On the stability of pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances.

*(English)*Zbl 0579.76054Summary: The stability of fully developed pipe-Poiseuille flow to finite-amplitude axisymmetric disturbance and non-axisymmetric disturbances has been studied using the equilibrium-amplitude method of W. C. Reynolds and M. C. Potter [ibid. 27, 465-492 (1967; Zbl 0166.461)]. In both the cases the least-stable centre modes were investigated. Also, for the non-axisymmetric case the mode investigated was the one with azimuthal wavenumber equal to one. Many higher-order Landau coefficients were calculated, and the Stuart-Landau series was analysed by the D. Shanks method [J. Math. Phys. 34, 1-42 (1955; Zbl 0067.286)] and by using Padé approximants to look for the existence of possible equilibrium states. The results show in both cases that, for each value of the Reynolds number R, there is a preferred band of spatial wavenumbers \(\alpha\) in which equilibrium states are likely to exist. Moreover, in both cases it was found that the magnitude of the minimum threshold amplitude for a given R decreases with increasing R. The scales of the various quantities obtained agree very well with those deduced by A. Davey and H. P. F. Nguyen [J. Fluid Mech. 45, 701-720 (1971; Zbl 0245.76038)].

##### MSC:

76E30 | Nonlinear effects in hydrodynamic stability |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M99 | Basic methods in fluid mechanics |

##### Keywords:

existence of equilibrium states; pipe-Poiseuille flow; non-axisymmetric disturbances; equilibrium-amplitude method; least-stable centre modes; higher-order Landau coefficients; Stuart-Landau series; Padé approximants; band of spatial wavenumbers; minimum threshold amplitude
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