Abstract
A new approach to the investigation of the nonlinear development of disturbances, based on the theory of invariant manifolds, is outlined. A method of obtaining projections of the Navier-Stokes equations on finite-dimensional invariant manifolds is proposed. The behavior of the disturbances is finally described by a system of ordinary differential equations with right sides in the form of power series in the amplitudes. It is an important property of the method that these series are convergent. A two-dimensional invariant projection is calculated numerically for plane Poiseuille flow. As a result, it is possible to clarify the nature of the change from subcritical to supercritical bifurcation and investigate new bifurcations of periodic flow regimes.
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 9–15, January–February, 1990.
The author is grateful to A. Zharilkasinov for carrying out the numerical calculations.
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Skobelev, B.Y. Nonlinear theory of hydrodynamic stability and bifurcations of solutions of the Navier-Stokes equations. Fluid Dyn 25, 6–11 (1990). https://doi.org/10.1007/BF01051290
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DOI: https://doi.org/10.1007/BF01051290