On the Secondary Bifurcation of Three Dimensional Standing Waves.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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This paper contains and analysis of the complex set of periodic solutions which may occur in a fluid filled vessel of rectangular cross section. A previous analysis by Verma and Keller found only simple eigenvalues for the linearized problem. It is shown herein that at critical values of the vessel aspect ratio double eigenvalues occur. Eight non-linear solution branches are emitted from these double eigenvalues. The solutions along the various branches are derived, and the results displayed graphically. It is shown that irregular waves occur along some of these branches. In an interesting development, Bauer, Keller, and Reiss, in their analysis of shell buckling, showed that the splitting of multiple eigenalues may lead to secondary bifurcation. This theory is applied to the non-linear standing wave problem herein, and it is shown that secondary bifurcation does occur in the neighborhood of the double eigenvalue. A perturbation method is used to find the secondary bifurcation points, and the solutions along the secondary branches, in the neighborhood of their respective branch points, are found. The neighborhood around the critical aspect ratios is substantial, suggesting that secondary branching may occur in a variety of vessels with rectangular cross section. This theory offers an explanation of the irregular waves often observed in the sloshing of fluid in a rectangular vessel.
- Numerical Mathematics
- Fluid Mechanics