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Chemical Reactor Modeling pp 369–536Cite as

Multiphase Flow

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Abstract

In this chapter the pertinent multiphase modeling concepts established in fluid mechanics are examined.

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Notes

  1. 1.

    For non-spherical rigid particles the particle rotation may become important. This requires that an angular momentum equation for each particle has to be solved [42].

  2. 2.

    It is noticed that after some re-evaluation Tryggvason and co-workers [236, 242] classified their front tracking method [241] as an embedded interface method, since it is best described as a hybrid between a front tracking and a front capturing method.

  3. 3.

    The work of Brenner [28] also contains a micro-mechanical derivation of the differential equation of interface statics that clearly distinguish between micro-scale and macro-scale viewpoints. From thermodynamic analysis it is concluded that the surface tension manifests itself in the normal direction as a force that drives surfaces towards a minimum energy state characterized by a configuration of minimum surface area.

  4. 4.

    If we define \({\mathbf {n}}_I\) positive out of the curvature instead, the curvature itself must be defined in a consistent manner with sign defining its orientation. Note that several variations of sign conventions may be chosen. The choice of conventions is to a large extent a matter of convenience.

  5. 5.

    In the case of two immiscible fluids, a characteristic phase indicator function, \(X_I\), may be defined that is equal to 1 in one of the phases and 0 in the other phase. Then \(X_I\) and \({\mathbf {n}}_I\) are related by \({\mathbf {n}}_I \delta _I = \nabla X_I\) analogue to the relations used in standard volume averaging procedures [58, 177]. An averaged representation of this relation may be given as \(\frac{1}{\varDelta V} {\underset{\delta A_I}{\int }} {\mathbf {n}}_1 \delta _I da' = - \nabla \tilde{\alpha }_1\)

  6. 6.

    These droplet-droplet collisions are simulated by use of an in-house VOF code called FS3D developed at University of Stuttgart and Institut für Thermodynamik der Luft-und Raumfahrt, ITLR.

  7. 7.

    The reformulation is based on the relationship: \(\nabla H_\epsilon (\varphi ) = \frac{\partial H_\epsilon }{\partial x}+\frac{\partial H_\epsilon }{\partial y}+\frac{\partial H_\epsilon }{\partial z}=\frac{\partial H_\epsilon }{\partial \varphi }(\frac{\partial \varphi }{\partial x}+\frac{\partial \varphi }{\partial y}+\frac{\partial \varphi }{\partial z}) = \frac{\partial H_\epsilon }{\partial \varphi } \nabla \varphi \).

  8. 8.

    The 2D dividing surface model was originally proposed by Gibbs [89] (p. 219).

  9. 9.

    The notation used in the generic equation is strictly only valid for scalar properties. In the particular case when a vector property is considered the tensor order of the corresponding variables is understood to be adjusted accordingly. Hence, the quantities \(\psi _k\), \(\phi _k\) and \(\phi _I\) may be vectors or scalars, while \(\mathbf {J}_k\) and \(\varvec{\varphi }_I\) may be vectors, or second order tensors.

  10. 10.

    This frame is named the Frenet frame after Jean-Frédéric Frenet (1816–1900).

  11. 11.

    Note that other sign conventions exist as well.

  12. 12.

    The surface tension parameter (a scalar) has the same role as the pressure term in the three dimensional case (e.g., [46], p. 398; [213], p. 163).

  13. 13.

    It is noted that the requirement of proper separation of scales represents the main drawback of the volume averaging method. The constitutive equations used generally depend strongly on this assumption which is hardly ever fulfilled performing simulations of laboratory, pilot and industrial scale reactor units.

  14. 14.

    It is noted that the requirement of proper separation of scales represents the main drawback of the time averaging method. The constitutive equations used generally depend strongly on this assumption which is hardly ever fulfilled performing simulations of turbulent reactive flows.

  15. 15.

    For some turbulent flows, the boundary conditions and initial conditions cannot be controlled sufficiently to allow repeatable experiments. In this case, although turbulent flows are not really deterministic, a useful conceptualization of the ensemble average assumes that the flow is deterministic but that randomness may arise through the uncertainty in the initial and boundary conditions [62].

    Another possible conceptualization of the ensemble average imagines that the process is affected by small random forces through the motion. Particulate flows can then be described by distributions of positions, velocities and sizes adopting the basic principles of kinetic theory [91, 182, 262–264]. This alternative ensemble averaging approach is examined in relation to granular flows in Chap. 4.

  16. 16.

    It is noted that the original Reynolds axioms are not applicable to discontinuous functions as normally occur across the interfaces in multiphase flow. As a remedy, Drew [58] extended these functions making them continuous by use of the generalized function concept connecting the functions of the continuous phases on each side of the interface across the interface. Hence the discontinuous functions are modified to be continuous but locally very steep functions across the interface. Formally the averaging axioms can then be extended to include the interfaces, giving rise to the modified formulations of the axioms.

  17. 17.

    For comparison we note that if we reverse the order in which we apply the averaging operators to the generalized quantity \(\psi \), the deviation \(\widehat{\overline{\psi }}_k\) between the un-smoothed local time averaged \(\overline{\psi }_k\) and the time- and volume averaged property value \(\langle \overline{\psi }_k \rangle _V\) is defined by:

    $$\begin{aligned} \widehat{\overline{\psi }}_k = \overline{\psi }_k - \langle \overline{\psi }_k \rangle _V \end{aligned}$$
    (3.326)
  18. 18.

    To relate the classical mixture theory to the more familiar volume averaging method we may assume that the mixture CV, which is larger than a phase element but smaller than the characteristic domain dimension, coincides with the averaging volume used in the volume averaging approach.

  19. 19.

    Area averaging can be considered to be a limiting case of local volume averaging [47, 51, 202]. Thus the phrase limiting form refers to the modified forms of the averaging theorems which are applicable to the governing 3D equations to derive a set of equations valid for 1D problems.

