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MIプレプリント



2017-2
Title:On the spectrum for the artificial compressible system
Author : Yoshiyuki Kagei, Takaaki Nishida & Yuka Teramoto
Abstract: Stability of stationary solutions of the incompressible Navier-Stokes system and the corresponding artificial compressible system is considered. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number $\ep$ which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small $\ep$. The result is applied to the Taylor problem.

File: 2017-2pdf

2017-1
Title:Gyroscopic Analogy of Coriolis Effect of Rotating Stratified Flows Confined in a Spheroid
Author : Yuki Miyachi & Yasuhide Fukumoto
Abstract: An insight is gained into the mechanism of system rotation for suppressing the Rayleigh-Taylor instability (RTI) by drawing analogy with the gyroscopic effect. A rotating flow of a stratified fluid confined in a spheroid, subject to gravity force, whose velocity field is linear in coordinates, is equivalent, in the Boussinesq approximation, to the motion of the Lagrange top, a heavy symmetrical rigid body. The sleeping top corresponds to the state in which a heavy fluid lies on top of a lighter fluid. Specifically, we investigate the incompressible two-layer RTI confined in the lower-half of a spheroid rotating about the axis of symmetry oriented parallel to the vertical direction. We derive the dispersion relation and the critical rotation rate for suppressing the axisymmetric mode of RTI. The gyroscopic analogy accounts for decrease of the critical rotation rate with oblateness of the spheroid. The stabilizing effect of rotation is enhanced for the half spheroid as compared with a circular cylinder of finite length.

File: 2017-1pdf