-Hill, New York
Hirt CW (1968) Heuristic stability theory for finite difference equations. J Comput Phys 2:339-355
Hirt CW, Nichols BD (1980) Adding Limited Compressibility to Incompressible Hydrocodes. J Comput Phys 34:390-300
Hirt CW, Nichols BD (1981) Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J Comput Phys 39:201-225
Holm DD, Kupershmidt BA (1986) A multipressure regulation for multiphase flow. Int J Multiphase Flow, 12(4):681-697
Howes FA, Whitaker S (1985) The Spatial Averaging Theorem Revisited. Chem Eng Sci 40(8): 1387-1392
Hyman JM (1984) Numerical Methods for Tracking Interfaces. Physica 12D:396-407.
Ishii M (1975) Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris
Ishii M, Chawla TC (1979) Local drag laws in dispersed two-phase flows. Argonne National Laboratory Report NUREG/CR-1230, ANL-79-105, Argonne, Illinois, USA
Ishii M, Mishima K (1981) Study of two-fluid model and interfacial area. Argonne National Laboratory Report ANL-80-111, Argonne, Illinois, USA
Ishii M, Mishima K (1984) Two-Fluid Model and hydrodynamic Constitutive Equations. Nuclear Engineering and Design 82:107-126
Ishii M (1990) Two-fluid model for two-phase flow. Multiphase Science and Technology 5 (chap 1) Hemisphere, New York
Issa RI, Oliveira PJ (1995) Numerical Prediction of Turbulent Dispersion in Two-Phase Jet Flows. In: Celata GP, Shah RK (eds) Two-Phase Flow Modelling and Experimentation. pp. 421-428
Ivanov IB (1988) Thin Liquid Films. Fundamentals and Applications. Marcel Dekker Inc, New York and Basel
Jakobsen HA (1993) On the modelling and simulation of bubble column reactors using a two-fluid model. Dr Ing Thesis, Norwegian Institute of Technology, Trondheim, Norway.
Jakobsen HA, Sannæs BH, Grevskott S, Svendsen HF (1997) Modeling of Vertical Bubble Driven Flows. Ind Eng Chem Res 36 (10):4052-4074.
Jakobsen HA (2001) Phase Distribution Phenomena in Two-Phase Bubble Column Reactors. Chem Eng Sci 56(3):1049-1056
Jakobsen HA, Lindborg H, Handeland V (2002) A numerical study of the interactions between viscous flow, transport and kinetics in fixed bed reactors. Computers and Chemical Engineering 26:333-357
Jakobsen HA (2003) Numerical Convection Algorithms and Their Role in Eulerian CFD Reactor Simulations. International Journal of Chemical Reactor Engineering A1:1-15.
Jayatilleke CLV (1969) The Influence of Prandtl Number and Surface Roughness on the Resistance of the Laminar Sublayer to Momentum and Heat Transfer. Prog Heat Mass Transfer 1:193-329
Johansen ST (1990) On the Modelling of Dispersed Two-Phase Flows. Dr Techn thesis, The Norwegian Institute of Technology, Trondheim.
Joseph DD, Lundgren TS, Jackson R, Saville DA (1990) Ensemble Averaged and Mixture Theory Equations for Incompressible Fluid-Particle Suspensions. Int J Multiphase Flow 16 (1):35-42
Juric D, Tryggvason G (1996) A Front-Tracking Method for Dendritic Solidification. J Comput Phys 123:127-148
Juric D, Tryggvason G (1998) Computations of Boiling Flows. Int J Multiphase Flow. 24(3):387-410
Kolev NI (2002) Multiphase Flow Dynamics 1: Fundamentals. Springer, Berlin
Kuipers JAM, van Swaaij WPM (1997) Application of Computational Fluid Dynamics to Chemical Reaction Engineering. Reviews in Chemical Engineering 13 (3):1-118.
