The Couette-Taylor Problem pp 59-104 | Cite as

# Mode Interactions

## Abstract

Taylor vortices are commonly observed in experiments when the cylinders are co-rotating and Couette flow becomes unstable. Similarly, when the cylinders are counter-rotating, the instability that is usually seen first is the spiral flow. However, the analysis of Chapter III does not take into account the higher bifurcations, that are observed when the Reynolds number is increased and lead to more complicated spatio-temporal patterns. This is appearent in Figure IV. 1, which was established experimentally for an apparatus with radius ratio 0.883. This diagram shows the critical curves in the plane (ℜ_{o},ℜ), where ℜ_{o} is proportional to the outer angular velocity ℜ_{0} = ℜΩ/η. It is remarkable that some of these complicated regimes, for example, the wavy vortex flow or the interpenetrating spirals, can occur very close to the primary bifurcation curve. This suggests that we could reach such regimes by looking at higher codimension bifurcation points on the primary bifurcation curve. By codimension-two points, we mean situations where, for example, two or more different critical modes are competing, giving rise to secondary and higher-order bifurcations. In order to study these points we need to consider additional parameters, which we allow to vary. In Chapter II, when analyzing the linear stability of Couette flow, we saw that mode interaction could occur when varying the angular velocity Ω, i.e., when considering Ω as a *second* parameter (the first one being the Reynolds number ℜ).

## Keywords

Normal Form Hopf Bifurcation Mode Interaction Imaginary Axis Couette Flow## Preview

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