  20. 20.

    Whitaker [249] (Chap. 8) explains the convention normally used to distinguish between these two types of parameters. The friction factors for dispersed bodies immersed in a flowing fluid is traditionally referred to as dimensionless drag coefficients, whereas the drag force for flow inside closed conducts is generally expressed in terms of a dimensionless friction factor.

References

  1. Adamson AW (1967) Physical chemistry of surfaces. 2nd edn. Interscience Publishers/Wiley, New York

    Google Scholar 

  2. Ahmadi G (1987) On the mechanics of incompressible multiphase suspensions. Adv Water Res 10:32–43

    Google Scholar 

  3. Ahmadi G, Ma D (1990) A thermodynamical formulation for dispersed multiphase turbulent flows-I Basic theory. Int J Multiph Flow 16:323–340

    MATH  Google Scholar 

  4. Albråten PJ (1982) The dynamics of two-phase flow. PhD thesis, Chalmers University of Technology, Göteborg, Sweden

    Google Scholar 

  5. 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. Lake Tahoe, USA, pp 13–18

    Google Scholar 

  6. Amsden AA, Harlow FH (1970) A simplified MAC technique for incompressible fluid flow calculations. J Comput Phys 6:322–325

    MATH  Google Scholar 

  7. Andersson TB, Jackson R (1967) A fluid mechanical descriptin of fluidized beds: equations of motion. Ind Engng Chem Fundam 6(4):527–539

    Google Scholar 

  8. Andersson DM, McFadden GB (1997) A diffuse-interface description of internal waves in a near-critical fluid. Phys Fluids 9(7):1870–1879

    MathSciNet  Google Scholar 

  9. Andersson DM, McFadden GB, Wheeler AA (1998) Diffuse-interface methods in fluid mechanics. Annu Rev Fluid Mech 30:139–165

    Google Scholar 

  10. Ashgriz N, Poo JY (1991) FLAIR: flux line-segment model for advection and interface reconstruction. J Comput Phys 93:449–468

    MATH  Google Scholar 

  11. Aris R (1962) Vectors, Tensors, and the basic equations of fluid mechanics. Dover Inc, New York

    MATH  Google Scholar 

  12. Banerjee S, Chan AMC (1980) Separated flow models-I: analysis of the averaged and local instantaneous formulations. Int J Multiph Flow 6:1–24

    MATH  Google Scholar 

  13. Banerjee S (1999) Multifield formulations. In: Modelling and computation of multiphase flows, short course, Zurich, Switzerland, March 8–12, 14B:1–49

    Google Scholar 

  14. Barkhudarov MR, Chin SB (1994) Stability of a numerical algorithm for gas bubble modelling. Int J Numer Meth Fluids 19:415–437

    MATH  MathSciNet  Google Scholar 

  15. Batchelor GK (1970) An introduction to fluid dynamics. Cambridge University Press, Cambridge

    Google Scholar 

  16. Beckermann C, Viskanta R (1993) Mathematical modeling of transport phenomena during alloy solidification. Appl Mech Rev 46(1):1–27

    MathSciNet  Google Scholar 

  17. Bedford A, Drumheller DS (1983) Theories of immiscible and structured mixtures. Int J Engng Sci 21(8):863–960

    MATH  MathSciNet  Google Scholar 

  18. Bendiksen K, Malnes D, Moe R, Nuland S (1991) The dynamic two-fluid model OLGA: theory and applications. SPE production engineers, pp 171–180

    Google Scholar 

  19. 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

    Google Scholar 

  20. Bertola F (2003) Modelling of bubble columns by computational dynamics. PhD thesis, Politecnico Di Torino, Torino

    Google Scholar 

  21. 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. Multiph Sci Technol 17(1–2):123–145

    Google Scholar 

  22. Beux F, Banerjee S (1996) Numerical simulation of three-dimensional two-phase flows by means of a level set method. ECCOMAS 96 proceedings, Wiley

    Google Scholar 

  23. Biesheuvel A, van Wijngaarden L (1984) Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J Fluid Mech 168:301–318

    Google Scholar 

  24. Bouré 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

    Google Scholar 

  25. Bouré JA (1979) On the form of the pressure terms in the momentum and energy equations of the two-phase flow models. Int J Multiph Flow 5:159–164

    MATH  Google Scholar 

  26. Bouré JA, Delhaye JM (1982) General equations and two-phase flow modeling. In: Hetsroni G (ed) Handbook of multiphase systems, Sect. 1.2, McGraw-Hill, New York, pp 1–36 to 1–95

    Google Scholar 

  27. Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surfacetension. J Comput Phys 100:335–354

    MATH  MathSciNet  Google Scholar 

  28. Brenner H (1979) A micromechanical derivation of the differential equation of interfacial statics. J Colloid Interface Sci 68(3):422–439

    Google Scholar 

  29. Buyevich YA (1971) Statistical hydrodynamics of disperse systems. Part 1. physical background and general equations. J Fluid Mech 49(3):489–507

    MATH  Google Scholar 

  30. Buyevich YA, Shchelchkova IN (1978) Flow of dense suspensions. Prog Aerosp Sci 18:121–150

    Google Scholar 

  31. Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28(2):258–267

    Google Scholar 

  32. Carmo MP do (1976) Differential geometry of curves and surfaces. Pretice-Hall Inc, Englewood Cliffs

    Google Scholar 

  33. Celik I (1993) Numerical uncertainty in fluid flow calculations: needs for future research. ASME J Fluids Eng 115:194–195

    Google Scholar 

  34. Chandrasekhar S (1981) Hydrodynamic and hydromagnetic stability. Dover Publications, New York

    Google Scholar 

  35. 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

    MATH  MathSciNet  Google Scholar 

  36. Chen C (2005) Rarefied gas dynamics: fundamentals, simulations and micro flows. Heat and mass transfer series. In: Mewes D, Mayinger F (eds), Springer, Berlin