Kuo KK (1986) Principles of Combustion. John Wiley & Sons, New York
Lafaurie B, Nardone C, Scardovelli R, Zaleski S, Zanetti G (1994) Modelling Merging and Fragmentation in Multiphase Flows with SURFER. J Comput Phys 113:134-147.
Lafi AY, Reyes JN (1994). General particle transport equations. Final Report OSU-NE-9409. Department of Nuclear Engineering, Oregon State University
Lahey RT Jr, Cheng LY, Drew DA, Flaherty JE (1980) The effect of virtual mass on the numerical stability of accelerating two-phase flows. Int J Multiphase Flow 6:281-294
Lahey RT Jr, Drew DA (1989) The Three-Dimensional Time-and Volume Averaged Conservation Equations of Two -Phase Flow. Advances in Nuclear Science and Technology 20:1-69
Lahey RT Jr (1992) The prediction of phase distribution and separation phenomena using two-fluid models. In: Lahey RT Jr (ed) Boiling Heat Transfer. Elsevier Science Publishers BV
Lahey RT Jr, Drew DA (1992) On the Development of Multidimensional Two-Fluid Models for Vapor/Liquid Two-Phase Flows. Chem Eng Comm 118:125-139
Laplace PS (1806) Traité de Méchanique Céleste. Supplement to book 10, Vol. IV. Paris: Gauthier-Villars, 1806. Annotated English translation by Nathaniel Bowditch (1839). Reprinted by New York: Chelsea Publishing Company, 1996
Laux H (1998) Modeling of dilute and dense dispersed fluid-particle flow. Dr Ing Thesis, Norwegian University of Science and Technology, Trondheim, Norway
Leonard BP, Drummond JE (1995) Why you should not use ’Hybrid’, ’Power-Law’ or related exponential schemes for convective modelling - There are much better alternatives. Int J for Numerical Methods in Fluids 20:421-442.
Li Y, Zhang J, Fan L-S (1999) Numerical simulation of gas-liquid-solid fluidization systems using a combined CFD-VOF-DPM method: bubble wake behavior. Chem Eng Sci 54:5101-5107.
Liljegren LM (1997) Ensemble-Average Equations of a Particulate Mixture. J Fluids Engr 119:428-434
Lopez de Bertodano M (1992) Turbulent Bubbly Two-Phase Flow in a Triangular Duct. PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY
Lyckowski RW, Gidaspow D, Solbrig CW, Hughes ED (1978) Characteristics and stability analysis of transient one-dimensional two-phase flow equation and their finite difference approximations. Nucl Sci Engng 66:378-396
Manninen M, Taivassalo V, Kallio S (1996) On the mixture model for multiphase flow. Technical Research Center of Finland: VIT Publications, Espoo.
Mavrovouniotis GM, Brenner H (1993) A Micromechanical Investigation of Interfacial Transport Processes. I. Interfacial Conservation Equations. Phil Trans R Soc Lond A 345:165-207
Mavrovouniotis GM, Brenner H, Edwards DA, Ting L (1993) A Micromechanical Investigation of Interfacial Transport Processes. II. Interfacial Constitutive Equations. Phil Trans R Soc Lond A 345:209-228
Maxey MR, Riley JJ (1983) Equation of motion for a a mall rigid sphere in a non-uniform flow. Phys Fluids 26 (4):883-889.
Meier M, Andreani M, Smith B, Yadigaroglu G (1998) Numerical and Experimental Study of Large Stream-Air Bubbles Condensing in Water. Proc Third Int Conf Multiphase Flow, Lyon, June 8-12
Miller CA, Neogi P (1985) Interfacial Phenomena: Equilibrium and Dynamic Effects. Marcel Dekker, Inc., New York and Basel
Moeckel GP (1975) Thermodynamics of an Interface. Arch ration Mech Analysis 57:255-280
Mostafa AA, Mongia HC (1987) On the modeling of turbulent evaporating sprays: Eulerian versus Lagrangian approach. Int J Heat Mass Transfer 30 (12):2583-2593.
