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

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/v979v691f

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  • 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.
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  • description.provenance : Approved for entry into archive by Patricia Black(patricia.black@oregonstate.edu) on 2012-04-30T21:07:28Z (GMT) No. of bitstreams: 1 HagelbergCarlR1990.pdf: 520995 bytes, checksum: 1e8019def46ea6856d4947be2ce4edc6 (MD5)
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  • description.provenance : Approved for entry into archive by Patricia Black(patricia.black@oregonstate.edu) on 2012-04-30T21:09:07Z (GMT) No. of bitstreams: 1 HagelbergCarlR1990.pdf: 520995 bytes, checksum: 1e8019def46ea6856d4947be2ce4edc6 (MD5)

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