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## Stability analysis of homogeneous shear flow : the linear and nonlinear theories and a Hamiltonian formulation

The stability of steady-state solutions of the equations governing two-dimensional,

homogeneous, incompressible fluid flow are analyzed in the context

of shear-flow in a channel. Both the linear and nonlinear theories are

reviewed and compared. In proving nonlinear stability of an equilibrium,

emphasis is placed on using the stability algorithm developed in Holm et al.

(1985). It is shown that for certain types of equilibria the linear theory is

inconclusive, although nonlinear stability can be proven.

Establishing nonlinear stability is dependent on the definition of a norm

on the space of perturbations. McIntyre and Shepherd (1987) specifically

define five norms, two for corresponding to one flow state and three to a

different flow state, and suggest that still others are possible. Here, the

norms given by McIntyre and Shepherd (1987) are shown to induce the same

topology (for the corresponding flow states), establishing their equivalence as

norms, and hence their equivalence as measures of stability. Summaries of the

different types of stability and their mathematical definitions are presented.

Additionally, a summary of conditions on shear-flow equilibria under which

the various types of stability have been proven is presented.

The Hamiltonian structure of the two-dimensional Euler equations is

outlined following Olver (1986). A coordinate-free approach is adopted emphasizing

the role of the Poisson bracket structure. Direct calculations are

given to show that the Casimir invariants, or distinguished functionals, are

time-independent and therefore are conserved quantities in the usual sense. / Graduation date: 1990

Identifer | oai:union.ndltd.org:ORGSU/oai:ir.library.oregonstate.edu:1957/28933 |

Date | 17 October 1989 |

Creators | Hagelberg, Carl R. |

Contributors | Mahrt, Larry |

Source Sets | Oregon State University |

Language | en_US |

Detected Language | English |

Type | Thesis/Dissertation |

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