Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 027, 19 pages      arXiv:0802.3521      https://doi.org/10.3842/SIGMA.2008.027
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Applications of Group Analysis to the Three-Dimensional Equations of Fluids with Internal Inertia

Piyanuch Siriwat and Sergey V. Meleshko
School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand

Received October 31, 2007, in final form February 12, 2008; Published online February 24, 2008

Abstract
Group classification of the three-dimensional equations describing flows of fluids with internal inertia, where the potential function W = W(ρ,ρ·), is presented. The given equations include such models as the non-linear one-velocity model of a bubbly fluid with incompressible liquid phase at small volume concentration of gas bubbles, and the dispersive shallow water model. These models are obtained for special types of the function W(ρ,ρ·). Group classification separates out the function W(ρ,ρ·) at 15 different cases. Another part of the manuscript is devoted to one class of partially invariant solutions. This solution is constructed on the base of all rotations. In the gas dynamics such class of solutions is called the Ovsyannikov vortex. Group classification of the system of equations for invariant functions is obtained. Complete analysis of invariant solutions for the special type of a potential function is given.

Key words: equivalence Lie group; admitted Lie group; optimal system of subalgebras; invariant and partially invariant solutions.

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References

  1. Gavrilyuk S.L., Teshukov V.M., Generalized vorticity for bubbly liquid and dispersive shallow water equations, Contin. Mech. Thermodyn. 13 (2001), 365-382.
  2. Iordanski S.V., On the equations of motion of the liquid containing gas bubbles, Prikl. Mekh. Tekhn. Fiz. 3 (1960), 102-111.
  3. Kogarko B.S., On a model of a cavitating liquid, Dokl. Akad. Nauk SSSR 137, 1331-1333 (English transl.: Soviet Physics Dokl. 6 (1961), 305-306).
  4. van Wijngaarden L., On the equations of motion for mixtures of liquid and gas bubbles, J. Fluid Mech. 33 (1968), 465-474.
  5. Green A.E., Naghdi P.M., A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech. 78 (1976), 237-246.
  6. Salmon R., Lectures on geophysical fluid dynamics, Oxford University Press, New York, 1998.
  7. Ovsiannikov L.V., Group analysis of differential equations, Nauka, Moscow, 1978 (English transl.: Editor W.F. Ames, Academic Press, New York, 1982).
  8. Ibragimov N.H. (Editor), CRC Handbook of Lie group analysis of differential equations, Vols. 1, 2, 3, CRC Press, Boca Raton, 1994, 1995, 1996.
  9. Ovsyannikov L.V., The program "Submodels". Gas dynamics, Prikl. Mat. Mekh. 58 (1994), no. 4, 30-55 (English transl.: J. Appl. Math. Mech. 58 (1994), no. 4, 601-627).
  10. Meleshko S.V., Isentropic flows of an ideal gas, in Fragment, Vol. 401, Institute of Theoretical and Applied Mechanics, Institute of Hydrodynamics, Novosibirsk, 1989, 1-5.
  11. Hematulin A., Meleshko S.V., Gavrilyuk S.G., Group classification of one-dimensional equations of fluids with internal inertia, Math. Methods Appl. Sci. 30 (2007), 2101-2120.
  12. Ovsyannikov L.V., A singular vortex, Prikl. Mekh. Tekhn. Fiz. 36 (1995), no. 3, 45-52 (English transl.: J. Appl. Mech. Tech. Phys. 36 (1995), no.  3, 360-366).
  13. Popovych H.V., On SO(3)-partially invariant solutions of the Euler equations, in Proceedings of Third International Conference "Symmetry in Nonlinear Mathematical Physics" (June 12-18, 1999, Kyiv), Editors A.G. Nikitin and V.M. Boyko, Proceedings of Institute of Mathematics, Kyiv 30 (2000), Part 1, 180-183.
  14. Chupakhin A.P., Invariant submodels of a special vortex, Prikl. Mat. Mekh. 67 (2003), no. 3, 390-405 (English transl.: J. Appl. Math. Mech. 67 (2003), no. 3, 351-364).
  15. Cherevko A.A., Chupakhin A.P., A homogeneous singular vortex, Prikl. Mekh. Tekhn. Fiz. 45 (2004), no. 2, 75-89 (English transl.: J. Appl. Mech. Tech. Phys. 45 (2004), no. 2, 209-221).
  16. Pavlenko A.S., A projective submodel of the Ovsyannikov vortex, Prikl. Mekh. Tekhn. Fiz. 46 (2005), no. 4, 3-16 (English transl.: J. Appl. Mech. Tech. Phys. 46 (2005), no. 4, 459-470).
  17. Hematulin A., Meleshko S.V., Rotationally invariant and partially invariant flows of a viscous incompressible fluid and a viscous gas, Nonlinear Dynam. 28 (2002), 105-124.
  18. Golovin S.V., Invariant solutions of the singular vortex in magnetohydrodynamics, J. Phys. A: Math. Gen. 38 (2005), 8169-8184.
  19. Meleshko S.V., Methods for constructing exact solutions of partial differential equations, Springer, New York, 2005.
  20. Hearn A.C., REDUCE users manual, version 3.3, The Rand Corporation CP 78, Santa Monica, 1987.
  21. Bagderina Yu.Yu., Chupakhin A.P., Invariant and partially invariant solutions of the Green-Naghdi equations, Prikl. Mekh. Tekhn. Fiz. 46 (2005), no. 6, 26-35 (English transl.: J. Appl. Mech. Tech. Phys. 46 (2005), no. 6, 791-799).
  22. Ovsyannikov L.V., Optimal systems of subalgebras Dokl. Akad. Nauk 333 (1993), no. 6, 702-704 (English transl.: Russian Acad. Sci. Dokl. Math. 48 (1994), no. 3, 645-649).

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