Ni J, Beckermann C (1990) A Two-Phase Model for Mass, Momentum, Heat, and Species Transport during Solidification. In: Charmchi M, Chyu MK, Joshi Y, Walsh SM (eds) Transport Phenomena in Material Processing, New York. ASME HTD-VOL. 132:45-56
Ni J, Beckermann C (1991) A Volume-Averaged Two-Phase Model for Transport Phenomena during Solidification. Metallurgical Transactions B 22B:349-361
Nichols BD, Hirt CW (1975) Methods for Calculating Multi-Dimensional, Transient, Free Surface Flows Past Bodies. Proceedings First Intern Conf Num Ship Hydrodynamics, Gaithersburg, Md, October.
Nigmatulin RI (1979) Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int J of Multiphase Flow 5:353-385
Nigmatulin RI, Lahey RT Jr, Drew DA (1996) On the Different Forms of Momentum Equations and on the Intra- and Interphase Interaction in the Hydromechanics of a Monodispersed Mixture. Chem Eng Comm 141-142:287-302
Nobari MR, Jan Y-J, Tryggvason G (1996) Head-on collision of droplets - A numerical investigation. Phys Fluids 8(1):29-42
Pan Y, Suga K (2005) Numerical simulation of binary liquid droplet collision. Phys Fluids 17 (8):82105-082105-14
Patankar SV (1980) Numerical heat transfer and fluid flow. Series in Computational Methods in Mechanics and Thermal Sciences, Hemisphere publishing corporation, New York.
Pauchon C, Banerjee S (1986) Interface momentum interaction effects in the averaged multifield model. Part I: Void propagation in bubbly flows. Int J Multiphase Flow 12(4):559-573
Pauchon C, Banerjee S (1988) Interphase momentum interaction effects in the averaged multifield model, P art II: Kinematic waves and interfacial drag in bubbly flows. Int J Multiphase Flow 14(3):253-264
Poirier DR, Nandapurkar PJ, Ganesan S (1991) The Energy and Solute Conservation Equation for Dendritic Solidification. Metallurgical Transactions B 22B: 889-900
Popinet S, Zaleski S (1999) A Front-Tracking Algorithm for Accurate Representation of Surface Tension. Int J Numer Meth Fluids 30:775-793.
Prescott PJ, Incropera FP (1994) Convective Transport Phenomena and Macrosegregation During Solidification of a Binary Metal Alloy: I-Numerical Predictions. J Heat Transfer 116:735-749
Probstain RF (1994) Physicochemical Hydrodynamics: An Introduction, Second edition, John Wiley & Sons, Inc, New York
Prosperetti A, van Wijngaarden L (1976) On the characteristics of the equation of motion for bubbly flow and the related problem of critical flow. Journal of Engineering Math 10(2):153-162
Prosperetti A, Jones AV (1984) Pressure forces in dispersed two-phase flow. Int Journal of Multiphase Flow 10(4):425-440
Prosperetti A, Zhang DZ (1996) Disperse Phase Stress in Two-Phase Flow. Chem Eng Comm 141-142:387-398
Qian J, Tryggvason G, Law CK (1998) A Front Tracking Method for the Motion of Premixed Flames. J Comput Phys 144:52-69
Quintard M, Whitaker S (1993) Transport in Ordered and Disordered Porous Media: Volume-Averaged Equations, Closure Problems, and Comparisons with Experiments. Chem Eng Sci 48(14):2537-2564
Ramshaw JD, Trapp JA (1978) Characteristics, stability and short wavelength phenomena in two-phase flow equation systems. Nucl Sci Engng 66:93-102
Ransom VH, Ramshaw JD (1988) Discrete Modeling Considerations in Multiphase Fluid Dynamics. Japan - U.S. Seminar on Two-Phase Flow Dynamics, Kyoto, Japan, 15. July.