    Google Scholar 

  37. Chorin AJ (1968) Numerical solution of the Navier-Stokes equations. Math Comput 22:745–762

    MATH  MathSciNet  Google Scholar 

  38. Clift R, Grace JR, Weber ME (1978) Bubble, drops, and particles. Academic Press, New York

    Google Scholar 

  39. 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

    Google Scholar 

  40. Crowe CT (1982) Review: numerical models for dilute gas-particle flows. J Fluids Engng 104:297–303

    Google Scholar 

  41. Crowe CT, Troutt TR, Chung JN (1996) Numerical models for two-phase turbulent flows. Annu Rev Fluid Mech 28:11–43

    MathSciNet  Google Scholar 

  42. Crowe CT, Sommerfeld M, Tsuji Y (1998) Multiphase flows with droplets and particles. CRC Press, Boca Raton

    Google Scholar 

  43. Danckwerts PV (1953) Continuous flow systems: distribution of residence times. Chem Eng Sci 2(1):1–18

    Google Scholar 

  44. Danov KD, Gurkov TD, Dimitrova T, Ivanov IB, Smith D (1997) Hydrodynamic theory for spontaneously growing dimple in emulsion films with surfactant mass transfer. J Colloid Interface Sci 188:313–324

    Google Scholar 

  45. Deemer AR, Slattery JC (1978) Balance equations and structural models for phase interfaces. Int J Multiph Flow 4:171–192

    MATH  Google Scholar 

  46. Delhaye JM (1974) Jump conditions and entropy sources in two-phase systems: local instant formulation. Int J Multiph Flow 1:395–409

    MATH  Google Scholar 

  47. Delhaye JM, Achard JL (1977) On the averaging operators introduced in two-phase flow. In: Banerjee S, Weaver JR (eds) Transient two-phase flow. Proceedings CSNI specialists meeting, Toronto, 3–4 Aug

    Google Scholar 

  48. Delhaye JM (1977) Instantaneous space-averaged equations. In: Kakac S, Mayinger F (eds) Two-phase flows and heat transfer, vol 1, Hemisphere, Washington, pp 81–90

    Google Scholar 

  49. Delhaye JM (1977) Local time-averaged equations. In: Kakac S, Mayinger F (eds) Two-phase flows and heat transfer, vol 1. Hemisphere, Washington, pp 91–100

    Google Scholar 

  50. Delhaye JM (1977) Space/time and Time/space-averaged equations. In: Kakac S, Mayinger F (eds) Two-phase flows and heat transfer, vol 1. Hemisphere , Washington, pp 101–114

    Google Scholar 

  51. 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

    Google Scholar 

  52. 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

    Google Scholar 

  53. Delnoij E, Kuipers JAM, van Swaaij WPM (1997) Computational fluid dynamics applied to gas-liquid contactors. Chem Eng Sci 52(21/22):3623–3638

    Google Scholar 

  54. Delnoij E (1999) Fluid dynamics of gas-liquid bubble columns. PhD thesis, University of Twente, The Netherlands, Twente

    Google Scholar 

  55. Dobran F (1983) On the formulation of conservation, balance and constitutive equations for multiphase flows. In: Vezirogly TN (ed) Proceedings of condensed Papers 3rd multi-phase flow and heat transfer symposium-workshop, University of Miami, Coral Gables, 18–20 April

    Google Scholar 

  56. Drew DA (1971) Averaged field equations for two-phase media. Stud Appl Math 50(2):133–166

    MATH  Google Scholar 

  57. Drew DA, Lahey RT Jr (1979) Application of general constitutive principles to the derivation of multidimensional two-phase flow equations. Int J Multiph Flow 5:243–264

    MATH  Google Scholar 

  58. Drew DA (1983) Mathematical modeling of two-phase flow. Ann Rev Fluid Mech 15:261–291

    Google Scholar 

  59. Drew DA (1992) Analytical modeling of multiphase flows: modern developments and advances. In: Lahey RT Jr (ed) Boiling heat transfer, Elsevier Science Publishers BV, Amsterdam, pp 31–83

    Google Scholar 

  60. Drew DA, Lahey RT Jr (1993) Analytical modeling of multiphase Flow. In: Roco MC (ed) Particulate two-phase flow, Chap. 16, Butterworth-Heinemann, Boston, pp 509–566

    Google Scholar 

  61. Drew DA, Wallis GB (1994) Fundamentals of two-phase flow modeling. Multiph Sci Technol 8:1–67

    Google Scholar 

  62. Drew DA, Passman SL (1999) Theory of multicomponent fluids. Springer, New York

    Google Scholar 

  63. Drew DA (2005) Probability and repeatibility: one particle diffusion. Nucl Eng Des 235:1117–1128

    Google Scholar 

  64. 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

    Google Scholar 

  65. Duducovič MP (1999) Trends in catalytic reaction engineering. Catal Today 48(1–4):5–15

    Google Scholar 

  66. Dupuy PM (2010) Droplet deposition in high-pressure natural-gas streams. PhD thesis, the Norwegian University of Science and Technology (NTNU), Trondheim, Norway

    Google Scholar 

  67. Edwards CH Jr, Penny DE (1982) Calculus and analytic geometry. Prentice-Hall Inc, Englewood Cliffs, New Jersey

    Google Scholar 

  68. Edwards DA, Brenner H, Wasan DT (1991) Interfacial transport processes and rheology. Butterworth-Heinemann, Boston

    Google Scholar 

  69. Elghobashi SE, Abou-Arab TW (1983) A two-equation turbulence model for two-phase flows. Phys Fluids 26(4):931–938

    MATH  Google Scholar 

  70. Elghobashi SE (1994) On predicting particle laden turbulent flows. Appl Sci Res 52:309–329

    Google Scholar 

  71. Enwald H, Peirano E, Almstedt AE (1996) Eulerian two-phase flow theory applied to fluidization. Int J Multiph Flow 22:21–66

