Nonlinear analysis of instability modes in the Taylor–Dean system.

The linear and weakly nonlinear stability of flow in the Taylor-Dean system is investigated. The base flow far from the boundaries, is a superposition of circular Couette and curved channel Poiseuille flows. The computations provide for a finite gap system, critical values of Taylor numbers, wave numbers and wave speeds for the primary transitions. Moreover, comparisons are made with results obtained in the small gap approximation. It is shown that the occurrence of oscillatory nonaxisymmetric modes depends on the "anisotropy" coefficient in the dispersion relation, and that the critical Taylor number changes slightly with the azimuthal wave number for large absolute values of rotation ratio. The weakly nonlinear analysis is made in the framework of the Ginzburg-Landau equations for anisotropic systems. The primary bifurcation towards stationary or traveling rolls is supercritical when Poiseuille component of the base flow is produced by a partial filling. An external pumping can induce a subcritical bifurcation for a finite range of rotation ratio. Special attention is also given to the influence of anisotropy properties on the phase dynamics of bifurcated solution (Eckhaus and Benjamin-Feir conditions). [ABSTRACT FROM AUTHOR]