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Multiphase Flow Modeling via Hamilton’s Principle

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Variational Models and Methods in Solid and Fluid Mechanics

Part of the book series: CISM Courses and Lectures ((CISM,volume 535))

Abstract

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.

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Bibliography

  1. R. Abgrall (1996) How to prevent pressure oscillations in multicomponent flow calculations: A quasi-conservative approach. J. Comput. Phys. 125, 150–160.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Alvarez-Samaniego and D. Lannes (2008) Large time existence for 3D water waves and asymptotics, Invent. math. v. 171, 485–541.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  4. V. L. Berdichevsky (2009) Variational Principles of Continuum Mechanics, Springer-Verlag Berlin Heidelberg.

    Google Scholar 

  5. A. D. Drew and S. L. Passman (1996) Theory of multicomponent fluids, Springer-Verlag New York Inc.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  12. S.L. Gavrilyuk, N. Favrie and R. Saurel (2008) Modeling wave dynamics of compressible elastic materials. Journal of Computational Physics, 227, 2941–2969.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. K. Godunov (1978) Elements of Continuum Mechanics, Nauka, Moscow (in Russian).

    Google Scholar 

  14. S. K. Godunov and E. I. Romenskii (2003) Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic Plenum Publishers, NY.

    MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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. M. Ishii and T. Hibiki (2006), Thermo-fluid dynamics of two-phase flow, Springer.

    Google Scholar 

  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.

    Article  Google Scholar 

  20. S. Karni (1994) Multi-component flow calculations by a consistent primitive algorithm, Journal of Computational Physics, 112, 31–43.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. S. Kogarko (1961) On the model of cavitating liquid. Dokl. AN SSSR 137, 1331–1333 (in Russian).

    MathSciNet  Google Scholar 

  22. Y. A. Li (2001) Linear stability of solitary waves of the Green-Naghdi equations, Comm. Pure Appl. Math. 54:501–536.

    Article  MathSciNet  MATH  Google Scholar 

  23. L.-H. Luu and Y. Forterre (2009) Drop impact of yield-stress fluids, J. Fluid Mechanics, 632, 301–327.

    Article  MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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. J. Miles and R. Salmon (1985), Weakly dispersive nonlinear gravity waves, J. Fluid Mech. 157:519–531.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. N. Plohr and B. J. Plohr (2005) Linearized analysis of Richtmyer-Meshkov flow for elastic materials, J. Fluid Mech., 537, 55–89.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Saurel and R. Abgrall (2001), A multiphase Godunov method for compressible multifluid and multiphase flows, J. Comput. Physics, 150, 425–467.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  31. R. Salmon (1988), Hamiltonian Fluid Mechanics, Ann. Rev. Fluid Mech 20:225–256.

    Article  Google Scholar 

  32. R. Salmon (1998), Lectures on Geophysical Fluid Dynamics, Oxford University Press, New York, Oxford.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  34. V. M. Teshukov and S. L. Gavrilyuk (2006), Three-Dimensional Nonlinear Dispersive waves on Shear Flows, Stud. Appl. Math. 116:241–255.

    Article  MathSciNet  MATH  Google Scholar 

  35. L. van Wijngaarden (1968) On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33, 465–474.

    Article  MATH  Google Scholar 

  36. G. B. Whitham (1974), Linear and Nonlinear Waves, John Wiley & Sons, New York.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. CNRS UMR 6595, IUSTI, Aix-Marseille University, Marseille, France

    Sergey Gavrilyuk

  2. International Center for Mechanical Sciences, CISM, Udine, Italy

    Sergey Gavrilyuk

Authors
  1. Sergey Gavrilyuk
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Editors and Affiliations

  1. Università “La Sapienza”, Roma, Italy

    Francesco dell’Isola

  2. Polytech Marseille, France

    Sergey Gavrilyuk

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© 2011 CISM, Udine

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Gavrilyuk, S. (2011). Multiphase Flow Modeling via Hamilton’s Principle. In: dell’Isola, F., Gavrilyuk, S. (eds) Variational Models and Methods in Solid and Fluid Mechanics. CISM Courses and Lectures, vol 535. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0983-0_4

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  • DOI: https://doi.org/10.1007/978-3-7091-0983-0_4

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-0982-3

  • Online ISBN: 978-3-7091-0983-0

  • eBook Packages: EngineeringEngineering (R0)

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