    Google Scholar 

  72. Ervin EA, Tryggvason G (1997) The rise of bubbles in a vertical shear flow. ASME J Fluid Eng 119:443–449

    Google Scholar 

  73. 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

    Google Scholar 

  74. Esmaeeli A, Tryggvason G (1996) An inverse energy cascade in two-dimensional low reynolds number bubbly flows. J Fluid Mech 314:315–330

    MATH  Google Scholar 

  75. Esmaeeli A, Tryggvason G (2004) Computations of film boiling. Part I: numerical method. Int J Heat Mass Transf 47:5451–5461

    MATH  Google Scholar 

  76. Esmaeeli A, Tryggvason G (2004) A front tracking method for computations of boiling in complex geometries. Int J Multiph Flow 30:1037–1050

    MATH  Google Scholar 

  77. Fan F-S, Tsuchiya K (1990) Bubble wake dynamics in liquids and solid-liquid suspensions. Butterworth-Heinemann, Boston

    Google Scholar 

  78. Fan F-S, Zhu C (1998) Principles of gas-solid flows. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  79. Favre A (1965) Equations des gaz turbulents compressibles. J Mechanique 4(3):361–390

    Google Scholar 

  80. Favre A (1969) Statistical equations of turbulent gases. Problems of hydrodynamics and continuum mechanics. SIAM, Philadelphia, pp 231–266

    Google Scholar 

  81. Fix GJ (1983) Phase-field models for free boundary problems. In: Fasano F, Primicerio A (eds) Free boundary problems: theory and applications, vol 2, PitmanBoston, pp 580–589

    Google Scholar 

  82. Fogler Scott H (2006) Elements of chemical reaction engineering, 4th edn. Prentice-Hall International Inc, New Jersey

    Google Scholar 

  83. Freitas J (1993) Editorial. Trans ASME J Fluids Eng New York American Soc Mech Eng 115:339–340

    Google Scholar 

  84. Froment GF, Bischoff KB (1990) Chemical reactor analysis and design. Wiley, New York

    Google Scholar 

  85. Ganesan S, Poirier DR (1990) Conservation of mass and momentum for the flow of interdendritic liquid during solidification. Metall Trans B 21B:173–181

    Google Scholar 

  86. Ganesan V, Brenner H (2000) A diffuse interface model of two-phase flow in porous media. Proc R Soc Lond A 456:731–803

    MATH  MathSciNet  Google Scholar 

  87. Gauss CF (1830) Principia generalia Theoriae Figurae Fluidorum in statu Aequilibrii, Gottingen, 1830, or Werke, v 29, Grottingen, 1867

    Google Scholar 

  88. Gibbs JW (1876) On the equilibrium of heterogeneous substances, Trans. Conn. Acad. 3, (1876) 108–248, reprinted in The Scientific Papers of J Willard Gibbs. Longmans, Green, London 1906

    Google Scholar 

  89. Gibbs JW (1928) The collected works of J Willard Gibbs. Longmans, Green & Co, New York

    Google Scholar 

  90. Gidaspow D (1974) Introduction to modeling of two-phase flow. Round table discussion (RT-1-2). In: Proceedings 5th International Heat Transfer Conference, vol VII, p 163

    Google Scholar 

  91. Gidaspow D (1994) Multiphase flow and fluidization-continuum and kinetic theory descriptions. Academic Press, Harcourt Brace & Company, Publishers, Boston

    Google Scholar 

  92. Gosman AD, Pun WM, Runchal AK, Spalding DB, Wolfshtein M (1969) Heat and mass transfer in recirculating flows. Academic Press, New York

    Google Scholar 

  93. 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

    Google Scholar 

  94. 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

    Google Scholar 

  95. Gray WG (1975) A derivation of the equations for multi-phase transport. Chem Eng Sci 30:229–233

    Google Scholar 

  96. Gray WG, Lee PCY (1977) On the theorems for local volume averaging of multiphase systems. Int J Multiph Flow 3:333–340

    MATH  Google Scholar 

  97. Gray WG (1983) Local volume averaging of multiphase systems using a non-constant averaging volume. Int J Multiph Flow 9(6):755–761

    MATH  Google Scholar 

  98. 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

    MATH  Google Scholar 

  99. Hagesæther L, Jakobsen HA, Svendsen HF (1999) Theoretical analysis of fluid particle collisions in turbulent flow. Chem Eng Sci 54:4749–4755

    Google Scholar 

  100. Heinbockel JH (2001) Introduction to tensor calculus and continuum mechanics. Trafford Publishing, Canada (ISBN 1553691334)

    Google Scholar 

  101. Han J, Tryggvason G (1999) Secondary breakup of axisymmetric liquid drops. I. Acceleration by a constant body force. Phys Fluids 11(12):3650–3667

    MATH  Google Scholar 

  102. Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys Fluids 8:2182–2189

    MATH  Google Scholar 

  103. Harlow FH, Amsden AA (1975) Numerical calculation of multiphase fluid flow. J Comput Phys 17:19–52

    MATH  Google Scholar 

  104. Hassanizadeh M, Gray WG (1979) General conservation equations for multi-phase systems: 1. averaging procedure. Adv Water Resour 2:131–144

    Google Scholar 

  105. Hassanizadeh M, Gray WG (1979) General conservation equations for multi-phase systems: 2. Mass, momenta, energy, and entropy equations. Adv Water Resour 2:191–203

