The onset of auto-oscillations in a fluid: PMM Vol. 35. No. 4. 1971. pp.638–655

Abstract

The onset of auto-oscillations at transition of the Reynolds number (or any other parameter defining the steady motion of a viscous incompressible fluid) through its critical value is investigated. Landau in [1] (see, also, [2, 3]) considered the onset of the periodic autooscillation mode to be the first stage of transition from a laminar to a turbulent flow of a fluid. His method, developed also by Meksyn,Stuart and Watson (see [4–7]), implies the knowledge of the eigenvectors of the linearized (with respect to the basic laminar mode at a given Reynolds number) Navier-Stokes operator to which (according to the linear theory) correspond increasing perturbations. A system of ordinary nonlinear differential equations is derived for the determination of the Fourier coefficients of the velocity field. The calculation of the right-hand sides of equations of this system is, however, somewhat involved. Owing to this, this method had not, so far, provided final results in specific cases, such as, for example, the Poiseuille flow in a channel. The Landau method is clearly more suitable for investigating the onset of a periodic mode rather than for the calculation of a stabilized one. Here the onset of auto-oscillations is analyzed by the Liapunov-Schmidt method described in [8, 9], The branching out of periodic solutions of systems of ordinary differential equations is considered in [10], where references to earlier works are cited. The generation of a cycle is considered in [10, 11] for a system of ordinary differential equations, while [12–14] deal with the special case of Galerkin equations approximating the Navier-Stokes system. Certain statements related to the complete Navier-Stokes equations are also formulated in [13, 14], A comprehensive statement of the problem and basic definitions are given in Sect. 1 ; an a priori estimate of possible auto-oscillation modes is presented (Lemma 1.2), and it is shown that only the critical value of a parameter can be a point of branching out of the system (Lemma 1.3). This is followed by the analysis of supplementary conditions for the actual generation of a cycle. Theorem 2.1, which is an analogy of Krasnosel'skii's theorem on bifurcation [15], is proved in Sect. 2. The existence of periodic auto-oscillatory motion under conditions of Theorem 2.1 is established by the analysis of linear equations only, independently of the form of nonlinear terms. A more detailed analysis of generated cycles, of their number and analytic properties is given in Theorem 2.2 and related notes in terms of parameter γ. Since the proofs of Theorems 2.1 and 2.2 are based on the most general properties of Navier-Stokes equations, these theorems can be readily extended to a wide class of ordinary differential equations in a Banach space (see Theorems 3.1 and 3.2 in Sect. 3) which comprise, in particular, problems involving equations of the parabolic kind, equations of convection, magnetohydrodynamics, etc.