Dynamical instability of laminar axisymmetric flows of ideal incompressible fluid
The instability of axisymmetric flows of ideal incompressible fluid with respect to infinitesimal perturbations with the nonconservation of angular momentum is investigated by numerically integrating the differential equations of hydrodynamics. The problem has been solved for two types of rotation profiles of an unperturbed flow: with zero and nonzero pressure gradients at the flow boundaries. Both rigid and free boundary conditions have been considered. The stability of axisymmetric flows with free boundaries is of great importance in disk accretion problems. Our calculations have revealed a crucial role of the flow pattern near the boundaries in the instability of the entire main flow. When the pressure gradient at the boundaries is zero, there is such a limiting scale of perturbations in azimuthal coordinate that longer-wavelength perturbations grow, while growing shorter-wavelength perturbations do not exit. In addition, for a fixed radial flow extent, there exists a nonzero minimum amplitude of the deviation of the angular velocity from the Keplerian one at which the instability vanishes. For a nonzero pressure gradient at the boundaries, the flow is unstable with respect to perturbations of any scale and at any small deviation of the angular velocity from the Keplerian one.
Key wordshydrodynamics instability disk accretion
PACS numbers97.10.Gz 47.15.Fe
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- 13.N. E. Kochin, Vector Calculus and the Elements of Tensor Calculus (Nauka, Moscow, 1965) [in Russian].Google Scholar
- 15.G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968; Nauka, Moscow, 1984).Google Scholar
- 16.L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Nauka, Moscow, 1986; Pergamon Press, Oxford, 1986).Google Scholar
- 18.C.-C. Lin, The Theory of Hydrodynamic Stability (Inostrannaya Literatura, Moscow, 1958; Cambridge University Press, Cambridge, 1966).Google Scholar
- 24.V. V. Petkevich, Fundamentals of Continuum Mechanics (Éditorial, URSS, 2001) [in Russian].Google Scholar
- 26.L. Rayleigh, Proc. R. Soc. London, Ser. A 93, 143 (1916).Google Scholar
- 28.J.-L. Tassoul, Theory of Rotating Stars (Mir, Moscow, 1978; Princeton Univ. Press, Princeton, 1979).Google Scholar