    Google Scholar 

  106. Hassanizadeh M, Gray WG (1980) General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv Water Resour 3:25–40

    Google Scholar 

  107. Hassanizadeh M, Gray WG (1987) High velocity flow in porous media. Transp Porous Media 2:521–531

    Google Scholar 

  108. Hassanizadeh M, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resour 13(4):169–186

    Google Scholar 

  109. Hidy GM, Broch JR (1970) The Dyn Aerocolloidal Syst. Pergamon, Oxford

    Google Scholar 

  110. Hinch EJ (1977) An average-equation approach to particle interactions in a fluid suspension. J Fluid Mech 83(4):695–720

    MATH  Google Scholar 

  111. Hinze JO (1975) Turbulence, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  112. Hirt CW (1968) Heuristic stability theory for finite difference equations. J Comput Phys 2:339–355

    MATH  Google Scholar 

  113. Hirt CW, Nichols BD (1980) Adding limited compressibility to incompressible hydrocodes. J Comput Phys 34:300–390

    MathSciNet  Google Scholar 

  114. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201–225

    MATH  Google Scholar 

  115. Holm DD, Kupershmidt BA (1986) A multipressure regulation for multiphase flow. Int J Multiph Flow 12(4):681–697

    MATH  MathSciNet  Google Scholar 

  116. Howes FA, Whitaker S (1985) The spatial averaging theorem revisited. Chem Eng Sci 40(8):1387–1392

    Google Scholar 

  117. Hyman JM (1984) Numerical methods for tracking interfaces. Physica 12D:396–407

    Google Scholar 

  118. Ishii M (1975) Thermo-fluid dynamic theory of two-phase flow. Eyrolles, Paris

    MATH  Google Scholar 

  119. 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

    Google Scholar 

  120. Ishii M, Mishima K (1981) Study of two-fluid model and interfacial area. Argonne National Laboratory Report ANL-80-111, Argonne, Illinois, USA

    Google Scholar 

  121. Ishii M, Mishima K (1984) Two-fluid model and hydrodynamic constitutive equations. Nucl Eng Des 82:107–126

    Google Scholar 

  122. Ishii M (1990) Two-fluid model for two-phase flow. Multiphase science and technology 5 (Chap 1). Hemisphere, New York

    Google Scholar 

  123. Ishii M, Takashi H (2011) Thermo-fluid dynamics of two-phase flow. Springer, New York

    MATH  Google Scholar 

  124. 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

    Google Scholar 

  125. Ivanov IB (1988) Thin liquid films. Fundamentals and applications. Marcel Dekker Inc, New York and Basel

    Google Scholar 

  126. Jacqmin D (1999) Calculation of two-phase Navier-Stokes flows using phase-field modeling. J Comput Phys 155:96–127

    MATH  MathSciNet  Google Scholar 

  127. 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

    Google Scholar 

  128. Jakobsen HA, Sannæs BH, Grevskott S, Svendsen HF (1997) Modeling of vertical bubble driven flows. Ind Eng Chem Res 36(10):4052–4074

    Google Scholar 

  129. Jakobsen HA (2001) Phase distribution phenomena in two-phase bubble column reactors. Chem Eng Sci 56(3):1049–1056

    MathSciNet  Google Scholar 

  130. Jakobsen HA, Lindborg H, Handeland V (2002) A numerical study of the interactions between viscous flow, transport and kinetics in fixed bed reactors. Comput Chem Eng 26:333–357

    Google Scholar 

  131. Jakobsen HA (2003) Numerical convection algorithms and their role in eulerian CFD reactor simulations. Int J Chem Reactor Eng A1:1–15

    MathSciNet  Google Scholar 

  132. 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 Transf 1:193–329

    Google Scholar 

  133. Johansen ST (1990) On the modelling of dispersed two-phase flows. Dr Techn thesis, The Norwegian Institute of Technology, Trondheim

    Google Scholar 

  134. Joseph DD, Lundgren TS, Jackson R, Saville DA (1990) Ensemble averaged and mixture theory equations for incompressible fluid-particle suspensions. Int J Multiph Flow 16(1):35–42

    MATH  Google Scholar 

  135. Juric D, Tryggvason G (1996) A front-tracking method for dendritic solidification. J Comput Phys 123:127–148

    MATH  MathSciNet  Google Scholar 

  136. Juric D, Tryggvason G (1998) Computations of boiling flows. Int J Multiph Flow 24(3):387–410

    MATH  Google Scholar 

  137. Kolev NI (2002) Multiph Flow Dyn 1 Fundam. Springer, Berlin

    Google Scholar 

  138. Korteweg DJ (1901) Sur la forme que prennent les equations du mouvements des fluides si l’on tient compte des forces capillaires causees par des variations de densite considerables mais continues et sur la theorie de la capillarite dans l’hypothese d’une variation continue de la densite. Arch Neerl Sci Exactes Nat Ser II 6:1–24

    MATH  Google Scholar 

  139. Kuipers JAM, van Swaaij WPM (1997) Application of computational fluid dynamics to chemical reaction engineering. Rev Chem Eng 13(3):1–118

    Google Scholar 

  140. Kuo KK (1986) Principles of combustion. Wiley, New York

    Google Scholar 

  141. 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

    MATH  MathSciNet  Google Scholar 

  142. Lafi AY, Reyes JN (1994) General particle transport equations. Final Report OSU-NE-9409. Department of nuclear engineering, Oregon State University

    Google Scholar 

  143. 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 Multiph Flow 6:281–294

    MATH  Google Scholar 

  144. Lahey RT Jr, Drew DA (1989) The three-dimensional time-and volume averaged conservation equations of two -phase flow. Adv Nucl Sci Technol 20:1–69

    Google Scholar 

  145. 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

    Google Scholar 

  146. 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

    Google Scholar 

  147. Laplace PS (1806) Traité de Méchanique Céleste. Supplement to book 10, vol. IV. Paris: Gauthier-Villars, (1806) Annotated English translation by Nathaniel Bowditch (1839). Chelsea Publishing Company, Reprinted by New York, p 1996