Raupach MR, Shaw RH (1982) Averaging Procedures for Flow within Vegetation Canopies. Boundary Layer Meteorology 22:79-90
Reeks MW (1991) On a kinetic equation for the transport of particles in turbulent flows. Physics of Fluids A 3:446-456
Reyes Jr JN (1989) Statistically derived conservation equations for fluid particle flows. Proc ANS Winter Meeting. Nuclear Thermal Hydraulics, 5th Winter Meeting
Reynolds O (1895) On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos Trans Roy Soc London A186:123-164
Richtmyer RD, Morton KW (1957) Difference Methods for Initial-Value Problems. Second Edition, Interscience Publishers (John Wiley & Sons), New York
Rider WJ, Kothe DB (1995) Stretching and Tearing Interface Tracking Methods. AIAA paper 95-1717, pp. 806-816.
Rider WJ, Kothe DB (1998) Reconstructing Volume Tracking. J Comput Phys 141:112-152.
Roberts IF (1997) Conservation Equations, Two-Phase. Int Encyclopedia of Heat and Mass Transfer, pp. 223-230
Sanyal J, Vásquez S, Roy S, Dudukovic MP (1999) Numerical simulation of gas-liquid dynamics in cylindrical bubble column reactors. Chem Eng Sci 54:5071-5083
Scardovelli R, Zaleski S (1999) Direct Numerical Simulation of Free-Surface and Interfacial Flow. Annu Rev Fluid Mech 31:567-603
Schwartz MP, Turner WJ (1988) Applicability of the standard k-ε turbulence model to gas-stirred baths. Appl Math Modelling 12:273-279
Scriven LE (1960) Dynamics of a fluid interface. Chem Eng Sci 12:98-108
Sethian JA (1996) Level Set Methods. Cambridge University Press, Cambridge.
Sha WT, Soo SL (1978) Multidomain Multiphase Fluid Mechanics. Int J Heat Mass Transfer 21:1581-1595
Sha WT, Soo SL (1979) Brief Communication: On the effect of pΔα term in multiphase mechanics. Int J Multiphase Flow 5:153-158
Sha WT, Slattery JC (1980) Local Volume-Time Averaged Equations of Motion for Dispersed, Turbulent, Multiphase Flows. NUREG/CR-1491, ANL-80-51
Sha WT, Chao BT, Soo SL (1983) Averaging Procedures of Multiphase Conservation Equations. Transactions of the American Nuclear Society 45:814-816
Sha WT, Chao BT, Soo SL (1983) Time Averaging of Volume-Averaged Conservation Equations of Multiphase Flow. AIChE Symposium Series 79(225):420-426
Sha WT, Chao BT, Soo SL (1984) Porous-Media Formulation for Multiphase Flow with Heat Transfer. Neclear Engineering and Design 82:93-106
Shyy W, Thakur S, Ouyang H, Liu J, Blosch E (1997) Computational Techniques for Complex Transport Phenomena. Cambridge University Press, Cambridge
Slattery JC (1967) Flow of Viscoelastic Fluids Through Porous Media. AIChE J 13(6):1066-1071
Slattery JC (1969) Single-Phase Flow through Porous Media. AIChE J 15(6):866-872
Slattery JC (1972) Momentum, Energy, and Mass Transfer in Continua. Second Edition, McGraw-Hill Kogakusha, LTD, Tokyo
Slattery JC (1980) Invited Review: Interfacial Transport Phenomena. Chem Eng Sci 4:149-166
Slattery JC, Flumerfelt RW (1982) Interfacial Phenomena. In: Hetsroni G (ed) Handbook of Multiphase Systems, Section 1.4, pp. (1-224) - (1-2246), McGraw-Hill, New York
Slattery JC (1990) Interfacial Transport Phenomena. Springer-Verlag, New York
Slattery JC (1999) Advanced Transport Phenomena. Cambridge University Press, New York
Sokolichin A, Eigenberger G, Lapin A, Lübbert A (1997) Dynamic numerical simulations of gas-liquid two-phase flows. Euler/Euler versus Euler/Lagrange. Chem Eng Sci 52 (4):611-626.
