Loss of stability and bifurcation of auto-oscillatory regimes close to Poiseuille flow

Abstract

Bifurcation of Poiseuille flow in a flat channel is used as an example to analyze the problem of determining variables that permit study of bifurcation of a main steady flow of a viscous incompressible liquid for parameters close to the values of the coordinates of a point on the curve of neutral stability at which the first Lyapunov exponent d₀ vanishes and there is a changeover from subcritical to supercritical bifurcation. For Poiseuille flow, such a point (R₂,α₂, where R₂ is the Reynolds number, and α₂ is the wave number, occurs on the lower branch of the neutral curve. In this paper, it is shown by the Lyapunov-Schmidt method that for α < α₂ the stable time-periodic solution that bifurcates into the subcritical region loses stability in the case of slight supercriticality, and a fold singularity is formed in the amplitude surface. The nature of this additional bifurcation is determined by the sign of the second Lyapunov exponent d₁. For its calculation, the value of α₂ is fixed, and the bifurcation that occurs when the Reynolds number is changed is considered. A solution is sought in the form of a convergent series in powers of δ = (θ(R – R₀)^1/4, θ = ±1. The condition of solvability, which serves to determine the coefficient of δ⁴, makes it possible to determine the value of d₂. This procedure is entirely general and makes it possible to study bifurcation in the neighborhood of a point of degeneracy on the neutral curve in other hydrodynamic problems too.