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One class of partially invariant solutions of the Navier-Stokes equations

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Abstract

A family of partially invariant solutions of the Navier-Stokes equations of rank 2 and defect 2 is considered. These solutions describe the three-dimensional unsteady motions of a viscous incompressible fluid in which the vertical velocity component and the pressure are independent of the horizontal coordinates. In particular, they can be interpreted as flows in a horizontal layer, one boundary of which is the free surface.

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References

  1. V. O. Bytev, “Group properties of the Navier-Stokes equations,” in:Numerical Methods in Mechanics of Continuous Media (collected scientific papers) [in Russian], Novosibirsk, bf 3, No. 3 (1972), pp. 13–17.

  2. V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachev, and A. A. Radionov,Application of Group-Theoretic Methods in Hydrodynamics [in Russian], Nauka, Novosibirsk (1994).

    MATH  Google Scholar 

  3. L. V. Ovsyannikov,Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1994).

    MATH  Google Scholar 

  4. T. von Kármán, “Uber laminare und turbulente reibung,”ZAMM, bf 1, No. 4, 233–252 (1921).

    Google Scholar 

  5. V. V. Pukhnachev, “Nonstationary motions of a viscous fluid with a free boundary described by partially invariant solutions of the Navier-Stokes equations,” in:Dynamics of Continuous Media (collected scientific papers) [in Russian], Novosibirsk,10 (1972), pp.125–137.

  6. L. V. Ovsyannikov, “Partial invariance,”Dokl. Akad. Nauk SSSR,186, No. 1, 22–25 (1969).

    MATH  Google Scholar 

  7. A. F. Sidorov, “Two classes of solutions of fluid and gas mechanics equations and their connection to the theory of traveling waves,”Prikl. Mekh. Tekh. Fiz., No. 2, 34–40 (1989).

    MathSciNet  Google Scholar 

  8. O. M. Lavrent’eva, “Flow of a viscous fluid in a layer on a rotating plane,”Prikl. Mekh. Tekh. Fiz., No. 5, 41–48 (1989).

    MathSciNet  Google Scholar 

  9. O. M. Lavrent’eva and G. B. Volkova, “Limiting regimes of spreading of a layer on a rotating plane,” in:Dynamics of Continuous Media (collected scientific papers) [in Russian], Novosibirsk,111 (1994), pp. 68–77.

  10. V. V. Pukhnachov, “Nonstationary viscous flows with a cylindrical free surface,” in:Progress in Nonlinear Differential Equations and Their Applications, V. 35:Topics in Nonlinear Analysis: The Herbert Amann Anniversary Volume, Birkhäuser Verlag, Basel (1998), pp. 553–569.

    Google Scholar 

  11. V. A. Galaktionov and J. L. Vazquez, “Blow-up of a class of solutions with free boundary for the Navier-Stokes equations,”Advan. Differ. Equations (in press).

  12. L. V. Ovsyannikov, “Regular and nonregular, partially invariant solutions,”Dokl. Akad. Nauk SSSR,343, No. 2, 156–159 (1995).

    MATH  MathSciNet  Google Scholar 

  13. L. V. Ovsyannikov and A. P. Chupakhin, “Regular, partially invariant submodels of the gas dynamics equations,”Prikl. Mat. Mekh.,60, No. 6, 990–999 (1996).

    MATH  MathSciNet  Google Scholar 

  14. E. Goursat,Cours d’Analyse Mathematique, Paris (1927).

  15. A. C. Hearn,REDUCE Users Mannual, Ver. 3.3., CP 78, Rand Corporation, Santa Monica (1987).

    Google Scholar 

  16. L. V. Ovsyannikov, “Group Stratification of boundary-layer equations,” in:Dynamics of Continuous Media (collected scientific papers) [in Russian], Novosibirsk,1 (1969), pp. 24–35.

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Authors and Affiliations

  1. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, 630090, Novosibirsk

    S. V. Meleshko

  2. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, 630090, Novosibirsk

    V. V. Pukhnachev

Authors
  1. S. V. Meleshko
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  2. V. V. Pukhnachev
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 24–33, March–April, 1999.

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Meleshko, S.V., Pukhnachev, V.V. One class of partially invariant solutions of the Navier-Stokes equations. J Appl Mech Tech Phys 40, 208–216 (1999). https://doi.org/10.1007/BF02468516

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  • DOI: https://doi.org/10.1007/BF02468516

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