Multiphase Flow Modeling via Hamilton’s Principle

  • Sergey Gavrilyuk
Part of the CISM Courses and Lectures book series (CISM, volume 535)


We present here a variational approach to derivation of multiphase flow models. Two basic ingredients of this method are as follows. First, a conservative part of the model is derived based on the Hamilton principle of stationary action. Second, phenomenological dissipative terms are added which are compatible with the entropy inequality. The variational technique is shown up, and mathematical models (classical and non-classical) describing fluid-fluid and fluid-solid mixtures and interfaces are derived.


Solitary Wave Sound Speed Multiphase Flow Entropy Inequality Hamilton Action 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Abgrall (1996) How to prevent pressure oscillations in multicomponent flow calculations: A quasi-conservative approach. J. Comput. Phys. 125, 150–160.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    B. Alvarez-Samaniego and D. Lannes (2008) Large time existence for 3D water waves and asymptotics, Invent. math. v. 171, 485–541.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. Baer and J. Nunziato (1986) A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials. Int. J. Multiphase Flows, 12, 861–889.zbMATHCrossRefGoogle Scholar
  4. [4]
    V. L. Berdichevsky (2009) Variational Principles of Continuum Mechanics, Springer-Verlag Berlin Heidelberg.Google Scholar
  5. [5]
    A. D. Drew and S. L. Passman (1996) Theory of multicomponent fluids, Springer-Verlag New York Inc.Google Scholar
  6. [6]
    N. Favrie, S. L. Gavrilyuk and R. Saurel (2009) Diffuse solid-fluid interface model in cases of extreme deformations, Journal of Computational Physics, 228, 6037–6077.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    G. A. El, R. H. J. Grimshaw and N. F. Smyth (2006), Unsteady undular bores in fully nonlinear shallow-water theory, Physics of Fluids 18:027104 (17 pages).MathSciNetCrossRefGoogle Scholar
  8. [8]
    S. L. Gavrilyuk and H. Gouin (1999), A new form of governing equations of fluid arising from Hamilton’s principle, Int. J. Eng. Sci. 37:1495–1520.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    S. L. Gavrilyuk and V. M. Teshukov (2001), Generalized vorticity for bubbly liquid and dispersive shallow water equations, Continuum Mech. Thermodyn. 13:365–382.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    S.L. Gavrilyuk and R. Saurel (2002), Mathematical and Numerical Modeling of Two-Phase Compressible Flows with Microinertia, Journal of Computational Physics, 175, 326–360.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    S. L. Gavrilyuk and V. M. Teshukov (2004), Linear stability of parallel inviscid flows of shallow water and bubbly fluid, Stud. Appl. Math. 113:1–29.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S.L. Gavrilyuk, N. Favrie and R. Saurel (2008) Modeling wave dynamics of compressible elastic materials. Journal of Computational Physics, 227, 2941–2969.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    S. K. Godunov (1978) Elements of Continuum Mechanics, Nauka, Moscow (in Russian).Google Scholar
  14. [14]
    S. K. Godunov and E. I. Romenskii (2003) Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic Plenum Publishers, NY.zbMATHGoogle Scholar
  15. [15]
    A. E. Green, N. Laws and P. M. Naghdi (1974), On the Theory ofWater Waves, Proceedings of the Royal Society of London A 338: 43–55.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    A. E. Green and P. M. Naghdi (1976), A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78:237–246.zbMATHCrossRefGoogle Scholar
  17. [17]
    S. V. Iordansky (1960) On the equations of motion of the liquid containing gas bubbles. Zhurnal Prikladnoj Mekhaniki i Tekhnitheskoj Fiziki N3, 102–111 (in Russian).Google Scholar
  18. [18]
    M. Ishii and T. Hibiki (2006), Thermo-fluid dynamics of two-phase flow, Springer.Google Scholar
  19. [19]
    A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son, D. S. Stewart (2001) Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Phys. Fluids, 13(10), 3002–3024.CrossRefGoogle Scholar
  20. [20]
    S. Karni (1994) Multi-component flow calculations by a consistent primitive algorithm, Journal of Computational Physics, 112, 31–43.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    B. S. Kogarko (1961) On the model of cavitating liquid. Dokl. AN SSSR 137, 1331–1333 (in Russian).MathSciNetGoogle Scholar
  22. [22]
    Y. A. Li (2001) Linear stability of solitary waves of the Green-Naghdi equations, Comm. Pure Appl. Math. 54:501–536.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    L.-H. Luu and Y. Forterre (2009) Drop impact of yield-stress fluids, J. Fluid Mechanics, 632, 301–327.zbMATHCrossRefGoogle Scholar
  24. [24]
    N. Makarenko (1986) A second long-wave approximation in the Cauchy-Poisson problem, Dynamics of Continuous Media, v. 77, pp. 56–72 (in Russian).MathSciNetzbMATHGoogle Scholar
  25. [25]
    J.-C. Micaelli (1982), Propagation d’ondes dans les écoulements diphasiques à bulles à deux constituants. Etude théorique et expérimentale. Th`ese, Université de Grenoble, France.Google Scholar
  26. [26]
    J. Miles and R. Salmon (1985), Weakly dispersive nonlinear gravity waves, J. Fluid Mech. 157:519–531.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    J. N. Plohr and B. J. Plohr (2005) Linearized analysis of Richtmyer-Meshkov flow for elastic materials, J. Fluid Mech., 537, 55–89.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    R. Saurel and R. Abgrall (2001), A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Physics, 150, 425–467.MathSciNetCrossRefGoogle Scholar
  29. [29]
    R. Saurel, S. L. Gavrilyuk and F. Renaud (2003) A multiphase model with internal degrees of freedom: application to shock-bubble interaction, Journal of Fluid Mechanics, 495, 283–321.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    R. Saurel, O. Le Metayer, J. Massoni and S. Gavrilyuk (2007), Shock jump relations for multiphase mixtures with stiff mechanical relaxation, Shock Waves, 16, 209–232.zbMATHCrossRefGoogle Scholar
  31. [31]
    R. Salmon (1988), Hamiltonian Fluid Mechanics, Ann. Rev. Fluid Mech 20:225–256.CrossRefGoogle Scholar
  32. [32]
    R. Salmon (1998), Lectures on Geophysical Fluid Dynamics, Oxford University Press, New York, Oxford.Google Scholar
  33. [33]
    C. H. Su and C. S. Gardner (1969), Korteweg-de Vries Equation and Generalizations. III. Derivation of the Korteweg-de Vries Equation and Burgers Equation, J. Math. Phys. 10:536–539.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    V. M. Teshukov and S. L. Gavrilyuk (2006), Three-Dimensional Nonlinear Dispersive waves on Shear Flows, Stud. Appl. Math. 116:241–255.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    L. van Wijngaarden (1968) On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33, 465–474.zbMATHCrossRefGoogle Scholar
  36. [36]
    G. B. Whitham (1974), Linear and Nonlinear Waves, John Wiley & Sons, New York.zbMATHGoogle Scholar

Copyright information

© CISM, Udine 2011

Authors and Affiliations

  • Sergey Gavrilyuk
    • 1
    • 2
  1. 1.CNRS UMR 6595, IUSTIAix-Marseille UniversityMarseilleFrance
  2. 2.International Center for Mechanical SciencesCISMUdineItaly

Personalised recommendations