Sokolichin A, Eigenberger G (1999) Applicability of the standard ĸ-ε turbulence model to the dynamic simulation of bubble columns: Part I. Detailed numerical simulations. Chem Eng Sci 54:2273-2284.
Sommerfeld M (2001) Validation of a stochastic Lagrangian modelling approach for inter-particle collisions in homogeneous isotropic turbulence. Int J Multiphase Flow 27:1829-1858
Soo SL (1967) Fluid Dynamics of Multiphase Systems. Blaisdell Publishing Company, Waltham, Massachusetts
Soo SL (1989) Particles and Continuum: Multiphase Fluid Dynamics. Hemisphere Publishing Corporation, New York
Soo SL (1990) Multiphase Fluid Dynamics. Science Press, Beijing and Gower Technical, Aldershot
Spalding DB (1977) The Calculation of Free-Convection Phenomena in Gas-Liquid Mixtures. ICHMT Seminar 1976. Published in: Turbulent Buoyant Convection, Hemisphere, Washington, pp. 569-586.
Spalding DB (1980) Numerical Computation of Multi-Phase Fluid Flow and Heat Transfer. In: Taylor C. et al. (eds) Recent Advances in Numerical Methods in Fluids. Pineridge Press, pp. 139-167.
Spalding DB (1980) Mathematical Methods in Nuclear Reactor Thermal Hydraulics. In: Lahey RT (ed) Proceedings of ANS Meeting on Nuclear Reactor Thermal Hydraulics, Saratoga, N. Y., pp. 1979-2023.
Spalding DB (1981) IPSA 1981; New Developments and Computed Results. HTS/81/2, Imperial College of Science and Technology, London, 1981.
Spalding DB (1985) Computer Simulation of Two-Phase Flows, with Special Reference to Nuclear-Reactor Systems. In: Lewis RW, Morgan K, Johanson JA, Smith WR (eds) Computational Techniques in heat Transfer, pp. 1-44.
Stewart HB, Wendroff B (1984) Two-Phase Flow: Models and Methods. J Comput Phys 56:363-409
Stuhmiller JH (1977) The Influence of interfacial pressure forces on the character of two-phase flow model equations. Int J Multiphase Flow 3:551-560
Sussman M, Smereka P, Osher S (1994) A Level-Set Approach for Computing Solutions to Incompressible Two-Phase Flow. J Comput Phys 114:146-159.
Sussman M, Smereka P (1997) Axisymmetric free boundary problems. J Fluid Mech 341:269-294.
Tayebi D, Svendsen HF, Jakobsen HA, Grislingås A (2001) Measurement Techniques and Data Interpretations for Validating CFD Multiphase Reactor Models. Chem Eng Comm 186:57-159
Thomas GB Jr, Finney RL (1996) Calculus and Analytic Geometry. Addison-Wesley Publishing Company, 9th Edition, Reading, Massachusetts
Tomiyama A, Miyoshi K, Tamai H, Zun I, Sakaguchi T (1998) A bubble tracking method for the prediction of spatial evolution of bubble flow in a vertical pipe. Third International Conference on Multiphase Flow, Lyon, France
Trapp JA (1986) The mean flow character of two-phase flow equations. Int J Multiphase Flow 12(2):263-276
Travis JR, Harlow FH, Amsden AA (1976) Numerical calculations of two-phase flows. Nucl Sci Engng 61:1-10
Tryggvason G, Bunner B, Esmaeeli A, Mortazavi S (1998) Direct numerical simulations of dispersed flows. Third International Conference on Multiphase Flow, ICMF’98, Lyon, France, June 8-12.
Tryggvason G (1999) Embedded Interface Methods Applications. In: Modelling and Computation of Multiphase Flows, Short Course, Zurich, Switzerland, March 8-12, 16B:1-27.
Tryggvason G (1999) Embedded Interface Methods Applications. In: Modelling and Computation of Multiphase Flows, Short Course, Zurich, Switzerland, March 8-12, 18B:1-24.
Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, Tauber W, Han J, Nas S, Jan Y-J (2001) A Front-Tracking Method for the Computations of Multiphase Flow. J Comput Phys 169:708-759
Tryggvason G, Esmaeeli A, Al-Rawahi N (2005) Direct numerical simulations of flows with phase change. Computers and Structures 83:445-453
Ubbink O, Issa I (1999) A method for capturing sharp fluid interfaces on arbitrary meshes. J Comput Phys 153:26-50
Unverdi SO, Tryggvason G (1992) A Front-Tracking Method for Viscous, Incompressible, Multi-Fluid Flows. J Comput Phys 100:25-37.
Unverdi SO, Tryggvason G (1992) Computations of multi-fluid flows. Physica D60:70-83, North-Holland
Vernier P, Delhaye JM (1968) General two-phase flow equations applied to the thermo-hydrodynamics of boiling water nuclear reactors. Energie Primare 4(1-2)
Voller VR, Brent AD, Prakash C (1989) The modelling of heat, mass and solute transport in solidification systems. Int J Heat Transfer 32(9):1719-1731
Wallis GB (1969) One-dimensional Two-phase Flow. McGraw-Hill Book Company, New York
Weatherburn CE (1927) Differential Geometry of Three Dimensions. Cambridge University Press, Cambridge
Whitaker S (1967) Diffusion and Dispersion in Porous Media. AIChE J 13(3):420-427
Whitaker S (1968) Introduction to Fluid Mechanics. Prentice-Hall, Inc., Englewood Cliffs
Whitaker S (1969) Fluid Motion in Porous Media. Industrial and Engineering Chemistry. 61(12): 15-28
Whitaker S (1973) The Transport Equations for Multiphase Systems. Chem Eng Sci 28:139-147
Whitaker S (1985) A Simple Geometrical Derivation of the Spatial Averaging Theorem. Chemical Engineering Education, pp. 18-21 and pp. 50-52
Whitaker S (1992) The species mass jump condition at a singular surface. Chem Eng Sci 47(7):1677-1685
Whitaker S (1999) The Method of Volume Averaging. Kluwer Academic Publishers, Dordrecht
White FM (1974) Viscous Fluid Flow. McGraw-Hill, New York
Willmore TJ (1961) An Introduction to Differential Geometry. Oxford University Press, Glasgow
Wright K, Cygan RT, Slater B (2001) Structure of the (1014) surfaces of calcite, dolmolite and magnesite under wet and dry conditions. Phys Chem Chem Phys 3:839-844
Yadigaroglu G, Lahey RT Jr (1976) On the Various Forms of the Conservation Equations in Two-Phase Flow. Int J Multiphase Flow 2:477-494
Young T (1805) An essay on the cohesion of fluids. Phil Trans Roy Soc London 95:65-87
Zaleski S (1999) Multiphase-Flow CFD with Volume of Fluid (VOF) Methods. In: Modelling and Computation of Multiphase Flows, Short Course, Zurich, Switzerland, March 8-12, 15B/17B:1-43.
Zapryanov Z, Tabakova S (1999) Dynamics of Bubbles, Drops and Rigid Particles. Kluwer Academic Publishers, Dordrecht
Zhang DZ, Prosperetti A (1994) Ensemble phase-average equations for bubbly flows. Phys Fluids 6(9):2956-2970
Zhang DZ, Prosperetti A (1994) Averaged equations for inviscid disperse two-phase flow. J Fluid Mech 267:185-219
Zhang DZ, Prosperetti A (1997) Momentum and Energy Equations for Disperse Two-Phase Flows and Their Closure for Dilute Suspensions. Int J Multiphase Flow 23(3):425-453
Zuber N, Findlay JA (1965) Average Volumetric Concentration i Two-Phase Flow Systems. J Heat Transfer 87:453-468
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Jakobsen, H.A. (2009). Multiphase Flow. In: Chemical Reactor Modeling. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68622-4_3
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