Introduction to Hamiltonian Fluid Dynamics and Stability Theory by Swaters Gordon E.;

Author:Swaters, Gordon E.;
Language: eng
Format: epub
Publisher: Routledge

4.8 Exercises

Exercise 4.1 Working from (4.116) show that a necessary condition for the insta-bility of a circular flow φs = φs (r) in the domain (4.105) subject to the boundary condition (4.106) is that there exists r* ∈ (0, a) such that , or conversely, a sufficient condition for stability is that for all r ∈ (0, a). This is Fjortoft’s Theorem applied to circular flows.

Exercise 4.2 Show that the minimum eigenvalue, denoted λmin, of the eigenvalue problem

where Ω is the circular domain (4.105), is given by

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