    Google Scholar 

  148. Laux H (1998) Modeling of dilute and dense dispersed fluid-particle flow. Dr Ing Thesis, Norwegian University of Science and Technology, Trondheim, Norway

    Google Scholar 

  149. 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 Numer Methods Fluids 20:421–442

    MATH  Google Scholar 

  150. 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

    Google Scholar 

  151. Liljegren LM (1997) Ensemble-average equations of a particulate mixture. J Fluids Engr 119:428–434

    Google Scholar 

  152. Lopez de Bertodano M (1992) Turbulent bubbly two-phase flow in a triangular duct. PhD Thesis, Rensselaer Polytechnic Institute, Troy

    Google Scholar 

  153. Rayleigh Lord (1892) On the theory of surface forces - II. Compressible Fluids Phil Mag 33:209–220

    MATH  Google Scholar 

  154. 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

    Google Scholar 

  155. Manninen M, Taivassalo V, Kallio S (1996) On the mixture model for multiphase flow. Technical research center of finland. VIT Publications, Espoo

    Google Scholar 

  156. 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

    MATH  Google Scholar 

  157. 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

    MATH  Google Scholar 

  158. Maxey MR, Riley JJ (1983) Equation of motion for a mall rigid sphere in a non-uniform flow. Phys Fluids 26(4):883–889

    MATH  Google Scholar 

  159. Maxwell JC (1876) Capillary action. In: Encyclopaedia Britannica, 9th ed. Reprinted in 1952. The Scientific Papers of James Clerk Maxwell, 2:541–591. New York, Dover

    Google Scholar 

  160. Meier M, Andreani M, Smith B, Yadigaroglu G (1998) Numerical and experimental study of large stream-air bubbles condensing in water. In: Proceedings of Third International Conference Multiphase Flow, Lyon, June 8–12

    Google Scholar 

  161. Miller CA, Neogi P (1985) Interfacial phenomena: equilibrium and dynamic effects. Marcel Dekker Inc, New York

    Google Scholar 

  162. Moeckel GP (1975) Thermodynamics of an interface. Arch Ration Mech Anal 57:255–280

    MATH  MathSciNet  Google Scholar 

  163. 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

    Google Scholar 

  164. Nadiga BT, Zaleski S (1996) Investigations of a two-phase fluid model. Eur J Mech B Fluids 15:885–896

    MATH  Google Scholar 

  165. 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

    Google Scholar 

  166. Ni J, Beckermann C (1991) A volume-averaged two-phase model for transport phenomena during solidification. Metall Trans B 22B:349–361

    Google Scholar 

  167. Nichols BD, Hirt CW (1975) Methods for calculating multi-dimensional, transient. Free surface flows past bodies, Proceedings first International Conference on Numerical Ship Hydrodynamics, Gaithersburg, Md, October

    Google Scholar 

  168. Nigmatulin RI (1979) Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int J Multiph Flow 5:353–385

    MATH  Google Scholar 

  169. 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

    Google Scholar 

  170. Nobari MR, Jan Y-J, Tryggvason G (1996) Head-on collision of droplets - A numerical investigation. Phys Fluids 8(1):29–42

    MATH  Google Scholar 

  171. Pan Y, Suga K (2005) Numerical simulation of binary liquid droplet collision. Phys Fluids 17 (8):82105–082105-14

    Google Scholar 

  172. Patankar SV (1980) Numerical heat transfer and fluid flow. Series in computational methods in mechanics and thermal sciences. Hemisphere Publishing Corporation, New York

    Google Scholar 

  173. Pauchon C, Banerjee S (1986) Interface momentum interaction effects in the averaged multifield model. Part I: Void propagation in bubbly flows. Int J Multiph Flow 12(4):559–573

    Google Scholar 

  174. 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 Multiph Flow 14(3):253–264

    Google Scholar 

  175. Poirier DR, Nandapurkar PJ, Ganesan S (1991) The energy and solute conservation equation for dendritic solidification. Metall Trans B 22B:889–900

    Google Scholar 

  176. Poisson SD (1831) Nouvelle Theorie de l’Action Capillaire. Bachelier, Paris

    Google Scholar 

  177. Popinet S, Zaleski S (1999) A front-tracking algorithm for accurate representation of surface tension. Int J Numer Meth Fluids 30:775–793

    MATH  Google Scholar 

  178. Prescott PJ, Incropera FP (1994) Convective transport phenomena and macrosegregation during solidification of a binary metal alloy: I-numerical predictions. J Heat Transf 116:735–749

    Google Scholar 

  179. Probstain RF (1994) Physicochemical hydrodynamics: an introduction, 2nd edn. Wiley, New York

    Google Scholar 

  180. Prosperetti A, van Wijngaarden L (1976) On the characteristics of the equation of motion for bubbly flow and the related problem of critical flow. J Eng Math 10(2):153–162

    MATH  Google Scholar 

  181. Prosperetti A, Jones AV (1984) Pressure forces in dispersed two-phase flow. Int J Multiph Flow 10(4):425–440

    MATH  Google Scholar 

  182. Prosperetti A, Zhang DZ (1996) Disperse phase stress in two-phase flow. Chem Eng Comm 141–142:387–398

    Google Scholar 

  183. Qian J, Tryggvason G, Law CK (1998) A front tracking method for the motion of premixed flames. J Comput Phys 144:52–69

    Google Scholar 

  184. 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

    Google Scholar 

  185. Ramshaw JD, Trapp JA (1978) Characteristics, stability and short wavelength phenomena in two-phase flow equation systems. Nucl Sci Engng 66:93–102

