On Instability of Steady–State Three–Dimensional Flows of an Ideal Compressible Fluid

Yuriy G. Gubarev^{1, 2,}

¹Laboratory for Fluid and Gas Vortical Motions, Lavrentyev Institute for Hydrodynamics, Novosibirsk, 630090, Russian Federation

²Department for Differential Equations, Novosibirsk State University, Novosibirsk, 630090, Russian Federation

Cite this paper:
Yuriy G. Gubarev. On Instability of Steady–State Three–Dimensional Flows of an Ideal Compressible Fluid. International Journal of Partial Differential Equations and Applications. 2017; 5(1):10-18. doi: 10.12691/ijpdea-5-1-2

Abstract

The problem on linear stability of stationary spatial flows of an inviscid compressible fluid entirely occupying a certain volume with quiescent solid impenetrable boundary in absence of external mass forces is studied. Applying the direct Lyapunov method, such flows are proved to be absolutely unstable under small three–dimensional (3D) perturbations. Constructive conditions for linear practical instability are obtained. The a priori exponential lower estimate for the growth of the considered perturbations in time is found.

Keywords:
an ideal compressible fluid steady–state spatial flows small 3D perturbations the direct Lyapunov method absolute linear theoretical instability conditions for linear practical instability a priori exponential lower estimate

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Dikii, , “On the nonlinear theory of hydrodynamic stability,” J. Appl. Math. Mech., 29 (5), 1009-1012, 1965.
[2]	Friedman, J.L., Schutz B.F., “Lagrangian perturbation theory of nonrelativistic fluids,” Astrophys. J., 221 (3), 937-957, 1978.
[3]	Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A., “Nonlinear stability of fluid and plasma equilibria,” Phys. Reports, 123 (1 & 2), 1-116, 1985.
[4]	Larsson, J., “A new Hamiltonian formulation for fluids and plasmas. Part 1. The perfect fluid,” J. Plasma Phys., 55 (2), 235-259, 1996.
[5]	Isichenko, M.B., “Nonlinear hydrodynamic stability,” Phys. Rev. Lett., 80 (5), 972-975, 1998.
[6]	Gubarev, Yu.G., “On stability of steady–state three–dimensional flows of an ideal compressible medium,” in Youth Conf. “Stability and Turbulence of Flows of Homogeneous and Heterogeneous Fluids,” Parallel, Novosibirsk, 107-110 (in Russian).
[7]	Gubarev, Yu.G., The direct Lyapunov method. The stability of quiescent states and steady–state flows of fluids and gases, Palmarium Academic Publishing, Saarbrücken, 2012 (in Russian).
[8]	Gubarev, Yu.G., “Stability of steady–state shear jet flows of an ideal fluid with a free boundary in an azimuthal magnetic field against small long–wave perturbations,” J. Appl. Mech. Techn. Phys., 45 (2), 239-248, 2004.
[9]	Gubarev, Y.G., “On instability of hydrodynamical flows,” in Zababakhin Scientific Talks. 2005: Intern. Conf. High Energy Density Physics, American Institute of Physics, 335-340.
[10]	Gubarev, Yu.G., “On stability of steady–state plane–parallel shearing flows in a homogeneous in density ideal incompressible fluid,” Nonlinear Anal. Hybrid Syst., 1 (1), 103-118, 2007.
[11]	Gubarev, Yu.G., Progress in nonlinear analysis research, Nova Science Publishers, New York, 2009, 137-181.
[12]	Gubarev, Yu.G., “On the stability of jetlike magnetohydrodynamic flows,” J. Appl. Ind. Math., 4 (3), 318-331, 2010.
[13]	Gubarev, Yu.G., “Linear stability criterion for steady screw magnetohydrodynamic flows of ideal fluid,” T & A, 16 (3), 407-418, 2009.
[14]	Gubarev, Yu.G., “Sufficient conditions for linear long–wave instability of steady–state axisymmetric flows of an ideal liquid with a free boundary in an azimuthal magnetic field,” Tech. Phys., 56 (3), 345-350, 2011.
[15]	Gavrilieva, A.A. and Gubarev, Yu.G., “Stability of steady–state plane–parallel shear flows of an ideal stratified fluid in the gravity field,” Vestnik of the NEFU named after M.K. Ammosov, 9 (3), 15-21, 2012 (in Russian).
[16]	Gubarev, Yu.G., “The problem of adequate mathematical modeling for liquids fluidity,” AJFD, 3 (3), 67-74, 2013.
[17]	Gavrilieva, A.A., Gubarev, Yu.G., Lebedev, M.P., “Rapid approach to resolving the adequacy problem of mathematical modeling of physical phenomena by the example of solving one problem of hydrodynamic instability,” IJTMP, 3 (4), 123-129, 2013.
[18]	Gubarev, Yu.G. and Latkin, I.E., “On stability of spherically symmetric dynamic equilibrium states of self–gravitating Vlasov–Poisson gas,” in All–Russian Conf. with Participation of Foreign Scientists “Modern Problems of Rarefied Gas Dynamics,” Institute for Thermophysics SB RAS, 82-84 (in Russian).
[19]	Gubarev, Yu.G., “On stability of steady–state three–dimensional flows of an ideal incompressible fluid,” JMSA, 3 (1), 12-21, 2015.
[20]	Gubarev, Yu.G. and Svetonosov, A.I., “On instability of 1D dynamic equilibrium states of Vlasov–Poisson plasma,” in Intern. Conf. “Actual Problems of Computational and Applied Mathematics – 2015,” Abwei, Novosibirsk, 196-202 (in Russian).
[21]	Lyapunov, A.M., The general problem of the stability of motion, Taylor & Francis, London, 1992.
[22]	Chetaev, N.G., Stability of motion, Nauka, Moscow, 1990 (in Russian).
[23]	Demidovich, B.P., Lectures on the mathematical stability theory, Nauka, Moscow, 1967 (in Russian).
[24]	Karacharov, K.A. and Pilyutik, A.G., Introduction in technical theory of motion stability, Fizmatgiz, Moscow, 1962 (in Russian).
[25]	La Salle, J. and Lefschetz, S., Stability by Liapunov's direct method with applications, Academic Press, New York, 1961.
[26]	Landau, L.D. and Lifshitz, E.M., Course of theoretical physics. Vol. 6. Fluid mechanics, Butterworth–Heinemann, Oxford, 1987.
[27]	Ovsyannikov, L.V., Lectures on the fundamentals of gas dynamics, Nauka, Moscow, 1981 (in Russian).
[28]	Chandrasekhar, S., Ellipsoidal figures of equilibrium, Yale University Press, New Haven, 1969.
[29]	Bekkenbach, E.F. and Bellman, R., Inequalities, Springer–Verlag, Berlin, 1961.
[30]	Chaplygin, S.A., New method of approximate integration of differential equations, GITTL, Moscow, 1950 (in Russian).
[31]	Il'in, V.A., Sadovnochii, V.A., Sendov, Bl.Kh., Mathematical analysis, Nauka, Moscow, 1979 (in Russian).
[32]	Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, Clarendon Press, Oxford, 1961.
[33]	Godunov, S.K., Equations of mathematical physics, Nauka, Moscow, 1979 (in Russian).