, Volume 2, Issue 1, pp 37-68

Primary and secondary bifurcations in the Couette-Taylor problem

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Abstract

We study the preturbulent transitions for the Couette-Taylor flows via bifurcation theory in the presence of symmetry. The difficulty is that the linearized stability analysis leads to multiple eigenvalues for the most simple flows. Only a consideration of the symmetry-group action on the critical eigenvectors allows us to derive and to solve the bifurcation equations. We recover through this analysis the different patterns which are observed in experiments as the Reynolds number is increased: Steady Taylor vortices and bifurcation of either wavy, or twisted vortices from the Taylor vortex flow in the case of co-rotating cylinders; spiral vortices in the case of (strongly) counterrotating cylinders, and ribbon-cells, which have not yet been observed in experiments. Then we show that, under natural assumptions on the loss of stability of these oscillatory flows, the next bifurcation leads to quasi-periodic flows without frequency locking, whose different patterns are studied.