    Google Scholar 

  186. 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

    Google Scholar 

  187. Raupach MR, Shaw RH (1982) Averaging procedures for flow within vegetation canopies. Bound Layer Meteorol 22:79–90

    Google Scholar 

  188. Reeks MW (1991) On a kinetic equation for the transport of particles in turbulent flows. Phys Fluids A 3:446–456

    MATH  Google Scholar 

  189. Reyes Jr JN (1989) Statistically derived conservation equations for fluid particle flows. In: Proceedings of nuclear thermal hydraulics, American Nuclear Society, San Francisco, 12–19 November

    Google Scholar 

  190. 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

    Google Scholar 

  191. Richtmyer RD, Morton KW (1957) Difference methods for initial-value problems. 2nd edn, Interscience Publishers/Wiley, New York

    Google Scholar 

  192. Rider WJ, Kothe DB (1995) Stretching and tearing interface tracking methods. AIAA Paper 95–1717:806–816

    Google Scholar 

  193. Rider WJ, Kothe DB (1998) Reconstructing volume tracking. J Comput Phys 141:112–152

    MATH  MathSciNet  Google Scholar 

  194. Roberts IF (1997) Conservation equations. Two-phase. In: International Encyclopedia of Heat and Mass Transfer, pp 223–230

    Google Scholar 

  195. 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

    Google Scholar 

  196. Scardovelli R, Zaleski S (1999) Direct numerical simulation of free-surface and interfacial flow. Annu Rev Fluid Mech 31:567–603

    MathSciNet  Google Scholar 

  197. Schwartz MP, Turner WJ (1988) Applicability of the standard k-\(\epsilon \) turbulence model to gas-stirred baths. Appl Math Model 12:273–279

    Google Scholar 

  198. Scriven LE (1960) Dynamics of a fluid interface. Chem Eng Sci 12:98–108

    Google Scholar 

  199. Sethian JA (1996) Level set methods. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  200. Sha WT, Soo SL (1978) Multidomain multiphase fluid mechanics. Int J Heat Mass Transfer 21:1581–1595

    MATH  Google Scholar 

  201. Sha WT, Soo SL (1979) Brief communication: on the effect of \(p\nabla \alpha \) term in multiphase mechanics. Int J Multiph Flow 5:153–158

    MATH  Google Scholar 

  202. Sha WT, Slattery JC (1980) Local volume-time averaged equations of motion for dispersed, turbulent, multiphase flows. NUREG/CR-1491, ANL-80-51

    Google Scholar 

  203. Sha WT, Chao BT, Soo SL (1983) Averaging procedures of multiphase conservation equations. Trans American Nucl Soc 45:814–816

    Google Scholar 

  204. Sha WT, Chao BT, Soo SL (1983) Time averaging of volume-averaged conservation equations of multiphase flow. AIChE Symp Ser 79(225):420–426

    Google Scholar 

  205. Sha WT, Chao BT, Soo SL (1984) Porous-media formulation for multiphase flow with heat transfer. Neclear Eng Des 82:93–106

    Google Scholar 

  206. Shyy W, Thakur S, Ouyang H, Liu J, Blosch E (1997) Computational techniques for complex transport phenomena. Cambridge University Press, Cambridge

    Google Scholar 

  207. Singer-Loginova I, Singer HM (2008) The phase field technique for modeling multiphase materials. Rep Prog Phys 71:106501 (32pp)

    Google Scholar 

  208. Slattery JC (1967) Flow of viscoelastic fluids through porous media. AIChE J 13(6):1066–1071

    Google Scholar 

  209. Slattery JC (1969) Single-phase flow through porous media. AIChE J 15(6):866–872

    Google Scholar 

  210. Slattery JC (1972) Momentum, energy, and mass transfer in continua, 2nd edn. McGraw-Hill Kogakusha LTD, Tokyo

    Google Scholar 

  211. Slattery JC (1980) Invited review: interfacial transport phenomena. Chem Eng Sci 4:149–166

    Google Scholar 

  212. Slattery JC, Flumerfelt RW (1982) Interfacial phenomena. In: Hetsroni G (ed) Handbook of multiphase systems, Sect. 1.4, pp (1–224) – (1–2246), McGraw-Hill, New York

    Google Scholar 

  213. Slattery JC (1990) Interfacial transport phenomena. Springer, New York

    Google Scholar 

  214. Slattery JC (1999) Advanced transport phenomena. Cambridge University Press, New York

    MATH  Google Scholar 

  215. 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

    Google Scholar 

  216. Sokolichin A, Eigenberger G (1999) Applicability of the standard \(k-\epsilon \) turbulence model to the dynamic simulation of bubble columns: Part I. Detailed numerical simulations. Chem Eng Sci 54:2273–2284

    Google Scholar 

  217. Sommerfeld M (2001) Validation of a stochastic lagrangian modelling approach for inter-particle collisions in homogeneous isotropic turbulence. Int J Multiph Flow 27:1829–1858

    MATH  Google Scholar 

  218. Soo SL (1967) Fluid dynamics of multiphase systems. Blaisdell Publishing Company, Waltham, Massachusetts

    MATH  Google Scholar 

  219. Soo SL (1989) Particles and continuum: multiphase fluid dynamics. Hemisphere Publishing Corporation, New York

    Google Scholar 

  220. Soo SL (1990) Multiphase Fluid Dynamics. Science Press, Beijing and Gower Technical, Aldershot

    Google Scholar 

  221. Spalding DB (1977) The calculation of free-convection phenomena in gas-liquid mixtures. ICHMT Seminar 1976. Turbulent Buoyant Convection, Hemisphere, Washington, pp 569–586

    Google Scholar 

  222. 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

    Google Scholar 

  223. Spalding DB (1980) Mathematical methods in nuclear reactor thermal hydraulics. In: Lahey RT (ed) Proceedings of ANS meeting on nuclear reactor thermal hydraulics, Saratoga, pp 1979–2023

