Abstract
New exact 3-D steady translationally invariant (or z-invariant) flows of isentropic gas and ideal incompressible fluid are constructed. New exact solutions for steady magnetohydrodynamics equations are derived. We show that Euler equations for steady z-invariant flows of isentropic gas are equivalent to a coupled system of a partial differential equation of second order for the streamfunction ψ(x, y) which contains an arbitrary differentiable function H(ψ) and a transcendental equation connecting gas density ρ(x, y) with function ψ(x, y) and depending on equation of state p(ρ) = cρ γ and on the function H(ψ). We prove that functions ψ(x, y) and ρ(x, y) satisfy a universal nonlinear equation that is independent of equation of state p = p(ρ) and of the function H(ψ).
Acknowledgments
The author thanks the reviewers for thoughtful reading of my paper and important remarks.
-
Author contribution: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: None declared.
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] H. Lamb, Hydrodynamics, New York, Dover Publications, 1945.Search in Google Scholar
[2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge, Cambridge University Press, 1981.Search in Google Scholar
[3] G. G. Stokes, “On the steady motion of incompressible fluid,” Trans. Cambridge Philos. Soc., vol. 7, pp. 439–453, 1842.10.1017/CBO9780511702242.002Search in Google Scholar
[4] L. M. Milne-Thomson, Theoretical Aerodynamics, 4th ed. New York, St Martin’s Press, 1966.Search in Google Scholar
[5] R. Von Mises, Mathematical Theory of Compressible Fluid Flow, New York, Academic Press, 1958.Search in Google Scholar
[6] R. Von Mises and K. O. Friedrichs, Fluid Dynamics, New York, Berlin, Springer-Verlag, 1971.10.1007/978-1-4612-6406-4Search in Google Scholar
[7] R. Courant and K. O. Friedrichs, Supersonic Flows and Shock Waves, New York, Interscience Publishers, Inc., 1967.Search in Google Scholar
[8] G. Birkhoff, Hydrodynamics, Princeton, Princeton University Press, 1960.Search in Google Scholar
[9] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, New York, Berlin, Springer-Verlag, 1998.Search in Google Scholar
[10] H. W. Liepmann and A. Roshko, Elements of Gasdynamics, Mineola, New York, Dover Publications, Inc., 2001.Search in Google Scholar
[11] O. I. Bogoyavlenskij, “Counterexamples to Parker’s theorem,” J. Math. Phys., vol. 41, pp. 2043–2057, 2000. https://doi.org/10.1063/1.533225.Search in Google Scholar
[12] O. Bogoyavlenskij, “Unsteady equipartition MHD solutions,” J. Math. Phys., vol. 45, pp. 381–390, 2004. https://doi.org/10.1063/1.1629137.Search in Google Scholar
[13] O. Bogoyavlenskij, “Counterexamples to Moffatt’s statements on vortex knots,” Phys. Rev. E, vol. 95, pp. 043104–043114, 2017. https://doi.org/10.1103/PhysRevE.95.043104.Search in Google Scholar
[14] O. I. Bogoyavlenskij, “Exact unsteady solutions to the Navier-Stokes and viscous MHD equations,” Phys. Lett. A, vol. 307, pp. 281–286, 2003. https://doi.org/10.1016/s0375-9601(02)01732-2.Search in Google Scholar
[15] O. I. Bogoyavlenskij, “Invariants of the gas dynamics inside the mushroom clouds,” Phys. Fluids, vol. 32, pp. 106103–106115, 2020. https://doi.org/10.1063/5.0023495.Search in Google Scholar
[16] N. A. Lange, Lange’s Handbook of Chemistry, New York, McGraw Hill, 1967.Search in Google Scholar
[17] D. J. Struik, A Concise History of Mathematics, New York, Courier Corporation Dover Publications, 2012.Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
