Flow Induced Bifurcations to Three-Dimensional Motions of Tubes with an Elastic Support
The fascinating behavior of tubes conveying fluid has attracted great interest among engineers as well as mathematicians. A nice and rather complete review of the relevant literature is given in . In a fundamental paper Bajaj and Sethna () studied the Hopf bifurcation (flutter instability) of the downhanging equilibrium position of a cantilever tube. The mathematical interest in this problem stems from the fact that due to the rotational and reflectional symmetry (O(2)-symmetry) each eigenvalue of the linearized problem, associated with the loss of stability, occurs with multiplicity two. The consequence of this double multiplicity is that the system of bifurcation equations (amplitude equations of the critical modes) is rather high dimensional and hence has a richer solution set. However, on the other hand the bifurcation equations must obey the same symmetry properties as the equations of motion. Therefore the number of terms which are consistent with the symmetry requirements is pretty small. Thus the complication resulting from the high dimension is to some extend compensated. In  it is shown that depending on the mass ratio β between tube and fluid (see eq. (14) below) two distinct oscillatory motions occur after loss of stability. Both follow from symmetry breaking bifurcations. One is a planar oscillation which breaks the rotational symmetry and the other is a rotating motion which breaks the reflectional symmetry. Both solutions can be easily observed in experiments.
KeywordsHopf Bifurcation Amplitude Equation Imaginary Eigenvalue Reflectional Symmetry Bifurcation Equation
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