    Google Scholar 

  224. Spalding DB (1981) IPSA 1981; New developments and computed results. HTS/81/2, Imperial college of science and technology, London

    Google Scholar 

  225. 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

    Google Scholar 

  226. Stewart HB, Wendroff B (1984) Two-phase flow: models and methods. J Comput Phys 56:363–409

    MATH  MathSciNet  Google Scholar 

  227. Stuhmiller JH (1977) The influence of interfacial pressure forces on the character of two-phase flow model equations. Int J Multiph Flow 3:551–560

    MATH  Google Scholar 

  228. 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

    MATH  Google Scholar 

  229. Sussman M, Smereka P (1997) Axisymmetric free boundary problems. J Fluid Mech 341:269–294

    MATH  MathSciNet  Google Scholar 

  230. 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

    Google Scholar 

  231. Thomas GB Jr, Finney RL (1996) Calculus and analytic geometry. Addison-Wesley Publishing Company, 9th edn, Reading, Massachusetts

    Google Scholar 

  232. 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. In: Third international conference on multiphase flow, Lyon, France

    Google Scholar 

  233. Trapp JA (1986) The mean flow character of two-phase flow equations. Int J Multiph Flow 12(2):263–276

    MATH  Google Scholar 

  234. Travis JR, Harlow FH, Amsden AA (1976) Numerical calculations of two-phase flows. Nucl Sci Engng 61:1–10

    Google Scholar 

  235. Tryggvason G, Bunner B, Esmaeeli A, Mortazavi S (1998) Direct numerical simulations of dispersed flows. In: Third international conference on multiphase flow, ICMF’98, June 8–12, Lyon, France

    Google Scholar 

  236. Tryggvason G (1999) Embedded interface methods applications. In: Modelling and computation of multiphase flows, short course, Zurich, Switzerland, March 8–12, 16B:1–27

    Google Scholar 

  237. Tryggvason G (1999) Embedded interface methods applications. In: Modelling and computation of multiphase flows, Short Course, Zurich, Switzerland, March 8–12, 18B:1–24

    Google Scholar 

  238. 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

    MATH  Google Scholar 

  239. Tryggvason G, Esmaeeli A, Al-Rawahi N (2005) Direct numerical simulations of flows with phase change. Comput and Struct 83:445–453

    Google Scholar 

  240. Ubbink O, Issa I (1999) A method for capturing sharp fluid interfaces on arbitrary meshes. J Comput Phys 153:26–50

    MATH  MathSciNet  Google Scholar 

  241. Unverdi SO, Tryggvason G (1992) A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys 100:25–37

    MATH  Google Scholar 

  242. Unverdi SO, Tryggvason G (1992) Computations of multi-fluid flows. Physica D60:70–83, North-Holland

    Google Scholar 

  243. van der Waals JD (1893) The thermodynamic theory of capillarity flow under the hypothesis of a continuous variation of density. Verhandel Konink Akad Weten 1. Translation Published by Rowlinson JS. J Stat Phys 20:200–244

    Google Scholar 

  244. 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):1–46

    Google Scholar 

  245. Voller VR, Brent AD, Prakash C (1989) The modelling of heat, mass and solute transport in solidification systems. Int J Heat Transf 32(9):1719–1731

    Google Scholar 

  246. Wallis GB (1969) One-dimensional two-phase flow. McGraw-Hill, New York

    Google Scholar 

  247. Weatherburn CE (1927) Differential geometry of three dimensions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  248. Whitaker S (1967) Diffusion and dispersion in porous media. AIChE J 13(3):420–427

    Google Scholar 

  249. Whitaker S (1968) Introduction to fluid mechanics. Prentice-Hall Inc, Englewood Cliffs

    Google Scholar 

  250. Whitaker S (1969) Fluid motion in porous media. Ind Eng Chem 61(12):15–28

    Google Scholar 

  251. Whitaker S (1973) The transport equations for multiphase systems. Chem Eng Sci 28:139–147

    Google Scholar 

  252. Whitaker S (1985) A simple geometrical derivation of the spatial averaging theorem. Chem Eng Educ 19:18–21 and 50–52

    Google Scholar 

  253. Whitaker S (1992) The species mass jump condition at a singular surface. Chem Eng Sci 47(7):1677–1685

    MathSciNet  Google Scholar 

  254. Whitaker S (1999) The method of volume averaging. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  255. White FM (1974) Viscous Fluid Flow. McGraw-Hill, New York

    MATH  Google Scholar 

  256. Willmore TJ (1961) An introduction to differential geometry. Oxford University Press, Glasgow

    Google Scholar 

  257. 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

    Google Scholar 

  258. Yadigaroglu G, Lahey RT Jr (1976) On the various forms of the conservation equations in two-phase flow. Int J Multiph Flow 2:477–494

    MATH  Google Scholar 

  259. Young T (1805) An essay on the cohesion of fluids. Phil Trans Roy Soc London 95:65–87

    Google Scholar 

  260. 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

    Google Scholar 

  261. Zapryanov Z, Tabakova S (1999) Dynamics of bubbles, drops and rigid particles. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  262. Zhang DZ, Prosperetti A (1994) Ensemble phase-average equations for bubbly flows. Phys Fluids 6(9):2956–2970

    MATH  MathSciNet  Google Scholar 

  263. Zhang DZ, Prosperetti A (1994) Averaged equations for inviscid disperse two-phase flow. J Fluid Mech 267:185–219

    MATH  MathSciNet  Google Scholar 

  264. Zhang DZ, Prosperetti A (1997) Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions. Int J Multiph Flow 23(3):425–453

    MATH  Google Scholar 

  265. Zuber N, Findlay JA (1965) Average volumetric concentration i two-phase flow systems. J Heat Transf 87:453–468

    Google Scholar 

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Jakobsen, H.A. (2014). Multiphase Flow. In: Chemical Reactor Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-05092-8_3

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