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Partial Invariance and Problems with Free Boundaries

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Symmetries and Applications of Differential Equations

Part of the book series: Nonlinear Physical Science ((NPS))

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Abstract

The foundations of group analysis of differential equations were laid by S. Lie. This theory was essentially developed in works of L. V. Ovsiannikov, N. H. Ibragimov, their students, and followers. Notion of the partially invariant solution to the system of differential equations (Ovsiannikov 1964) substantially extended possibilities of exact solutions construction for multidimensional systems of differential equations admitting the Lie group. It is important to note that fundamental equations of continuum mechanics and physics fall in this class a priori as invariance principle of space, time, and moving medium there with respect to some group (Galilei, Lorenz, and others) are situated in the base of their derivation. It should be noticed that classical group analysis of differential equations has a local character. To apply this approach to initial boundary value problems, one need to provide the invariance properties of initial and boundary conditions. Author (1973) studied these properties for free boundary problems to the Navier–Stokes equations. Present chapter contains an example of partially invariant solution of these equations describing the motion of a rotating layer bounded by free surfaces.

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References

  1. Ovsyannikov LV (1982) Group analysis of differential equations. Academic, New York, p 432

    Google Scholar 

  2. Ovsyannikov LV (1964) Partly invariant solutions of the equations admitting a group. In: Proceedings of the \(11^{\text{th}}\) international congress of applied mechanics, Münich, pp 868–870

    Google Scholar 

  3. Ovsyannikov LV (1969) Partial invariance. Dokl Akad Nauk SSSR 186(1):22–25 (in Russian)

    MathSciNet  MATH  Google Scholar 

  4. Ovsyannikov LV (1994) Program SUBMODELS. Gas dynamics. J Appl Math Mech 58(4):30–55

    Article  MathSciNet  Google Scholar 

  5. Ovsyannikov LV (1995) Special vortex. J Appl Mech Tech Phys 36(3):45–52

    Article  MathSciNet  Google Scholar 

  6. Yanenko NN (1956) Traveling waves of the system of differential equations. Dokl Akad Nauk SSSR 109(1):44–47 (in Russian)

    MathSciNet  MATH  Google Scholar 

  7. Sidorov AF, Shapeev VP, Yanenko NN (1984) Method of differential constrains and its applications in gas dynamics. Nauka (Sibirskoe Otdelenie), Novosibirsk, 272 (in Russian)

    Google Scholar 

  8. Serrin J (1959) Mathematical principles of classical fluid mechanics. Handbuch der Physik. Band VIII/1. Strömungsmechanik I. Berlin/Göttingen/Heidelberg, pp 125–263

    Google Scholar 

  9. Kochin NE, Kibel IA, Rose NV (1963) Theoretical hydrodynamics. Part 2. Interscience Publication Co, New York, p 736

    Google Scholar 

  10. Danilov YuA (1967) Group properties of Maxwell and Navier–Stokes equations. Inst Atomic Energy, 1452, Moscow, 15. (in Russian)

    Google Scholar 

  11. Bytev VO (1972) Group properties of Navier - Stokes equations. Chislennye Metody Mekhaniki Sploshnoi Sredy, Novosibirsk 3(3):13–17 (in Russian)

    Google Scholar 

  12. Lloyd SP (1981) The infinitesimal group of the Navier - Stokes equations. Acta Mech 38(1–2):85–98

    Article  MathSciNet  Google Scholar 

  13. Napolitano LG (1979) Thermodynamics and dynamics of surface phases. Acta Astron 6(9):1093–1112

    Article  Google Scholar 

  14. Andreev VK, Gaponenko YuA, Goncharova ON, Pukhnachev VV (2020) Mathematical models of convection, 2nd edn. De Gruyter, Berlin/Boston, p 418

    Google Scholar 

  15. Pukhnachev VV (1972) Invariant solutions of the Navier - Stokes equations, which describe motions with free boundaries. Dokl Akad Nauk SSSR 202(2):302–305 (in Russian)

    Google Scholar 

  16. Andreev VK, Kaptsov OV, Pukhnachov VV, Rodionov AA (1998) Applications of group-theoretical methods in hydrodynamics. Kluwer Academic Publishers, Dordrecht, p 396

    Google Scholar 

  17. Hiemenz K (1911) Die Grenzschicht an einem in den gleichfoermigen Fluessigkeitstrom eingetauchten geraden Kreiszylinder. Dingl Polytech J 326:321–410

    Google Scholar 

  18. Karman Th (1921) Ueber laminare und turbulentne Reibung. Z Angew Mat Mech 1(4):233–252

    Article  Google Scholar 

  19. Petrova AG, Pukhnachev VV, Frolovskaya OA (2016) Analytical and numerical investigation of unsteady flow near a critical point. J Appl Math Mech 80(3):215–224

    Article  MathSciNet  Google Scholar 

  20. Meleshko SV, Pukhnachev VV (1999) One class of partially invariant solutions of the Navier–Stokes equations. J Appl Mech Tech Phys 40(2):208–216

    Article  MathSciNet  Google Scholar 

  21. Pukhnachev VV (1972) Unsteady motions of a viscous liquid with a free boundary described by partially invariant solutions to the Navier–Stokes equations. Dinamika Sploshnoi Sredy, Novosibirsk 10:125–137 (in Russian)

    Google Scholar 

  22. Matsumoto S, Saito K, Takashima Y (1973) Thickness of liquid film on a rotating disk. J Chem Eng Jpn 6(6):503–507

    Article  Google Scholar 

  23. Lavrenteva OM, Volkova GB (1996) Limit regimes of the spreading of a layer on a rotating plane. Dinamika Sploshnoi Sredy, Novosibirsk 111:58–74 (in Russian)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk State University, Novosibirsk, 630090, Russia

    V. V. Pukhnachev

Authors
  1. V. V. Pukhnachev
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Editors and Affiliations

  1. Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA

    Albert C. J. Luo

  2. Department of High Performance Computing, Ufa State Aviation Technical University, Ufa, Russia

    Rafail K. Gazizov

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Pukhnachev, V.V. (2021). Partial Invariance and Problems with Free Boundaries. In: Luo, A.C.J., Gazizov, R.K. (eds) Symmetries and Applications of Differential Equations. Nonlinear Physical Science. Springer, Singapore. https://doi.org/10.1007/978-981-16-4683-6_9

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