K. Fujimura: Centre manifold reduction and the Stuart-Landau equation for fluid motions, Proc. R. Soc. Lond. A 453 (1997) 181-203.

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.

Related papers

  1. K. Fujimura: The equivalence between two perturbation methods in weakly nonlinear stability theory for parallel shear flows, Proc. R. Soc. Lond. A424 (1989) 373-392.
  2. K. Fujimura: Methods of centre manifold and multiple scales in the theory of weakly nonlinear stability for fluid motions, Proc. R. Soc. Lond. A434 (1991) 719-733.