Centre manifold reduction and the Stuart‐Landau equation for fluid motions

Abstract

The centre manifold reduction to derive the Stuart-Landau equation is examined. A double expansion in terms of the Fourier series and linear eigenfunctions is introduced in hydrodynamic equations. A centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the Stuart–Landau equation. Through a formal expansion in linear eigenfunctions, the latter equation is shown to be equivalent with the one derived by the method of multiple scales. Numerical coefficients involved in the quintic Stuart–Landau equation are evaluated for plane Poiseuille flow, convection in a vertical slot, and Rayleigh-Benard convection. In all the cases, the coefficients converge as a dimension of the ODE system increases and approach the numerical values obtained by the method of multiple scales.