Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations

ABSTRACT

The purpose of this paper is to find the admitted Lie group of the reduction of the Navier-Stokes equationswhere using the basic Lie symmetry method. This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Two-dimensional optimal system is determined for symmetry algebras obtained through classification of their subalgebras. Some invariant solutions are also found.

The purpose of this paper is to find the admitted Lie group of the reduction of the Navier-Stokes equationswhere using the basic Lie symmetry method. This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Two-dimensional optimal system is determined for symmetry algebras obtained through classification of their subalgebras. Some invariant solutions are also found.

KEYWORDS

Optimal System; Invariant Solutions; Partially Invariant Solutions; Navier-Stokes Equations

Optimal System; Invariant Solutions; Partially Invariant Solutions; Navier-Stokes Equations

Cite this paper

S. Khamrod, "Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 124-134. doi: 10.4236/am.2013.41022.

S. Khamrod, "Optimal System of Subalgebras for the Reduction of the Navier-Stokes Equations,"

References

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[23] A. M. Grundland and L. Lalague, “Invariant and partially Invariant Solutions of the Equations Describing A Non-Stationary and Isotropic Flow for an Ideal and Compressible Fluid in (3+1) Dimensions,” Journal of Physics A: Mathematical and General, Vol. 29, No. 8, 1996, pp. 1723-1739. doi:10.1088/0305-4470/29/8/019

[24] L. V. Ovsiannikov, “Group Analysis of Differential Equations,” Nauka, Moscow, 1978.

[25] L. V. Ovsiannikov, “Regular and Irregular Partially Invariant Solutions,” Doklady Academy of Sciences of USSR, Vol. 2, No. 343, 1995, pp. 156-159.

[26] L. V. Ovsiannikov, “Program SUBMODELS. Gas Dynamics,” Journal of Applied Mathematics and Mechanics, Vol. 58, 1994, pp. 30-55.

[1] L. V. Ovsiannikov and A. P. Chupakhin, “Regular Partially Invariant Submodels of the Equations of Gas Dynamics,” Journal of Applied Mechanics and Technics, Vol. 6, No. 60, 1996, pp. 990-999.

[2] S. Lie, “On General Theory of Partial Differential Equations of an Arbitrary Order,” German, No. 4, 1895, pp. 320-384.

[3] L. V. Ovsiannikov, “Partly Invariant Solutions of the Equations Admitting a Group,” Proceedings of the 11th International Congress of Applied Mechanics, Springer-Verlag, Berlin, 1964, pp. 868-870.

[4] V. V. Pukhnachov, “Group Properties of the Navier-Stokes Equations in Two-Dimensional Case,” Journal of Applied Mechanics and Technical Physics, Vol. 44, No. 3, 1960, pp. 317-323. doi:10.1023/A:1023472921305

[5] V. O. Bytev, “Group Properties of Navier-Stokes Equations,” Chislennye Metody Mehaniki Sploshnoi Sredy, Vol. 3, No. 3, 1972, pp. 13-17.

[6] S. V. Khabirov, “Partially Invariant Solutions of Equations of Hydrodynamics,” Exact Solutions of Differential Equations and Their Assymptotics, Ufa, 1992.

[7] B. J. Cantwell, “Introduction to Symmetry Analysis,” Camridge University Press, Camridge, 2002.

[8] B. J. Cantwell, “Similarity Transformations for the Two-Dimensional, Unsteady, Stream-Function Equation,” Journal of Fluid Mechanics, Vol. 85, No. 2, 1978, pp. 257-271. doi:10.1017/S0022112078000634

[9] S. P. Lloyd, “The Infinitesimal Group of the Navier-Stokes Equations,” Acta Mathematica, Vol. 38, 1981, pp. 85-98.

[10] R. E. Boisvert, W. F. Ames and U. N. Srivastava, “Group Properties and New Solutions of Navier-Stokes Equations,” Journal of Engineering Mathematics, Vol. 17, 1983, pp. 203-221. doi:10.1007/BF00036717

[11] A. Grauel and W. H. Steeb, “Similarity Solutions of the Euler Equation and the Navier-Stokes Equations in Two Space Dimensions,” International Journal of Theoretical Physics, No. 24, No. 3, 1985, pp. 255-265. doi:10.1007/BF00669790

[12] N. H. Ibragimov and G. Unal, “Equivalence Transformations of Navier-Stokes Equation,” Bulletin of the Technical University of Istanbul, Vol. 1-2, No. 47, 1994, pp. 203-207.

[13] R. O. Popovych, “On Lie Reduction of the Navier-Stokes Equations,” Nonlinear Mathematical Physics, Vol. 3-4, No. 2, 1995, pp. 301-311. doi:10.2991/jnmp.1995.2.3-4.10

[14] W. I. Fushchich and R. O. Popovych, “Symmetry Reduction and Exact Solution of the Navier-Stokes Equations,” Nonlinear Mathematical Physics, Vol. 1, No. 1, 1994, pp. 75-113. doi:10.2991/jnmp.1994.1.1.6

[15] W. I. Fushchich and R. O. Popovych, “Symmetry Reduction and Exact Solution of the Navier-Stokes Equations,” Nonlinear Mathematical Physics, Vol. 2, No. 1, 1994, pp. 158-188. doi:10.2991/jnmp.1994.1.2.3

[16] D. K. Ludlow, P. A. Clarkson and A. P. Bassom, “Similarity Reduction and Exact Solutions for the Two-Dimensional Incompressible Navier-Stokes Equations,” Studies in Applied Mathematics, Vol. 103, 1999, pp. 183-240. doi:10.1111/1467-9590.00125

[17] V. V. Pukhnachov, “Free Boundary Problems of the Navier—Stokes Equations,” Doctoral Thesis, Novosibirsk, 1974.

[18] A. F. Sidorov, V. P. Shapeev and N. N. Yanenko, “The Method of Differential Constraints and Its Applications in Gas Dynamics,” Nauka, Novosibirsk, 1984.

[19] S. V. Meleshko, “Classification of the Solutions with Degenerate Hodograph of the Gas Dynamics and Plasticity Equations,” Doctoral Thesis, Ekaterinburg, 1991.

[20] S. V. Meleshko, “One Class of Partial Invariant Solutions of Plane Gas Flows,” Differential Equations, Vol. 10, No. 30, 1994, pp. 1690-1693.

[21] L. V. Ovsiannikov, “Isobaric Motions of a Gas,” Differential Equations, Vol. 10, No. 30, 1994, pp. 1792-1799.

[22] A. P. Chupakhin, “On Barochronic Motions of a Gas,” Doklady Rossijskoj Akademii Nauk, Vol. 5, No. 352, 1997, pp. 624-626.

[23] A. M. Grundland and L. Lalague, “Invariant and partially Invariant Solutions of the Equations Describing A Non-Stationary and Isotropic Flow for an Ideal and Compressible Fluid in (3+1) Dimensions,” Journal of Physics A: Mathematical and General, Vol. 29, No. 8, 1996, pp. 1723-1739. doi:10.1088/0305-4470/29/8/019

[24] L. V. Ovsiannikov, “Group Analysis of Differential Equations,” Nauka, Moscow, 1978.

[25] L. V. Ovsiannikov, “Regular and Irregular Partially Invariant Solutions,” Doklady Academy of Sciences of USSR, Vol. 2, No. 343, 1995, pp. 156-159.

[26] L. V. Ovsiannikov, “Program SUBMODELS. Gas Dynamics,” Journal of Applied Mathematics and Mechanics, Vol. 58, 1994, pp. 30-55.