Abstract
The exact solution of the Euler equations of relativistic hydrodynamics of an compressible fluid—a relativistic analog of the Ovsyannikov vortex (singular vortex) in the classical gas dynamics—was found and investigated. A theorem was proved which shows that the factor system can be represented as a union of a noninvariant subsystem for the function defining the deviation of the velocity vector from the meridian and an invariant subsystem for the function defining thermodynamic parameters, the Lorentz factor, and the radial component of the velocity vector. Compatibility conditions of the overdetermined noninvariant system were obtained. The stationary solution was studied in detail. It was proved that the invariant subsystem reduces to an implicit differential equation. The branching manifold of the solutions of this equations was studied, and many singular points were found. It is proved that there exist two flow regimes, i.e., the solutions describing the vortex source of a relativistic gas, was proved. One of these solutions is defined only at a finite distance from the source, and the other is an analog of supersonic gas flow from the surface of a sphere.
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References
L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon Press, Oxford-Elmsford, New York, 1987).
P. G. LeFloch and S. A. Ukai, “A Symmetrization of the Relativistic Euler Equations in Several Spatial Variables,” Kinet. Rel. Models 2, 275–292 (2009).
S. K. Godunov, “Thermodynamic Formalization of Hydrodynamic Equations for a Charged Dielectric in an Electromagnetic Field,” Zh. Vychisl. Mat. Mat. Fiz. 52 (5), 916–929 (2012).
E. Gourgoulhon, “An Introduction to Relativistic Hydrodynamics, Stellar Fluid Dynamics and Numerical Simulations: from the Sun to Neutron Stars,” EAS Publ. Ser. 21, 43–79 (2006).
L. V. Ovsyannikov, “Singular Vortex,” Prikl. Mekh. Tekh. Fiz. 36 (3), 45–52 (1995) [J. Appl. Mekh. Tech. Phys. 36 (3), 360–366 (1995)].
L. V. Ovsyannikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978; Academic Press, New York, 1982).
M. S. Borshch and V. I. Zhdanov, “Exact Solutions of the Equations of Relativistic Hydrodynamics Representing Potential Flows,” Symmetry, Integrability Geometry: Methods Appl. 3 (116), 1–11 (2007).
C. Alexa and D. Vrinceanu, “On the Symmetries of the 1+1 Dimensional Relativistic Fluid Dynamics,” Preprint No. 9710004-1997 (Europ. Organizat. Nuclear Res., 1997).
M. Perucho, J. M. Marti, and M. Hanasz, “Nonlinear Stability of Relativistic Sheared Planar Jets,” Arch. Rational Mech. Anal. 139, 377–398 (1997).
N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical Hydromechanics (Fizmatgiz, Moscow, 1963; John Wiley and Sons, 1964).
A. A. Cherevko and A. P. Chupakhin, “Stationary Ovsyannikov Vortex,” Preprint No. 1-2005 (Inst. of Hydrodynamics, Sib. Branch, Russian Acad. of Sci., Novosibirsk, 2005).
S. V. Golovin, “Singular Vortex in Magnetohydrodynamics,” J. Phys. A. Math. General. 38, 4501–4516 (2005).
K. P. Stanyukovich, Unsteady Motion of Continuum (Nauka, Moscow, 1971) [in Russian].
V. I. Arnold, Ordinary Differential Equations (Springer-Verlag, Berlin-Heidelberg, 2006).
A. O. Remizov, “Multidimensional Poincar’e Construction and Singularities of Lifted Fields for Implicit Differential Equations,” Sovr. Mat. Fund. Napr. 19, 131–170 (2006).
A. A. Davydov, “Normal Form of the Differential Equation Not Resolved with Respect to the Derivative in the Neighborhood of Its Singular Point,” Funkts. Anal. Ego Pril. 19 (2), 1–10 (1985).
A. M. Barlukova and A. P. Chupakhin, “Partially Invariant Solutions in Gas Dynamics and Implicit Equations,” Prikl. Mekh. Tekh. Fiz. 53 (6), 11–24 (2012) [J. Appl. Mekh. Tech. Phys. 53 (6), 812–824 (2012)].
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2 (Wiley-VCH, 1989).
A. P. Mishina and I. V. Proskuryakov, Higher Algebra: Linear Algebra, Polynomials, General Algebra (Nauka, Moscow, 1965) [in Russian].
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Original Russian Text © A.P. Chupakhin, A.A. Yanchenko.
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 2, pp. 5–18, March–April, 2019.
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Chupakhin, A.P., Yanchenko, A.A. Ovsyannikov Vortex in Relativistic Hydrodynamics. J Appl Mech Tech Phy 60, 187–199 (2019). https://doi.org/10.1134/S0021894419020019
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DOI: https://doi.org/10.1134/S0021894419020019